Find the LCM & HCF of 120 & 144 by fundamental theorem of …€¦ · CLASS – X CHAPTER – 1 1....

CLASS – X CHAPTER – 1

1. Use Euclid’s division algorithm to find the HCF of 10224 & 9648.


3. Find the HCF of 455 & 84using division algorithm


5. Find the LCM of 120 & 70 by fundamental theorem of arithmetic.

6. Find the LCM & HCF of 120 & 144 by fundamental theorem of arithmetic.

7. Use fundamental theorem of arithmetic, find the HCF of 26, 51 & 91.

8. Find the LCM of 336, 54 by the prime factorization method.

9. Find the LCM & HCF of 15, 18, 45 by the prime factorization method.

10. Find LCM (306, 1314), if HCF (306, 1314) = 18. 11. If HCF (6,a ) = 2& LCM(6, a) = 60, then find a.

12. The HCF & LCM of two numbers are 9 & 90 respectively. If one number is 18, find the other.

13. Find the LCM & HCF of 312&27.Verify that HCFLCM = Product of the two numbers.

14. An army contingent of 616 members is to march behind an army band of 32 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?

15. Show that every positive even integer is of the form 2q & that every positive odd integer is of the

form 2q + 1, where q is some integer.

16. Show that any positive even integer is of the form 6m, 6m + 2 or 6m + 4 where m is some integer. 17. Show that any positive odd integer is of the form 4q + 1 & 4q + 3, where q is some integer.

18. Show that any positive odd integer is of the form 6p + 1, 6p + 3 or 6p + 5 where p is some integer.

19. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m,

3m + 1 for some integer m.

20. Show that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

21. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1

or 9m + 8.

22. Check whether 6 n can end with the digit 0 for any natural number n?

23. Can a number 6 n, n being a natural number end with the digit 5? Give reasons.

24. Show that 9 n can’t end with 2 for any integer n.

25. Is 7×11×13 + 11 a composite number? Justify your answer. 26. Explain why 7532 +3 is a composite number.

27. Is 7×11×13 + 13 a composite number? Justify your answer.

28. Explain why 11131517 +17 is a composite number.

29. Prove that 3 is an irrational number.

30. Prove that 11 is an irrational number.

31. Prove that 3 + 5 is an irrational number.


33. Prove that 5 – 2 is an irrational number.

34. Prove that 3 +5 2 is an irrational number.

35. Prove that 5 +7 3 is an irrational number.

36. Prove that 2 3 – 7 is an irrational number.

37. Prove that 3

25 is irrational number.

38. Prove that 52

3 is irrational number.



41. Prove that 32

1

is an irrational number.

42. In the adjoining factor tree, find the numbers m, n:

43. In the adjoining factor tree, find the numbers m, n:

44. Without actually performing the long division, state whether the following number has a

terminating decimal expansion or non terminating recurring decimal expansion225

543.

CHAPTER – 2

1. ,β are the roots of the quadratic polynomial p(x) = x2 – (k + 6)x + 2(2k – 1). Find the value of k, if

+ β = 2

1 β.

2. ,β are the roots of the quadratic polynomial p(x) = x2 – (k – 6)x + 2(2k + 1). Find the value of k, if

+ β = β.

3. If , β are the two zeros of the polynomial x2 – 6x + a, then find the value of ‘a’ if 3 + 2β =20.

4. ,β are the roots of the quadratic polynomial x2 – 5x + k such that – β = 1, find the value of k .

5. If &

1 are the zeros of the polynomial 4x

2 – 2x + (k – 4). Find the value of k.

6. If , & are zeros polynomial 6x3 + 3x

2 – 5x + 1, then the value of

111 .

7. If , β are the two zeros of the polynomial 21 y2 – y – 2, find a quadratic polynomial whose zeros

are 2 & 2β.

8. If , β are the two zeros of the polynomial x2 – 2x – 8,then find a quadratic polynomial whose zeros

are 3 & 3β.

9. If , β are the two zeros of the polynomial x2 – 4x + 3,then find a quadratic polynomial whose

zeros are 3 & 3β.

10. If , β are the two zeros of the polynomial 6 y2 – 7y + 2, find a quadratic polynomial whose zeros

are

1&

1.

11. If , β are the two zeros of the polynomial 25 p2 –15 p + 2, find a quadratic polynomial whose zeros

are 2

1&

2

1.

12. If 2 & 3 are zeros of polynomial 3x2 – 2kx + 2m, find the values of k & m.

13. Find all the zeros of the polynomial 4x4 – 20 x

3 + 23x

2 + 5x – 6, it two of its zeros are 2 & 3.

14. Find all the zeros of x4 – 3x

3 + 6x – 4, if two of its zeros are 2 , – 2 .

15. Find all the zeros of the polynomial x4 + x

3 – 9x

2 – 3x + 18, if two of its zeros are 3 , – 3 .


3 – 5x

2 + 9x – 3, it being given that two of its zeros are

3 & – 3 .


3 + 5x

2 + 15x – 12 , if two of its zeros are

2

3& –

2

3.

18. If two zeros of the polynomial p(x) = x4 – 6x

3 – 26x

2 + 138x – 35 are 2 3 , find the other zeros.

19. Find all other zeros of the polynomial p(x) = 2x3 + 3x

2 – 11x – 6, if one of its zero is –3.

20. If being given that 1 is one of the zeros of the polynomial 7x – x3 – 6. Find its other zeros.

21. If the remainder on division of x3 + 2x

2 + kx + 3 by x – 3 is 21, find the quotient & the value of k.

Hence, find the zeros of the cubic polynomial x3 + 2x

2 + kx – 18.

22. What must be added to the polynomial p(x) = 5 x4 + 6x

3 – 13x

2 – 44x + 7 so that the resulting

polynomial is exactly divisible by the polynomial Q(x) = x2 + 4x + 3 & the degree of the polynomial

to be added must be less then degree of the polynomial Q(x).

23. Find the zeros of the quadratic polynomial x2 + 7x + 12 & verify the relationship between the zeros

& its coefficients.

24. Find the zeros of the quadratic polynomial 6x2 – 7x – 3 & verify the relationship between the zeros

& its coefficients.

25. Find the zeros of 4 5 x2 – 17x – 3 5 & verify the relation between the zeros & coefficients of the

polynomial.

26. Find the zeros of 4 3 x2 + 5x – 2 3 & verify the relation between the zeros & coefficients of the

polynomial.

27. Find the zeros of the quadratic polynomial 3 x2 – 8x + 4 3 .

28. Divide 2x4 – 9 x

3 + 5x

2 + 3x – 8 by x

2 – 4x + 1 &verify the division algorithm.

29. Divide 3x2– x

3 – 3x + 5 by x – 1 – x

2&verify the division algorithm.

30. Divide x4 – 3x

2 + 4x + 5 by x

2 – x + 1, find quotient & remainder.

31. On dividing x3 – 3x

2 + x + 2 by a polynomial g(x) the quotient & the remainder were x – 2 & – 2x +

4 respectively. Find g(x).

32. If the polynomial x4 – 6x

3 + 16x

2 – 25x + 10 is divided by another polynomial x

2 – 2x + k, the

remainder comes out to be x + a, find the values of k & a.

33. Check whether x2 + 3x + 1 is a factor of 3x

4 + 5x

3 – 7x

2 + 2x + 2

34. Check whether x2 – x + 1 is a factor of x

3 – 3x

2 + 3x – 2

35. Can (x – 3) be the remainder on the division of a polynomial p(x) by (2x + 5)? Justify your answer.

36. Find a quadratic polynomial with zeros 3 + 2 & 3 – 2 .

37. Form a quadratic polynomial p(y) with sum & product of zeros are 2 &5

3 respectively.

38. Find the zeros of the polynomial 100x2 – 81.

39. Find the zeros of the polynomial 4x2 – 7.

40. Find all the zeros of the polynomial x4 – 3x

3 – x

2 + 9x – 6, if two of its zeros are 3 , – 3 .

41. What must be added to polynomial f(x) = x4 + 2x

3 – 2x

2 + x – 1 so that the resulting polynomial is

exactly divisible by x2 + 2x – 3.

42. If one solution of the equation 3x2 =8x +2k + 1 is seven times the other. Find the solution & the

value of k.

43. Check whether the polynomial g(x) = x3 – 3x + 1 is a factor of polynomial p(x) = x

5 – 4x

3 + x

2 + 3x

+ 1

44. Find a quadratic polynomial, the sum of whose zeros is 7 & their product is 12. Hence find the

zeros of the polynomial.

45. Find a quadratic polynomial whose zeros are 2 & – 6. Verify the relation between the coefficient &

zeros of the polynomial.

CHAPTER – 3

1. In fig. ABCD is a rectangle. Find the values of x & y.

2. In fig. ABCD is a rectangle. Find the values of x & y.

3. In fig, ABCD is a parallelogram. Find the values of x & y.

4. Solve graphically x + 2y = 5 & 2x – 3y = – 4. Also find the points where lines meet the x axis.

5. Solve graphically: 2(x – 1) = y & x + 3y = 15. Also find the points where lines meet the y axis.

6. Solve graphically; 2x + 3y = 11 & 2x – 4y = – 24. Hence find the value of coordinates of the vertices

of the triangle so formed.

7. Draw the graph of 2x + y = 4 & 2x – y = 4. Find the coordinates of the triangle formed by these

lines with y axis. Also shade this triangle.

8. Solve graphically: x – y + 1 = 0 & 3x + 2y = 12. Find the solution from the graph. Shade the

triangular area formed by two lines with x axis

9. The age of a father is equal to sum of the ages of his 6 children. After 15 years, twice the age of the

father will be the sum of ages of his children. Find the age of the father.

10. The auto fare for the first kilometer is fixed & is different from the rate per km for the remaining

distance. A man pays Rs. 57 for the distance of 16 km &Rs. 92 for a distance of 26 km. Find the

auto fare for the first km & each successive km.

11. The taxi charges in a city consist of a fixed charge together with the charge for the distance

covered. For a distance of 10 km, the charge paid is Rs. 105 & a journey of 15 km, the charge paid

Rs. 155. What are the fixed charges & the charge per km?

12. Four years ago a father was six times as old as his son. Ten years later, the father will be two &

half times as old as his son. Determine the present age of father & his son.

13. Find a fraction, that becomes 2

1 when 2 is added to its numerator & while 1 is subtracted from its

denominator it becomes3

1.

14. Father’s age is 3 times the sum of ages of his 2 children. After 5 years, his age will be twice the

sum of ages of his children. Find the age of the father.

15. Places A& B are 100 km apart on a highway. One car starts from A & another from B at the

same time. If the car travels in the same direction at different speeds, they meet in 5 hours. If they

towards each other they meet in 1 hour. What are the speeds of the two cars?

16. A lending library has a fixed charged for first three days & an additional charge for each day

there after. Bhavya paid Rs. 27 for a book for seven days, while Vrinda paid Rs 21 for a book kept

for five days. Find the fixed charge & the charge for each extra day.

17. A part of monthly hostel charges is fixed & the remaining depends on the number of days one has

taken food in the mess. When a student A takes food for 20 days she has to pay Rs. 1000 as hostel

charges where as a student B, who takes food for 26 days, pays Rs. 1180 as hostel charges. Find

the fixed charges & the cost of the food per day.

18. Half the perimeter of a rectangular garden, whose length is 4 m more then its breadth is 36 m.

Find the dimensions of the garden.

19. The sum of digits of a two digit number is 5. On reversing the digits of the number, it exceeds the

original number, by 9. Find the original number.

20. The sum of digits of a two digit number is 13. On reversing the digits of the number, it exceeds

the original number, by 27. Find the original number.

21. The sum of a 2 digit number & number obtained by reversing the order of digits is 66.If the digits

of the number differ by 2. Find the number.

22. Seven times a two digit number is equal to four times the number obtained by reversing the order

of its digits. If the digits of the number differ by 3. Find the number.

23. Nine times a two digit number is equal to two times the number obtained by reversing the order of

its digits. If one digits of the number exceeds the other number by 7. Find the number.

24. A sailor goes 8 km downstream in 40 minutes & returns in 1 hour. Find the speed of sailor in still

water & the speed of current.

25. A man travels 370 km partly by the train & partly by car. If he covers 250 km by train & rest by

car, it takes him 4 hours. But if he travels 130 km by train & rest car, he takes 18 minutes longer.

Find speed of the train & that of car.

26. A man travels 600 km partly by the train & partly by car. If he covers 400 km by train & rest by

car, it takes him 6 hours 30 minutes. But if he travels 200 km by train & rest car, he takes half an

hour longer. Find speed of the train & that of car.

27. Rekha’s mother is five times as old as her daughter Rekha. Five years later, Rekha’s mother will

be three times as old as her daughter Rekha. Find the present age of Rekha& her mother’s age.

28. Two numbers are in the ratio 5:6. If 8 is subtracted from each of the number, the ratio becomes

4:5. Find the numbers.

29. Six years hence a man’s age will be three times his son’s age & three years ago, he was nine

times as old as his son. Find their present ages.

30. A boat goes 24 km upstream & 28 km downstream in 6 hours. It goes 30 km upstream & 21km

downstream in 62

1hours. Find the speed of boat in still water & also speed of the stream.

31. 8 men & 12 boys can finish a piece of work in 10 days while 3 women &6 men can finish it in 3

days. Find the time taken by one woman alone & one man alone to finish the work. After getting

the answer what value is depicted by you?

32. Solve 2x + 3y = 11 & 2x – 4y = - 24 & hence find the value of m for which y = mx – 7.

33. 2women &5 men can finish a piece of work in 4 days while 6 men & 8 boys can finish it in 14

days. Find the time taken by one man alone & one boy alone to finish the work.

34. The sum of the numerator & denominator of a fraction is 8. If 3 are added to both the numerator

& denominator the fraction becomes4

3. Find the fraction.

35. Yash scored 40 marks in a test, getting 3 marks for each right answer & losing 1 mark for each

wrong answer. Had 4 marks been awarded for each correct answer & 2 marks been deducted for

each incorrect answer, then Yash would have scored 50 marks. How many questions were there in

the test?

36. The ratio of incomes of two persons is 9:7 & the ratio of their expenditure is 4:3. If each of them

mangoes to save Rs. 2000 per month, find their monthly incomes.

37. If 4 times the area of a smaller square is subtracted from the area of a larger square, the result is

144 m2. The sum of the areas of the two squares is 464 m

2. Determine the sides of the two squares.

38. The annual incomes of A & B are in the ratio 5:4 & their annual expenditure is in the ratio 7:5.

If each saves Rs. 3000 month. Find their monthly incomes.

39. The annual incomes of A & B are in the ratio 3:4 & their annual expenditure are in the ratio 5:7.

If each saves Rs. 15000 annually. Find their annual incomes.

40. Solve for x & y: 47x + 31y = 63; 31x + 47y = 15

41. Solve: 99x + 101y = 499; 101x + 99y = 501

42. Solve for x & y: 37x + 43y = 123; 43x + 37y = 117

43. Solve for x & y: 148x + 231y = 527; 231x + 148y = 610

44. Solve for x & y

1

5

x +

2

1

y = 2;

1

6

x –

2

3

y= 1

45. Solve for x & y

(a – b)x + (a + b)y = a2 – 2ab – b

2; (a + b)(x + y) = a

2 + b

2

46. Solve for x & y: 4x + y

6 = 15; 3x –

y

4= 7

47. Solve for x & y: x

4 + 5y = 7;

x

3 + 4y = 5

48. Solve for x & y: 4x + 3

y =

3

8;

2

x +

4

3y=

2

5

49. Solve for x & y: 3x + 4y = 10; 2x – 2y = 2

50. Solve for x & y: mx – ny = m2 + n

2; x – y = 2n

51. Solve for x & y: a

x +

b

y = 2; ax – by = a

2 – b

2

52. For what value of k, 2x + 3y = 4 & (k + 2)x + 6y = 3k + 2 will have infinitely many solutions.

53. For which values of p does the pair of equations given below have unique solution? 4x + py + 8 =

0; 2x + 2y + 2 = 0

54. For what values of a & b does the following pairs of linear equations have infinitely number of

solutions. 2x + 3y = 7; a(x + y) – b(x – y) = 3a + b – 2

55. Find values of a & b for which the system of linear equations has infinite number of solutions.

(a + b)x – 2by = 5a + 2b + 1; 3x – y = 14

56. Find values of a & b for which the system of linear equations has infinite number of solutions.

2x – (a – 4)y = 2b + 1; 4x – (a – 1)y = 5b – 1.

57. For which values of k will the following pair of equations have no solution? 3x + y = 1; (2k

– 1)x + (k – 1)y = 2k + 1

58. Is the system of linear equations 2x + 3y – 9 = 0 & 4x + 6y – 18 = 0 consistent? Justify your

answer.

59. Solve for u & v by changing into linear equations 2(3u – v) = 5uv; 2(u + 3v) = 5uv.

60. Solve the following system of linear equations by cross multiplication method: 2(ax

– by) + (a + 4b) = 0 & 2(bx + ay) + (b – 4a) = 0

61. For which values of k does the pair of equations given below have unique solution? 2x + ky = 1;

3x – 5y = 7.

62. Determine whether the following system of linear equations has a unique solution, no solution or

infinitely many solutions: 4x – 5y = 3 & 8x – 10y = 6.

CHAPTER – 4

1. Solve the following quadratic equation:(i) 6x2 + 7x – 10 = 0 (ii) x2 – 3x – 10 = 0 (iii) 3x2 – 14x + 8 = 0 (iv)

7 x2 – 6x – 13 7 = 0 (v) 4x2 + 4 3 x + 3 = 0 (vi) 6x2 – 2 x – 2 = 0 (vii) 3x2 – 2 6 x + 2 = 0

2. Find the nature of roots of the quadratic equation 2 x2 –2

3x +

2

1 =0

3. Solve: (i) 2x – x

3 = 1(ii)

2

11

xx = 3 (iii)

2

3,3;0

)32)(3(

93

32

1

3

2

x

xx

x

xx

x (iv)

xx

x

x

x 171

2

3

(v)

2

3,1;3

32

)1(4

1

32

x

x

x

x

x (vi) 7,4,

30

11

7

1

4

1

x

xx(vii)

4,2,1;4

4

2

2

1

1

x

xxx (viii) 2,1;3

2

2

1

1

x

x

x

x

x

4. Using quadratic formula, determine the roots of the following equation: x – x

1 = 3

5. Solve : (i) 10ax2 – 6x + 15ax – 9 = 0, a ≠ 0 (ii) 9x2 – 9(a + b)x + (2a2 + 5ab + 2b2) = 0 (iii) a(a2 + b2) x2 + b2x – a = 0 (iv) 9x2 – 3(a + b)x + ab = 0 (v) P2x2 + (p2 – q2)x – q2 = 0 (vi) a2b2x2 + b2x – a2x – 1 = 0

(vii) .1111

baxxba

6. Solve for x by using quadratic formula: 36x2 – 12ax + (a2 – b2) = 0 7. Three consecutive positive integers are taken such that the sum of the square of the first & the product of

the other two is 154. Find the integers. 8. The sum of the squares of two consecutive natural numbers is 421. Find the numbers. 9. Find two consecutive odd positive integers, sum of whose squares is 290. 10. Find two positive numbers whose squares have the difference 48 & the sum of the numbers is 12. 11. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger

number. Find the two numbers.

12. The sum of a number & its positive square root is25

6. Find the number.

13. Two positive numbers differ by 3 & their product is 504. Find the numbers.

14. The sum of a number & its reciprocal is 3

10. Find the number.

15. The sum of two natural numbers is 8. Determine the numbers, if the sum of their reciprocal is 15

8.

16. A two digit number is such that product of its digit is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

17. The product of the digits of a two digit positive number is 24. If 18 is added to the number then the digits of the number are interchanged. Find the number.

18. The product of Tanay’s age (in years) five years ago & his age ten year later is 16. Determine Tanay’s present age.

19. The sum of the reciprocals of Rehman’s ages (in years) 3 years ago & 5 years from now is 3

1. Find his

present age.

20. The sum of ages of father & his son is 45 years. 5 years ago, the product of their ages was 124. Determine their present ages.

21. The hypotenuse of right angled triangle is 6 cm more than twice its shortest side. If third side is 2 cm less than hypotenuse, find the sides of this triangle.

22. The hypotenuse of a right triangle is 3 5 cm. If the smaller side is tripled & the larger side is doubled,

the new hypotenuse will be 15 cm. Find the length of each side. 23. By increasing the speed of a bus by 10 km/hr, it takes one & half hours less to cover a journey of 450 km.

Find the original speed of the bus. 24. A fast train takes 3 hours less than a slow train for a journey of 600 km. If speed of the slow train was 10

km/hr less than that of the fast train, find the speeds of the trains. 25. A plane left 30 minutes later than the schedule time & in order to reach its destination 1500 km away in

time, it has to increase its speed by 250 km/h from its usual speed. Find its usual speed. 26. An express train takes 1 hour less than a passenger train to travel 132 km between stations A & B

(without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the trains.

27. A motor boat whose speed is 18 km/h in still water takes 1hr. more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

28. The speed of a boat in still water is 11 km/h. It can go 12 km upstream & return downstream to the original point in 2 hours 45 minutes. Find the speed of the stream.

29. The speed of a boat in still water is 15 km/h. It can go 30 km upstream & return downstream to the original point in 4 hours 30 minutes. Find the speed of the stream.

30. The numerator of fraction is 3 less than its denominator. If 2 is added to both numerator as well as

denominator, the sum of new & original fraction is 20

29, find the fraction.

31. The denominator of a fraction is one more than twice the numerator. If the sum of the fraction & its

reciprocal is 221

16, find the fraction.

32. The length of a rectangular plot is greater than thrice its breadth by 2 m. The area of the plot is 120 m2. Find the length & breadth of the plot.

33. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.

34. A farmer wishes to start a 100 m2 rectangular vegetable garden. Since, he has only 30 m barbed wire, he fences three sides of the rectangular garden letting his house compound wall act as the fourth side of the fence. Find the dimensions of his garden.

35. Some students planned a picnic. The budget for food was Rs. 480. But 8 of them failed to go, the cost of food for each member increased by Rs. 10. How many students attended the picnic?

36. A man bought a certain number of toys for Rs. 180. He kept one for his own use & sold the rest for one rupee each more than he gave for them. Besides getting his own toy for nothing, he made a profit of Rs. 10. Find the number of toys, he initially bought.

37. A factory produces certain pieces of pottery in a day. It was observed on a particular day that the cost of production of each piece (in Rs.) was 3 more than twice the number of articles produced in the day. If the total cost of production on that day was Rs. 90, find the number of pieces produced & cost of each piece.

38. If two pipes function simultaneously, a reservoir will be filled in 12 hours. First pipe fills the reservoir 10 hours faster than the second pipe. How many hours will the second pipe take to fill the reservoir?

39. Two pipes running together can fill a tank in 6 minutes. If one pipe takes 5 minutes more than the other to fill the tank, find the time in which each pipe would fill the tank separately.

40. Two pipes together fill a tank in 98

3hours. The tap of larger diameter takes 10 hours less than the smaller

one to fill the tank separately. Find the time in which each tap can separately fill the tank. 41. A takes 6 days less than the time taken by B to finish a piece of work. If both A & B together can finish it

in 4 days, find the time taken by B to finish the work. 42. Find the value of k for which the equation has equal roots.(i) kx(x – 2) + 6 = 0 (ii) x2 – 2x + k = 0 (iii) 9x2 +

8kx + 16 = 0 (iv) 4a2x2 – 4abx + k = 0 (v) x2 – 8kx + 2k = 0 (vi) (k – 12)x2 + 2(k – 12)x + 2 = 0 (vii) (k + 4)x2 + (k + 1)x + 1 = 0

43. Write all the values of k for which the quadratic equation 2x2 + kx + 8 = 0, has equal roots. Also find the roots.

44. One root of the equation 2x2 – 8x – m = 0 is 2

5

. Find the other root & the value of m.

45. If the equations 5x2 + (9 + 4p)x + 2p2 = 0 & 5x + 9 = 0 are satisfied by the same value of x, find the value of p.

46. If one of the quadratic equation x2 – 5x + 6k = 0 is reciprocal of other, find the value of k. Also find the roots.

47. If – 4 is a root of the quadratic equation x2 + px – 4 = 0 & the equation 2x2 + px + k = 0 has equal roots, find the value of k.

CHAPTER – 5

1. Find the 12th term of the A.P. 2 , 3 2 , 5 2 , ………….. 2. Which term of an AP 21, 18, 15, …………… is zero? 3. Find the number of terms of the series: ( – 5) + ( – 8) + ( – 11) + ………+ ( – 230) 4. Find the 10th term from the end of the A.P. 4, 9, 14, …….. 254. 5. Find the 20th term from the last term (end) of the A.P: 3, 8, 13, ……….. 253.

6. Which term is the first negative term in the given AP: 23, 212

1, 20, ………..?

7. Which term of the A.P. 3, 15, 27, 39, ……is 132 more than its 54th term? 8. Which term of the A.P. 3, 15, 27, 39, ……is 132 more than its 60th term? 9. Find the value of the middle most term(s) of the arithmetic progression: –11, –7, –3, ………49.

10. Find the common difference of an A.P. whose first term is 2

1& the 8th term is

6

17.

11. If 6th term of an A.P. is –10 & its 10th term is –26, then find the 15th term of the A.P. 12. The 4th term of an A.P. is equal to 3 times the first term & the 7th term exceeds twice the 3rd term by 1.

Find the A.P. 13. If m times the mthterm of an A.P. is same as n times the nth term, find its (m + n)th term. 14. If a, b & c be the sums of first p, q & r terms respectively of an AP, show that

0)()()( qpr

cpr

q

brq

p

a

15. For an A.P. show that ap + ap + 2q = 2 ap + q 16. For what value of p are 2p, p + 10 & 3p + 2 in A.P.? 17. Find the value of p, if the numbers x, 2x + p, 3x + 6 are three consecutive terms of an A.P. 18. Find the sum of all two digit odd positive numbers. 19. Find the sum of first tweleve multiples of 7. 20. Find the sum of all natural numbers between 200 & 1000 exactly divisible by 6. 21. Find the number of all 2-digit numbers divisible by 3. 22. Find the sum of all 2-digit natural numbers which are divisible by 4. 23. Find the sum of all three digit numbers which leave the same remainder 2 when divided by 5. 24. Find the sum of the first 25 terms of an AP whose nth term is given by tn = 7 – 3n. 25. If the nth term of an A.P. is (2n + 1), find the sum of first n terms of the A.P.

26. If an A.P, the sum of first n terms is given by sn = 2

5

2

3 2 nn . Find the 25th term of the A.P.

27. How many terms of the A.P. 9,17, 25, ………, must be taken to get a sum of 450? 28. How many terms of the A.P. 78, 71, 64, ………are needed to give the sum 465? Also find the last term of

this A.P. 29. If the sum of all the terms of an A.P. 1, 4, 7, 10, ……, x is 287, find x. 30. In an A.P. the first term is –4, the last term is 29 & the sum of all its terms is 150. Find the common

difference of the A.P. 31. If the sum of first fourteen terms, of an AP is 1050 & its first term is 10, find its 20th term. 32. The sum of first 7 terms of an A.P. is 49 & that of first 17 terms is 289. Find the sum of first n terms. 33. The sum of the 5th& 7th terms of an AP is 52 & its 10th term is 46. Find the AP. 34. In an AP the sum of first ten terms is –80 & the sum of next ten terms is –280. Find the A.P. 35. If Sn denotes the sum of n terms of an AP whose common difference is d & first term is a, find Sn – 2Sn – 1 +

Sn – 2 . 36. If the sum of first m terms of an A.P. is n & the sum of first n terms is m, then show that the sum of its first

(m + n) terms is – (m + n).

37. Find three numbers in A.P. whose sum is 15 & product is 80. 38. If sum of three numbers in A.P. is 21 & their product is 231. Find the numbers. 39. The sum of the first three terms of an AP is 33. If the product of the first & the third term exceeds the

second term by 29, find the AP. 40. The angles of a triangle are in AP. The greatest angle is twice the least. Find all angles of the triangle. 41. The sum of the third & seventh term of an AP is 6 & their product is 8. Find the sum of the first sixteen

terms of the AP. 42. The sum of first six terms of an AP is 42. The ratio of its 10th term to its 30th term is 1:3. Find the first & the

13th term of the AP. 43. A sum of Rs. 700 is to be used for giving 7 cash prizes to students of a school for their academic

performance. If each prize is Rs. 20 less than its preceding prize, find the value of each of the prizes. 44. Neera saves Rs. 1600 during the first year, Rs. 2100 in the second year, Rs. 2600 in the third year. If she

continues her savings in this pattern, in how many years will she save Rs. 38500? 45. A contractor on construction job specifies a penalty for delay of completion beyond a certain date as

follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day etc., the penalty for each succeding day being Rs. 50 more than for the preceeding day. How much money the contractor has to pay as penalty if he has delayed the work by 30 days?

46. Jaipal singh repays the total loan of Rs. 118000 by paying every month starting with the first instalment of Rs. 1000. If he increases the instalment by Rs. 100 every month. What amount will be paid by him in the 30thinstalment? What amount of loan does he still have to pay after 30thinstalment?

47. The ticket receipts at the show of a film amounted to Rs. 6500 on the first day & showed a drop of Rs. 110 every succeeding day. If the operational expenses of the show are Rs. 1000 a day, find on which day the show ceases to be profitable.

48. 228 logs are to be stacked in a store in the following manner: 30 logs in the bottom, 28 in the next row, then 26 & so on. In how many rows can these 228 logs be stacked? How many logs are there in the last row?

49. In November 2009, the number of visitors to a zoo increased daily by 20. If a total of 12300 people visited the zoo in that month, find the number of visitors on 1st November 2009 .

50. The houses of a row are numbered consecutively from 1 to 49. There is a value of x such that the

sum of the numbers of the houses preceding the house numbered x is equal to the sum of the

numbers of the houses following it. Find this value of x.

51. A spiral is made up of successive semicircles, with centers alternatively at A & B, starting with

center at A of radii 0.5 cm, 1 cm, 1.5 cm, 2 cm …….as shown in the figure. What is the total

length of such spiral made up of thirteen consecutive semicircles? ( = 7

22)

CHAPTER – 6

1. State & prove proportionality theorem.

2. If a line drawn parallel to one side of a triangle to intersect the other sides in distinct points, then

prove that the line drawn, divides the two sides in the same ratio.

3. Prove that the line joining the mid points of any two sides of triangle is parallel to third side.

4. In the fig, ABCD is a trapezium in which AB ׀׀ DC. The diagonals AC & DB intersect at O.

Prove that OD

OB

OC

OA

5. In fig, PQ ׀׀ CD & PR ׀׀ CB. Prove thatRB

AR

QD

AQ .

6. In fig, DE ׀׀ AC & DF ׀׀ AE. Prove thatBE

EC

BF

EF

7. In fig, A, B, C are points on OP, OQ & OR respectively such that AB ׀׀ PQ & AC ׀׀ PR. Show

that BC ׀׀ QR.

8. In fig., DE ׀׀ OQ & , DF ׀׀ OR. Show that , EF ׀׀ QR.

9. In ABC fig, D & E are two points lying on side AB such that AD = BE. If DP ׀׀ BC & EQ ׀׀

AC, then prove that PQ ׀׀ AB.

10. In fig. if A = B & AD = BE show that DE ׀׀ AB inABC

11. In fig, XY ׀׀ QR, XQ

PQ=

3

7& PR = 6.3 cm. Find YR.

12. If D, E are points on the sides AB & AC of ABC such that AD = 6 cm, BD = 9 cm, AE = 8 cm,

EC = 12 cm. Prove that DE ׀׀ BC.

13. In fig, AB ׀׀ DC. Find the value of x.

14. In fig, DE ׀׀ BC. Find x.

15. In ABC, AB = AC & D is a point on side AC such that BC 2 = AC.CD. Prove that BD = BC.

16. In the fig, PR

QT

QS

QR &TQR = PRS. Show thatPQS ~PQR.

17. In fig, D is a point on the side BC of ABC such that ADC = BAC. Prove that CA

CB

CD

CA

18. In fig, E is a point on side CB produced of an isoscelesABC with AB = BC. If AD BC & EF

AC. Prove that ABD ~ ECF.

19. If AD & PM are medians of ABC &PQR respectively where ABC ~PQR. Prove that PQ

AB

= PM

AD

20. In fig, ABE ACD. Prove that ADE ~ABC

21. In fig, AB BC, DE AC &GF BC. Prove that ADE ~GCF.

22. In fig, sides XY & YZ median XA of a triangle XYZ are respectively proportional to sides DE, EF

& median DB of DEF, then prove that XYZ ~ DEF.

23. In fig, ACB ~APQ, if BC = 8 cm, PQ = 4 cm, BA = 6.5 cm, AP = 2.8 cm, find CA & AQ.

24. In fig, OA.OB = OC.OD. Show that A = C &B = D

25. If one diagonal of a trapezium divides the other diagonal in the ratio 1:2. Prove that one of the

parallel sides is double the other.

26. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their

corresponding sides.

27. If the areas of two similar triangles are equal, prove that they are congruent.

28. Two Isosceles triangles have equal vertical angles & their areas in the same ratio 16:25. Find of

their corresponding heights.

29. In fig, DE ׀׀ BC & AD: DB = 5:4. Find)(

)(

CFBar

DEFar

30. In fig,ABC is an isosceles triangle right angled at B. Two equilateral triangles are constructed

with side BC & AC. prove that arBCD = 2

1arACE

31. In fig, D, E, F are mid points of sides BC, CA, AB respectively of ABC. Find ratio of areas of

DEF to area of ABC.

32. The diagonals of a trapezium ABCD, in which AB ׀׀ DC, intersect at O. If AB = 2CD, then find

the ratio of areas of triangles AOB & COD.

33. P & Q are points on sides AB & AC respectively of ABC. If AP = 3 cm, PB = 6 cm, AQ = 5 cm

& QC = 10 cm, show that BC = 3PQ.

34. State & prove of Pythagoras theorem.

35. Prove that in a right triangle the square of the hypotenuse is equal to the sum of the squares of

the other two sides.

36. State & prove converse of Pythagoras theorem.

37. Prove that in a triangle if the square of one side is equal to the sum of the squares of the other

two sides then the angle opposite to the first side is a right angle.

38. ABC is an isosceles triangle, right angled at C. Prove that AB2 = 2BC

2.

39. ABC is an isosceles triangle with AC = BC. If AB2 = 2AC

2. Prove that ABC is an right angled

triangle.

40. In ABC, AD BC such that AB 2 = BD.CD. Prove that ABC is right angled triangle, right

angle at A.

41. In ABC, AD BC such that AD 2 = BD.CD. Prove that ABC is right angled triangle, right

angle at A.

42. In figure, ACB = 900& CD AB. Prove that

AD

BD

AC

BC

2

2

.

43. In fig. ABD is a triangle in which DAB = 90

0& AC BD. Prove that AD

2 = BD.CD.

44. In fig. ABD is a triangle in which DAB = 90

0& AC BD. Prove that AC

2 = BC.DC.

45. In fig, PQR is a triangle in which PM PR & PR

2 – PQ

2 = QR

2. Prove that QM

2 = PM.MR

46. Prove that the area of the equilateral triangle described on the side of a square is half the area of

the equilateral triangle described on its diagonal.

47. Prove that the equilateral triangles described on the two sides of a right-angled triangle are

together equal to the equilateral triangle described on the hypotenuse in terms of their areas.

48. Prove that the sum of the squares of the side of a rhombus is equal to the sum of the squares of its

diagonals.

49. The perpendicular AD on the base BC of ABC intersects BC in D such that BD = 3CD. Prove

that 2AB 2 = 2AC

2 + BC

2

50. In an isosceles triangle ABC with AB = AC. BD AC. Prove that BD 2 – CD

2 = 2CD.AD

51. In an equilateral triangle ABC, D is the point on side BC such that 3BD = BC. Prove that 9AD 2

= 7 AB 2.

52. In ABC, if AD is the median, then prove that AB2 + AC

2 = 2[AD

2 + BD

2].

53. In a quadrilateral ABCD,B = 900. If AD

2 = AB

2 + BC

2 + CD

2, prove that ACD = 90

0.

54. In fig, PQR is a right angled triangle in which Q = 900. If QS = SR, show that PR

2 = 4PS

2 –

3PQ2

55. In fig, P & Q are the midpoints of the sides CA & CB respectively of ABC right angle at C.

Prove that 4(AQ2 + BP

2) = 5AB

2.

56. In fig, if AD BC. Prove that AB2 + CD

2 = BD

2 + AC

2.

57. In fig, ABCD is a rhombus. Prove that 4AB2 = AC

2 + BD

2

58. In fig. if AD BC, then prove that AB2 + CD

2 = AC

2 + BD

2

59. In fig, two triangles ABC & DBC are on the same base BC in which A = D = 900. If CA &

BD meet each other at E, show that AE.CE = BE.DE.

60. Two right triangles ABC & DBC are drawn on the same hypotenuse BC & on the same side of

BC. If AC & BD intersect at P, prove that AP PC = BP PD

61. In a right angled triangle if hypotenuse is 20 cm & the ratio of the other two sides is 4:3, find the

sides.

62. In an isosceles triangle ABC, if AB = AC = 13 cm & the altitude from A on BC is 5 cm. Find BC.

63. In fig. QPR = 900, PMR = 90

0, QR = 26 cm, PM = 6 cm. Find area (PQR).

64. In fig. PM =6 cm, MR = 8 cm & QR = 26 cm, find the length of PQ.

65. In fig, ABC is a right triangle right angled at C. Let BC = a, CA = b, AB = c & let p be the length

of perpendicular from C on AB. Prove that.

(i) cp = ab (ii) 222

111

bap

66. In fig, PA, QB & RC are perpendicular to AC. Prove that x

1+

z

1=

y

1

67. In fig, l ׀׀ m & line segments AB, CD & EF are concurrent at p. prove that: 𝑨𝑬

𝑩𝑭=

𝑨𝑪

𝑩𝑫=

𝑪𝑬

𝑭𝑫.

68. In fig. PQR & SQR are two triangles on the same base QR. If PS intersect QR at O, then show

that SO

PO

SQRar

PQRar

69. In fig, XY ׀׀ AC & XY divides triangular region ABC into two parts equal in area. Find the ratio

of AB

AX

CHAPTER – 7

1. Show that the points (– 4, 0), (4, 0) & (0, 3) are vertices of an isosceles triangle. 2. Show that P (1, –1) is the Centre of the circle circumscribing the triangle whose angular points are A (4, 3),

B (–2, 3) & C (6, –1). 3. Show that the points (7, 3), (3, 0), (0, –4) & (4, –1) are the vertices of a rhombus. 4. If the points A (4, 3) & B (x, 5) are on the circle with Centre O (2, 3); find the value of x. 5. Find the value of x such that PQ = QR where the coordinates of P, Q &R are (6, –1), (1, 3) & (x, 8)

respectively. 6. Find a relation between x & y such that the point P (x, y) is equidistant from the points A (3, 6) & B (–3, 4) 7. Find a point on X-axis which is equidistant from (–3, –4) & (4, –3). 8. Find a point on Y-axis which is equidistant from (2, 2) & (9, 9).

9. Find the points on the X-axis which are at a distance of 2 5 from the point (7, –4). How many such points

are there? 10. Find those points on the X-axis which are at a distance of 5 units from the point (5, –3) 11. Find the value of x, if the distance between the points (x, –1) & (3, –2) is x + 5. 12. Find the values of y for which the distance between the points P (2, –3) & Q (10, y) is 10 units. 13. In figure, in ABC, D & E are the mid-points of the sides BC & AC respectively. Find the length of DE. Prove

that DE = 2

1AB.

14. If A & B are the points (–2, –2) & (2, –4) respectively find the coordinates of P on the line segment AB such

that AP = 7

3AB.

15. Find the coordinates of the point B, if the point P (–4, 1) divides the line segment joining the points A (2, –2) & B in the ratio 3:5.

16. Find the ratio in which the Y-axis divides the join of (5, –6) & (–1, –4). 17. Determine the ratio in which the point (x, 2) divides the line segment joining the points (–3, –4) & (3, 5).

Also find x.

18. In what ratio does the point

6,

2

1divide the line segment joining the points (3, 5) & (–7, 9)?

19. Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points (2, –2) & (3, 7).

20. The line segment joining the points A(2, 1) & B (5, –8) is trisected at the points P & Q where P is nearer to A. If point P lies on the line 2x – y + k = 0, find the value of k.

21. One end of a diameter of a circle is at (2, 3) & the centre is (–2, 5). What are the coordinates of the other end of its diameter?

22. Three consecutive vertices of a parallelogram ABCD are A (1, 2), B (1, 0) &C(4, 0). Find the fourth vertex D. 23. If points A (–2, –1), B (a, 0), C (4, b) & D (1, 2) are the vertices of a parallelogram ABCD, find the values of

a & b. 24. Find the lengths of the medians AD & BE of the triangle ABC whose vertices are A (1, –1), B (0, 4) & C (–5,

3). 25. The mid points of the sides AB, BC & CA of a triangle ABC are D (2, 1), E (1, 0) & F (–1, 3) respectively. Find

the coordinates of the vertices of the triangle ABC. 26. Find the area of ABC whose vertices are A (4, 4), B (0, 0) & C (6, 2). 27. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices

are (0, –1), (2, 1) & (0, 3). 28. The points A (2, 9), B (a, 5) & C (5, 5) are the vertices of aABC, right angled at B. Find the value of a &

hence the area of ABC. 29. The area of a triangle whose vertices are (–2, –2), (–1, –3) & (x, 0) is 3 square units. Find the value of x. 30. Find the area of the quadrilateral ABCD formed by the joining of the following points in order: A(2, 9), B

(3, 5), C (5, 5), D(7, 9) 31. Find the area of a rhombus if the vertices are (3, 0), (4, 5), (–1, 4) & (–2, –1) taken in order. 32. Show that the point P (–4, 2) lies on the line segment joining the points A (–4, 6) & B (–4, –6). 33. Show that the points (1, –1), (5, 2) & (9, 5) are collinear. 34. Find the value of k for which the points A (–1, 3), B (2, k) & C (5, –1) are collinear.

CHAPTER – 8

1. In ABC, B = 900, AB = 3 cm & BC = 4 cm. Find (i) sinC (ii) cosC (iii) secA (iv) cosecA

2. If tan = 12

5, find the value of:

sincos

sincos

3. If 3 tan = 3sin , then prove that sin2 – cos

2 = 3

1

4. If tanA = 2 – 1. Show that sinAcosA = 4

2

5. If 3cotA = 4, find the value of 1cos

1cos2

2

Aec

Aec

6. If tanA = 2 Evaluate secA.sinA + tan2A – cosecA

7. In PQR, right angled at Q, PR + QR = 25 cm & PQ = 5 cm. Find the value of sinP.

8. If sin = n

m, find the value of

1cot4

4tan

.

9. If sec – tan = 4 then prove that cos = 17

8

10. If tanA = 3

1,ABC is right angled at B. Find the value of sinA.cosC + cosA.sinC

11. If 7sin2 + 3cos

2 = 4, then prove that sec + cosec = 2 + 3

2

12. If sec = x + x4

1 , then prove that sec + tan = 2x or

x2

1

13. If sin ( A + B ) = cos ( A – B ) = 2

3& A, B ( A > B ) are acute angles, find the values of A & B.

14. If tan ( A + B ) = 3 , tan ( A – B ) = 1, where A > B & A, B are acute angles. Find the values of

A & B.

15. If sin ( A – B ) = 2

1, cos ( A + B ) =

2

1& 0 < A + B < 90

0 & A > B then find the value of A & B.

16. If tan ( A + B ) = 3 , tan ( A – B ) = 3

1, where A > B & A, B are acute angles. Find the values

of A & B.

17. If A & B are acute angles such that cosA = cosB. Show that A = B.

18. If sin(2A + 450) = cos(30

0 – A), find A.

19. If sec4A = cosec(A – 200), where 4A is an acute angle, find the value of A.

20. If sin3A = cos(A – 260) where 3A is an acute angle, find the value of A.

21. Find the value of the expression 00

00

30sin60cos1

60sin30cos

22. Evaluate: 0200

02020202

30cot60sec30cos

90cos530sec345sin460tan

ec

23. In ABC, ABC = 900, AB =5 cm &ACB = 30

0, find BC & AC.

24. Given that sin(A+B) = sinAcosB + cosAsinB, find the value of sin750

25. Find the value of 0

0

70sin

20cos +

0

0

20sin

70cos – 8 sin

230

0

26. Find the value of sin25

0 + sin

210

0 + sin

280

0 + sin

285

0

27. Evaluate: 0

0

51cos

39sin + 2tan11

0tan31

0tan45

0tan59

0tan79

0 –3( sin

221

0 + sin

269

0)

28. Evaluate: 0

0

22cos

68sin2 –

0

0

75tan5

15cot2 –

5

70tan.50tan.40tan.20tan.45tan3 00000

29. Evaluate: sin(500+ ) – cos(40

0 – ) + tan1

0tan10

0tan20

0tan70

0tan80

0 tan89

0 + sec(90

0 –

).cosec – tan(900 – )cot

30. Evaluate: )65sin25(sin2

cot)90(sec0202

202

+

)47cot43(sec3

62tan28tan60cos20202

020202

+

0

0

50tan

40cot

31. Evaluate: )42cos48(cos4

tan)90(cos0202

202

ec –

0202

020202

20tan70cos

38sin52sec30tan2

ec

32. Without using trigonometric tables, evaluate )42cos48(cos4

tan)90(cos0202

202

ec –

0202

020202

70cos20tan

38sin52sec30tan2

ec

33. Find the value of k, if 0

0

70sin

30cos +

)90sin(

cos20

=

2

k

34. Evaluate: 0

0

20sin

70cos +

00000

00

85tan65tan45tan25tan5tan

35cos.55cot ec

35. Evaluate: 00000

02020

80tan70tan30tan.20tan.10tan

55sin35sin)90(cos.sec)90cot(.tan ec

36. Evaluate: )42cos48(cos4

tan)90(cos0202

202

ec +

5

2sin48

0.sec42

0 –

5

1tan

260

0

37. If sin + cos = 2 sin(900 – ), show that cot = 2 + 1

38. If sin + cos = 2 sin(900 – ), then find the value of tan

39. Pt cosec2(90

0 – ) – tan

2 = cos2(90

0 – ) + cos

2

40. Pt sec2 + cot

2(90 – ) = 2cosec

2(90 – ) – 1

41. Pt)90sin(

sin0

+

)90cos(

cos0

= sec cosec

42. Pt)90sin(1

)90cos(0

0

+

)90cos(

)90sin(10

0

= 2cosec

43. Pt

tan

)90cot( 0 +

)90tan(

sin).90(cos0

0

ec = sec

2

44. If A + B = 900, then prove that

A

B

BA

BABA2

2

cos

sin

secsin

cot.tantan.tan

= tanA

45. If A, B & C are interior angles of ABC, then show that: tan2

BA = cot

2

C

46. If A, B & C are interior angles of ABC, then show that: cosec2

BA = sec

2

C

47. If A, B, C are interior angles of ABC, show that: cosec2

2

CB – tan

2

2

A = 1

48. If A, B, C are interior angles of ABC, show that: sec2

2

CB – 1 = cot

2

2

A

49. If A, B, C are interior angles of ABC, show that:cos2

2

A + cos

2

2

CB = 1

50. Simplify: (sec + tan )(1 – sin )

51. Pt (1 + cot – cosec )(1 + tan + sec ) = 2

52. Pt 2sec2 – sec

4 – 2cosec2 + cosec

4 = cot4 – tan

4

53. Pt sin6 + cos

6 = 1 – 3sin2 cos

2

54. Pt sec4 – tan

4 = 1 + 2 tan2 .

55. Pt tan2 + cot

2 + 2 = sec2 cosec

2

56. Pt cos4 – cos

2 = sin4 – sin

2

57. Pt sec ( 1 – sin )( sec + tan ) = 1

58. If sin + sin2 = 1, then find the value of cos

2 + cos4 .

59. Pt( 1 + tan A.tanB)2 + ( tanA – tanB)

2 = sec

2A.sec

2B.

60. PtsinA(1 + tanA) + cosA(1 + cotA) = secA + cosecA

61. Pt (cosecA – sinA)(secA – cosA)(tanA + cotA) = 1

62. Pt 2(sin6 + cos

6 ) – 3(sin4 + cos

4 ) + 1 = 0

63. Pt (cosecA – sinA)(secA – cosA) = AA cottan

1

64. Prove that ( cosec – cot )2 =

cos1

cos1

65. If A is an acute angle of a right angled triangle, prove that A

A

sin1

sin1

= sec A + tan A

66. Pt1cos

1cos

ecA

ecA = secA + tanA

67. Pt1sec

1sec

+

1sec

1sec

= 2cosec

68. PtA

A

tan1

cos

+

A

A

cot1

sin

= sinA + cosA

69. Pt tansec

1

–

cos

1 =

cos

1 –

tansec

1

70. Pt

cot1

tan

+

tan1

cot

= 1 + sec .cosec

71. Pt cotcos

1

ec –

sin

1 =

sin

1 –

cotcos

1

ec

72. Pt

cot1

tan

+

tan1

cot

= 1 + tan + cot

73. PtA

A

sin

cos1+

A

A

cos1

sin

= 2cosecA

74. PtA

A

sin1

cos

+

A

A

cos

sin1 = 2secA

75. PtA

A

sin1

cos

+

A

A

sin1

cos

= 2secA

76. PtA

A

sin1

cos

+

A

A

cos

sin1 = 2secA

77. Pt

cossin

cossin

+

cossin

cossin

=

22 cossin

2

78. Pt

cossin

cossin

+

cossin

cossin

=

1sin2

22

79. Pt 1 +

eccos1

cot 2

=

sin

1

80. Pt

sincos1

sincos1

=

cos

sin1

81. Pt

sin1cos

sincos1

= cosec + cot

82. Pt

sec1tan

sec1tan

=

tansec

1

83. Pt 4242 sin

1

sin

2

cos

1

cos

2 = cot

4 – tan4

84. PtA

A

sec

sec1 =

A

A

cos1

sin 2

85. Pt

coscos2

sin2sin3

3

= tan

86. Pt sec2 –

24

42

coscos2

sin2sin

= 1

87. PtA

AA

sin1

)1(seccot 2

= sec

2A

A

A

sec1

sin1

88. Pt

cos.sin

cottan = tan

2 –cot2

89. Simplify:

cos

sin1

cos

sin

cos

1

90. Simplify: sin

eccos

1

sin

1

91. If x = rsinA.cosC, y = rsinA.sinC, z = rcosA, prove that r2 = x

2 + y

2 + z

2

92. If msin + ncos = p &mcos – n sin = q then prove that m2 + n

2 = p

2 + q

2

93. If x = asec + btan & y = atan + bsec . Prove that x2 – y

2 = a

2 – b

2

94. If a

xcos +

b

ysin = 1 &

a

xsin –

b

ycos = 1, prove that

2

2

a

x +

2

2

b

y = 2

95. If

cos

cos = m &

sin

cos = n, then show that(m

2 + n

2)cos

2 = n

2.

96. If tan + sin = m & tan – sin = n, prove that m2 – n

2 = 4 mn

97. If tan + sin = m & tan – sin = n, prove that (m2 – n

2)2 = 16mn

98. If tanA = ntanB&sinA = msinB, prove that cos2A =

1

12

2

n

m

99. If sin + cos = p & Sec + cosec = q then prove that q( p2 – 1) = 2p.

100. If sec + tan = p, show that 1

12

2

p

p = sin

101. If asin + bcos = c, then prove that acos – bsin = 222 cba .

102.

103. If 2cos – sin = x &cos – 3sin = y. Prove that 2x2 + y

2 – 2xy = 5.

104. In fig, ABC is right angled triangle, right angled at C, D is mid point of BC. Show that

tan

tan

= 2

1

CHAPTER – 9

1. Find the elevation of the sun at the moment when the length of the shadow of a vertical tower is

just equal to the height of the tower.

2. An electric pole is 10 m high. If its shadow is 10 𝟑 m in length. Find the angle ofelevation of

the sun at that time.

3. A player sitting on the top of a tower of height 20 m observes the angle of depression of a ball

lying on the ground as 600. Find the distance between the foot of the tower and the ball.

4. A kite is flying at a height of 90 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 600. Find the length

of the string assuming that there is no slack in the string. (𝒖𝒔𝒆 𝟑 = 𝟏.𝟕𝟑𝟐 )

5. The upper part of tree is broken over by the wind makes an angle of 300 with the ground and the

horizontal distance from the root of tree to the point where the top of tree meets the ground is 25

m. find the height of tree before it was broken.

6. A tree breaks due to storm and the broken part bends so that the top of the tree touches the

ground making an angle 300 with it. The distance between the foot of the tree to the point where

the top touches the ground is 8 m. Find the height of the tree.

7. If the shadow of a tower is 30 m long, when the sun’s elevation is 300. What is the length of the

shadow, when sun’s elevation is 600?

8. Find the height of a mountain if the elevation of its top at an unknown distance from the base is

600 and at a distance 10 km further off from the mountain, along the same line, the angle of

elevation is 300.

9. From the top of a hill the angles of depression of two consecutive kilometer stones due east are

found to be 300 and 60

0. Find the height of the hill.

10. A boy 2 m tall is standing at some distance from a 30 m tall building. The angle of elevation from

his eyes of the top of the building increases from 300 to 60

0 as he walks towards the building.

Find the distance he walked towards the building.

11. A person standing on the bank of a river observes that the angle of elevation of the top of the tree

standing on the opposite bank is 600. When he moves 30 m away from the bank, he finds the

angle of elevation to be 300. Find the height of the tree and the width of the river.

12. From the top of a building 60 m high the angles of depression of the top and the bottom of a

tower are observed to be 300 and 60

0 respectively. Find the height of the tower.

13. Two pillars of equal heights are on either side of a road, which is 100 m wide. The angles of

elevation of the top of the pillars are 600 and 30

0 at a point on the road between the pillars. Find

the position of the point between the pillars on the road and the height of the pillars.

14. From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are

observed to be 300 and 45

0 respectively. Find the height of the pole.(𝒖𝒔𝒆 𝟑 = 𝟏.𝟕𝟑 )

15. Two men on either side of a cliff, 60 m high, observe the angles of elevation of the top of the cliff

to be 450 and 60

0 respectively. Find the distance between two men.

16. From the top and foot of a tower 40 m high, the angle of elevation of the top of a light house are

found to be 300 and 60

0 respectively. Find the height of the light house. Also find the distance of

the top of the light house from the foot of the tower.

17. The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-

storied building are 300 and 45

0 respectively. Find the height of the multi storied building.

18. An aero plane at an altitude of 200 m observes the angles of depression of two opposite points on

two banks of the river to be 450 and 60

0. Find, in metres, the width of the river.( use 𝟑 = 𝟏.𝟕𝟑𝟐

)

19. An aircraft is flying at a constant height with a speed of 360 km/hour. From a point on the

ground, the angle of elevation at an instant was observed to be 450. After 20 seconds, the angle of

elevation was observed to be 300. Determine the height at which the aircraft is flying. (Take

𝟑 = 𝟏.𝟕𝟑2 )

20. The angle of elevation of an aeroplane from a point on the ground is 450. After flying for 15

seconds, the angle of elevation changes to 300. If the aeroplane is flying at a constant height of

2500 m, find the average speed of the aeroplane.

21. From a point on the ground, the angles of elevation of the bottom and top of a transmission tower

fixed at the top of 20 m high building are 450 and 60

0 respectively. Find the height of the

transmission tower.

22. A vertical tower is surmounted by a flag staff of height 5 metres. At a point on the ground, the

angles of elevation of bottom and top of flag staff are 450 and 60

0 respectively. Find the height of

the tower.

23. The angle of elevation of a cloud from a point 60 m above the lake is 300 and the angle of

depression of its reflection in the lake is 600. Find the height of the cloud above the lake.

24. The angle of elevation of a cloud from a point 200 m above the lake is 300 and the angle of

depression of its reflection in the lake is 600, find the height of the cloud above the lake.

25. From the top of a light house the angle of depression of a ship sailing towards it was found to be

300. After 10 seconds the angle of depression changes to 60

0. Assuming that the ship is sailing at

uniform speed, find how much time it will take to reach the light house.

26. A man on the top of a vertical tower observes a car moving toward the tower. If it takes 12

minutes for the angle of depression to change from 300 to 45

0, how soon after this car will reach

the tower?

27. The angle of elevation of top of a tower from two points at a distance of 4 m and 9 m from the

base of tower and in the same straight line with it are 600 and 30

0 respectively. Prove that height

of tower is 6m

28. A straight highway leads to foot of a tower. A man standing at the top of the tower observes a car

at an angle of depression of 300, which is approaching the foot of the tower with a uniform speed.

Six seconds later the angle of depression of the car is found to be 600. Find the time taken by the

car to reach the foot of the tower from this point.

29. The angles of elevation of the top of a tower, as seen from two points A and B situated in the

same line and at distances p and q respectively, from the foot of the tower, are 300 and 60

0

respectively. Prove that the height of the tower is 𝒑𝒒.

30. The length of the shadow of a tower standing on level ground is found to be 2x metres longer

when the sun’s altitude is 300 than when it was 45

0. Prove that the height of tower is ( 𝟑+

1)xmetres.

CHAPTER – 10

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

contact.

2. Prove that the tangents to a circle from an external point are equal.

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

chord.

4. Prove that the tangents drawn at the ends - points of a diameter of a circle are parallel.

5. Prove that the parallelogram circumscribing a circle is rhombus.

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

contact.

7. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles

at the centre of the circle.

8. Prove that the angle between the two tangents drawn from any external point to a circle is

supplementary to the angle subtended by the line segment joining the points of contact at the

centre

9. Prove that the line segment joining the points of contact of two parallel tangents to a circle is a

diameter of the circle.

10. If all the sides of a parallelogram touch a circle, prove that the parallelogram is a rhombus

11. In the figure, a quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD

+ BC.

12. In Fig, XY and X’ Y’ are two parallel tangents to a circle with centre O and another tangent AB,

with point of contact C intersects XYat A and X’ Y’ at B. Prove thatAOB = 900.

13. In the figure, O is the centre of the circle. PA and PB are tangents to the circle from the point P.

Prove that AOBP is a cyclic quadrilateral.

14. In Fig. two tangents TP and TQ are drawn to a circle with centre O from an external point T.

Prove that PTQ = 2OPQ

15. In fig. triangle ABC is isosceles in which AB = AC, circumscribed about a circle. Prove that

base is bisected by the point of contact.

16. If from an external point B of a circle with centre O, two tangents BC and BD are drawn such

that ∠ DBC = 1200. Prove that BO = 2BC

17. In Fig, O is the centre of a circle and BCD is tangent to it at C. Prove that BAC +

ACD = 900.

18. In the figure PO⊥QO. The tangents to the circle with centre O at P and Q intersect ata point T.

Prove that PQ and OT are right bisectors of each other.

19. In the figure, OP is equal to the diameter of the circle. Prove that ABP is an equilateral triangle.

20. In fig. XP and XQ are tangents from an external point X to the circle with centre O. R is a point

on the circle where another tangent ARB is drawn to the circle. Prove that XA + AR = XB + BR.

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

of contact. Using the above, find PTQ in Fig. if TP and TQ are the two tangents to a circle

with centre O so that POQ = 1100.

22. In Fig. two circles touch each other externally at C. Prove that the common tangent at C bisects

the other two common tangents.

23. A circle is touching the side BC of ∆ABC at P and touching AB and AC produced at Q and R

respectively. Prove that AQ = 𝟏

𝟐× perimeter of ∆ABC.

24. Find the length of the tangent drawn from a point, whose distance from the centre of the circle is

5 cm and radius of the circle is 3 cm.

25. From and external point T, tangent PT is drawn to a circle whose center is O. If OT =29 cm and

PT = 21 cm, determine the radius of the circle.

26. Two concentric circles are of radii 10 cm and 6 cm. Find the length of the chord of the larger

circle which touches the smaller circle.

27. In Fig, PT and PS are tangents to a circle from a point P such that PT = 5 cm and TPS = 600.

Find the length of chord TS

28. In fig., from an external point P, PA and PB are tangents to the circle with centre O. If CD is

another tangent at point E to the circle and PA = 12 cm. Find the perimeter of ∆PCD.

29. Two concentric circles have centre at O, OP = 4 cm and OB = 5 cm. AB is a chord of the outer

circle and a tangent to the inner circle at P. Find the length of AB.

30. In Fig., AB = 12 cm, BC = 8 cm and AC = 10 cm. Find AD, BE and CF.

31. In fig. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at point P&Q

intersect at point T. Find the length of tangent TP.

32. In Fig. ABC is a right angled triangle, right angled at A, with AB = 6 cm and AC = 8 cm, a circle

with centre O has been inscribed inside the triangle. Calculate the radius of the inscribed circle.

33. In the fig, ADC =90

0, BC =38 cm, CD = 28 cm and BP = 25 cm. Find the radius of the circle.

34. In Fig. all three sides of a triangle touch the circle. Find the value of x.

35. In Fig. a circle touches the side BC of ∆ABC at P and touches AB and AC produced at Q and R

respectively. If AQ = 5 cm. Find the perimeter of ∆ABC.

36. In Fig. a circle is inscribed in a quadrilateral ABCD in which B = 90

0. If AD = 23 cm, AB =

29 cm and DS = 5 cm, find the radius ‘r’ of the circle.

37. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and

DC into which BC is divided by the point of contact are of lengths 8 cm and 6 cm respectively. If

area of ∆ABC is 84 cm2, then find the sides AB and AC.

CHAPTER – 11

1. Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and AC = 7 cm. Construct another

triangle whose sides are 3/5 times the corresponding sides of triangle ABC.

2. Draw a right triangle with sides of length 5 cm and 4 cm making an right angle.

Construct another triangle whose sides are 𝟑

𝟓 times the corresponding sides of the first triangle.

3. Construct a triangle ABC in which AB = 4cm, B = 1200 and BC = 5 cm . Construct another

triangle, whose sides are𝟒

𝟓times the corresponding sides of ∆ABC.

4. Construct a triangle ABC in which AB = 5 cm, B = 600 and the altitude CD = 3 cm. Then

Construct another triangle whose sides are 𝟒

𝟓times the corresponding sides of ∆𝑨𝑩𝑪.

5. Construct a triangle ABC, in which base BC = 6 cm, B = 600 and BAC = 90

0. Then

construct another triangle whose sides are 𝟑

𝟒of the corresponding sides of ∆ABC.

6. Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6 cm from its centre

O.

7. Construct a pair of tangents to a circle of radius 4 cm inclined at an angle of 450.

8. Construct two circles of radii 3 cm and 4 cm whose centres are 8 cm apart. Draw the pair of

tangents from the centre of each circle to the other circle.

9. Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of

600.

CHAPTER – 12

1. A wire when bent in the form of a square encloses an area 121 sq.cm. If the wire were bent in the

form of a circle, find the area enclosed by the circle. 22

7

use

2. The difference between circumference and diameter of a circle is 135 cm. Find the radius

of the circle.22

7

use

3. The sum of circumferences of two circles is 132 cm. If the radius of one circle is 14 cm, find the

radius of the second circle.

4. The circumference of a circle exceeds its diameter by 16.8 cm. Find the circumference of the

circle. 22

7

use

5. If the diameter of a semicircular protractor is 14 cm, then find its perimeter.

6. A race track is in the form of a ring whose inner circumference is 352 m and outer circumference

is 396 m. Find the width of the track. 22

7

use

7. How many times will the wheel of a car rotate in a journey of 2002 m, if the radius of the wheel is

49 cm?

8. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel

make in 10 minutes when the car is travelling at the speed of 66 km/hour?

9. If the area and circumference of a circle are numerically equal, then find the radius of the circle.

10. Find the area of the quadrant of that circle whose circumference is 22 cm .22

7

use

11. The radii of two circles are 4 cm and 3 cm. Find the radius of the circle whose area is equal to the

sum of the areas of the two circles. Also find the circumference of this circle.

12. A horse is tied to a peg at one corner of a square shaped grass field of side 25 m by means of a 14

m long rope. Find the area of the part of the field in which the horse can graze.

13. A piece of wire that has been bent in the form of a semicircle including the bounding diameter is

straightened and then bent in the form of a square. The diameter of the semicircle is 14 cm.

Which has a larger area, the semi-circle or the square? Also find the difference between them.

14. Two concentric circles are of radii 7 cm and 5 cm. Find the area of the portion between two

circles.

15. Area of a sector of a circle of radius 36 cm is 54𝝅 cm2. Find the length of corresponding arc of

sector.

16. The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand from

6.00 pm to 6.10 pm.

17. Find the area of the sector of a circle with radius 10 cm and of central angle 600. Also, find the

area of the corresponding major sector.

18. A chord of a circle of radius 12 cm subtends an angle of 1200 at the centre. Find the area of the

corresponding minor segment of the circle. (Use𝝅 = 𝟑.𝟏𝟒)

19. In a circle of radius 12 cm, an arc subtends an angle of 600 at the centre. Find

i) Area of sector formed by the arc

ii) Area of the segment formed by the corresponding chord

20. The perimeter of a sector of a circle of radius 5.6 cm is 27.2 cm. Find the area of the sector.

21. A square of side 4 cm is inscribed in a circle. Find the area enclosed between the circle and the

square. 22

7

use

22. Find the area of the shaded region, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

23. In Fig., AB is a diameter of the circle with centre O and OA = 7 cm. Find the area of the shaded

region. 22

7

use

24. In fig. AB and CD, the two diameters of a circle with centre O are perpendicular to each other

and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of shaded region.

25. In fig. a circle of radius 7 cm is inscribed in a square.Find the area of the shaded portion

22

7use

26. In Fig., a circle of radius 7 cm is inscribed in a square. Find the area of the shaded region.

27. Find the area of the shaded region in Fig. where ABCD is a square of side 14 cm and four circles

are each of same radius

28. AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O.

If ∠ AOB = 300, find the area of the shaded region.

29. In Fig. sectors of two concentric circles of radii 7 cm and 3.5 cm are given. Find the area of the

shaded region. 22

7

use

30. In fig. ABC is a triangle right angled at A. Find the area of the shaded region if AB = 6 cm, BC =

10 cm and Iis the centre of incircle of ∆ABC.

31. PQRS is a rectangle in which length is two times the breadth and L is mid point of PQ. With P

and Q as centres, draw two quadrants as shown in fig. Find the ratio of the area of rectangle

PQRS to the area of shaded portion.

32. In fig. find the area of the shaded region, where a circular arc of radius 6 cm is drawn with a

vertex O of an equilateral tringle OAB of side 12 cm as centre.

33. A round table cover has six equal designs as shown in fig. If the radius of the cover is 28 cm find

the cost of making the designs at the rate of Rs. 0.35 per cm2.

34. The area of an equilateral triangle ABC is 17320.5 cm

2. With each vertex of the triangle as

centre, a circle is drawn with radius equal to half the length of the side of the triangle. Find the

area of the shaded region. (use𝝅 = 𝟑.𝟏𝟒 & 𝟑 = 𝟏.𝟕𝟑𝟐𝟎𝟓 )

35. In the fig, OPQR is a rhombus, three of whose vertices lie on the circle with centre O. If the area

of the rhombus is 32 𝟑 cm2, find the radius of the circle.

36. In fig., OPQR is a rhombus whose three vertices P, Q, R lie on a circle of radius 8 cm. Find the

area of the shaded region.

37. In Fig., two circular flower beds have been shown on two sides of a square lawn ABCD of side 56

m. If the centre of each circular flower bed is the point of intersection O of the diagonals of the

square lawn, find the sum of the areas of the lawn and flower beds

38. In the figure, ABC is a triangle right angled at A. Semicircles are drawn on AB, AC and BC as

diameters. Find the area of the shaded region.

39. In Fig., find the area of the shaded region 22

7

use

40. Fig. depicts a racing track whose left and right ends are semi - circular. The distance between the

two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide,

everywhere find the area of the track.

41. Find the area of shaded region in Fig. in term of 𝝅.

42. In the given fig, diameter AB is 12 cm long. AB is trisected at points P and Q. Find the area of the

shaded region.

43. Find the perimeter of the shaded region in the figure given below.

44. In the given fig, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm,

find the area of the:

i) Quadrant OACB

ii) Shaded region

45. Calculate the area of the shaded region in the Fig. common between two quadrants of circle of

radius 8 cm each.

46. In Fig. the shape of the top of a table in a restaurant is that of a sector of a circle with centre O

and angle BOD = 900. If OB = OD = 60 cm, find the perimeter of the table top. (Use 𝝅 = 𝟑.𝟏𝟒 )

47. In Fig. ABC is a quadrant of a circle of radius 14 cm and a semi circle is drawn with BC as

diameter. Find the area of shaded region.

48. In the fig, ABC is a right–angled triangle, ∠ B = 90

0, AB = 28 cm and BC = 21 cm. With AC as

diameter, a semi–circle is drawn and with BC as radius a quarter circles is drawn. Find the area

of the shaded region.

49. In Fig. find the area of the shaded design, where ABCD is a square of side 10 cm and semi circles

are drawn with each side of the square as diameter. 3.14use

50. In Fig, arcs are drawn by taking vertices A, B and C of an equilateral triangle of side 10 cm, to

intersect the sides BC, CA and AB at their respective mid-points D, E and F. Find the area of the

shaded region. 3.14use

51. With the vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 5 cm each as

shown fig. If AB = 14 cm, BC = 48 cm and CA = 50 cm, then find the area of the shaded region

3.14use

52. In Fig. ABCDEF is any regular hexagon with different vertices A, B, C, D, E and F as the

centres of circles with same radius ‘r’ are drawn. Find the area of the shaded portion.

CHAPTER – 13

1. The radius and slant height of a right circular cone are in the ratio of 7: 13 and its curved surface

area is 286 cm2. Find its radius.

22

7use

2. A tent is of the shape of a right circular cylinder up to a height of 3 metres and conical above it.

The total height of the tent is 13.5metres above the ground. Calculate the cost of painting the

inner side of the tent at the rate of Rs. 2 per square metre, if the radius of the base is 14 metres.

3. A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total

height of the toy is 31 cm. Find the total surface area of the toy.

4. A toy is in the form of a cone of radius 3.5 cm surmounted on a hemisphere of same radius. The

total height of the toy is 15.5 cm. Find the total surface area of the toy. 22

7

use

5. A toy is in the form of a cone mounted on a hemisphere. The diameter of the base of the cone and

that of hemisphere is 18 cm and the height of cone is 12 cm. Calculate the surface area of the toy.

3.14use

6. Three cubes each of side 15 cm are joined end to end. Find the total surface area of the resulting

cuboid.

7. The total surface area of a right circular cone is 90 𝝅 cm2. If the radius of base of the cone is 5

cm, find the height of the cone.

8. A solid sphere of diameter 14 cm is cut into two halves by a plane passing through the centre.

Find the combined surface area of the two hemispheres so formed

9. A solid cone of radius 4 cm and vertical height 3 cm has to be painted from outside except the

base. Find the surface area to be painted.

10. Decorative block shown in Fig. is made of two solids a cube and a hemisphere. The base of the

block is a cube with edge 5 cm and hemisphere fixed on the top has a diameter 4.2 cm. Find total

surface area of the block.22

7

use

11. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same

height and same diameter is hollowed out. Find the total surface area of the remaining solid to

the nearest cm 2.

22

7use

12. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder of same radius. The

diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner

surface area of the vessel.

13. A solid is composed of a cylinder with hemispherical ends. If the whole height of the solid is 100

cm and the diameter of cylindrical part and the hemispherical ends is 28 cm, find the cost of

polishing the surface of the solid at the rate of 5 paise per sq cm.

14. Two cubes each of edge 4 cm are joined face to face. Find the surface area of the resulting

cuboid.

15. The volume of a right circular cylinder of height 7 cm is 567𝝅cm3. Find its curved surface area.

22

7use

16. The radius of the base and the height of a right circular cylinder are in the ratio of 2: 3 and its

volume is 1617 cu.cm. Find the curved surface area of the cylinder. 22

7

use

17. The perimeters of the ends of a frustum of a right circular cone are 44 cm and 33 cm. If the

height of frustum is 16 cm, find its volume and total surface area.

18. Find the volume of the largest right circular cone that can be cut out of a cube of side 4.2 cm.

22

7use

19. From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and

of base radius 6 cm, is hollowed out. Find the volume of the remaining solid. (Take 𝝅 = 𝟑.𝟏𝟒16).

Also find the slant height of the cone.

20. An ice cream cone consisting of a cone is surmounted by a hemisphere. The common radius of

hemisphere and cone is 3.5 cm and the total height of ice–cream cone is12.5 cm. Calculate the

volume of ice cream in the cone.

21. A rectangular sheet of paper 44 cm × 18 cm is rolled along its length (44 cm) and a cylinder is

formed. Find the volume of the cylinder.

22. The circumference of the circular end of a hemispherical bowl is 132 cm. Find the capacity of the

bowl.

23. A right triangle, whose sides other than hypotenuse, are 3 cm and 4 cm is made to revolve about

its hypotenuse. Find the volume of the double cone so formed.

24. The circumference of the base of a 9 m high wooden solid cone is 44 m. Find the volume of the

cone.

25. A solid toy is in the form of a hemisphere surmounted by a right circular cone of the same base

radius. The height of the cone is 2 cm and diameter of the base is 4 cm. Determine the volume of

the toy. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the

cylinder and the toy. 3.14use

26. The radii of the circular bases of a right circular cylinder and a cone are in the ratio of 3 : 4 and

their heights are in the ratio of 2 : 3. What is the ratio of their volumes?

27. If the radius of the base of a right circular cylinder is halved, keeping the height same, find the

ratio of the volume of the reduced cylinder to that of the original cylinder.

28. The difference between the outer and inner curved surface areas of a hollow right circular

cylinder 14 cm long, is 88 cm2. If the volume of metal used in making the cylinder is 176 cm

3, find

the outer and inner diameter of the cylinder.

29. A sphere and a cube have same surface. Show that the ratio of the volume of sphere to that of the

cube is 𝟔 : 𝝅

30. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If

its volume be 𝟏

𝟐𝟕th volume of the cone, at what height above the base is thesection made?

31. A right triangle, whose sides are 6 cm and 8 cm (other than hypotenuse) is made to revolve about

its hypotenuse. Find the volume and surface area of the double cone so formed.

32. 50 circular plates, each of radius 7 cm and thickness 0.5 cm, are placed one above another to

form a solid right circular cylinder. Find the total surface area and the volume of the cylinder so

formed.

33. A solid wooden toy is in the form of cone mounted on a hemisphere. If the radii of hemisphere

and base of cone are 4.2 cm each and the total height of toy is 10.2 cm, find the volume of wood

used in the toy. Also find the total surface area of toy.

34. A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the

solid is 19 cm and the diameter of the cylinder and the hemispheres is 7 cm. Find the volume and

total surface area of the solid.

35. A building is in the form of a right circular cylinder surmounted by a hemispherical dome both

having the same base radii. The base diameter of the dome is equal to 𝟐

𝟑 of the total height of the

building. Find the height of the building, if it contains 67𝟏

𝟐𝟏m

3 of air.

36. A vessel in the form of a hemispherical bowl is full of water. Its water is emptied in to a cylinder.

The internal radii of bowl and the cylinder are 10 𝟏

𝟐cm and 7 cm respectively. Find the height of

water in the cylinder.

37. An iron pillar has some part in the form of a right circular cylinder and the remaining in the

form of a right circular cone. The radius of the base of each of the cone and the cylinder is 8 cm.

The cylindrical part is 240 cm high and conical part is 36 cm high. Find the weight of the pillar if

1 cu cm of iron weighs 7.5 grams.

38. A gulabjamun, contains sugar syrup up to about 30% of its volume. Find approximately how

much syrup would be found in 45 such gulabjamuns, each shaped like a cylinder with two

hemispherical ends with total length 5 cm and diameter 2.8 cm.

39. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is

surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole,

given that 1 cm3 of iron has 8 gm mass.

40. Find the mass of a 3.5 m long lead pipe, if the external diameter of pipe is 2.4 cm, thickness of the

metal is 2 mm and 1 cm3 of lead weighs 11.4 kg.

41. A well of diameter 7 m is dug 22.5 m deep. The earth taken out of it is spread evenly all around it

to width of 10.5 m to form an embankment. Find the height of embankment.

42. A well whose diameter is 7 m has been dug 22.5 m deep and the earth dugout is used to form an

embankment around it. If the height of the embankment is 1.5 m. find the width of the

embankment.

43. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to

form a platform 22 m by 14 m. Find the height of the platform.

44. A cone of height 24 cm and diameter of base 12 cm is made up of modeling clay. A child reshapes

it in the form of a sphere. Find the total surface area of the sphere.

45. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform

thickness. Find the thickness of the wire.

46. A spherical copper shell, of external diameter 18 cm, is melted and recast into a solid cone of base

radius 14 cm and height 4 𝟑

𝟕 cm. Find the inner diameter of the shell.

47. The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into a cone of height

28 cm. Find the diameter of the base of the cone so formed.

48. A right circular cone of height 8.4 cm and the radius of its base is 2.1 cm is melted and recast into

a sphere. Find the diameter of the sphere.

49. A solid sphere of radius 3 cm is melted and drawn into a long wire of uniform circular cross-

section. If the length of the wire is 36 m, find its radius.

50. The internal and external radii of a hollow spherical shell are 3 cm and 5 cm respectively. If it is

melted to form a solid cylinder of height 10𝟐

𝟑cm, find the diameter of the cylinder.

51. The diameter of a metallic solid sphere is 9 cm. It is melted and drawn into a wire having

diameter of cross-section as 0.2 cm. Find the length of the wire.

52. A solid cuboidal slab of iron of dimensions 66 cm ×20 cm×27 cm is used to cast an iron pipe. If

the outer diameter of the pipe is 10 cm and thickness is 1 cm, then calculate the length of the pipe.

53. A sphere of radius 6 cm is dropped into a cylindrical vessel partly filled with water. The radius of

the vessel is 8 cm. If the sphere is submerged completely, then find the increase in level of the

water.

54. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his

field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3

km/hr, in how much time will the tank be filled?

55. A cylindrical pipe has inner diameter of 4 cm and water flows through it at the rate of 20 m per

minute. How long would it take to fill a conical tank, with diameter of base as 80 cm and depth 72

cm?

56. Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 20 km/h, how much

area it will irrigate in 30 minutes if 9 cm of standing water is desired?

57. The rain water from a roof 22 m×20 m drain in to a conical vessel having diameter of base as 2

m and height 3.5 m. If the vessel is just full, find the rain fall in cm.

58. A hemispherical bowl of internal diameter 36 cm. is full of liquid. This liquid is to be filled in

cylindrical bottles of radius 3 cm and height 6 cm. How many such bottles are required to empty

the bowl?

59. A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones,

each of diameter 7 cm and height 3 cm. find the number of cones so formed.

60. How many solid spheres of diameter 6 cm are required to be melted to form a solid metal cylinder

of height 45 cm and diameter 4 cm?

61. How many coins 1.75 cm in diameter and 2 mm thick must be melted to form a cuboid of

dimensions 11 cm × 10 cm × 7 cm? 22

7

use

62. How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm,

each bullet being 4 cm in diameter?

63. Solid spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water and

are fully submerged. If the diameter of the beaker is 18 cm and the water rises by 40 cm in the

beaker, find the number of solid spheres dropped in the water.

64. Find the number of solid cylindrical structures of radius 7 cm and height 10 cm which can be

made from a solid cylinder of radius 7 m and height 10 m.

65. Find the number of spherical bullets of radii is 1 mm each that can be made out of a cylindrical

solid of radius 4 cm and height 6 cm.

66. Right circular cylinder having diameter 12 cm and height 15 cm is full of ice–cream. This ice–

cream is to be filled in cones of height 12 cm and diameter 6 cm having a hemispherical shape on

the top. Find the number of such cones which can be filled with ice–cream.

67. A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones,

each of diameter 3.5 cm and height 3 cm. Find the number of cones so formed.

68. Solid spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water and

are fully submerged. If the diameter of the beaker is 18 cm and the water rises by 40 cm, find the

number of solid spheres dropped in the water.

69. A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones

each of diameter 7cm and hight 3 cm. Find the number of cones so formed.

70. A spherical solid ball of diameter 21 cm is melted and recast into cubes, each of side 1 cm. Find

the number of cubes thus formed.22

7

use

71. A cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its

axis, parallel to its base. Compare the volume of two parts.

72. The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two

circular ends is 4 cm, find the height of the frustum.

73. The radii of the ends of a bucket of height 45 cm are 28 cm and 7 cm. Find its volume and the

curved surface area.

74. A container open at the top and made up of a metal sheet, is in the form of a frustum of cone of

height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost

of milk which can completely fill the container, at the rate of Rs. 20 per litre. 3.14use

75. A drinking glass open at the top is in the shape of a frustum of a cone of height 24 cm. The

diameters of its top and bottom circular ends are 18 cm and 4 cm respectively. Find the capacity

and total surface area of the glass.

76. The internal radii of the ends of a bucket, full of milk and of internal height 16 cm, are 14 cm

and 7 cm. If this milk is poured into a hemispherical vessel, the vessel is completely filled. Find

the internal diameter of the hemispherical vessel.

77. A milk container is made of metal sheet in the shape of frustum of cone whose volume is

10459𝟑

𝟕cm

3. The radii of its lower and upper circular ends are 8 cm and 20 cm. Find the cost of

metal sheet used in making the container at the rate of Rs. 1.40 per square cm.

78. A bucket is 18 cm in diameter at the top and 6 cm in diameter at the bottom. If it is 8 cm high,

find its capacity. Also find the area of sheet used in making the bucket.

79. A metallic bucket is in the shape off a frustum of a cone. If the diameter of two circular ends of

the bucket are 45 cm and 25 cm respectively and the total vertical height is 24 cm find the area of

the metallic sheet used to make the bucket. Also find the volume of water of it can hold.

80. A shuttle cock used for playing Badminton has the shape of a frustum of a cone mounted on a

hemisphere (see figure). The diameters of the ends of the frustrum are 5 cm and 2 cm, the height

of the entire shuttle cock is 7 cm. Find the external surface area.

CHAPTER – 15

1. Find the probability that a non-leap year chosen at random has

i) 52 Sundays ii) 53 Sundays

2. In a leap year, find the probability that there are 53 Tuesdays in the year. 3. Two coins are tossed together. Find the probability of getting at least one tail. 4. Three coins are tossed together. Find the probability of getting at least two heads. 5. Three coins are tossed simultaneously. Find the probability of getting

i) three heads

ii) exactly 2 heads

iii) at least 2 heads.

6. A bag contains cards numbered from 1 to 25, one card is drawn at random from the bag. Find the probability that this card has a number which is divisible by both 2 & 3.

7. Cards marked with numbers 13, 14, 15, ......... 60 are placed in a box & mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card Is

i) divisible by 5

ii) a number which is a perfect square.

8. 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 equally likely outcomes. What is the

probability that it will point at

i) a prime number ii) a factor of 8

9. A box contains 17 cards numbered 1, 2, 3, ....... 16, 17. A card is drawn at random from

the box. Find the probability that the number on the drawn card is:

i) Odd

ii) even and prime

iii) divisible by 3

10. A bag contains cards numbered from 2 to 26. One card is drawn from the bag at

random. Find the probability that it has a number divisible by both 2 & 3.

11. A box contains 20 balls bearing numbers 1, 2, 3, 4, ........20. A ball is drawn at random

from the box. What is the probability that the number on the drawn ball is

i) An odd number

ii) Divisible for 2 or 3

iii) Prime number

iv) Not divisible by 10

12. Cards numbered from 1 to 64 are placed in a box. A card is drawn at random from the box. Find the probability that the card number on the card drawn is a perfect cube.

13. A box contains cards numbered from 1 to 17. If one card is drawn at random from the box, find the probability that it bears a prime number.

14. A bag contains 19 cards, bearing numbers 1, 2, 3, ....... , 19. A card is drawn at random

from the bag. Find the probability that number on the drawn card is (i) prime (ii)

divisible by 3

15. A die is thrown once. Find the probability of getting:

i) a prime number

ii) a number less than 6

16. A dice is thrown once. Find the probability of getting:

i) a prime number

ii) a number divisible by 2

17. A pair of dice is thrown once.

i) Write sample space for the experiment.

ii) Find the probability of getting an odd number on each dice.

18. Dice are thrown simultaneously. Find the probability that

i) Sum is less than 5. ii) Doublet appears.

19. Two dice are thrown together. Determine the probability of two coming on the first die

and multiple of three on other die.

20. Two dice are thrown once. Find the probability of obtaining

i) a total of 6 of numbers on both dice.

ii) the same number on both dice.

21. Two dice are thrown at the same time. Find the probability of getting (i) different

numbers on both the dice (ii) sum of the numbers as 9 or 11.

22. Two dice are thrown at the same time. Find the probability of

i) Same number on both the dice. ii) Different number on both the dice.

23. In a single throw of two dice, find the probability of getting

i) a total of 7

ii) a total of 11

iii) six as product 24. An urn contains 8 red, 6 white, 4 black balls. A ball is drawn at random from the urn.

Find the probability that the drawn ball is (i) red or white (ii) neither black nor white

25. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue

ball from the bag is four times that of a red ball, find the number of blue balls in the

bag.

26. A box contains 3 blue, 2 white & 4 red marbles. If a marble is drawn at random from

the box, what is the probability that it will be

i) White

ii) Blue

iii) Red

27. A bag contains 5 red, 8 green & 7 white balls. One ball is drawn at random from the

bag. Find the probability of getting neither a green ball nor a red ball.

28. A bag contains 4 green, 5 white, 7 black & 3 red balls. A ball is taken out of the bag at

random. Find the probability that the ball taken out is (i) red (ii) not black

29. A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at

random from the jar, the probability that it is green is 2/3. Find the number of blue

marbles.

30. A bag contains 5 red balls & some blue balls. If the probability of drawing a blue ball

from the bag is thrice that of red ball, find the number of blue balls in the bag.

31. A bag contains 14 balls of which x are white. If 6 more white balls are added to the

bag, the probability of drawing a white ball is 𝟏

𝟐. Find the value of x.

32. A bag contains 18 balls out of which x balls are red.

i) If one ball is drawn at random from the bag, what is probability that it is red ball. ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be

𝟗

𝟖times that of probability of red ball coming in part (i). Find x.

33. A child has a die whose six faces show the letters as given below:

A B C D E A

The die is thrown at random once. What is the probability of getting (i) A (ii) E. 34. A letter is chosen at random from the English alphabet. Find the probability that it is

i) a vowel ii) a consonant

35. From a group of 2 boys & 2 girls, two children are selected at random. What is the sample space representing the event. Find the probability that one boy and one girl is selected.

36. Geeta&Sita are friends. What is the probability that both will have

i) different birthdays?

ii) the same birthday ? (ignoring a leap year)

37. Two customers Shyam&Ekta are visiting a particular shop in the same week

(Tuesday to Saturday). Each is equally likely to visit the shop on any day. What is the

probability that both will visit the shop on

i) the same day?

ii) consecutive day?

38. All the face cards of spades are removed from a pack of 52 playing cards & then the

pack is shuffled well. A card is then drawn at random from the remaining pack of

cards. Find the probability of getting (i) a black face card, (ii) a queen.

39. From a well shuffled pack of 52 cards, two black kings & two black jacks are

removed. From the remaining cards, a card is drawn at random. Find the probability

that the drawn card is not a king.

40. A card is drawn at random from a pack of well shuffled deck of playing cards. Find the

probability that the card is

i) a king or a jack ii) a card of spade or an ace

41. King, queen & jack of hearts are removed from a pack of 52 playing cards & then the

pack is well shuffled. A card is drawn from the remaining cards. Find the probability

of getting a card of

i) Hearts ii) a queen iii) not a king.

42. All the three face cards of spades are removed from a well – shuffled pack of 52

cards. A card is then drawn at random from the remaining pack. Find the probability

of getting

i) a black face card

ii) a queen

iii) a black card

43. From a well shuffled pack of 52 playing cards, black jacks, black kings & black aces

are removed. A card is then drawn at random from the remaining pack. Find the

probability of getting

i) a red card ii) not a diamond card

44. A card is drawn from a well shuffled pack of 52 cards. Find the probability that the

card drawn is neither a black card nor a queen.

45. From a deck of playing cards all aces & clubs are removed, a card is drawn at random

from the remaining cards. Find the probability that it is:

i) A black face card.

ii) A red card.

46. A card is drawn at random from a well shuffled pack of 52 playing cards. Find the

probability that card drawn is:

i) spade or an ace ii) neither king nor queen

47. A card is drawn at random from a well shuffled pack of 52 cards. Find the probability

that the card drawn is neither a red card nor a queen.

48. A card is drawn at random from a well shuffled pack of playing cards. Find the

probability of getting a red face card.

49. One card is drawn from a well shuffled deck of 52 cards. Find the probability of

getting a red face card.

PREPARED BY – K. K. ARYA

Find the LCM & HCF of 120 & 144 by fundamental theorem of …€¦ · CLASS – X CHAPTER – 1 1....

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