MATHS HOLIDAY HOMEWORK

Chapter: polynomials

1. Find the quotient and remainder when p(x) is divided by q(x).

(i) p(x) = 6x³ + 11x² – 39x – 65, q(x) = x² – 1 + x

(ii) p(x) = 4 + 9x³ – 4x², q(x) = x + 3x² – 1

(iii) p(x) = 30x⁴ – 82x² + 11x³ + 48 – 12x, q(x) = 3x² + 2x - 4

2. What must be subtracted from 8x⁴ + 14x³ – 2x² + 7x – 8 so that the resulting polynomial is exactly divisible by 4x² + 3x – 2?

3. What must be added to 4x⁴ + 2x³ – 2x² + x – 1, so that the resulting polynomial is divisible by x² + 2x – 3?

4. If -2 is a zero of f(x) = x³ + 13x² + 32x + 20, find its other zeroes.

5. 3 and - 3 are zeroes of f(x) = x⁴ – 3x³ – x² + 9x – 6. Find all the zeroes of p(x).

6. Obtain all zeroes of the polynomial p(x) = 2x⁴ + x³ – 14x² – 19x – 6, if two of its zeroes are -1 and -2.

7. Find all the zeroes of f(x) = 2x⁴ – 2x³ – 7x² + 3x + 6 if two of its zeroes are  and -  .

8. Find all values of p and q so that 1, -2 are zeroes of the polynomial f(x) = x³ + 10x² + px + q.

9. If p(x)=2x⁴ + 3x³ - 3x² – 2x + 5 is divided by 2x² + 3x – 1,then the remainder is x – a. Find a.

10. On dividing f(x) = 2x³ – 5x² + 4x – 8 by g(x), the quotient and the remainder are (2x – 9) and 24x – 17, respectively. Find g(x).

                                                                                Chapter: REAL NUMBERS

1. Show that only one out of n, n + 4, n + 8, n + 12, n + 16 is divisible by 5, when n is a positive integer.

2. Use Euclid’s division algorithm to find the HCF of: (a) 135 & 225       (b) 196 & 38220

(c) 867 & 255.

3. Find the HCF of the following pairs of integers by the prime factorization method.

(a) 963 & 657   (b) 506 & 1155               (c) 1288 & 575

4. Find the greatest number which divides 285 and 1245 leaving remainders 9 & 7 respectively.

5. The length, breadth and height of a room are 8m 25cm, 6m 75cm and 4m 50cm, respectively. Find the longest rod which can measure the three dimensions of the room exactly.

6. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

7. A rectangular courtyard is 18m 72cm long and 13m 20cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such square tiles required.

8. HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other.

9. Prove that 3 + 2 5 is irrational.

10. Prove that 5 + 3 is irrational.

11. What can you say about prime factors of denominators of following real numbers?

(i):34.12345      (ii): 25. 567        (iii): 2.5055055505555….

Chapter: Trigonometry

1.   If cot Θ = 15/8 , evaluate  .

2.   If 7 sin2 Ѳ + 3 cos2 Ѳ = 4 , show that tan Ѳ = 1/√3 .

3. Evaluate: tan2 60˚ - 2 cos2 60˚ - ¾ sin2 45˚ - 4 sin2 30˚.                                                                                                                                                                                                                 (9/8)

4. Evaluate:         + 2sin2 38˚ sec2 52˚ - sin2 45˚

                                                                                                                                                                                          (5/2)

5. Evaluate : √2 tan2 45˚ + cos2 30˚ - sin2 60˚                                                                                                                                                                                                                           (√2)

6. If sec2 Ѳ ( 1 + sin Ѳ ) ( 1 – sin Ѳ ) = k , find the value of k.                                                                                                                                                                                                                                    (k=1)

7. Evaluate : ( sin 90˚ + cos 45˚ + cos 60˚ ) ( cos 0˚ - sin 45˚ + sin 30˚ ).                                                                                                                                                                                                                      (7/4)

8.   Find  the value of :   +    + .

                                                                                                                                                                                                               (1)9.                          Ifsin(A+B)=1,cos(A –B)=1,findAandB              (45˚,45˚)

9.       If cos (40˚+ x ) = sin 30˚ , find the value of x.                                                                                                                                                                                                               (20˚)

10.    Sin 4A = cos ( A - 20˚ ) , where 4A is an acute angle , find the value of A.                                                                                                                                                                                                              (22˚)

12.    Find the acute angles A and B , A > B , if sin ( A + 2B ) = √3/2 and cos ( A + 4B ) = 0.                                                                                                                                                                                                                     (30˚,15˚)

13.    If sin A – cos B = 0 , prove that A + B = 90˚.

14.    If  =   , evaluate   .

                                           Chapter: Application of Trigonometry     

1.     The angle of elevation of the top of the building from the foot of the tower is 30°

and the angle of the top of the tower from the foot of the building is 60°. If the

tower is 50 m high, find the height of the building.

1.      A ladder rests against a wall at an angle 𝜶 to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle 𝛃 with the horizontal, Show that  = .

2.      A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 𝟑𝟎°with it. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the height of the tree.

3.      A contractor plans to install two slides for the children to play in a park. For the

children below the age of 5 years, she prefers to have a slide whose top is at a height of

4.      1.5m, and is inclined at an angle of 𝟑𝟎° to the ground, where ad for the elder children she wants to have a steep side at a height of 3m, and inclined at an angle of 𝟔𝟎° to the ground. What should be the length of the slide in each case?

5.      The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 𝟑𝟎°. Find the height of the tower.

6.      A 1.5 m tall boy is standing at some distance from a 30m tall building. The angle of elevation from his eyes to the top of the building increases from 𝟑𝟎° to 𝟔𝟎° as he walks towards the building. Find the distance he walked towards the building.

7.      The angle of elevation of the top of a building from the foot of the tower is 𝟑𝟎° and the angle of elevation of the top of the tower from the foot of the building is 60° . If the tower is 50m high, find the height of the building.

8.      Two poles of equal heights are standing opposite each other an either side of the road, which is 80m wide. From a point between them on the road, the angles of elevation of the top of the poles are 𝟔𝟎° 𝒂𝒏𝒅 𝟑𝟎°, respectively. Find the height of poles and the distance of the point from the poles.

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

10.  As observed from the top of a 75m high lighthouse from the sea-level, the angles of depression of two ships are 𝟑𝟎° 𝒂𝒏𝒅 𝟒𝟓°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

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