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Mathematics

Q. 1. Find the HCF and LCM

1. 12(9x² - 4) and 18 (6x² - 5x - 6)
2. (18x³ + 45 x² - 27x) and (15x⁴ - 135 x²)
3. 18(x³- x²+ x - 1) and 12(x⁴ - 1)
4. 42(2x³ - 5x² - 3x) and 60(8x⁴ + x)
5. 8(x⁴-16) and 12(x³-8)
6. (2x⁴ - 2y4) and (3x³ + 6x² - 3xy² - 6y3)
7. (x⁴ - 1 ) and (x³ + x² + x + 1)
8. 36(3x⁴+5x³-2x²) and 54(27x⁴-x)
9. (x³ - 1) and (x⁴ + x² + 1)
10. (x² + x - 2) and (x³ + 4x² + x - 6)
11. (x⁴ - y⁴) and (x⁶ - y⁶)
12. (6x⁴ - 13x³ + 6x²) and (8x⁴ - 36x³ + 54x² - 27x)

Q. 2. For what value of k,the HCF of x² + x - 2(k + 1)and (2x²+ kx - 12) is (x + 4).

Q. 3. if (x - k) is the HCF of (x² + x - 12) and (2x² - kx - 9) find the value of k.

Q. 4. Find the value of a and b so that x³ + ax² + bx - 6 is completely divisible by x² - 4x + 3.

Q. 5. If (x - 3) is the HCF of (x³ - 2x² + px + 6) and (x² - 5x + q) find the value of (6p +5 q).

Q. 6. Find the value of a and b so that f(x) = 3x³ + ax² - 13x + b is divisible by (x² - 2x - 3).

Q. 7. If (x + 1) (x - 4) is the HCF of the polynomial (x - 4) (2x² + x - a) and (x + 1) (2x² + bx - 12), find a and b.

Q. 8. The HCF and LCM of two polynomial p(x) and q(x) are 56(x⁴ + x) and 4(x² - x + 1) respectively. If p(x) = 28 (x³ + 1) find q(x).

Q. 9. The HCF and LCM of the polynomial P(x) and Q(x) are respectively 5(x + 3) (x - 1) and 20 x (x² - 9)(x² - 3x + 2). If P(x) = 10 (x² - 9) (x - 1) find q(x).

Q. 10. The HCF of the polynomial P(x) = (x - 3)(x² + x - 2) and Q(x)=(x² - 5x + 6), find the LCM of P(x) and Q(x).

Q. 11. (x² + x - 2) is the HCF of the expression (x - 1)(2x² + ax + 2) and (x + 2)(3x² + bx + 1). Find the value of a and b.

Q. 12. If ( x + 3) (x - 2) is the G.C.D of f(x)=(x + 3)(2x² - 3x + a) and g(x)=(x - 2)(3x² + 10x - b) Find the value of a and b.

Q. 13. Find the L.C.M of the polynomials: x(8x² + 27) and 2x²(2x² + 9x + 9).

Q. 14. If (x - 1)(x + 4) is the HCF of the polynomials P(x)= (x² + 2x - 3) (2x² + 5x + a) and Q(x) = (x² + x - 12) ( 3x² - x + b) find the value of a and b

Q. 15. . If the HCF of P(x) = ( 2x² - x - 1) (px² + 8x - 3) and Q(x) = (x² + x - 6) (3x² + qx - 1)is (x² + 2x - 3) , find the values of p and q.

Q. 16. Find the HCF of the polynomials: f(x) = 6 (x³ + 3x²) (x² - 16) (x² + 9x + 18) and g(x) = 8 (x⁴ + 4x³) (x² + 6x + 9)².

Q. 17. Find the HCF of the polynomials: f(x) = 9 (x⁴ – 1) (x + 5) and g(x) = 6 (x² – 1) (x²+ 2x + 1) (x² + 25).

Q. 18. Find the HCF of the polynomials: f(x) = 8 (x³ – x2 + x) (x³ – 8) and g(x) = 28 (x³ + 1) (x² – 4).

Q. 19. Find the G.C.D. of the polynomials: f(x) = 2x⁴ – 2y⁴ and g(x) = 3x³ + 6x²y – 3xy² – 6y³.

Q. 20. Find the L.C.M. of the polynomials: f(x) = 11x³ (x + 1)³ and g(x) = 121x (2x² + 3x + 1).

Q. 21. Find the L.C.M. of the polynomials: f(x) = 35 (x⁴ – 27x) and g(x) = 40 (2x³ - 5x² - 3x).

Q. 22. Find the L.C.M. of the polynomials: f(x) = 15x³ - 75x² - 90x and g(x) = 6x⁴ – 18x³ – 108x²

Q. 23. The GCD and LCM of two polynomials are (x – 7) and (x³ - 10x²+ 11x + 70). If one is (x² - 5x - 4), find the other.

Q. 24. If (x + 3) (x - 2) is the HCF of the polynomials f(x) = (x + 3) (2x²– 3x + p) and g(x) = (x - 2) (3x² + 10x - q), find p, q.

Q. 25. For what value of k the HCF of x² + x - (2k + 2) and (2x² + kx - 12) is (x + 4)?

Q. 26. Simplify the following rational expressions: (i) (x4 - 6x³ + 36x²) / (x³ + 216). (ii) (x⁴ - 1) / (x - 1).

Q. 27. Which rational expression should be subtracted from (2x² + 2x - 7) / (x² + x - 6) to get (x - 1) / (x - 2).

Q. 28. Simplify: [ (2x – 1) / (x – 1) – (x + 1) / (2x + 1) ] + [ (x - 1) / (x + 2) – (x + 1) / (x – 2)].

Q. 29. If A = x² – 7x + 12 and B = x² – 5x + 6, then evaluate the sum their reciprocals to the lowest form.

Q.30. Simplify the rational expression to the lowest terms :