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ICSE Guess > ICSE/ISC eBooks > Class X > Maths by Mr. M. P. Keshari

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Exercise - 5

1. Find the HCF of the following:

2x⁴ - 2y⁴ and 3x³ + 6x²y - 3xy² - 6y³
12(x³ + x² + x + 1) and 18(x⁴ - 1)
x³ + 2x² - 3x and 2x³ + 5x² - 3x
2(x⁴ - y⁴) and 3(x³ + 2x²y - xy² - 2y³)
18(x³ - x² + x - 1) and 12(x⁴ - 1)
45(x⁴ - x³ - x²) and 75(8x⁵ + x²)
36(3x⁴ + 5x³ - 2x²) and 54(27x⁴ - x)
42(2x³ - 5x² - 3x) and 60(8x⁴ + x)
4(x⁴ - 1) and 6(x³ - x² - x + 1)

2. Find the LCM of the following polynomials:

35(x⁴ - 27x) and 40(2x³ - 5x² - 3x)
20(2x³ + 3x² - 2x) and 45(x⁴ + 8x)
15(4x³ - 4x² + x) and 35(2x² - 7x + 3)
25(x² + 7x + 12) and 15x(x² - 16)
x(8x³ + 27) and 2x² (2x² + 9x + 9)

3. Find the GCD and LCM of the polynomials P(x) and Q(x), where

P(x) = (x³ - 27) (x² - 3x + 2) and

Q(x) =( x² + 3x + 9) (x² - 5x + 6)

4. The LCM and GCD of two polynomials, P(x) and Q(x) are 56(x⁴ + x) and 4(x² - x + 1) respectively. If P(x) = 28(x³ + 1), find Q(x) .

5. For what value of k, the g.c.d. of x² + x - (2k + 2) and 2x² + kx - 12 is x + 4?

6. Find the value of K for which the g.c.d. of x² - 2x - 24 and x² - kx - 6 is x - 6.

7. (x² - x - 6) is the GCD of the expression (x + 2) (2x² + ax + 3) and (x - 3) (3x² + bx + 8). Find the value of a and b.

8. (x + 1) ( x - 4) is the g.c.d. of the polynomials ( x - 4) (2x² + x - a) and ( x + 1) (2x² + bx - 12) find a and b .

9. (x - 3) is the g.c.d. of (x³ -2x² + px + 6) and ( x² - 5x + q) . Find (6p + 5q) .

10. Find the value of a and b so that the polynomials P(x) and Q(x) have (x - 1) (x + 4) as their HCF:

P(x) = (x² - 3x + 2) (x² + 7x + a)

Q(x) = (x² + 5x + 4) (x² - 5x + b)

11. Find the value of a and b so that the polynomial x³ + ax² + bx + 15 is divisible by x² + 2x - 15.

12. Find the value of p and q so that the polynomial f(x) = px³ + 2x² - 19x + 9 is divisible by x² + x - 6.

13. Determine the value of k such that x + 3 is a factor of the polynomial

f(x) = kx³ + x² - 22x - 21

14. If x - 2 is a factor of x² + ax - 6 = 0 and x² - 9x + b = 0, find the value of a and b.

Answers

1. (i) x² - y²                          (ii) 6(x + 1) (x² + 1)
    (iii) x(x + 3)                      (iv) x² - y²
    (v) 6(x - 1) (x² + 1)         (vi) 15x²(2x + 1)
    (vii) 18x(3x - 1)               (viii) 6x(2x + 1)
    (ix) 2(x² - 1)

3. G. C. D. = (x³ - 27) (x - 2)
LCM = (x - 1) (x - 2) (x - 3) (x² + 3x + 9)

5. K = 5

7. a = -7, b = 10

9. 0

11. a = 1, b = -17

13. K = 2

2. (i) 280x(x - 3) (2x + 1) (x² + 3x + 9)
    (ii) 180x(x + 2) (2x - 1) (x² - 2x + 4)
    (iii) 105x((x - 3) (2x - 1)²
    (iv) 75x(x² - 16) (x + 3)
    (v) 2x²(8x² + 27) (x + 3)

4. 8(x³ - x² + x)

6. K = 5

8. a = 1, b= -5

10. a = 12, b = 4

12. p = 3, q = 6

14. a = 1, b = 14

Maths by Mr. M. P. Keshari
	Chapter - 1	Chapter - 2	Chapter - 3
	Chapter - 4	Chapter - 5	Chapter - 6
	Chapter - 7	Chapter - 8	Chapter - 9
	Chapter - 10	Chapter - 11