Most Important
Science Questions With Solution

CH1

Real Number

CH2

Polynomial

CH3

Linear Equation In 2 Variable

CH4

Quadratic Equation

CH5

Arithmetic Progressions

CH6

Triangle

CH7

Coordinate Geometry

CH8

Introduction To Trigonometry

CH9

Application Of Trigonometry

CH10

Circle

CH11

Area Related To Circle

CH12

Surface Area & Volume

CH13

Statistics

CH13

Probability

1. Two positive integers 𝛼 and 𝛽 can be written as 𝑎 = 𝑥3𝑦3𝑎𝑛𝑑 𝑏 = 𝑥𝑦3. 𝑥, 𝑦 are prime number, then LCM (a, b). (CBSE 2019)

2. What is the HCF of smallest prime number and the smallest composite number? (CBSE 2018)

3. Two tankers contain 850 liters and 680 liters of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times. (CBSE 2016)

4. If HCF of 144 and 180 is expressed in the from 13m – 3, find the value m. (CBSE 2014)

5. Three bells toll at intervals of 12 minutes, 15 minutes and 18 minutes respectively. If they start tolling together, after what time will they next toll together? (CBSE 2015)

6. An army contingent of 1000 members is to march behind an army band of 56 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? (CBSE 2011)

7. The HCF of 45 and 105 is 15. Write their LCM. (CBSE 2010)

8. Show that 9𝑛 cannot end with digit 0 for any natural number 𝑛. (CBSE 2014)

9. Determine the value of 𝑝 and 𝑞 so that the prime factorization of 2520 is expressible as 23 × 3p × q × 7.

   (CBSE 2014)

10. Find the LCM and HCF of 120 and 144 using fundamental theorem of arithmetic. (CBSE 2011)

11. Find HCF and LCM of 404 and 96 and verify that HCF × LCM = product of the two given number. (CBSE 2018)

12. Explain whether the number (3 × 5 × 13 × 46 + 23) is a prime number or a composite number.

    (CBSE 2016)

13. The LCM of two number is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, then find the other number. (CBSE 2012)

14. Find HCF of 378, 180 and 420 by prime factorization method. Is HCF × LCM of three numbers equal to the three numbers? (CBSE 2014)

15. Given that √2 is irrational, prove that (5 + 3√2) is an irrational number. (CBSE 2017)

16. Show that (√3 + √5)2 is an irrational number. (CBSE 2015)

17. Show that 2√2 is an irrational number. (CBSE 2014)

18. Prove that 15 + 17√3 is an irrational number. (CBSE 2011)

19. Prove that √2 is an irrational number. (CBSE 2019)

20. Prove that 2+ √3 is an irrational number, given that √3 is an irrational number. (CBSE 2019)

    5

21. Prove that 2 + 5√3 is an irrational number, given that √3 is an irrational number. (CBSE 2019)

22. Show that reciprocal of 3 + 2√2 is an irrational number. (CBSE 2014)

23. Prove that is 2√3 irrational. (CBSE 2011)

    5

24. Prove that 2 − 3√5 is an irratiofrnal number. (CBSE 2010)

25. Prove that 2√3 − 1 is an irrational number. (CBSE 2010)

26. Prove that 7 − 2√3 is an irrational number. (CBSE 2010)

27. Show that 5 + 3√2 is an irrational number. (CBSE 2010)

28. Find a rational number between √2 and √3. (CBSE 2019)

29. Write whether 2√45+3√20 on simplification given a rational or an irrational number. (CBSE 2010)

    2√5

    1

30. Find the value of: (−1)𝑛 + (−1)2𝑛 + (−1)2𝑛+1 + (−1)4𝑛+2, where 𝑛 is any positive odd integer.

    (CBSE 2016)

31. Express the number 0. 3 ̅1̅7̅̅8̅ in the from of rational number 𝑎.

    (CBSE 2012)

𝑏

1. Find the zeroes of the quadratic polynomial √3𝑥2 − 8𝑥 + 4√3. (CBSE 2013)

2. For what value of k, is 3 a zero of the polynomial 2𝑥2 + 𝑥 + 𝑘?. (CBSE 2010)

3. If 𝛼 𝑎𝑛𝑑 𝛽 are the zeroes of the polynomial 𝑎𝑥2 + 𝑏𝑥 + 𝑐, find the value of a2 + b2. (CBSE 2013)

4. Find the zeroes of the quadratic polynomial 9𝑡2 − 6𝑡 + 1 and verify the relationship between the zeroes and the coefficients? (CBSE 2014)

5. Find the zeroes of the quadratic polynomial 6𝑥2 − 3 − 7𝑥 and verify the relationship between the zeroes and the coefficients of the polynomial. (CBSE 2015, 2016 OD)

6. Find the zeroes of the quadratic polynomial 3𝑥2 − 2 and verify the relationship between the zeroes and the coefficients. (CBSE 2014, 2016)

7. Show that 1 and −3 are the zeroes of the polynomial 4𝑥2 + 4𝑥 − 3 and verify the relationship between

   2 2

   zeroes and coefficient of polynomial. (CBSE 2012)

8. Find the zeroes of the quadratic polynomial 𝑓(𝑥) = 𝑥2 − 3𝑥 − 28 and verify the relationship between the zeroes and the co-efficient of the polynomial. (CBSE 2012, 2017 D)

9. If one zero of the polynomial x2 − 4x + 1is 2 + √3, write the other zero. (CBSE 2010)

10. Find the condition that zero of polynomial 𝑝(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 are reciprocal of each other.

    (CBSE 2017 OD)

11. If 2 and -3 are the zeroes of the quadratic polynomial x2 + (a + 1)x + b, then find the value of 𝛼 𝑎𝑛𝑑 𝛽.

    (CBSE 2011)

12. If 𝛼, 𝛽 are the zeroes of the polynomial 2𝑦2 + 7𝑦 + 5, write the value of 𝛼 + 𝛽 + 𝛼𝛽.

    ‌(CBSE 2010)

13. Find the value of k such that the polynomial 𝑥2 − (𝑘 + 6)𝑥 + 2(2𝑘 − 1) has sum of its zeroes equal to half of their product. (CBSE 2019)

14. If 𝛼 𝑎𝑛𝑑 𝛽 are the zeroes of the polynomial 6𝑦2 − 7𝑦 + 2, find a quadratic polynomial whose zeroes are

    1 𝑎𝑛𝑑

    𝛼

    1. (CBSE 2012)

    𝛽

15. Find a quadratic polynomial, the sun and product of whose zeroes are 0 and − 3 respectively. Hence find

    5

    the zeroes. (CBSE 2015)

16. If 𝛼, 𝛽 are the zeroes of a polynomial, such that 𝛼 + 𝛽 = 6 and 𝛼𝛽 = 4, then write the polynomial.

    (CBSE 2010)

    
    

17. Find a quadratic polynomial whose zeroes are 3 + √2 𝑎𝑛𝑑 3 − √2. (CBSE 2013)

18. Find a quadratic polynomial whose zeroes are 3+√5 𝑎𝑛𝑑 3−√5. (CBSE 2013)

    5 5

19. Find the quadratic polynomial, sum and product of whose zeroes are -1 and -20 respectively. Also, find the zeroes of the polynomial so obtained. (CBSE 2019)

20. Quadratic polynomial 2𝑥2 − 3𝑥 + 1 has zeroes as 𝛼 𝑎𝑛𝑑 𝛽. Now from a quadratic whose zeroes are

    3𝛼 𝑎𝑛𝑑 3𝛽. (CBSE 2016)

21. If 𝛼 𝑎𝑛𝑑 𝛽are zeroes of a polynomial x2 + 6x + 9, then from a polynomial whose zeroes are – 𝛼 𝑎𝑛𝑑 −

    𝛽. (CBSE 2016)

22. If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of 2x2 − 5x − 3, find the value of p and q. (CBSE 2012)

23. If the product of zeroes of the polynomial ax2 − 6x − 6 is 4, find the value of 𝛼. Find the sum of zeroes of the polynomial. (CBSE 2014)

24. If 𝛼, 𝛽 are the zeroes of x2 = px + q,find the value of (𝛼 + 2) . ( 𝛽 + 2). (CBSE 2013)

    𝛽 𝛼

25. If one zero of the quadratic polynomial 𝑓(𝑥) = 4𝑥2 − 8𝑘𝑥 + 8𝑥 − 9 is negative of the other, then find the zeroes of kx2 + 3kx + 2. (CBSE 2015)

26. If 𝛼 𝑎𝑛𝑑 𝛽 are zeroes of 𝑝(𝑥) = 𝑘𝑥2 + 4𝑥 + 4, such that α2 + β2 = 24, find k. (CBSE 2013)

27. If 𝛼 𝑎𝑛𝑑 𝛽 are the zeroes of the polynomial 𝑝(𝑥) = 2𝑥2 + 5𝑥 + 𝑘, satisfying the relation, α2 + β2 +

αβ = 21, then find the value of k. (CBSE 2017 OD)

4

1. For what value of k, the following system of equations 𝑘𝑥 + 2𝑦 = 3, 3𝑥 + 6𝑦 = 10 has a unique solution. (CBSE 2019)

2. Find 𝑐 if the system of equation 𝑐𝑥 + 3𝑦 + (3 − 𝑐) = 0; 12𝑥 + 𝑐𝑦 − 𝑐 = 0 has infinitely many solutions?

   (CBSE 2019)

3. Find the value of k for which the following pair of linear equations have infinitely many solutions.

   2𝑥 + 3𝑦 = 7, (𝑘 + 1)𝑥 + (2𝑘 − 1)𝑦 = 4𝑘 + 1. (CBSE 2019)

4. Find the value (s) of k so that the pair of equations 𝑥 + 2𝑦 = 5 𝑎𝑛𝑑 3𝑥 + 𝑘𝑦 + 15 = 0 has a unique solutions. (CBSE 2019)

5. Find the value of k for which the following pair of linear equations have infinitely many solutions:

   2𝑥 + 3𝑦 = 7, (𝑘 − 1)𝑥 + (𝑘 + 2)𝑦 = 3𝑘. (CBSE 2010)

6. Find the value of m for which the pair of linear equations 2𝑥 + 3𝑦 − 7 = 0 𝑎𝑛𝑑 (𝑚 − 1)𝑥 + (𝑚 + 1)𝑦 = (3𝑚 − 1) has infinitely many solution. (CBSE 2010)

7. For what value of k will the following pair of linear equations have no solution?

   2𝑥 + 3𝑦 = 9; 6𝑥 + (𝑘 − 2)𝑦 = (3𝑘 − 2). (CBSE 2010)

8. For what value of p will the following pair of linear equations have infinitely many solutions?

   (CBSE 2010)

9. Find the value of 𝛼 𝑎𝑛𝑑 𝛽, for which the following pair of linear equations has infinitely many solutions:2𝑥 + 3𝑦 = 7; (𝑎 + 𝑏)𝑥 + (2𝑎 − 𝑏)𝑦 = 21 (CBSE 2010)

10. If the systems of equations 6𝑥 + 2𝑦 = 3 𝑎𝑛𝑑 𝑘𝑥 + 𝑦 = 2 has a unique solution, find the value of k.

    (CBSE 2013)

11. Given a linear equation 3𝑥 − 5𝑦 = 11. From another linear equation in these variables such that the geometric representation of the pair of formed is: (CBSE 2015)

    (a) Intersecting lines (b) Coincident lines (c) Parallel lines.

12. Determine the value of m and n so that the following pair of linear equations has infinite number of solutions: (2𝑚 − 1)𝑥 + 3𝑦 = 5; 3𝑥 + (𝑛 − 1)𝑦 = 2. (CBSE 2013)

13. For what values of p and q will the following pair of linear equations has infinitely many solutions?

    4𝑥 + 5𝑦 = 2; (2𝑝 + 7𝑞)𝑥 + (𝑝 + 8𝑞)𝑦 = 2𝑞 − 𝑝 + 1 (CBSE 2013)

14. For what value of k will the pair of equations has no solution? 3𝑥 + 𝑦 = 1; (2𝑘 − 1)𝑥 + (𝑘 − 1)𝑦 = 2𝑘 + 1 (CBSE 2012)

15. For what value of p will the following system of equations has no solution: (2𝑝 − 1)𝑥 + (𝑝 − 1)𝑦 = 2𝑝 + 1; 𝑦 + 3𝑥 − 1 = 0 (CBSE 2011)

16. Check graphically whether the pair of equations 3𝑥 − 2𝑦 + 2 = 0 𝑎𝑛𝑑 3 𝑥 − 𝑦 + 3 = 0, is consistent.

    2

    Also find the coordinates of the points where the graphs of the equations meet the Y-axis.

    (CBSE 2012)

17. Draw the graphs of equations: 2𝑥 − 𝑦 = 1; 𝑥 + 2𝑦 = 13 (CBSE 2013)

18. Amit bought two pencils and three chocolates for ₹11 and Sumeet bought one pencil and two chocolate for ₹ 7. Represent this situation in the from of a pair of linear equations. Find the price of one pencil and that of one chocolate graphically. (CBSE 2017)

19. Solve the following pair of linear equations: 𝑦 − 4𝑥 = 1 𝑎𝑛𝑑 6𝑥 − 5𝑦 = 9 (CBSE 2019)

20. Solve: 99𝑥 + 101𝑦 = 499; 101 𝑥 + 99𝑦 = 501 (CBSE 2011)

21. Solve for 𝑥 𝑎𝑛𝑑 𝑦: 𝑥 + 2𝑦 − 3 = 0; 3𝑥 − 2𝑦 + 7 = 0. (CBSE 2015)

22. Solve for 𝑥 𝑎𝑛𝑑 𝑦: 6(𝑎𝑥 + 𝑏𝑦) = 3𝑎 + 2𝑏; 6(𝑏𝑥 − 𝑎𝑦) = 3𝑏 − 2𝑎. (CBSE 2014)

23. Solve the following pair of equations: 4 + 3𝑦 = 8; 6 − 4𝑦 = −5. (CBSE 2010)

    𝑥 𝑥

24. A father’s age is three times the sum of the ages of his two children. After 5 year his age will be two times the sum of their ages. Find the present age of the father. (CBSE 2019)

25. A fraction becomes 1 when 2 is subtracted from the numerator and become 1 when 1 is subtracted from

    3 2

    the denominator. Find the fraction. (CBSE 2019)

26. A part of monthly Hostel charge is fixed and the remaining depends on the number of days one has taken food in the mess. When Swati takes food for 20 days, she has to pay ₹ 3000 as hostel charges, whereas Mansi takes food for 25 days and pays ₹ 3500 as hostel charges. Find the fixed charges and the cost of food per days. (CBSE 2016)

27. The sum of the numerator and denominator of a fraction is 12. If 1 is added to both the numerator and

    the denominator the fraction becomes 3, find the fraction. (CBSE 2011)

    4

28. 4 chairs and 3 tables cost ₹ 2100 and 5 chairs and 2 tables cost ₹ 1750. Find the cost of one chair and one table separately. (CBSE 2015)

29. The owner of a taxi company decides to run all the taxi on CNG fuels instead of petrol / diesel. The taxi for journey of 13 km, the charge paid is ₹ 129 and for journey of 22 km, the charge paid is ₹ 32 km?

    (CBSE 2014)

30. The area of a rectangle reduces by 160 m2 if its length is increased by 2 m, then its area is decreased by 100 m2 Find the dimensions of the rectangle. (CBSE 2014)

31. At a certain time in a zoo, the number of heads and the number legs of tiger and peacocks were counted, and it was found that were 47 heads and152 legs, Find the number of tigers and peacocks in the zoo.

    (CBSE 2014)

32. The age of the father is twice the sum of the ages of his 2 children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father. (CBSE 2013)

33. Place A and B are 80 km apart from each other on a highway. A car stats from A and another from B at the same time. If they move in same direction they meet in 8 hrs and if they move in opposite directions they meet in 1 hr 20 minutes. Find speeds of the cars. (CBSE 2013)

34. Find those integral value of m for which the x-coordinate of the point of intersection of lines represented by 𝑦 = 𝑚𝑥 + 1 𝑎𝑛𝑑 3𝑥 + 4𝑦 = 9 is an integer. (CBSE 2014)

35. Solve the following pair of equations for 𝑥 and y:  𝑎𝑥 − 𝑏𝑦 = 𝑎 + 𝑏; 𝑎𝑥 − 𝑏𝑦 = 2𝑎𝑏 (CBSE 2014)

𝑏 𝑎

1. If 𝑥 = 3 is one root of the quadratic equation 𝑥2 − 2𝑘𝑥 − 6 = 0, then find the value of k.

   (CBSE 2018)

2. If one root of the quadratic equation 6𝑥2 − 𝑥 − 𝑘 = 0 is 2, then find the value of k. (CBSE 2017)

   3

3. If 𝑥 = −1, is a solution of the quadratic equation 3𝑥2 + 2𝑘𝑥 − 3 = 0, find the value of k.

   2

   (CBSE 2015)

   
   

4. Find that root of the quadratic equation√2𝑥2 + 7𝑥 + 5√2 = 0. (CBSE 2013, 2017)

5. Solve for 𝑥: √3𝑥2 + 10𝑥 − 8√3 = 0. (CBSE 2017)

6. Solve for 𝑥: √2𝑥 + 9 + 𝑥 = 13. (CBSE 2016)

7. Solve for 𝑥: √3𝑥2 − 2√2𝑥 − 2√3 = 0. (CBSE 2016)

8. Solve the quadratic equation for 𝑥: 9𝑥2 − 6𝑏2𝑥 − (𝑎4 − 𝑏4) = 0. (CBSE 2015)

9. Solve the quadratic equation for 𝑥: 𝑥2 − 2𝑎𝑥 − (4𝑏2 − 𝑎2) = 0. (CBSE 2015)

   
   

10. Solve the quadratic equation for 𝑥: 4√3𝑥2 + 5𝑥 − 2√3 = 0 (CBSE 2013)

11. Solve for 𝑥: 2𝑥

    𝑥−3

    + 1

    2𝑥+3

    + 3𝑥+9 (𝑥−3)(2𝑥+3)

    = 0, 𝑥 ≠ 3, − 3.

    2

    (CBSE 2016)

12. Solve for 𝑥: 𝑥+1 + 𝑥−2 = 4 − 2𝑥+3 ; 𝑥 ≠ 1, −2, 2. (CBSE 2016)

    𝑥−1

13. Solve for 𝑥: 𝑎

    𝑥−𝑏

    𝑥+2

    + 𝑏

    𝑥−𝑎

    𝑥−2

    = 2, 𝑥 ≠ 𝑎, 𝑏. (CBSE 2016)

14. Solve for 𝑥: 𝑥2 + 5𝑥 − (𝑎2 + 𝑎 − 6) = 0. (CBSE 2015)

15. Solve for 𝑥: 𝑥2 − (2𝑏 − 1)𝑥 + (𝑏2 − 𝑏 − 20) = 0. (CBSE 2015)

16. Solve the equation: 4 − 3 = 5 ; 𝑥 ≠ 0, −3 , 𝑓𝑜𝑟 𝑥. (CBSE 2014)

    𝑥 2𝑥+3 2

17. Solve for 𝑥: 4𝑥2 − 4𝑎𝑥 + (𝑎2 − 𝑏2) = 0. (CBSE 2012)

    
    

18. Solve for 𝑥: 𝑥2 − 5√5𝑥 + 30 = 0. (CBSE 2012)

19. Find the roots of the quadratic equation: 2√3𝑥2 − 5𝑥 + √3 = 0. (CBSE 2011)

20. Solve for 𝑥: 3𝑥−4 + 7 = 5 , 𝑥 ≠ 4. (CBSE 2010)

    7 3𝑥−4 2 3

21. Solve the following for 𝑥: 1 = 1 + 1 + 1 . (CBSE 2019)

    2𝑎+𝑏+2𝑥 2𝑎 𝑏 2𝑥

22. Solve for 𝑥: 1

    + 1 = 1 1 , 𝑥 ≠ 3 , 5.

    (CBSE 2017)

    2𝑥−3

23. Solve for 𝑥 ∶ 3

    𝑥+1

    𝑥−5

    + 4

    𝑥−1

    9

    = 29

    4𝑥−1

    2

    , 𝑥 ≠ 1, −1, 1.

    4

    (CBSE 2015)

24. Solve for 𝑥: 𝑥−2 + 𝑥−4 = 10 ; 𝑥 ≠ 3, 5. (CBSE 2014)

    𝑥−3

25. Solve for 𝑥: 1

    𝑥−3

    𝑥−5

    + 2

    𝑥−2

    3

    = 8 ; 𝑥 ≠ 0, 2, 3.

    𝑥

    (CBSE 2013)

26. Find the roots of the equation:  1  −   1  = 11 , 𝑥 ≠ −4,7. (CBSE 2011)

    𝑥+4 𝑥−7 30

27. If the quadratic equation 𝑝𝑥2 − 2√5𝑝𝑥 + 15 = 0 has two equal roots, then find the value of p.

    (CBSE 2015)

28. For what values of k, the roots of the equation x2 + 4x + k = 0 are real? (CBSE 2019)

29. Find the value of k for which the equation x2 + k(2x + k − 1) + 2 = 0 has real and equal roots.

    (CBSE 2017)

30. If −5 is a roots of the quadratic 2x2 + px − 15 = 0 and the quadratic equation 𝑝(𝑥2 + 𝑥) + 𝑘 = 0 has equal roots, find the value of k. (CBSE 2014)

31. Find the value (S) of k for which the quadratic equation 9x2 − 3kx + k = 0 has equal roots.

    (CBSE 2011, 2014)

    
    

32. Find the value (s) of p so that the quadratic equation 𝑝𝑥(𝑥 − 3) + 9 = 0 has equal roots.

    (CBSE 2012)

33. Find the value (s) of k so that the quadratic equation 2x2 + kx + 3 = 0 has equal roots.

    (CBSE 2012)

34. Find the value (s) of m for which the roots of the equation 𝑚𝑥(6𝑥 + 10) + 25 = 0, are equal.

    (CBSE 2012)

35. Find the value (s) of p for which the roots of the quadratic equation (𝑝 + 3)𝑥2 + 2(𝑝 + 3)𝑥 + 4 = 0 are equal. (CBSE 2011)

36. Find the value (s) of k for which the roots of the quadratic equation (𝑘 − 4)𝑥2 + 2(𝑘 − 4)𝑥 + 2 = 0 are equal. (CBSE 2012)

37. For what value (s) of k does the quadratic equation (𝑘 − 5)𝑥2 + 2(𝑘 − 5)𝑥 + 2 = 0 has equal roots?

    (CBSE 2011)

38. If the equation (1 + 𝑚2)𝑥2 + 2𝑚𝑐𝑥 + 𝑐2 − 𝑎2 = 0 has equal roots, then show that c2 = a2(1 + m2).

    (CBSE 2017)

39. If the roots of the equation (𝑎2 + 𝑏2)𝑥2 − 2(𝑎𝑐 + 𝑏𝑑)𝑥 + (𝑐2 + 𝑑2) = 0 are equal, prove that 𝑎 = 𝑐 .

    𝑏 𝑑

    (CBSE 2017)

40. If 𝑎𝑑 ≠ 𝑏𝑐, then prove that the equation (𝑎2 + 𝑏2)𝑥2 + 2(𝑎𝑐 + 𝑏𝑑)𝑥 + (𝑐2 + 𝑑2) = 0 has no real roots. (CBSE 2017)

41. If the roots of the quadratic equation (𝑐2 − 𝑎𝑏)𝑥2 − 2(𝑎2 − 𝑏𝑐)𝑥 + 𝑏2 − 𝑎𝑐 = 0 in x are equal, then show that either a = 0 or a3 + b3 + c3 = 3abc. (CBSE 2017)

42. If the roots of the equation (𝑥 − 𝑎)(𝑥 − 𝑏) + (𝑥 − 𝑏)(𝑥 − 𝑐) + (𝑥 − 𝑐)(𝑥 − 𝑎) = 0 are equal, then show that 𝑎 = 𝑏 = 𝑐. (CBSE 2017)

43. If the roots of the quadratic equation (𝑎 − 𝑏)𝑥2 + (𝑏 − 𝑐)𝑥 + (𝑐 − 𝑎) = 0 are equal, prove that

    2𝑎 = 𝑏 + 𝑐. (CBSE 2017)

44. For what value (s) of k, are the roots of the quadratic equation 𝑘𝑥(𝑥 − 2) + 6 = 0 equal?

    (CBSE 2013)

45. For what value (s) of k, are the roots of the quadratic equation? (𝑘 + 4)𝑥2 + (𝑘 + 1)𝑥 + 1 = 0, equal?

    (CBSE 2013)

46. Find the value of k for which the roots of the quadratic equation 3x2 − 10x + k = 0 are reciprocal of each other. (CBSE 2019)

47. If 𝛼, 𝛽 are the zeroes of the quadratic equation 2y2 + 7y + 5, write the value of 𝛼 + 𝛽 + 𝛼𝛽.

    (CBSE 2010)

48. Find the value of p, for which one root of the quadratic equations 𝑝𝑥2 − 14𝑥 + 8 = 0 is 6 times the other. (CBSE 2017)

49. If 𝑥 = 2 𝑎𝑛𝑑 𝑥 = −3 are roots of the quadratic equations ax2 + 7x + b = 0, find the values of a and b.

    3

    (CBSE 2016)

50. A two-digit number if four times the sum of the digits. It is also equal to 3 times the product of digits. Find the number. (CBSE 2016)

51. A plane left 30 minutes late than its schedules time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.

    (CBSE 2018)

    
    

52. Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the square of the other two by 60. Find the number. (CBSE 2016)

53. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the integers. (CBSE 2010)

54. Two water taps together can fill a tank in 9 hours 36 minutes. The tap of larger diameter takes 8 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. (CBSE 2016)

55. If the sum of two natural numbers is 8 and their product is 15, find the numbers. (CBSE 2012)

56. The difference of square of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers. (CBSE 2010)

57. A girl is twice as old as her sister. Four years hence, the product of their ages (in years) Will be 160. Find their present ages. (CBSE 2010)

58. Some students planned a picnic. The total budget for food was ₹2,000. But 3 students failed to attend the picnic and thus the cost of food for each member increased by ₹20. How many students

    Attended the picnic and how much did each student pay for the food? (CBSE 2010)

59. Two water taps together can fill a tank in 1 7 hours. The tap with longer diameter takes 2 hours less than

    8

    the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately. (CBSE 2019)

60. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

    (CBSE 2019)

61. A motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to retune downstream to the same spot. Find the speed of the stream. (CBSE 2018)

62. Speed of a boat in still water is 15 km/h. it goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. (CBSE 2017)

63. A motor boat whose speed is 24 km/h in still water takes 1 hours more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream. (CBSE 2016)

64. A motor boat whose speed is 20 km/h in still water, takes 1 hour more to go 48 km upstream than to return downstream to the same spot. Find the speed of the stream. (CBSE 2011)

65. A train at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed? (CBSE 2018)

66. A train covers a distance of 300 km at a uniform speed. If the train is increased by 5 km/h, it takes 2 hours less in the journey. Find the original speed of the train. (CBSE 2017)

67. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed? (CBSE 2015)

68. A train travels 180 km at a uniform speed. If the speed had been 9 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train. (CBSE 2011)

69. A takes 6 days less than B do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it? (CBSE 2017)

70. A passenger, while boarding the plane, slipped from the stairs and got hurt. The pilot took the passenger to the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passenger could catch the connecting flights, the speed of the plane was increased by 250 km/hr than the usual speed. Find the usual speed of the plane.

    (CBSE 2016)

71. The time taken by a person to cover 150 km was 2 1

    2

    hours more than the time taken in the return

    journey. If he returned at a speed of 10 km/hr more than the speed while going, find the speed per hour in each direction. (CBSE 2016)

72. Two pipes running together can fill a tank in 11 1 minutes. If one pipes taken 5 minutes more than the

    9

    other to fill the tank separately, find the time in which each pipes would fill the tank separately.

    (CBSE 2016)

73. The denominator of a fraction is one more than twice its numerator. If the sum of the fraction and its reciprocal is2 16, find the fraction. (CBSE 2016)

    21

74. The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the

    denominator, then the sum of the new fraction and original fraction is 29. Find the original fraction.

    20

    (CBSE 2015)

75. The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction

    is decreased by 1 . Find the fraction. (CBSE 2012)

    15

76. To fill a swimming pool two pipes are to be used. If the pipes of larger diameter are used for 4 hours and the pipe of smaller diameter for 9 hours, only half the fool can be filled. Find how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool. (CBSE 2015)

77. A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck. (CBSE 2015)

78. The difference of two natural numbers is 5 and the difference of their reciprocal is  1 . Find the numbers.

    10

    (CBSE 2014)

79. The sum of two numbers is 9 and the sum of their reciprocal is 1. Find the numbers. (CBSE 2012)

    2

80. The sum of the square of two consecutive odd numbers is 394. Find the numbers. (CBSE 2014)

81. The sum of the square of two consecutive multiple of 7 is 637. Find the multiples. (CBSE 2013)

82. Sum of the area of two squares is 400cm2. If the difference of their perimeters is 16 cm, find the sides of the two square. (CBSE 2013)

83. The present age of a father is equal to the square of the present age of his son. One year ago, the age of the father was 8 times the age of his son. Find their present ages. (CBSE 2013)

84. A shopkeeper buys some books for ₹80. If he had bought 4 more books for the same amount, each book would have cost ₹1 less. Find the number of books he bought. (CBSE 2012)

85. In a fight of 2,800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.

    (CBSE 2012)

86. A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits interchange their places. Find the number. (CBSE 2012)

87. Find two consecutive natural numbers, the sum of whose square is 145.

    (CBSE 2012)

88. Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

    (CBSE 2011)

89. Write all the values of p for which the quadratic equation x2 + px + 16 = 0 has equal roots. Find the roots of the equation so obtained. (CBSE 2019)

90. If 𝑥 = 3 is root of the quadratic equation x2 − x + k = 0, Find the value of p so that the roots of the equation 𝑥2 + 𝑘(2𝑥 + 𝑘 + 2) + 𝑝 = 0 are equal. (CBSE 2015)

91. If 1 is a root of the quadratic equation 3𝑥2 + 𝑎𝑥 − 2 = 0 and the quadratic equation 𝑎(𝑥2 + 6𝑥) − 𝑏 = 0 has equal roots, find the value of b. (CBSE 2014)

92. Find the non-zero value of k, for which the quadratic equation 𝑘𝑥2 + 1 − 2(𝑘 − 1)𝑥 + 𝑥2 = 0 has equal roots. Hence, find the roots of the equation. (CBSE 2015)

93. Find the value of p for which the quadratic equation (2𝑝 + 1)𝑥2 − (7𝑝 + 2)𝑥 + 𝑥2 = 0 has equal roots. Also find these roots. (CBSE 2014)

94. Find the value of k for which the quadratic equation (3𝑘 + 1)𝑥2 + 2(𝑘 + 1)𝑥 + 1 = 0 has equal roots. Also find these roots. (CBSE 2014)

95. Solve for 𝑥: 3 (3𝑥−1) − 2 (2𝑥+3) = 5; 𝑥 ≠ 1 , − 3. (CBSE 2014)

2𝑥+3 3𝑥−1 3 2

1. What is the common difference of an A. P. in which a21 − a7 = 84? (CBSE 2017)

2. For what value of k will k + 9, 2k – 1 and 2k + 7 are the consecutive terms of an A. P.?

   
   

3. For what value of k will the consecutive terms 2k + 1, 3k + 3 and 5k – 1 from an A. P.?

   (CBSE 2016)

   
   

   (CBSE 2016)

4. If   1  ,   1  𝑎𝑛𝑑 1 are in A. P., find the value of x. (CBSE 2011)

   𝑥+2

   𝑥+3

   𝑥+5

5. How many two digit numbers are divisible by 3? (CBSE 2019)

6. Find how many two digit natural numbers are divisible by 7? (CBSE 2019)

7. Find how many integers between 200 and 500 are divisible by 8? (CBSE 2017)

8. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

   (CBSE 2014)

9. How many three digit natural numbers are divisible by 7? (CBSE 2013)

10. Find the number of all three-digit natural numbers which are divisible by 9. (CBSE 2013)

11. How many three digit numbers are divisible by 11? (CBSE 2012)

12. Find the number of all two-digit numbers which are divisible by 6. (CBSE 2011)

13. How many natural numbers are there between 200 and 500, which are divisible by 7?

    (CBSE 2011)

14. In an A. P. if the common difference (d) = − 4 and the seventh term (a7) is 4, and then find the first term. (CBSE 2018)

15. Find the 9th term from the end (towards the first term) of the A. P. 5, 9, 13… 185. (CBSE 2016)

16. Find the 25 th term of the A. P. −5, −5 , 0, 5 , …

    (CBSE 2015)

    2 2

17. Which term of the A. P. 3, 15, 27, 39,….will be 120 more than its 21st term? (CBSE 2019)

18. Which term of the progression 20, 19 1 , 18 1 , 17 3, …. Is the first negative term? (CBSE 2017)

    4 2 4

19. Which term of the A. P .8, 14, 20, 26 ….will be 72 more than its 41st term? (CBSE 2017)

20. For what value of n, are the n th term of two APs 63, 65, 67, ..and 3, 10, 17, …. equal?

    (CBSE 2017)

21. If seven times the 7th term of an A. P. is equal to eleven times the 11th term, then what will be its 18th term? (CBSE 2017)

22. The 4th term of an A. P. is zero. Prove that the 25th term of the A. P. is three times its 11th term, therm.

    (CBSE 2016)

23. Find the middle term of the A. P. 6, 13, 20, …, 37. (CBSE 2015)

24. Find the middle term of the A. P. 213, 205, 197, … , 37. (CBSE 2015)

25. The fourth term of an A. P. is 11. The sum of the fifth and seventh term of the A. P. is 34. Find its common difference. (CBSE 2015)

26. In an A. P., the first term is 12 and the common difference is 6. If the last term of the A. P. is 252, find its middle term. (CBSE 2012)

27. Is – 150 a term of the A. P. 17, 12, 7, 2…? (CBSE 2011)

28. Which term of the A. P. 3, 14, 25, 36 … will be 99 more than its 25th term? (CBSE 2011)

29. If the pth term of an AP is q and qth term is p, prove that is nth term is (p + q – n). (CBSE 2017)

30. If the 10th term of an AP is 52 and the 17th term is 20 more than the 13th term, find the AP.

    (CBSE 2017)

31. If the seventh term of an AP is  1 and its ninth term is1, find its 63th term. (CBSE 2014)

    9 7

32. The sum of the 2th and the 7th terms of an AP is 30. If its 15th term is 1 less than twice its 8th term, find the A. P. (CBSE 2014)

33. The sum of the 5th and the 9th term of an A. P. is 30. If its 25th term is three times its 8th term, find the A.

    P. (CBSE 2014)

34. The 8th term of an AP is equal to three times its 3rd term. If its 6th term is 22, find the AP.

    
    

35. The 9th term of an AP is equal to 6 times its 2nd term. If its 5th term is 22, find the AP.

    (CBSE 2013)

    
    

    (CBSE 2013)

36. The 19th term of an AP is equal to three times its 6th term. If its 9th term is 1s 19, find the AP.

    (CBSE 2013)

37. The 8th term of an AP is 31. If its 15th term exceeds its 11th term by 16, find the AP. (CBSE 2013)

38. The 18th term of an AP is 30 more than its 8th term. If the 15th term of the AP is 48, find the AP.

    (CBSE 2013)

39. The 5th term of an AP exceeds its 12th term by 14. If its 7th term is 4, find the AP. (CBSE 2013)

40. The 15th term of an AP is 3 more than twice its 7th term. If the 10th term of the AP is 41, then find its nth term. (CBSE 2012)

41. The 17th term of an AP is 5 more than twice is 8th term. If the 11th term of the AP is 43, then find its nth term. (CBSE 2012)

42. If 4 times the fourth term of an AP is equal to 18 times its 18th term, then find its 22nd term.

    (CBSE 2012)

43. Find the value of the middle term of the AP, -6, -2, 2, …., 58. (CBSE 2011)

44. Determine the AP whose fourth term is 18 and the difference of the ninth term from the fifteenth term is

    30. (CBSE 2011)

45. If the sum of first p term of an A. P. is ap2 + bp, find its common difference. (CBSE 2010)

46. If the sum of first m terms of an A. P. is 2m2 + 3m, then what is its second term? (CBSE 2010)

47. If sn the sum of first n terms of an A. P. is given by sn = 3n2 − 4n, find the nth term.

    
    

48. The sum of the first n terms of an AP is 3n2 + 6n. Find the nth term of this A. P.

    (CBSE 2019)

    
    

    (CBSE 2014)

49. If the sum of first n term of an AP is n2, then find its 10th term. (CBSE 2019)

50. Find the sum of first 8 multiples of 3. (CBSE 2018)

51. How many terms of the A. P. 18, 16, 14 … should be taken so that their sum is zero? (CBSE 2016)

52. How many terms of the A. P. 27, 24, 21 … should be taken so that their sum is zero? (CBSE 2016)

53. How many terms of the A. P. 65, 60, 55 … should be taken so that their sum is zero? (CBSE 2016)

54. In an AP if s5 + s7 = 167 and s10 = 235, then find the A. P., where sn denotes the sum of its first n terms. (CBSE 2015)

55. The first and the last term of an A. P. are 8 and 65 respectively. If sum of all its term is 730, find its common difference. (CBSE 2014)

56. The first and the last term of an A.P. are 7 and 49 respectively. If sum of all its terms is 420, find its common difference. (CBSE 2014)

57. The fist and the last term of an A. P. are 5 and 45 respectively. If the sum of all its terms is 400, fin its common difference. (CBSE 2014)

58. Find the sum of all three digit natural numbers, which are multiplies of 11. (CBSE 2013)

59. Find the sum of all three digit natural numbers, which are multiply of 9. (CBSE 2012)

60. In an A. P., the first term is 2, the last term is 29 and sum of n terms is 155. Find the common difference of the A. P. (CBSE 2010)

61. Find the common difference of an A. P. whose first term is 4, the last term is 49 and the sum of all its terms is 265. (CBSE 2010)

62. How many terms of an A. P. 9, 17, 25 … must be taken to give a sum of 636? (CBSE 2017)

63. If the sum of the first 7 terms of an AP is 49 and that of the first 17 terms is 289, find the sum of its first n terms. (CBSE 2017)

64. The first terms of an AP is 3, the last terms is 83 and the sum of all its terms is 903. Find the number of terms and the common difference of the AP. (CBSE 2018)

65. If the sum of first four terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms. (CBSE 2011, 2019)

66. If the sum of the first 7 terms of an AP is 199 and that of the first 17 terms is 714, find the sum of its first n terms. (CBSE 2012)

67. If the sum of first 9 terms of an AP is equal to sum of first 11 terms, then what is the sum of fist 20 terms? (CBSE 2012)

68. In an AP of 50 terms, the sum of the first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the AP. (CBSE 2014, 2016, 2017)

69. In an AP, the sum of first ten terms is – 150 and the sum of its next ten terms is – 550. Find the AP.

    (CBSE 2010)

70. The sum of the first sixteen terms of an AP is 112 and the sum of its next fourteen terms is 518.

    Find the AP. (CBSE 2010)

71. The first and last terms of an AP are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum? (CBSE 2011)

72. The sum of the first seven terms of an AP is 182. If its 4th and the 17th terms are in the ratio 1: 5, find the

    A. P. (CBSE 2014)

73. Find an AP, whose fourth term is 9 and the sum of its sixth terms and thirteen term is 40.

    (CBSE 2011)

74. If sn denotes the sum of first n-terms of an A. P., prove that S30 = 3[S20 − S10]. (CBSE 2015)

75. If Sn denotes the sum of first n-terms of an A. P., prove that:S12 = 3[S8 − S4]. (CBSE 2015)

76. Find the sum of all multiple of 7 lying between 500 and 900. (CBSE 2012)

77. Find the sum of all multiple of 8 lying between 201 and 950. (CBSE 2012)

78. Find the sum of first 40 positive integers divisible by 6. (CBSE 2012)

79. Find the sum of all odd integers between 1 and 100, which are divisible by 3. (CBSE 2011)

80. Find the sum of the first 30 positive integers divisible by 6. (CBSE 2011)

81. The sum of three numbers in AP is 12 and sum of their cubes is 288. Find the numbers.

    (CBSE 2016)

82. Find the 60th term of the AP 8, 10, 12 … if it has a total of 60 terms and hence find the sum of its last 10 term. (CBSE 2016)

83. The 14th term of an AP is twice its 8th term. If its 6th term is -8, then find the sum of its first 20 terms.

    (CBSE 2015)

84. The 16th term of an AP is five times its third term. If its 10th term is 41, then find the sum of its first fifteen terms. (CBSE 2015)

85. In an AP, if the 12th term is -13 and the sum of its first four terms is 24, find the sum of its first ten terms.

    (CBSE 2015)

86. The sum of the first 7 terms of an AP is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A. P. (CBSE 2015)

87. The nth term of an AP is given by (-4n + 15). Find the sum of first 20 terms of first 20 terms of this

    AP. (CBSE 2013)

88. The sum of first n-terms of an AP is 3n2 + 4n. Find the 25th terms of this AP. (CBSE 2013)

89. Find the sum of first-n-terms of an AP. whose nth term is 5n – 1. Hence find the sum of first 20

    terms. (CBSE 2011)

90. In an AP, if the sum of 4th and the 8th terms is 70 and its 15th terms is 80, then find the sum of its

    first 20 terms. (CBSE 2011)

91. An A. P. 5, 12, 19… has 50 terms, find its last terms. Hence find the sum of its last 15 terms.

    (CBSE 2015)

92. The sum of the first 15 terms of an AP is 750 and its first terms is 15. Find its 20th term.

    (CBSE 2012)

93. Sum of the first 20 terms of an AP is – 240, and its first term is 7. Find its 24th term. (CBSE 2012)

94. Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25th term. (CBSE 2012)

95. Find the common difference of an AP whose first term is 5 and the sum of its first four terms is

    half the sum of the next four terms. (CBSE 2012)

96. The sum of 4th and 8th terms of an AP is 24 and the sum of its 6th and 10 th terms is 44. Find the

    sum of first ten terms of the AP. (CBSE 2012)

97. The sum of the first five terms of an AP is 25 and the sum of its next five terms is -75. Find the

    10th terms of the AP. (CBSE 2012)

98. The sum of the third and seventh terms of an A.P. is 40 and the sum of its sixth and 14th terms is

    70. Find the sum of the first ten terms of the A.P. (CBSE 2012)

99. In an AP, if the sum of its 4th and 10th terms is 40, and the sum of its 8th and 16th terms is 70, then

    find the sum of its first twenty terms. (CBSE 2011)

100. The ratio of the sums of first m and fist m and first n terms of an AP is m2: n2, show that the ratio of its mth and nth terms is (2m – 1): (2n – 1). (CBSE 2017)

101. If the ratio of the sum of the first n terms of two APs is (7n + 1): (4n + 27), then find the ratio of

     their 9th terms. (CBSE 2017)

102. The digits of a positive number of three digits are in A. P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.

     (CBSE 2016)

103. Find the number of terms of the AP: 18, 15 1, 13 …(−49 1) and find the sum of all its terms.

     2 2

     (CBSE 2013)

104. If m times the nth term of an Arithmetic progression is equal to n times its nth term and m = n,

     show that the (𝑚 + 𝑛)𝑡ℎ term of the AP is zero. (CBSE 2011)

105. The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and

     the last term to the product of two middle terms is 7: 15. Find the numbers. (CBSE 2018)

106. Find the middle term of the sequence formed by all three-digit numbers which leave a remainder

     3, when divided by 4. Also find the sum of all numbers on both sides of the middle terms

     separately. (CBSE 2015)

107. Find the middle term of the sequence formed by all numbers between 9 and 95, which leave a remainder 1 when divided by 3. Also find the sum of the numbers on both sides of the middle

     term separately. (CBSE 2017)

108. The sum of first m terms of an AP is 4m2 − m. If its nth term is 107, find the value of n. Also,

     find the 21st term of this AP. (CBSE 2013)

109. If mth term of an A. P. is  1 and nth term is 1 , and then find the sum of its first mn term.

     𝑛 𝑚

     
     

110. Find the sum of n terms of the series (4 − 1) + (4 − 2) + (4 − 3) + ⋯

     (CBSE 2017)

     (CBSE 2017)

     𝑛 𝑛 𝑛

111. Find the sum of the following series:

     5 + (−41) + 9 + (−39) + 13 + (−37) + 17 + ⋯ + (−5) + 81 + (−3) (CBSE 2017)

112. The sum of first n terms of three arithmetic progression are S1, S2 and S3 respectively. The first

     Term of each AP is 1 and their common difference is 1, 2 and 3 respectively. Prove that S1 + S3 = 2S2. (CBSE 2016)

     
     

113. If the sum of the first n-terms of an AP is 1 (3𝑛2 + 7𝑛), then find its nth term. Hence write its 20th

     2

     Term. (CBSE 2015)

114. If the sum of first m terms of an AP is the same as the sum of its first n term, show that the sum

     of its first n terms, show that the sum of its first (m + n) terms is zoo. (CBSE 2017)

115. Find the number of terms of the AP −12, −9, −6… 21. If 1 is added to each term of this AP,

     them Find the sum of all terms of the AP thus obtained. (CBSE 2013)

116. Divide 56 in four parts in A. P. such that the ration of the product of their extremes (1st and 4th)

     to the product of means (2nd and 3rd ) is 5 : 6. (CBSE 2016)

117. The pth, qth and rth terms of an AP are a, b and c respectively, show that a (q – r) + b(r – p) + c (p

     – q) = 0. (CBSE 2015)

118. Which term of the Arithmetic progression −7, −12, −17, −22…. will be -82. Is – 100 any term

     of the AP? Give reason for your answer. (CBSE 2019)

119. The 24th term of an AP is twice its 10th term. Show that its 72nd term is four times its 15th term.

     (CBSE 2013)

120. A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after the Thief to catch him. He goes with a speed of 10 m/minutes in the first minutes and increases his Speed by 10 m/minutes every succeeding minutes. After how many minutes the policeman will

     catch the thief? (CBSE 2016)

121. The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of x such that sum of numbers of houses proceeding the house numbered x is equal to sum to the

     number of houses following x. (CBSE 2016)

122. Ramkali required ₹2,500 after 12 weeks to send her daughter to school. She saved ₹100 in the first week and increased her weekly saving by ₹20 every seek. Find whether she will be able to

     send her daughter to school after 12 weeks or not. (CBSE 2015)

123. In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class has two sections, find how many

     trees were planted by the students. (CBSE 2014)

124. A sum of ₹1600 is to be used to give ten cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of

the prizes. (CBSE 2012)

1. In ∆𝐷𝐸𝑊, 𝐴𝐵 || 𝐸𝑊. If AD = 4 cm, DE = 12 cm and DW = 24 cm, then find the value of DB.

   (CBSE 2015)

2. In figure, 𝐷𝐸 || 𝐵𝐶 in ∆𝐴𝐵𝐶 such that BC = 8 cm, AB = 6 cm and DA = 1.5 cm. Find DE.

   (CBSE 2010)

   
   

3. In figure, 𝑀𝑁 || 𝐴𝐵, BC = 7.5 cm, AM = 4cm and MC =2 cm. Find the length BN . (CBSE 2010)

   
   

   
   

4. R and S are points on the sides DE and EF respectively of ∆DEF such that ER = 5 cm, RD = 2.5 cm, SE =

   1.5 cm and FS = 3. 5 cm. Find whether 𝑅𝑆 || 𝐷𝐹 or not. (CBSE 2016)

5. In the figure, D and E are points on AB and AC respectively such that 𝐷𝐸 || 𝐵𝐶. If AD = 1/3 BD and AE

   = 4. 5 cm, Find AC. (CBSE 2014)

   
   

6. In the given figure, 𝑃𝑄 || 𝐵𝐴; 𝑃𝑅 ||𝐶𝐴. If PD = 12 cm, find BD ×CD. (CBSE 2012)

   
   

   
   

7. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then other two sides are divided in the same ratio. (CBSE 2014)

8. If ∆𝐴𝐵𝐶~∆𝑅𝑃𝑄, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm, and PQ = 10 cm, then find QR.

   (CBSE 2014)

9. In the figure, PQR and SQR are two left triangles with common hypotenuse QR. If PR and SQ intersect at M such that PM = 3 cm, MR = 6 cm SM = 4cm, find the length of MQ. (CBSE 2013)

10. In the given figure, if 𝐴𝐵 || 𝐷𝐶, find the value of x. (CBSE 2012)

    
    

    
    

11. If one diagonal of a trapezium divides the other diagonal in the ratio 1:3. Prove that one of the parallel sides is three times the other. (CBSE 2011)

12. In the figure, ABCD is a parallelogram and E divides BC in the ratio 1: 3, DB and AE intersect at F. Show that DF = 4FB and AF = 4FE (CBSE 2016)

    
    

13. In figure, ∠CAB = ∠CED, then prove that AB × DC = ED × BC. (CBSE 2015)

    
    

    
    

14. State whether the given pairs of triangles are similar or not. In case of similarity mention the criterion.

    (CBSE 2015)

    
    

    (a) (b)

    
    

15. left angled triangle BAC and BDC are left angled AT A and D and they are on same side of BC. If AC and BD intersect at P, then prove that AP × PC = PB × DP. (CBSE 2014)

16. In the figure, DB ⊥ BC, DE ⊥ AB and AC ⊥ BC. Prove that 𝐵𝐸 = 𝐴𝐶 . (CBSE 2013)

    𝐷𝐸 𝐵𝐶

17. In figure, 𝐴𝐵 ||𝑃𝑄 ||𝐶𝐷, 𝐴𝐵 = 𝑥 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 𝑃𝑄 = 𝑧 units, prove that 1 + 1

    𝑥 𝑦

    = 1.

    𝑧

    (CBSE 2011)

18. In figure, ABCD is an isosceles triangle in which AB = AC. E is a point on the side CB produced, such that FE ⊥ AC. If AD ⊥ CB, prove that AB × EF = AD × EF = AD × EC. (CBSE 2010)

    
    

19. In ∆𝐴𝐵𝐶, from A and B altitudes AD and BE are drawn. Prove that ∆𝐴𝐷𝐶~∆𝐵𝐸𝐶. Is ∆𝐴𝐷𝐵~∆𝐴𝐸𝐵 and

    ∆𝐴𝐷𝐵~∆𝐴𝐷𝐶? (CBSE 2015)

20. In ∆𝐴𝐵𝐶, if ∠ADE = ∠B, then prove that ∆𝐴𝐷𝐸~∆𝐴𝐵𝐶. Also, if AD = 7. 6 cm, AE = 7. 2 cm, BE = 4. 2 cm and BC = 8. 4 cm, then find DE. (CBSE 2015)

1. Find the distance of a point p(x, y) from the origin. (CBSE 2018)

2. What is the distance between the points A(c, 0) and B (0, − C)? (CBSE 2010)

3. If the distance between the point (4, k) and (1, 0) is 5, them what can be the possible values of k?

   (CBSE 2017)

4. If the distance of P(x, y) from A (5, 1) and B (−1, 5) are equal, then prove that 3x = 2y.

   (CBSE 2017)

5. If a point A (0, 2) is equidistant from the points B (3, p) and C (P, 5), then find the value of p.

   or

   Find the value of k for which the point (0, 2) is equidistant from two points (3, k) and (k, 5).

   (CBSE 2012)

6. Find the value of k, if the point P (2, 4) is equidistant from the points A (5, k) and B (k, 7).

   (CBSE 2013)

7. Write the coordinate of a point P on x-axis which is equidistant from the points A (-2, 0) and B (6, 0).

   (CBSE 2012)

8. Find the point on y-axis which is equidistant from the points (5, −2) and (−3, 2). (CBSE 2018)

9. The x-coordinate of a point p is twice its y-coordinate. If p is equidistant from Q (2, −5) and R (−3, 6), find the coordinate of p. (CBSE 2019)

10. Find the value of x for which the distance between the points P (x, 4) and Q (9, 10) is 10 units.

    (CBSE 2019)

11. Find the value of y for which the distance between the points A (3, −1) and B (11, y) is 10 units.

    (CBSE 2016)

12. Find a point on x-axis which is equidistant from A (4, −3) and B (0, 11). (CBSE 2012)

13. Find the points on y-axis which is equidistant from the points (−5, −2) and (3, 2). (CBSE 2012)

14. If the point P (x, y) is equidistant from the points A (a + b, b –a) and B (a-b, a + b), prove that bx = ay.

    (CBSE 2012)

15. If the point A (0, 2) is equidistant from the points B (3, p), and C(P, 5), and p. Also find the length of AB. (CBSE 2012)

16. If the point P (k – 1, 2) is equidistant from the points A (3, K) and B (k, 5), find the value of k.

    (CBSE 2016)

17. Find a point P on the y-axis which is equidistant from the points A (4, 8) and B (−6, 6). Also find the distance AP. (CBSE 2014)

18. Find the ratio in which the segment joining the points (1, −3) and (4, 5) is divided by x-axis? Also find the coordinate of this point on x-axis. (CBSE 2014)

19. The line segment joining the points A (2, 1) and B (5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lines on the line given by 2x – y + k = 0, find the value of k.

    (CBSE 2014)

20. Find the ratio in which P (4, m) divides the line segment joining the points A (2, 3) and (6, –3). Hence find m. (CBSE 2019)

21. Find the ratio in which y-axis divides the line segment joining the points A (5, –6) and B (–1, –4). Also find the coordinate of the point of division. (CBSE 2016)

22. Find the ratio in which the point (–3, k) divides the line-segment joining the points (–5, -4) and (–2, 3). Also find the value of k. (CBSE 2016)

23. Find the ratio in which the point 𝑃 (3 , 5 ) divides the line segment joining the points 𝐴 (1 , 3) and B (2, –

    4 12 2 2

    5). (CBSE 2015)

24. In what ratio does the point (24 , 𝑦) divides the line segment joining the points P (2, –2) and Q (3, 7)?

    11

    Also find the value of y. (CBSE 2017)

25. Find the coordinate of the points of trisection of the line segment joining the points (3, –2) and (–3, –4).

    (CBSE 2017)

26. If the point C (–1, 2) divides internally the line-segment joining the points A (2, 5) and B (x, y) in the ratio 3: 4, find the value ofx2 + y2. (CBSE 2016)

27. If the coordinate of points A and B are (–2, –2) and (2, –4) respectively, find the coordinate of P such that 𝐴𝑃 = 3 𝐴𝐵, where P lines on the line segment AB. (CBSE 2015)

    7

28. Find the coordinate of a point P on the line segment joining A (1, 2) and B (6, 7) such that AP = 2 AB.

    5

    (CBSE 2015)

29. Find the ratio in which the line segment joining the points A (3, –3) and B (–2, 7) is divided by x-axis. Also find the coordinate of the point of division. (CBSE 2014)

30. Points P, Q, R and S divide the line segment joining the points A (1, 2) and B (6, 7) in 5 equal parts. Find the coordinate of the points P, Q and R. (CBSE 2014)

31. Find the ratio in which the y-axis divides the line segment joining the points (−4, -6) and (10, 12). Also find the coordinates of the point of division. (CBSE 2013)

32. Find the ratio in which point P (−1, y) lying on the line segment joining points A (−3, 10) and B (6, −8) divides it. Also find the value of y. (CBSE 2013)

33. Find the ratio in which the y-axis divides the line segment joining the points (5, −6) and (−1, −4). Also find the coordinate of the point of intersection. (CBSE 2012)

34. Point P divides the line segment joining the points A (−1, 3) and B (9, 8) such that (𝐴𝑃

    𝑃𝐵

    = 𝐾). If P lies on

    1

    the line x – y + 2 = 0, find the value of k. (CBSE 2012)

35. Point M (11, y) lies on the segment joining the points P (15, 5) and Q (9, 20). Find the ratio in which point M divides the line segment PQ. Also find the value of y. (CBSE 2012)

36. A point P divides the line segment joining the points A (3, −5) and B (−4, 8) such that 𝐴𝑃 = 𝐾. If P lies

    𝑃𝐵 1

    on the line x + y = 0, then find the value of k. (CBSE 2011)

37. If R (x, y) is point on the segment joining the points P (a, b) and Q (b, a), then prove that x + y = a + b.

    (CBSE 2011)

38. If point 𝑃 (1 , 𝑦) lines on the line segment joining the points A (3, −5) and B (−7, 9), then find the ratio

    2

    in which P divides AB. Also find the value of y. (CBSE 2014)

39. Find the ratio in which the point P (x, 2) divides the line segment joining the points A (12, 5) and B (4,

    −3). (CBSE 2019)

40. Find the coordinate of a point A, where AB is diameter of a circle whose centre is (2, −3) and B is the point (1, 4). (CBSE 2014)

41. Points A (−1, y) and B (5, 7) lie on a circle with centre O (2, −3y). Find the value of y. Hence, find the radius of the circle. (CBSE 2019)

42. Find the coordinate of a point A, where AB is a diameter of the circle with centre (−2, 2) and B is the point with coordinate (3, 4). (CBSE 2017)

43. A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, −5) is the mid-point of PQ. Then find the coordinate of P and Q. (CBSE 2017)

44. If two adjacent vertices of a parallelogram are (3, 2) and (−1, 0) and the diagonals intersect at (2, −5), then find the coordinates of the other two vertices. (CBSE 2010)

45. If P (2, p)is the mid-point of the line segment joining the points A (6, −5) and B (−2, 11), find the value of p. (CBSE 2010)

46. If A (1, 2), B (4, 3) and C (6, 6) are three vertices of parallelogram ABCD, find co-ordinate of D.

    (CBSE 2018)

47. If A (−2, 1), B (a, 0), C (4, b) and D (1, 2) are the vertices of a parallelogram ABCD, find the value of a and b. Hence find the lengths of its sides. (CBSE 2012)

48. If (3, 3), (6, y), (x, 7) and (5, 6) are the vertices of a parallelogram taken in order, find the value of x and

    y. (CBSE 2018)

49. Show that triangle ABC, where A (−2, 0), B (2, 0), C (0, 2) and triangle PQR where P (−4, 0) Q (4, 0),

    R (0, 4) are similar triangles. (CBSE 2017)

50. If A (5, −2), B (2, −2) and C (−2, t) are the vertices of left angled triangle with ∠B = 90°, then find the value of t. (CBSE 2015)

51. The points A (4, 7), B (p, 3) and C (7, 3) are the value the vertices of a triangle, left-angled at B. Find the value of p. (CBSE 2015)

52. If A (4, 3), B (−1, y) and C (3, 4) are the vertices of left triangle ABC, left-angled at A, then find the value of y. (CBSE 2015)

53. The base BC of an equilateral triangle ABC lies on y-axis. The coordinate of points C are (0, −3). The origin is the mid-point of the base. Find the coordinates of the points A and B. Also find the coordinates of another point D such that BACD is a rhombus. (CBSE 2015)

54. Prove that the points (7, 10), (−2, 5) and (3, −4) are the vertices of an isosceles left triangle.

    (CBSE 2013)

55. Prove that the points A (0, −1), B (3, −4), C (6, 7) and D (8, 3) the vertices of a rectangle ABCD.

    (CBSE 2013)

56. Show that the points (−2, 3) (8, 3) and (6, 7) and D (8, 3) are the vertices of a left triangle.

    (CBSE 2013)

57. The mid-points P of the line segment joining the points A (−10, 4) and B (−2, 0) lies on the line segment joining the points C (−9, −4) and D (−4, y). Find the ratio in which P divides CD. Also find the value of y. (CBSE 2014)

1. If 𝑐𝑜𝑠𝑒𝑐 𝜃 = 5, then what is the value of cos 𝜃 + tan 𝜃. (CBSE 2014)

   3

2. Find the value of cos 𝜃 + sec 𝜃, when it is given that cos 𝜃 = 1.

   2

   (CBSE 2014)

   4 sin 𝜃−cos 𝜃+1

3. If 4 tan 𝜃 = 3, evaluate( ). (CBSE 2018)

   4 sin 𝜃+cos 𝜃−1

   12

   2 𝜃−𝑐𝑜𝑠 2 𝜃 1

4. If sin 𝜃 =     , 0° < 𝜃 < 90°, find the value of: 𝑠𝑖𝑛

   13 2 𝑠𝑖𝑛𝜃

   .𝑐𝑜𝑠𝜃

   ×

   𝑡𝑎𝑛 2𝜃

   . (CBSE 2016)

5. If tan (A – B) = 1

   √3

   and tan (A + B)

   
   

   = √3,

   find A and B. (CBSE 2014)

6. ∆𝐴𝐵𝐶𝐷 is left angles at B, BC = 7 cm an AC – AB = 1. Find the value of cos A + sin A.

   
   

7. If cosec (A – B) = 2, cot (A + B) =  1 , 0° < (𝐴 + 𝐵) ≤ 90°, 𝐴 > 𝐵, 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝐴 𝑎𝑛𝑑 𝐵.

   √3

   
   

   (CBSE 2011)

   
   

   (CBSE 2011)

8. If cosec 𝜃 + cot 𝜃 = q, show that cosec 𝜃 - cot 𝜃 = 1 and hence find the values of sin 𝜃 and sec 𝜃.

   𝑞

   (CBSE 2014)

9. ∆𝑅𝑃𝑄 Is a left angled at Q. If PQ = 5 cm and RQ = 10 cm, find: (CBSE 2014)

   (i) sin2 𝑃 (ii) cos2 𝑅 𝑎𝑛𝑑 tan 𝑅 (iii) sin P × cos P (iv) sin2P − cos2P

10. Evaluate: 3 cot260°+sec245°. (CBSE 2014)

    
    

11. If √3 sin 𝜃 − cos 𝜃 = 0 and 0° < 𝜃 < 90°, find the value of c 𝜃 . (CBSE 2014)

12. If sin A = √3, find the value of 2cot2A − 1. (CBSE 2012)

    2

13. Write the values of sec 0°, sec 30°, 𝑠𝑒𝑐 45°, sec 60° 𝑎𝑛𝑑 sec 90°. What happens to sec x when x increases from 0° to 90°? (CBSE 2012)

14. Evaluate: 𝑡𝑎𝑛 2 60°+4𝑠𝑖𝑛 2 45°+3𝑠𝑒𝑐 230°+5𝑐𝑜𝑠 2 90°

    𝑐𝑜𝑠𝑒𝑐 30°+sec 60°−𝑐𝑜𝑡 230°

    . (CBSE 2016)

15. Find the value of cosec 30° geometrically. (CBSE 2011)

16. Find the value of sec 60° geometrically. (CBSE 2010)

17. Find the value of cosec 60° geometrically. (CBSE 2010)

18. If sin A = cos A, find the value of 2𝑡𝑎𝑛2A + 𝑠𝑖𝑛2A + 1. (CBSE 2010)

    
    

19. ABC is a triangle left angled at C and AC = √3 BC. Prove that ∠ABC = 60°. (CBSE 2014)

20. Given that cos (A – B) = cos A. cos B + sin A. sin B, find the value of cos 15° in two ways:

    (CBSE 2013)

    1. Taking A = 60°, B = 45° and

    2. Taking A = 45°, 𝐵 = 30°.

21. Evaluate: 4𝑐𝑜𝑡 2 60°+𝑠𝑒𝑐 2 30°−2𝑠𝑖𝑛 245°

    𝑠𝑖𝑛 2 60°+𝑐𝑜𝑠 245°

    
    

    . (CBSE 2014)

22. Evaluate:     4     + 1 − 𝑐𝑜𝑠245°. (CBSE 2013)

    𝑐𝑜𝑡 2 30° 𝑠𝑖𝑛 2 60°

23. Evaluate: 4(𝑠𝑖𝑛430° + 𝑐𝑜𝑠460°) − 3(𝑐𝑜𝑠245° − 𝑠𝑖𝑛290°).

24. Determine the value of x such that 2 cosec230° + xsin260° − 3 tan2 30° = 10. (CBSE 2013)

    1 1 4

25. If tan 𝜃 +       = 2, find the value of: tan2θ +    . (CBSE 2013)

    𝑡𝑎𝑛𝜃 tan 2 θ

26. If (1 + cos A)(1 – cos A) = 3, find the value of sec A. (CBSE 2011)

    4

27. If 𝑐𝑜𝑠𝑒𝑐 𝜃 + 𝑐𝑜𝑡𝜃 = 𝑥, find the value of cosec 𝜃 − 𝑐𝑜𝑡𝜃. (CBSE 2016)

28. Find the value of (𝑐𝑜𝑠𝑒𝑐2𝜃 − 1). 𝑡𝑎𝑛2𝜃. (CBSE 2014)

    
    

29. If 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 = √2 𝑐𝑜𝑠𝜃, show that 𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛𝜃 = √2 𝑠𝑖𝑛𝜃. (CBSE 2016)

30. If 3𝑥 = 𝑐𝑜𝑠𝑒𝑐𝜃 𝑎𝑛𝑑 3 = 𝑐𝑜𝑡𝜃, find the value of 3 (𝑥2 − 1 ). (CBSE 2016)

    𝑥 𝑥2

31. If 2𝑥 = sec 𝐴 𝑎𝑛𝑑 2 = tan 𝐴, find the value of 2 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

32. If 𝑐𝑜𝑠𝑒𝑐𝜃 = 2𝑥 𝑎𝑛𝑑 𝑐𝑜𝑡𝜃 = 2, find the value of 2 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

33. If 5𝑥 = 𝑠𝑒𝑐𝜃 𝑎𝑛𝑑 5 = 𝑡𝑎𝑛𝜃, find the value of 5 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

34. If 7𝑥 = 𝑐𝑜𝑠𝑒𝑐𝜃 𝑎𝑛𝑑 7 = 𝑐𝑜𝑡𝜃, find the value of (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

35. If 6𝑥 = 𝑠𝑒𝑐𝜃 𝑎𝑛𝑑 6 = 𝑡𝑎𝑛𝜃, find the value of 9 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

36. If 8𝑥 = 𝑐𝑜𝑠𝑒𝑐 𝐴 𝑎𝑛𝑑 8 = cot 𝐴, find the value of 4 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

37. If 4𝑥 = 𝑠𝑒𝑐𝜃 𝑎𝑛𝑑 4 = 𝑡𝑎𝑛𝜃, find the value of 8 (𝑥2 − 1 ). (CBSE 2010)

    𝑥 𝑥2

38. Prove that:   cos 𝐴  + 1+sin 𝐴 = 2𝑎𝑒𝑐 𝐴. (CBSE 2016)

    1+sin 𝐴

    cos 𝐴

    3 𝜃𝑐𝑜𝑠 3 𝜃

39. Prove the identity:  𝑠𝑖𝑛            = 1 − 𝑠𝑖𝑛𝜃. 𝑐𝑜𝑠𝜃. (CBSE 2015)

    sin 𝜃+𝑐𝑜𝑠𝜃

40. Prove that: (𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝑒𝑐𝜃)2 + (𝑐𝑜𝑠𝜃 + 𝑠𝑒𝑐𝜃)2 = 7 + 𝑡𝑎𝑛2𝜃 + 𝑐𝑜𝑡2𝜃. (CBSE 2019)

41. Prove that:(1 + cot 𝐴 − 𝑐𝑜𝑠𝑒𝑐 𝐴)(1 + 𝑡𝑎𝑛 𝐴 + sec 𝐴) = 2. (CBSE 2019)

42. Prove that:        1         −   1   =   1  −       1        . (CBSE 2016)

    sec 𝐴−tan 𝐴 cos 𝐴 cos 𝐴 sec 𝐴+tan 𝐴

43. For any acute angle 𝜃, prove that: (CBSE 2015)

    1. sin2θ + cos2θ = 1 (b) 1+ cot2θ = cosec2θ

       
       

44. If 𝑡𝑎𝑛𝜃 + 𝑐𝑜𝑡𝜃 = 2, find the value of √𝑡𝑎𝑛2𝜃 + 𝑐𝑜𝑡2𝜃. (CBSE 2014)

45. If x = r 𝑐𝑜𝑠𝜃. 𝑠𝑖𝑛∅′𝑦 = 𝑟 sin 𝜃. sin ∅; 𝑧 = 𝑟 𝑐𝑜𝑠∅. Prove that x2 + y2 + z2 = r2. (CBSE 2014)

46. Prove that:  cot 𝐴−cos 𝐴 = 𝑐𝑜𝑠𝑒𝑐 𝐴−1. (CBSE 2013)

    cot 𝐴+cos 𝐴 𝑐𝑜𝑠𝑒𝑐 𝐴+1

47. Show that: √1+cos 𝑎 = 𝑐𝑜𝑠𝑒𝑐 𝑎 + cot 𝑎. (CBSE 2013)

    1−cos 𝑎

    
    

48. If cos 𝜃 − sin 𝜃 = √2 sin 𝜃, Prove that cos 𝜃 + sin 𝜃 = √2 cos 𝜃. (CBSE 2010)

49. Prove that: (𝑐𝑜𝑠𝑒𝑐 𝜃 − sin 𝜃). (sec 𝜃 − cos 𝜃 = 1 . (CBSE 2010)

    tan 𝜃+cot 𝜃

50. Prove that: (1 + cot 𝐴 − 𝑐𝑜𝑠𝑒𝑐 𝐴)(1 + tan 𝐴 + sec 𝐴) = 2. (CBSE 2019)

51. Prove that: sin 𝜃(1 + tan 𝜃) + (1 + cot 𝜃) = sec 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃. (CBSE 2019)

52. Prove that:  sin 𝐴−cos 𝐴+1 = 1 . (CBSE 2018)

    sin 𝐴+cos 𝐴−1 sec 𝐴−tan 𝐴

53. Prove that: 𝑡𝑎𝑛 2 𝐴

    𝑐𝑜𝑠𝑒𝑐 2 𝐴         1    

    𝑡𝑎𝑛 2 𝐴−1 + 𝑠𝑒𝑐 2 𝐴− 𝑐𝑜𝑠𝑒𝑐 2𝐴 = 1−2𝑐𝑜𝑠 2 𝐴. (CBSE 2015)

54. Prove that: sin 𝜃−2 𝑠𝑖𝑛3 𝜃

    
    

    2 𝑐𝑜𝑠 3 𝜃−cos 𝜃

    = tan 𝜃 (CBSE 2014)

    
    

55. Prove that: √𝑠𝑒𝑐2𝜃 + 𝑐𝑜𝑠𝑒𝑐2𝜃 = tan 𝜃 + cot 𝜃 (CBSE 2014)

56. Prove that: sin 𝐴+cos 𝐴 + sin 𝐴−cos 𝐴 = 2 . (CBSE 2013)

    sin 𝐴−cos 𝐴 sin 𝐴+cos 𝐴 1−2 𝑐𝑜𝑠 2 𝐴

    2 −1

57. If 𝑐𝑜𝑠𝑒𝑐 𝐴 + cot 𝐴 = 𝑚, show that 𝑚 = cos 𝐴. (CBSE 2011)

    𝑚 2 +1

58. Prove that: (sec 𝜃 + tan 𝜃)2 = 𝑐𝑜𝑠𝑒𝑐 𝜃+1. (CBSE 2014)

    𝑐𝑜𝑠𝑒𝑐 𝜃−1

59. Prove that:   tan 𝐴  + tan 𝐴

    = 2𝑐𝑜𝑠𝑒𝑐 𝐴. (CBSE 2014)

    sec 𝐴−1 sin 𝐴+1

60. Prove that:         1          −   1   =   1  −         1      

    (CBSE 2013)

    𝑐𝑜𝑠𝑒𝑐 𝐴+cot 𝐴 sin 𝐴 sin 𝐴 𝑐𝑜𝑠𝑒𝑐 𝐴−cot 𝐴

61. Prove that: √1+sin 𝐴 + √1−sin 𝐴 = 2(sin 𝐴 tan 𝐴 + cos 𝐴. (CBSE 2011)

    1−sin 𝐴 1+sin 𝐴

62. If sec 𝜃 = 𝑥 + 1 , 𝑥 ≠ 0, find (sec 𝜃 + 𝑡𝑎𝑛𝜃). (CBSE 2019)

    4𝑥

    Or If sec 𝐴 = 𝑥 + 1 , prove that sec 𝐴 + tan 𝐴 = 2𝑥 𝑜𝑟 1 (CBSE 2016)

    4𝑥 2𝑥

63. If sec 𝜃 − 𝑡𝑎𝑛𝜃 = 𝑥, show that: sec 𝜃 1 (𝑥 + 1) and tan 𝜃 = 1 (1 − 𝑥) . (CBSE 2016)

    2 𝑥 2 𝑥

64. If 7𝑠𝑖𝑛2𝐴 + 3𝑐𝑜𝑠2𝐴 = 4, show that tan 𝐴 = 1

    √3

    . (CBSE 2016)

65. If sin 𝜃 = 𝑥 𝑎𝑛𝑑 sec 𝜃 = 𝑦 then find the value of cot 𝜃. (CBSE 2015)

    = 3. (CBSE 2014)

66. Solve the equation for 𝜃: 𝑐𝑜𝑠 2 𝜃

    𝑐𝑜𝑡 2 𝜃−𝑐𝑜𝑠 2 𝜃

67. Express cos 𝐴 in terms of cot 𝐴. (CBSE 2014)

68. If sec 𝜃 + tan 𝜃 = 𝑝, then find the value of 𝑐𝑜𝑠𝑒𝑐 𝜃. (CBSE 2013)

69. In an acute angled triangle ABC, If sin (A + B + C) = 1 and cos(𝐵 + 𝐶 − 𝐴) = 1 , 𝑓𝑖𝑛𝑑 ∠A,

2 √2

∠B and ∠C. (CBSE 2013)

1. Two ratio of the height of a tower and length of its shadow on the ground is √3 ∶ 1. What is the angle of elevation of the sun? (CBSE 2017)

2. If a tower 30 m height, casts a shadow 10 √3 m long on the ground, then what is the angle of elevation of the sun? (CBSE 2017)

3. A ladder 15 m long makes an angle of 60° with the wall. Find the height of the point where the ladder touches the wall. (CBSE 2017)

4. If figure 1, AB is a 6 m height pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder. (CBSE 2016)

   
   

5. A ladder, leaning against a wall, makes an angle of 60° with the horizontal. If the fool of the ladder is 2.5 m away from the wall, find the length of the ladder. (CBSE 2016)

6. An observer, 1.7 m tall, is 20 √3 m away from a tower. The angle of elevation from the eye of observer to the top of tower is 30° . Find the height of tower. (CBSE 2015)

7. The tops of two tower of height x and y, standing on level ground, subtend angle of 30° and 60° respectively at the centre of the line joining their feet, then find x : y. (CBSE 2015)

8. In figure 2, a tower AB is 20 m high and BC, its shadow on the ground, is 20 √3 m long. Find the sun’s altitude. (CBSE 2017)

   
   

9. A pole casts a shadow of length 20 √3 m on the ground, when the sun’s elevation is 60°. Find the height of the pole. (CBSE 2017)

10. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the towed is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill. (CBSE 2017)

11. On a straight line passing through the foot of a tower, two points C and D are at distance of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then the height of the tower. (CBSE 2016)

12. The shadow of a tower at a time is three times as long as its shadow when the angle of elevation of the sun is 60°. Find the angle of elevation of the sun at the time of the longer shadow.

    (CBSE 2016)

13. The angle of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectvely. Find the height of the tower and the horizontal distance between the tower and the building. (CBSE 2016)

14. Two men on either side of a 75 m high building and in line with base of building observe the angles of elevation of the top of the building as 30° and 60°. Find the distance between the two men.

    (CBSE 2016)

15. A 7 m long flagstaff is fixed on the top of the tower standing on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 60° and 45° respectvely. Find the height of the tower correct to one place of decimal. (CBSE 2015)

16. An aeroplane, when flying at a height of 4000 m from the ground passes vertically above another aeroplane at an instant when the angle of elevation of the two planes from the same point on the ground and 60° and 45° respectvely. Find the vertical distance between the aeroplanes at the instant.

    (CBSE 2014)

    
    

17. From the top of a tower of tower 50 m, the angles of depression of the top and bottom of a pole are 30° and 45° respectvaly, find: (a) How far the poles is from the bottom of a tower? (b) The height of the pole. (CBSE 2015)

18. Two ships are three in the sea on either sides of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two shops as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. (CBSE 2014)

19. Two ships are approaching a lighthouse from opposite directions. The angles of depression of the ships from the top of the lighthouse are 30° and 45°. If the distance between the two ships is 100 m, find the height of the light house. (CBSE 2014)

20. As observed from the top of a 60 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (CBSE 2013)

21. The angled of depression of two ships from the top of a lighthouse and on the same side of it are found to be 45° and 30°. If the ships are 200 m apart, find the height of the lighthouse.

    (CBSE 2012)

22. The horizontal distance between two poles is 15m. The angle of depression of top of first pole as seen from the top of second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole. (CBSE 2013)

23. The angles of elevation of the top of a tower from two points at a distance of 6 m and 13.5 from the base of the tower and in the same straight line with it are commentary. Find the height of the tower.

    (CBSE 2013)

24. A kite is flying at a height of 45 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the lenth of the string assuming that there is no slack in the sting. (CBSE 2012)

25. The angle of depression of the top and bottom of tower as seen from the top of a 60 √3 m high cliff are 45° and 60° respectively. Find the height of the tower. (CBSE 2012)

26. From the top of a tower 50 m high, the angle of depression of the top of a pole is 45° and from the foot of the pole, the angle of elevation of the top of the tower is 60°. Find the height of the pole if the pole and tower stand on the same plane. (CBSE 2012)

27. The angle of depression from the top of a tower of a point A on the ground is 30°. On moving a distance of 20 m from the point A towards the foot on the tower to a point B, the angle of elevation of the top of the tower from the point B is 60°. Find the height of the tower and its distance from the point A.

    (CBSE 2012)

28. From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30° and 45° repectively. Find the distance between the cars.

    (CBSE 2017)

29. From the top of a vertical tower, the angles of depression of two cars in the same straight line with the base of the tower, at an instant are found to be 45° and 60°. If the cars are 100 m apart and are on the same side of the tower, find the height of the tower. (CBSE 2017)

30. A ladder of length 6 m makes an angle of 45° with the floor while leaning against one wall of a room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60° with the floor. Find the distance between these two walls of the room.

    (CBSE 2011)

31. As observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If on ships is exactly behind the other on the same side of the light house, find the distance between the two ships. (CBSE 2011)

32. The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of water. (CBSE 2017)

33. Two points A and B are on the same side of tower and in the same straight line with its base. The angles of depression of these points from the top of two towers are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between these points. (CBSE 2017)

34. An observer finds the angles of elevation of the top of the tower from a certain point on the ground as 30°. If the observer moves 20 m towards the base of the tower, the angle of elevation of the top increases by 15° find the height of the tower. (CBSE 2017)

35. An aeroplane is flying at a high of 300 m above the ground. Flying at this angles of depression from the aeroplane of two points on both banks of a river in opposite direction are 45° and 60° respectively. Find the width of the river. (CBSE 2017)

36. The angles of depression of two ships from an aeroplane flying at the height of 7500 m are 30° and 45°. If both are ships are in the same line and on the same side of the aeroplane such that one ship is exactly behind the other, find the distance between the ships. (CBSE 2017)

37. Form the top of a hill , the angles of depression of two consecutive kilometer stones due east are found to be 45° and 30° respectively. Find the height of the hill. (CBSE 2017)

38. The angle of elevation of the top of tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are 60° and 30° respectively. Find the height of the tower.

    (CBSE 2016)

39. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower is 45°. Find the height of the tower PQ and the distance PX. Also find the length of QX. (CBSE 2016)

40. As observes from the top of a light house, 100 m high above sea level, The angles of depression of a ship, sailing directly toward it, changes from 30° to 60°. Find the distance travelled by the ships, sailing directly PX. Also find the length of QX. (CBSE 2016)

41. From a point on the ground, the angle of elevation of the top of a tower is observed to be 60°. From a point 40 m vertically above the first point of observation, the angle of elevation of the top of the tower is 30°. Find the height of the tower and its horizontal distance from point of observation.

    (CBSE 2016)

42. A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 5 m. From a point on the ground the angles of elevation of the top and bottom of the flagstaff are 60° and 30° respectively. Find the height of the tower and the distance of the point from the tower.

    (CBSE 2016)

43. At a point A, 20 meters above the level of water in a lake, the angle of elevation of a cloud is 30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A. (CBSE 2015)

44. The angles of elevation and depression of the top and the bottom of a tower from the top of building, 60 m high, are 30° and 60° respectively. Find the distance between the heights of the building and the tower and the distance between them. (CBSE 2014)

    
    

45. From the top of a 60 m high building, the angle of depression of the top and the bottom of a tower are 45° and 60° respectively. Find the height of the tower. (CBSE 2014)

46. The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. (CBSE 2014)

47. From a point P on the ground, the angle of elevation of the top of a 10 m building is 30°. A flagstaff is fixed at the top of the building and the angle of elevation of the top of the flagstaff from point P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (CBSE 2017)

48. The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 m, find the height of the building. (CBSE 2014)

49. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building. (CBSE 2013)

50. The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m hight, find the height of the building. (CBSE 2012)

51. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 45°. If the height of the building.

    (CBSE 2015)

52. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. (CBSE 2013)

53. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower. (CBSE 2010)

54. From the top of a 15 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower. (CBSE 2011)

55. The angles of elevation and depression of the top and bottom of a lighthouse from the top of a 60 m high building are 30° and 60° respectively. Find

    1. The difference between the heights of the lighthouse and the building.

    2. The distance between the lighthouse and the building. (CBSE 2012)

56. Two poles of equal heights are standing opposite to each other on either side of the road, which is 100 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles. (CBSE 2011)

57. Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle to elevation of the top of a pole is 60° and the angle of depression from the top of another pole at point P is 30°.Find the height of the poles and the distance of the point P from the poles. (CBSE 2015)

58. From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 10 m high building are 30° and 60° respectively. Find the height of the tower.

    (CBSE 2011)

59. The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 30°. Find the height of the tower.

    (CBSE 2011)

    
    

60. The angles of depression of the top and bottom of a 12 m tall building, from the top of a multi-storeyed building are 30° and 60° repectively. Find the height of the multi-storeyed building. (CBSE 2011)

61. The shadow of a tower standing on a level ground is found to be 30 m longer when the sun’s altitude is 30° then when it is 60°. Find the height of the tower. (CBSE 2011)

62. The shadow of a tower standing on a level ground is found to be 20 m longer when the sun’s altitude is 45° than when it is 60°. Find the height of the tower. (CBSE 2012)

63. A man standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he move 40 meters away from the ban, he finds the angle of elevation to be 30°. Find the height of the tree. (CBSE 2011)

64. A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of the cliff from the ships and the height of the cliff. (CBSE 2010)

65. From a window (9 m above the ground) of a house in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are 30° and 60° respectively. Find the height of the opposite house and the width of the street. (CBSE 2010)

66. A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minuts. Find the speed of the boat in m/h.

    (CBSE 2017)

67. The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500 √3 m, find the speed of the plane in km/hr. (CBSE 2015)

68. The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 3000 √3 m, find the speed of aeroplane. (CBSE 2014)

69. A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60° to 30°. Find the speed of the boat in meters per minute.

    (CBSE 2019)

70. A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. It the angle of depression of the car changes from 30° to 45° in 12 minutes, find the time taken by the car now to rich the tower. (CBSE 2017)

71. A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Find the speed of flying of the bird. (CBSE 2016)

72. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. 10 second later, the angle of depression of the car id found to be 60°. Find the time taken by the car to reach the foot of the tower from this point. (CBSE 2011)

1. If the angle between two tangents drawn from an external from an external point P to a circle of radius r and center O, is 60°, then find the length of OP. (CBSE 2017)

2. Two concentric circle of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle. (CBSE 2015)

3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 13 cm. Find the length of PQ. (CBSE 2010)

4. Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in the given figure. If AP = 15 cm, then find the length of BP.

   (CBSE 2012)

   
   

5. Prove that in two concentric circles, the chord of the lager circle, which touches the smaller circle, is bisected at the point of contact. (CBSE 2012)

6. Two concentric circles are of radii 7 cm and r cm respectively, where r > 7. A chord of the larger circle, of length 48 cm, touches the smaller circle. Find the value of r. (CBSE 2011)

7. In figure, the chord AB of the larger of the two concentric circle, with centre O, touches the smaller circle at C. Prove that AC = CB. (CBSE 2012)

8. In figure, there are two concentric circles, with centre O and of radii 5 cm and 3 cm. From an external point P, tangent PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

   (CBSE 2010)

   
   

9. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

   (CBSE 2014, 2016)

10. PQ is a tangent drawn from an external point P to a circle with centre O, QOR is the diameter of the circle. If ∠PQR =120°, what is the measure of ∠OPQ? (CBSE 2017)

11. In given figure, PQ is a tangent at a point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA. (CBSE 2016)

    
    

12. In figure, AOB is a diameter of a circle with centre with centre O and AC is a tangent to the circle at A. If ∠BOC = 130°, then find ∠ACO. (CBSE 2016)

    
    

13. In figure, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If

    ∠TPQ = 70°, find ∠TRQ. (CBSE 2015)

    
    

14. From an external point P, and PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find

    ∠AOB. (CBSE 2016)

15. In figure, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB. (CBSE 2015)

    
    

16. In figure, CP and CQ are tangents from an external point C to a circle with centre O. AB is another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length of BC.

    (CBSE 2010)

17. If the given figure, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB = 60°. Find the length of chord AB. (CBSE 2016)

18. In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

    (CBSE 2017)

    
    

19. In the figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.

    (CBSE 2017)

20. Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

    (CBSE 2017)

21. A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD + = BC + DA.

    (CBSE 2017)

    Or

    In the figure, a quadrilateral ABCD is draw to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively, Prove that AB + CD = BC + DA. (CBSE 2016)

    
    

22. In the given figure, if AB = AC, prove that BE = EC. (CBSE 2017)

    
    

23. The in circle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC. (CBSE 2014)

24. In figure, a circle inscribed in ∆ABC, touches its side BC, CA and AB at the points P, Q and R respectively. If AB = AC, then prove that BP = CP. (CBSE 2012, 2013)

25. In given figure, a circle is inscribed in a ∆ABC, such that it touches the side AB, BC and CA at points D, E and F respectively, If the lengths of sides AB, BC and CA are 12 cm, 8cm and 10 cm respectively, find the lengths of AD, BE and CF. (CBSE 2013, 2014, 2016)

26. In figure, common tangents AB and CD to the two circle with centres O1and O2 intersect at E. Prove that AB = CD. (CBSE 2014)

    
    

27. In figure, XP and ZQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. (CBSE 2014)

    
    

28. In the given figure, two circle touch each other at the point C. Prove that the common tangent to the circles at C, bisects the common tangent at P and Q. (CBSE 2013)

    
    

29. In figure, a circle touches all the four sides of a quadrilateral ABCD whose side are AB = 6 cm, BC = 9 cm and CD = 8 cm, Find the length of side AD. (CBSE 2011)

    
    

30. In figure, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR = 12 cm, Find the lengths of QM, RN and PL. (CBSE 2012, 219)

    
    

31. Prove that the length of the tangents draw from an external point to a circle is equal. (CBSE 2014)

32. Prove that the lengths of tangents draw from an external point to a circle are equal. Using it, prove: quadrilateral ABCD is draw to circumscribe a circle. Such that AB + CD = AD + BC.

    (CBSE 2012, 2016)

    
    

33. Prove that a parallelogram circumscribing a circle is a rhombus. (CBSE 2014)

34. In the given figure, the sides AB, BC and CA of ∆ABC touch a circle with centre O and radius r at P, Q and R respectively.

    1. AB +CQ = AC + BQ

    2. Area (∆ABC) = 1 (Perimeter of ∆ABC) × r (CBSE 2013)

       2

       
       

35. In figure, a triangles ABC is drawn to circumscribe a circle of radius 2 cm such that the segment BD and DC into which BC is divided by the point of contact D are the lengths 4 cm and 3 cm respectively. If area of ∆ABC = 21 cm2, then find the lengths of sides AB and AC. (CBSE 2011)

    
    

36. In the given figure, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If PO = 2r, show that ∠OTS = ∠OST 30°. (CBSE 2016)

    
    

37. In the given figure, from point P, two tangents PT and PS are draw to a circle with centre O such that

    ∠SPT = 120°, Prove that OP = 2PS. (CBSE 2016, 2014)

    
    

38. In figure, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.

    (CBSE 2015)

    
    

39. From a point T outside a circle of centre O, tangents TP and TQ are draw to the circle. Prove that OT is the left bisector of the line segment PQ. (CBSE 2015)

40. In figure, two tangents RQ and RP are draw from an external point R to the circle with centre O. If

    ∠PRQ = 120°, then prove that OR = PR + RQ. (CBSE 2015)

    
    

41. In figure, PQ is a chord of length 8 cm of a circle of radius 5 cm, The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ. (CBSE 2015, 219)

    
    

    
    

42. Prove that the line segment joining the points of contact of two parallel tangents of a circle passes through its centre. (CBSE 2014)

43. In figure, a left triangle ABC circumscribes a circle of radius r. If AB and BC are of lengths 8 cm and 6 cm respectively, find the value of r. (CBSE 2012)

44. In the given figure, PA and PB are tangents to a circle from an external points P such that PA = 4 cm and

    ∠BAC = 135°. Find the length of cord AB. (CBSE 2017)

    
    

45. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. (CBSE 2017)

46. In figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and X’Y’ at B. prove that ∠AOB = 90°. (CBSE 2017)

    
    

47. In given figure, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle. (CBSE 2016)

    
    

    
    

48. In figure, two equal circle, with centre O and O’, touch each other at X. OO’ produced meets the circle with centre O’ at A. AC is tangent to the circle with the centre with centre O, at the point C. O’D is perpendicular to AC. Find the value of DO’ / CO. (CBSE 2016)

    
    

    
    

49. In figure, tangents PQ and PR are drew from an external point P to a circle with centre O, such that

    ∠RPQ = 30°. A chord RS id drawn parallel to the tangent PQ. Find ∠RQS. (CBSE 2015)

    
    

50. Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the end points of the arc. (CBSE 2015)

51. In figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If

    ∠PBT = 30°, prove that BA: AT = 2: 1. (CBSE 2015)

    
    

    
    

52. In figure, PQ is a chord of length 16 cm, of a circle of radius 10 cm. The tangents at P and Q intersect a point T. Find the length of TP. (CBSE 2014, 2016)

    
    

53. In the given figure, 𝑙 𝑎𝑛𝑑 𝑚are two parallel tangents to a circle with centre O, touching the circle at A and B respectively? Another tangent at C intersects the line 𝑙 𝑎𝑛𝑑 D and m at E. Prove that ∠DOE = 90°.

    (CBSE 2013)

54. In the given figure, PA and PB are two tangents drawn from an external point P to a circle with centre O. Prove that OP is the left bisector of line segment AB. (CBSE 2013)

1. Area of a sector to a circle of radius cm2. Find the length of the corresponding arc of the sector.

   (CBSE 2011)

2. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

   (CBSE 2011)

3. Find the area of a quadrant of a circle, where the circumference of circle is 44 cm. (CBSE 2019)

4. A car has two wipes which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle 120°. Find the total area cleaned at each sweep of the blades. (CBSE 2013)

5. In figure find the area of the shaded region, enclosed between two concentric circles of radii 7 cm and 114 cm where ∠AOC = 40°. (CBSE 2012)

   
   

6. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find (i) the length of the arc (ii) area of the sector formed by the arc. (CBSE 2011)

7. The length of the minute hand of a clock is 14 cm. Find the area swept by the minutes hand in 10 minutes.

   (CBSE 2011)

8. A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle. (CBSE 2019)

9. A chord of a circle of radius 21 cm subtends an angle 60° at the centre. Find the area of the corresponding minor segment of the circle. (CBSE 2019)

10. Find the area of the major segment APB in figures, of a circle of radius 35 cm and ∠AOB = 90°.

    (CBSE 2011)

    
    

11. Find the area of the segment AMB shown in figure. If radius of the circle is 21 cm and ∠AOB = 120°.

    (CBSE 2011)

12. In figure, AB is a chord of a circle, with centre O and radius 10 cm that subtends a left angle at the centre of the circle. Find the area of the minor segment AQBP. Hence, find the area of major segment ALBQA. (CBSE 2016)

    Or

13. A chord of a circle of radius 10 cm subtends a left angle at the centre. Find the area of the corresponding minor segment and hence find the area of major segment. (CBSE 2011)

14. Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 60°. Also find the area of the corresponding major segment. (CBSE 2015)

15. A chord of length 10 cm divides a circle of radius 5√2 cm in two segments. Find the area of the minor segment. (CBSE 2013)

16. A chord PQ of a circle of radius 10 cm subtends an angle of 60° at the centre of circle. Find the area of major and minor segment of the circle. (CBSE 2017)

    
    

    
    

17. In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region. (CBSE 2019)

    
    

18. In figure, find the area of the shaded region. (CBSE 2015)

19. The long and short hand of a clock is 6 cm and 4 cm long respectively. Find the sum of the distance travelled by their tips in 2 days. (CBSE 2015)

1. A solid is the shape of a cone mounted on a hemisphere of same radius. If the cured surface area of the hemispherical part the conical part are equal. Then find the ratio of the radius and the height of the conical part. (CBSE 2012)

2. Two cubes, each of side 8 cm are joined end to end. Find the surface area of the resulting cuboids.

   (CBSE 2011)

3. A toy is the shape of a solid cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 21 cm and 40 cm respectively, and the height of cone is 15 cm, then find the total surface area of the toy. (CBSE 2011)

4. A wooden article was made by scooping out a hemisphere of radius 7 cm, from each and of a solid cylinder of height 10 cm and diameter 14 cm. Find the total surface area of the article.

   (CBSE 2015)

5. A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of solid so formed, at the rate of ₹5 per 100 sq. cm. (CBSE 2018)

6. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 cm and its base of radius 3.5 cm. find the total surface area of the article. (CBSE 2012)

   
   

7. A toy is in the form of a cone of radius 8 cm mounted on a hemisphere of same radius on its circular face. The total height of the toy is 14 cm. Find the total surface area of the toy. (CBSE 2017)

8. A write of diameter 3 mm is wound about a cylinder whose height is 12 cm and radius 5 cm so as to cover the curved surface of the cylinder completely. Find the length of the wire. (CBSE 2016)

9. In figure, a decorative block, made up of two solids-a cube and a hemisphere. The base of the block is a cube of side 6 cm and the hemisphere fixed on the top has a diameter of 3. 5 cm. Find the total surface are of the block. (CBSE 2016)

10. In figure, a tent is in the shape of a cylinder surmounted by a conical top of same diameter, if the height and diameter of cylindrical part are 2. 1 m and 3 m respectively and the slant height of conical part is 2. 8 m, find the cost of canvas needed to make the tent if the canvas is available at the rate of ₹500 sq. meter. (CBSE 2014)

11. A 5 m wide cloth is used to make a conical tent of base diameter 14 cm and height 24 m. Find the cost of cloth used at the rate of ₹25 per meter. (CBSE 2013)

12. A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of same diameter, the diameter of the hemispherical bowl is 14 cm and the total height of the vassal is 13 cm. Find the total surface area of the vassal. (CBSE 2013)

13. A toy is in the form of a cone mounted on a hemisphere of same radius 7 cm. If the total height of the toy is 31 cm, find t\its total surface area. (CBSE 2017)

14. From a solid left circular cylinder of height 1. 2 cm and radius 0. 5 cm, a left circular cone of same height and same radius is cut out. Find the total surface area of the remaining solid.

    (CBSE 2013)

15. The total surface area of a solid cylinder is 231cm2. If the curved surface are of this solid cylinder is 2/3 of its total surface area, find its radius and height. (CBSE 2012)

16. Due to heavy floods in a state, thousands were rendered homeless, 50 schools collectively offered to the state Government to provide place and the canvas for 1500 tents to be fixed by the government and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius

    2. 8 m and height costs ₹120 per sq. m, find the amount shared by each school to set up the tents.

    (CBSE 2016)

17. From a solid cylinder of height 2. 8 cm and diameter 4. 2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid.

    (CBSE 2014)

18. A hemispherical depression is cut from one face of a cubical block of side 7 cm, such that the diameter of the hemisphere is equal to the edge of the cube. Find the surface area of the remaining solid.

    (CBSE 2014)

19. A military tent of height 8. 25 m is in the form of left circular cylinder of base diameter 30 m and height 5. 5m surmounted by a left circular cone of same base radius. Find the length of the canvas use in making the tent, if the breadth of the canvas is 1. 5m. (CBSE 2011)

20. A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in the

    cylinder. The conical hole has a radius of 3 cm and its depth is 8 cm. Calculate the ratio of the volume of

    2 9

    metal left. The cylinder to the volume of metal taken out in conical shape. (CBSE 2011)

21. A solid left-circular cone of height 60 cm and radius 30 cm is dropped in a left-circular cylinder full of water of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres. (CBSE 2015)

22. ​The largest possible sphere is carved out of a wooden solid cube of side 7 cm. Find the volume of the wood left. (CBSE 2014)

23. A wooden toy was made by scooping out a hemispherical of same radius from each end of a solid cylinder If the height of the cylinder is 10 cm, and its base is of radius 3. 5 cm, find the volume of wood in the toy. (CBSE 2013)

24. From a solid cylinder of height 14 cm and base diameter 7 cm, two equal conical holes of radius 2. 1 cm and height 4 cm are cut off. Find the volume of the remaining solid. (CBSE 2011)

25. A solid is in the form of a cylinder with hemispherical end. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (CBSE 2019)

26. A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them being 3. 5 cm and the total height of solid is 9. 5 cm. Find the volume of the solid. (CBSE 2012)

27. A toy is in the shape of a cone mounted on a hemisphere of same base radius. If the volume of the toy is 231 cm3 and its diameter is 7 cm, then find the height of the toy. (CBSE 2012)

28. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere? (CBSE 2017)

29. If the total surface area of a solid hemisphere is 462 cm2, find the volume. (CBSE 2014)

    2425 1 𝑐𝑚

    2

    3. Find its curved surface area. (CBSE 2011)

    
    

30. The volume of a hemisphere is

31. Two cubes each of volume 27 cm3 are joined end to end to form a solid. Find the surface area of the resulting cuboid. (CBSE 2018)

32. A heap of rice is in the form of a cone of base diameter 24 m and height 3. 5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap? (CBSE 2017)

33. The radius and height of a solid left circular cone are in the ratio of 5: 12. If its volume is 314 cm3, find its total surface area. (CBSE 2015)

34. A left circular cone of radius 3cm, has a curved surface area 47. 1 cm2. Find the volume of the cone.

    (CBSE 2017)

35. A solid wooden toy is in the form of a hemisphere surmounted by a cone of same radius. The radius of hemisphere is 3. 5 cm and the total wood used in making of toy is 166 5 𝑐𝑚3. Find the height of the toy.

    6

    Also, find the cost of painting the hemispherical part of the toy at the rate of ₹10 per cm2.

    (CBSE 2010)

36. The height of a cone is 30 cm. From its top-side a small cone is cut by a lane parallel to its base. If

volume of smaller cone is  1 of the given cone, then at what height it is cut from its base.

27

(CBSE 2012)

A toy is in the form of a hemisphere surmounted by a left circular cone of the same base radius as that of the hemisphere. It the radius of base of the cone is 21 cm and its volume is 2/3 of the volume of the hemisphere calculate the height of the cone and the surface area of the toy. (CBSE 2010)

1. Date reading heights of students of class X of model school, Dehradun is given below. Calculate the average height of students of the class. (CBSE 2015)

   Height (in cm)	150-156	156-162	162-168	168-174	174-180
   No. of Students	4	7	15	8	6

   
   

2. In the class test, marks obtained by 120 students are given in the following frequency distribution. If it is given that mean is 59, find the missing frequencies x and y. (CBSE 2015)

   Mark	0-10	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
   No. of Students	1	3	7	10	15	x	9	27	18	y

   
   

3. The average score of boys in the examination of a school is 71 and that of the girls is 73. The average score of the school in the examination is 71.8. Find the ratio of number of boys to the number of girls who appeared in the examination. (CBSE 2015)

4. Find the mean of the following distribution by Assumed Mean Method. (CBSE 2015)

   
   

   Class interval	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
   Frequency	8	7	12	23	11	13	8	6	12

   
   

5. In the table below, heart-beats of 30 women are recorded. If mean of the data is 76, find the missing frequencies x and y. (CBSE 2014)

   No. of heart-beats (per min.)	65-68	68-71	71-74	74-77	77-80	80-83	83-86
   No. of women	x	4	3	7	8	4	y

   
   

6. Monthly pocked money of students of a class is given in the following frequency distribution:

   (CBSE 2014)

   
   

   Pocket money (in ₹)	100-125	125-150	150-175	175-200	200-225
   No. of women	14	8	12	5	11

   
   

   Find mean pocket money using step derivation method.

7. If the mean of the following distribution is 50, find the value of p. (CBSE 2013)

   
   

   Class	0-20	20-40	40-60	60-80	80-100
   Frequency	17	p	32	24	19

   
   

8. The frequency distribution table of agricultural holdings in a village is given below: (CBSE 2015)

   
   

   Area (in hectares)	1-3	4-6	7-9	10-12	13-15
   No. of Families	25	22	52	45	16

   
   

   Find the Mean Area held by a family.

9. The mean of the following frequency distribution is 53. But the frequencies f1and f2 in the classes 20 – 40 and 60 – 80 are missing. Find the missing frequencies. (CBSE 2013)

   Class

   0-20

   20-40

   40-60

   60-80

   80-100

   Total

   Frequency

   15

   f1

   21

   f2

   17

   100

   
   

10. The mean of the following data is 18.75. Find the value of P. (CBSE 2017)

    
    

    Class Marks (𝒙𝒊)	10	5	p	25	30
    Frequency(𝒇𝒊)	5	10	7	8	2

    
    

11. The mean of the following frequency distribution is 62. 8. Find the missing frequency x.

    (CBSE 2012)

    
    

    Class	0-20	20-40	40-60	60-80	80-100	100-120
    Frequency	5	8	x	12	7	8

    
    

12. Find the mean of the following data. (CBSE 2012)

    
    

    Class	Less than 20	Less than 40	Less than 60	Less than 80	Less than 100
    Frequency	15	37	74	99	120

    
    

13. The following table given the literacy rate (in %) in 40 cities. Find the mean literacy rate.

    (CBSE 2012)

    
    

    Literacy rate (in %)	45-55	55-65	65-75	75-85	85-95
    No. of literacy	4	11	12	9	4

    
    

14. The mean of the following frequency distribution is 62. 8 and the sum of frequency is 50. Find the missing frequency f1and f2. (CBSE 2015)

    Class	0-20	20-40	40-60	60-80	80-100	100-120
    Frequency	5	f1	10	f2	7	8

    
    

15. Find the median of the data using an empirical formula, when it is given that mode = 35. 3 and mean =

    30. 5. (CBSE 2014)

16. In a continuous frequency distribution, the median of the data is 21. If each observation by 5, then find the new median. (CBSE 2013)

17. From the following frequency distribution, find the median class. (CBSE 2013)

    
    

    Cost of living index	1400-1550	1550-1700	1700-1850	1850-2000
    No. of weeks	8	15	21	8

    
    

18. Consider the following distribution; find the frequency of class 30 – 40. (CBSE 2013)

    
    

    Marks obtained	0 or more	10 or more	20 or more	30 or more	40 or more	50 or more
    No. of students	63	58	55	51	48	42

    
    

19. From the following cumulative frequency table, construct a frequency distribution table.

    (CBSE 2013)

    
    

    Marks	0 and above	10 and above	20 and above	30 and above	40 and above	50 and above
    No. of students	40	28	16	10	8	0

    
    

20. Find the mean and median for the following data. (CBSE 2015)

    
    

    Class	0-4	4-8	8-12	12-16	16-20
    Frequency	3	5	9	5	3

    
    

21. For helping poor girls of their class, students saved pocket money as shown in the following table.

    (CBSE 2014)

    
    

    Money saved (in ₹)	5-7	7-9	9-11	11-13	13-15
    No. of students	6	3	9	5	7

    
    

    Find mean and median for this data.

22. Find the median for the following distribution. (CBSE 2013)

    
    

    Class	0-10	10-20	20-30	30-40	40-50
    Frequency	6	10	12	8	8

    
    

23. Find the missing frequency f1and f2 in the following frequency distribution table, if N = 100 and median is 32.

    
    

24. (CBSE 2013)

    
    

    Class	0-10	10-20	20-30	30-40	40-50	50-60	Total
    Frequency	10	f1	25	30	f2	10	100

    
    

25. Weekly income of 600 families is given below: (CBSE 2012)

    
    

    Income in (₹)	0-1000	1000-2000	2000-3000	3000-4000	4000-5000	5000-6000
    No. of consumers	250	190	100	40	15	5

    
    

    Find the median.

26. Find the median of the following data. (CBSE 2012)

    
    

    Monthly consumption (in units)	Below 85	Below 105	Below 125	Below 145	Below 165	Below 185	Below 205
    No. of consumers	4	9	22	42	56	64	68

    
    

27. Find the value of x and y if the median for the following data is 31. (CBSE 2012)

    
    

    Class	0-10	10-20	20-30	30-40	40-50	50-60	Total
    Frequency	5	x	6	y	6	5	40

    
    

28. The median of the following data is 525. Find x and y if the sum of all frequency is 100.

    (CBSE 2012, 2017)

    
    

    Class	200-300	300-400	400-500	500-600	600-700	700-800
    Frequency	16	x	17	20	15	y

    
    

29. Show that the mode of the series obtained by combining the two series and s2 given below is different from that of s1and s2 taken separately. (CBSE 2015)

    s1= 3, 5, 8, 8, 9, 12, 13, 9, 9

    s2 = 7, 4, 7, 8, 7, 8, 13

30. Following table shows sale of shoes in a store during one month: (CBSE 2014)

    Size of shoe 3 4 5 6 7 8

    No. of pairs sold 4 18 25 12 5 1

    Find the model size of the shoes sold.

31. Weekly household expenditure of families in a housing society are shown below. (CBSE 2014)

    
    

    Weekly expenditure (in ₹)	Upto 3000	3000-6000	6000-9000	9000-12000	12000-15000
    No. of families (f)	4	25	31	48	10

    
    

    Find the upper limit of the modal class.

32. A medical camp was held on a school to impart health education and the importance of exercise to children During this camp, a medical check of 35 students was done and their weights were recorded as follow: (CBSE 2016)

    
    

    Weight (in kg)	Below 40	Below 42	Below 44	Below 46	Below 48	Below 50	Below 52
    No of students	3	5	9	14	28	31	35

    
    

    Computer the model weight.

33. Find the mode of all following frequency distribution. (CBSE 2015)

    
    

    Class interval	25-35	35-45	45-55	55-65	65-75	75-85
    f	7	31	33	17	11	1

    
    

34. Mode of the following frequency distribution is 65 and sum of all the frequencies is 70. Find the missing frequency x and y. (CBSE 2015)

    Class	0-20	20-40	40-60	60-80	80-100	100-120	120-140	140-160
    Frequency	8	11	x	12	y	9	9	5

    
    

35. Given below is the distribution of weekly pocket money received by students of a class. Calculate the pocket money that is received by most of the students. (CBSE 2014)

    Pocket money (in ₹)	0-20	20-40	40-60	60-80	80-100	100-120	120-140
    No of students	2	2	3	12	18	5	2

    
    

36. Height of students of class X are given in the following frequency distribution. (CBSE 2014)

    
    

    Height (in cm)	150-155	155-160	160-165	165-170	170-175
    No of students	15	8	20	12	5

    
    

    Find the model height.

37. Cost of Living Index for some period is given in the following frequency distribution.

    (CBSE 2014)

    
    

    Index	1500-1600	1600-1700	1700-1800	1800-1900	1900-2000	2000-2100	2100-2200
    No. of weeks	3	11	12	7	9	8	2

    
    

    Find the mode and median for above data.

38. Find the mode of the following frequency distribution: (CBSE 2013)

1. A game consists of tossing a coin 3 times and nothing the outcome each time. If getting the same result in all the tosses is a success, find the probability of losing the game. (CBSE 2019)

2. Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

   (CBSE 2014)

3. Two coins are tossed simultaneously. Find the probability of getting at least on head. (CBSE 2013)

4. Three coins are tossed simultaneously. Find the probability of getting exactly two heads.

   (CBSE 2013)

5. A coin is tossed two times. Find the probability of getting at least on head. (CBSE 2011)

6. A coin is tossed two times. Find the probability of getting both head and both tails. (CBSE 2011)

7. Three different coins are tossed together. Find the probability of getting: (CBSE 2016)

   1. exactly two heads

   2. at least two heads.

   3. at least two tails.

8. Three different coins are tossed together. Find the probability of getting: (CBSE 2015)

   1. atleast heads

   2. atleast two tails.

9. A game consists of tossing a one-rupee coin three times and noting its outcomes each time. Find the probability of getting (CBSE 2015)

   1. three heads

   2. atleast two tails.

10. A game consists of tossing a coin 3 times and noting its outcome each time. Hanif wins if he gets three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

    (CBSE 2011)

11. Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is 6. (CBSE 2015)

12. A die is thrown once. Find the probability of getting a number which (CBSE 2019)

    1. is a prime number.

    2. lies between 2 and 6.

13. A die is thrown once. Find the probability of getting: (CBSE 2019)

    1. a prime number

    2. number divisible by 2

    3. a composite number

14. Two different dice are tossed together. Find the probability: (CBSE 2019)

    1. of getting a doublet

    2. of getting a sum 10, of the number on the two dice.

15. Two different dice are thrown together. Find are probability that the product of the numbers appeared is less than 18. (CBSE 2017)

16. Two different dice are tossed together. Find the probability: (CBSE 2014)

    1. That the number on each die is even.

    2. That the sum of number appearing on the two dice is 5.

17. Two different dice are rolled simultaneously. Find the probability that the sum of number appearing on the two dice is 10. (CBSE 2014)

18. A die is tossed once. Find the probability of getting an even number of a multiple of 3.

    (CBSE 2013)

19. Two different dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is 7. (CBSE 2011)

20. A die is thrown once. What is the probability of getting a number greater than 4? (CBSE 2010)

21. A die id thrown twice. What is the probability that the same number will come up either time?

    (CBSE 2010)

22. Two different dice are thrown together. Find the probability that the number obtained

    1. have a some less than 6 (CBSE 2017)

    2. have a product less than 16

    3. is a double off odd number.

23. In a single throw of a pair of different dice, what is the probability that the number of getting:

    1. a prime number on each dice? (CBSE 2016)

    2. a total of 9 or 11?

24. Two different dice are thrown together. Find the probability of: (CBSE 2016)

    1. getting a number greater than 3 on each die.

    2. getting a total of 6 or 7 of the numbers on two dice.

25. Two different dice are rolled together. Find the probability of getting: (CBSE 2015)

    1. the sum of numbers one two dice to be 5.

    2. even number on both dice.

26. A die is thrown once. Find the probability of getting the following: (CBSE 2012)

    1. a prime number

    2. a number lying between 2 and 5

27. Two dice are rolled once. Find the probability of getting such numbers on two dice, whose product is a perfect square. (CBSE 2011)

28. Peter thrown two different dice together and finds the product of the two numbers obtained. Rina throws a die and square the number obtained. Who has the better chance to get the number 25?

    (CBSE 2017)

29. Two different dice are thrown together. Find the probability that the numbers obtained have

    1. even sum. (CBSE 2017)

    2. even product

30. A die is thrown twice. Find the probability that (CBSE 2013)

    1. 5 may not come either time.

    2. same number may not come on the die thrown two times.

31. Two dice are thrown simultaneously. Determine the probability that the different of the numbers on the two dice is 2. (CBSE 2013)

32. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen. (CBSE 2016)

33. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a jack nor an ace. (CBSE 2013)

34. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn is neither a king nor a queen. (CBSE 2013)

35. Three cards of spades are lost from a pack of 52 playing cards. The remaining cards were well shuffled and then a card was drawn at random from them. Find the probability that the drawn card is of black colour. (CBSE 2013)

36. A card is drawn at random from a well-shuffled pack of 52 playing cards. Find the probability of getting

    (CBSE 2012)

    1. a red king

    2. a queen or a jack

37. A card is drawn at random from a well-shuffled pack of 52 playing cards. Find the probability of getting a red face card. (CBSE 2010)

38. All red face cards are removed from a pack of playing cards. The remaining cards were well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is (a) a red card.

    (CBSE 2010)

    1. a face card.

    2. a card of clubs.

39. A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting

    1. a king of red colour (CBSE 2015)

    2. a face card

    3. the queen of diamonds.

40. All kings, queen and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn from it. Find the probability that the drawn card is (CBSE 2012)

    1. a black face card.

    2. a red card.

41. From a well-shuffled pack of 52 playing cards, black kings and black aces are removed. A card is then drawn at random from the pack. Find the probability of getting (CBSE 2012)

    1. a red card

    2. not a diamond card.

42. A card is draw at random from a well-shuffled deck of playing card. Find the probability that the card drawn is (CBSE 2015)

    1. a card of spade or an ace.

    2. a black king.

    3. neither a jack nor a king

    4. either a king or a queen

43. All the black face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a (CBSE 2014)

    1. face card.

    2. red card.

    3. black card.

    4. king.

44. Five cards – the ten, jack, queen, king and ace of diamonds, are well shuffled with their faces downwards. One card is then picked up at random. (CBSE 2014)

    1. what is the probability that the drawn card is the queen?

    2. If the queen is drawn and put aside, and a second card is drawn, find the probability that the second card is (i) an ace, (ii) a queen.

45. A bag contains 15 white and some balck balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag. (CBSE 2017)

46. A box contains 90 discs which have are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it beard (i) a two-digit number, (ii) a number divisible by 5.

    (CBSE 2017)

47. Cards marked with number 3, 4, 5… 50 are placed in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the selected card bears a perfect square number.

    (CBSE 2016)

48. 20 tickets, on which number 1 to 20 are written, are mixed thoroughly and then a ticket is drawn at random out of them. Find the probability that the number on the drawn ticket is a multiple of 3 or 7.

    (CBSE 2016)

49. A box consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Ramesh, a shopkeeper will buy only those shirts which are good but ‘Kewal’ another shopkeeper will not buy shirts with major defects. A shirts is taken out of the box at random. What is the probability that.

    (CBSE 2016)

    1. Ramesh will buy the selected shirt?

    2. ‘Kewal’ will buy the selected shirt?

50. There are 100 cards in a bag on which numbers from 1 to 100 are written. A cards is thaken out from the bag at random. Find the probability that the number on the selected card: (CBSE 2016)

51. A bag contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is: (CBSE 2015)

    1. divisible by 3 or 5.

    2. a perfect square number.

52. A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the box, find the probability that it bears (CBSE 2015)

    1. a one digit number.

    2. a number divisible by 5.

    3. an odd number less than 30.

    4. a composite number between 50 and 70.

53. A bag contains cards numbered from 1 to 49. A and C is drown from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is

    1. an odd number. (CBSE 2014)

    2. a multiply of 5.

    3. a perfect square.

    4. an even prime number.

54. Cards numbered from 11 to 60 are are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn card is (CBSE 2014)

    1. an odd number

    2. a perfect square number.

    3. divisible by 5.

    4. a prime number less than 20.

55. Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is (CBSE 2014)

    1. not divisible by 3.

       4

    2. a prime number greater than 7.

    3. not a perfect square number.

56. A piggy bank contains hundred 50 paise coins, fifty ₹1 coins, twenty ₹2 coins and ten ₹5 coins. If it equally likely that one of the coins will fall out when the bank is turned upside down, find the probability that the coin which fell. (CBSE 2014)

    1. will be a 50 paise coin.

    2. will be of value more than ₹1.

    3. will be of value less than ₹5.

    4. will be a ₹1 or ₹2 coin.

57. A bag contains 12 balls, out of which x are white. (CBSE 2013)

    1. If one ball is drawn at random, find the probability that it is a white ball.

    2. If 6 more white balls are put in the bag, the probability of drawing a white ball is double than that in (a), find x.

       
       

58. A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the probability that it will be (i) a blue card (ii) not a yellow card (iii) neither yellow nor a blue card. (CBSE 2012)

59. A box contain 35 blue, 25 white and 40 red marbles. If a marble is drawn at random from the box, find the probability that the drawn marble is (i) white (ii) not blue (iii) neither white nor blue.

    (CBSE 2012)

60. Cared marked with number 1, 3, 5….101 are placed in a bag and mixed thoroughly. A card is then drawn at random from the bag. Find the probability that the number on the drawn card is (i) less than 19

    (ii) a prime number less than 20. (CBSE 2012)

62. A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. Find the probability that the selected ticket has a number which is a multiple of 5. (CBSE 2011)

63. A box contains 80 discs which are numbered from 1 to 80. If one disc is drawn at random from the box, find the probability that it bears a perfect square number. (CBSE 2011)

64. Cards marked with numbers 5, 6, 7…74 are placed in a bag and mixed thoroughly. One card is drawn at random from the bag. Find the probability that the number on the card is a perfect square.

    (CBSE 2011)

65. Card bearing number 1, 3, 5…35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing: (CBSE 2010)

    1. a prime number less than 15.

    2. a number divisible by 3 and 5.

66. An integer is chosen at random between 1 and 100. Find the probability that it is (CBSE 2018)

    1. divisible by 8.

    2. not divisible by 8.

67. A number is chosen at random from the numbers -3, -2, -1, 0, 1, 2, 3. What will be the probability that square of this number is less than or equal to 1? (CBSE 2017)

68. The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. what is the number of rotten apples in the heap? (CBSE 2017)

69. Find the probability that in a leap year there will be 53 Tuesdays. (CBSE 2017)

70. A game consist of tossing a one-rupee coin 3 times and noting the outcomes each time. Ramesh will win the game if all the tosses show the same result, (i,e., either all three heads all three tail) and loses the game otherwise. Find the probability that Ramesh will lose the game. (CBSE 2016)

71. A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

    (CBSE 2016)

72. A number x is selected at random from the numbers 1, 4, 9, 16 and another number y is selected at random from the numbers 1, 2, 3, 4. Find the probability that the value of xy is more than 16.

    (CBSE 2016)

73. A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is a consonant. (CBSE 2015)

74. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equality likely outcomes. Find the probability that the arrow will point at any factor of 8. (CBSE 2015)

75. The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is 1/4. The probability of selecting a blue ball at random from the same jar is 1/3. If the jar contains 10 orange balls, find the total number of balls in the jar. (CBSE 2015) Out of cards numbered from 1 to 20, which are mixed thoroughly, a card is drawn at random. Find the probability that the drawn card bears a number which is a multiple of 3 or 7. (CBSE 2013)

76. A group consists of 12 persons, of which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selected a person who is (CBSE 2013)

    1. extremely patient.

    2. extremely kind or honest.

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