M.C.Q : Quadratic Equations & Expressions (1 to 10)

1. Let f(x) = x2 + bx + c, where b, c ∈ R. If f(x) is a factor of both x4 + 6x2 + 25 and 3x4 + 4x2 + 28x + 5, then the least value of f(x) is

(A) 2

(B) 3

(C) 5/2

(D) 4

2. Let a, b, c be the sides of a triangle. No two of them are equal and λ ∈ R. If the roots of the equation x2 + 2(a + b+ c) x + 3λ (ab + bc + ca) = 0 are real, then

(A) λ < 4/3

(B) λ > 5/3

(C) λ ∈(1/3 , 5/3)

(D) λ ∈(4/3 , 5/3)

3. Let f(x) = x2 + ax + b be a quadratic polynomial in which a and b are integers. If for a given integer n, f(n) f(n + 1) = f(m) for some integer m, then the value of m is

(A) n(a + b) + ab

(B) n2 + an + b

(C) n(n + 1) + an + b

(D) n2 + n + a + b

4. If the equations x2 + ax + b=0 and x2 + bx + a = 0 have exactly one common root, then the numerical value of a + b is

(A) 1

(B) –1

(C) 0

(D) none of these

5. The number of ordered pairs of positive integers x, y such that x2 + 3y and y2 + 3x are both perfect squares is

(A) 2

(B) 3

(C) 4

(D) 5

6. For the equations x2 + bx + c = 0 and 2x2 + (b + 1)x + c + 1 = 0 select the correct alternative

(A) both the equations can have integral roots

(B) both the equations can’t have integral roots simultaneously

(C) none of the equations can have integral roots

(D) nothing can be said

7. If x2 +ax +b is an integer for every integer x then

(A) ‘ a ‘ is always an integer but ‘ b ‘ need not be an integer.

(B) ‘ b ‘ is always an integer but ‘ a ‘ need not be an integer.

(C) a + b is always an integer.

(D) none of these.

8. If a , b , c be the sides of ΔABC and equations ax2 + bx + c=0 and 5x2 + 12x + 13=0 have a common root, then ∠C is

(A) 60°

(B) 90°

(C) 120°

(D) 45°

9. The equation x2 + nx + m = 0, n, m ∈ I, can not have

(A) integral roots

(B) non-integral rational roots

(B) irrational roots

(D) complex roots

10. If then x lies in the interval

(A) (-4/3 , -20/11)

(B) (-4/3 , -23/22)

(C) (-5/3 , -23/22)

(D) None of these

ANSWER:

1. D   2. A   3. C   4. B   5. B   6. B   7. C   8. B   9. B   10. B  

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M.C.Q : Quadratic Equations & Expressions ( 11 to 20 )

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