Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 11

Question. If \( \alpha, \beta \) are the roots of the equation \( 8x^2 - 3x + 27 = 0 \), then the value of \( \left(\frac{\alpha^2}{\beta}\right)^{1/3} + \left(\frac{\beta^2}{\alpha}\right)^{1/3} \) is
(a) \( \frac{1}{3} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{7}{2} \)
(d) 4
Answer: (b) \( \frac{1}{4} \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 - px + q = 0 \) then the equation whose roots are \( \alpha\beta + \alpha + \beta \), \( \alpha\beta - \alpha - \beta \) is
(a) \( x^2 + 2qx + p^2 + q^2 = 0 \)
(b) \( x^2 - 2qx + p^2 + q^2 = 0 \)
(c) \( x^2 - 2qx + q^2 - p^2 = 0 \)
(d) \( x^2 + 2qx + q^2 - p^2 = 0 \)
Answer: (c) \( x^2 - 2qx + q^2 - p^2 = 0 \)

Question. If \( \sin\theta, \cos\theta \) are the roots of the equation \( ax^2 + bx + c = 0 \) then
(a) \( a^2 - b^2 + 2ac = 0 \)
(b) \( a^2 + b^2 + 2ac = 0 \)
(c) \( a - b + 2ac = 0 \)
(d) \( a + b + 2c = 0 \)
Answer: (a) \( a^2 - b^2 + 2ac = 0 \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 - x + 1 = 0 \) then \( \alpha^5 + \beta^5 = \)
(a) 2
(b) 1
(c) 12
(d) 4
Answer: (b) 1

Question. If \( \alpha, \beta \) are the roots of \( 3x^2 + 5x - 7 = 0 \) then the value of \( \left(\frac{1}{3\alpha+5}\right)^2 + \left(\frac{1}{3\beta+5}\right)^2 \) is
(a) 67/63
(b) 67/441
(c) 109/441
(d) 109/63
Answer: (b) 67/441

Question. If the sum of the roots of the equation \( ax^2 + bx + c = 0 \) is equal to sum of their squares, then
(a) \( ab + b^2 + 2ac = 0 \)
(b) \( ab + a^2 + 2ac = 0 \)
(c) \( ab + b^2 - 2ac = 0 \)
(d) \( ab + a^2 - 2ac = 0 \)
Answer: (c) \( ab + b^2 - 2ac = 0 \)

Question. If \( \alpha, \beta \) are the roots of \( 6x^2 - 6x + 1 = 0 \) then \( \frac{1}{2}(a+b\alpha+c\alpha^2+d\alpha^3) + \frac{1}{2}(a+b\beta+c\beta^2+d\beta^3) = \)
(a) \( a + b + c + d \)
(b) \( a + 2b + 3c + 4d \)
(c) \( a + (b/2) + (c/3) + (d/4) \)
(d) 0
Answer: (c) \( a + (b/2) + (c/3) + (d/4) \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 + ax - b = 0 \) and \( \gamma, \delta \) are the roots of \( x^2 + ax + b = 0 \) then \( (\alpha-\gamma)(\alpha-\delta)(\beta-\delta)(\beta-\gamma) = \)
(a) \( 4b^2 \)
(b) \( b^2 \)
(c) \( 2b^2 \)
(d) \( 3b^2 \)
Answer: (a) \( 4b^2 \)

Question. If the arithmetic mean of the roots of a quadratic equation is 8/5 and the arithmetic mean of their reciprocal is 8/7 then the equation is
(a) \( 5x^2 + 16x + 7 = 0 \)
(b) \( 5x^2 - 16x + 7 = 0 \)
(c) \( 7x^2 + 16x + 5 = 0 \)
(d) \( 7x^2 - 16x + 5 = 0 \)
Answer: (b) \( 5x^2 - 16x + 7 = 0 \)

Question. If one root of the equation \( 8x^2 - 6x + k = 0 \) is the square of the other, then \( k = \)
(a) 0, 3
(b) -1, 27
(c) 0, -2
(d) 1, -27
Answer: (d) 1, -27

Question. If the ratio of the roots of \( ax^2 + bx + c = 0 \) is same as the ratio of the roots of \( px^2 + qx + r = 0 \) then
(a) \( \frac{b^2}{ac} = \frac{p^2}{qr} \)
(b) \( \frac{b}{ac} = \frac{q}{pr} \)
(c) \( \frac{b^2}{ac} = \frac{q^2}{pr} \)
(d) \( \frac{b}{ac} = \frac{q^2}{pr} \)
Answer: (c) \( \frac{b^2}{ac} = \frac{q^2}{pr} \)

Question. If the difference of the roots of the equation \( x^2 - bx + c = 0 \) is equal to the difference of the roots of the equation \( x^2 - cx + b = 0 \) and \( b \neq c \) then \( b+c = \)
(a) 0
(b) 2
(c) 4
(d) -4
Answer: (d) -4

Question. Which one of the following Means of the roots of the equation \( x^2 - 2bx + a^2 = 0 \) is the A.M. of the roots of \( x^2 - 2ax + b^2 = 0 \)?
(a) A.M
(b) G.M
(c) H.M
(d) A.G.P
Answer: (b) G.M

Question. If \( \tan A, \tan B \) are the roots of \( x^2 - 2x + 2 = 0 \) then \( \sin^2(A+B) = \)
(a) 4/5
(b) 1/2
(c) 3/5
(d) 1/4
Answer: (a) 4/5

Question. If the roots of \( \frac{x^2-bx}{ax-c} = \frac{k-1}{k+1} \) are numerically equal but opposite in sign then \( k = \)
(a) c
(b) 1/c
(c) \( \frac{a+b}{a-b} \)
(d) \( \frac{a-b}{a+b} \)
Answer: (d) \( \frac{a-b}{a+b} \)

Question. If \( pr = 2(q+s) \) then among the equations \( x^2 + px + q = 0 \) and \( x^2 + rx + s = 0 \)
(a) both have real roots
(b) both have imaginary roots
(c) at least one has real roots
(d) at least one has imaginary roots.
Answer: (c) at least one has real roots

Question. The roots of the equation \( (b-c)x^2 + 2(c-a)x + (a-b) = 0 \) are always
(a) real and distinct
(b) real and equal
(c) real
(d) imaginary
Answer: (a) real and distinct

Question. The roots of the equation \( (a+c-b)x^2 + 2cx + (b+c-a) = 0 \, (a \neq b) \) are
(a) real and distinct
(b) real and equal
(c) real
(d) imaginary
Answer: (a) real and distinct

Question. If the roots of the equation \( x^2 - 2cx + ab = 0 \) be real and unequal, the roots of the equation \( x^2 - 2(a+b)x + (a^2+b^2+2c^2) = 0 \) are
(a) real and distinct
(b) real and equal
(c) real
(d) imaginary
Answer: (d) imaginary

Question. If a,b,c,d are real and no two of them are simultaneously zero, then the equation \( (x^2+ax-3b)(x^2-cx+b)(x^2-dx+2b) = 0 \) has
(a) at least two real roots
(b) at least four real roots
(c) all roots real
(d) at least two imaginary roots
Answer: (a) at least two real roots

Question. a,b,c are rational. Then the roots of \( (a+b+c)x^2 - 2(a+c)x + (a-b+c) = 0 \) are
(a) real
(b) equal
(c) rational
(d) imaginary
Answer: (c) rational

Question. If the equation \( (\lambda^2-5\lambda+6)x^2 + (\lambda^2-3\lambda+2)x + (\lambda^2-4) = 0 \) is satisfied by more than two values of x then \( \lambda = \)
(a) 1
(b) -2
(c) 3
(d) 2
Answer: (d) 2

Question. The number of real roots of the equation \( (\sqrt{3}+1)^{2x} + (\sqrt{3}-1)^{2x} = 2^{3x} \) is equal to
(a) 0
(b) 1
(c) 2
(d) more than 2
Answer: (b) 1

Question. If \( a \in Z \) and the equation \( (x-a)(x-10)+1=0 \) has integral roots, then values of 'a' are
(a) 10,8
(b) 12,10
(c) 12,8
(d) 10,12
Answer: (c) 12,8

Question. The number of real values of x for which \( \left(\frac{a}{b}\right)^x = x - 2x^2 - 4 \), where \( a, b > 0 \)
(a) 1
(b) 2
(c) 0
(d) infinite
Answer: (c) 0

Question. Number of solutions of the equation \( |x| = \cos x \)
(a) 1
(b) 2
(c) 0
(d) 3
Answer: (b) 2

Question. The value of \( \lambda \) in order that the equations \( 2x^2 + 5\lambda x + 2 = 0 \) and \( 4x^2 + 8\lambda x + 3 = 0 \) have a common root is given by
(a) 1
(b) -1
(c) \( \pm 1 \)
(d) 3
Answer: (c) \( \pm 1 \)

Question. If every pair from among the equations \( x^2 + ax + bc = 0 \), \( x^2 + bx + ca = 0 \) and \( x^2 + cx + ab = 0 \) has common root, then the sum of three common roots is
(a) abc
(b) 2abc
(c) \( 3(a+b+c) \)
(d) \( (a+b+c) \)
Answer: (d) \( (a+b+c) \)

Question. If \( x^2 - hx - 21 = 0 \), \( x^2 - 3hx + 35 = 0 \, (h > 0) \) have a common root, then the value of h is
(a) \( \pm 8 \)
(b) \( \pm 4 \)
(c) 4
(d) 2
Answer: (c) 4

Question. The conditions that a root of the equation \( ax^2 + bx + c = 0 \) may be reciprocal to a root of \( a_1x^2 + b_1x + c_1 = 0 \) is
(a) \( (bb_1 - aa_1)^2 = (ab_1 - bc_1)(ba_1 - b_1c) \)
(b) \( (cb_1 - ba_1)^2 = (ac_1 - cd_1)(ab_1 - bc_1) \)
(c) \( (cc_1 - aa_1)^2 = (ab_1 - bc_1)(ba_1 - b_1c) \)
(d) \( a + b + c = 0 \)
Answer: (c) \( (cc_1 - aa_1)^2 = (ab_1 - bc_1)(ba_1 - b_1c) \)

Question. If both roots of the equation \( x^2 + x + a = 0 \) exceed 'a' then
(a) \( 2 < a < 3 \)
(b) \( a > 3 \)
(c) \( -3 < a < 3 \)
(d) \( a < -2 \)
Answer: (d) \( a < -2 \)

Question. If both roots of the equation \( x^2 - 2ax + a^2 - 1 = 0 \) lie in the interval \( (-3, 4) \) then sum of the integral parts of 'a' is
(a) 0
(b) 2
(c) 4
(d) -1
Answer: (a) 0

Question. The expression \( (a-2)x^2 + 2(2a-3)x + (5a-6) \) is positive for all real values of x, then
(a) 'a' can be any real number
(b) a > 1
(c) a > 3
(d) a = 3
Answer: (c) a > 3

Question. If \( x^2 + 2ax + b \geq c \, \forall x \in R \) then
(a) \( b - c \geq a^2 \)
(b) \( c - a \geq b^2 \)
(c) \( a - b \geq c^2 \)
(d) \( a + b + c \geq 0 \)
Answer: (a) \( b - c \geq a^2 \)

Question. The roots of the equation \( (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 \) are real and equal if
(a) \( a > b > c \)
(b) \( a = b = c \)
(c) \( a < b < c \)
(d) \( a + b + c = 0 \)
Answer: (b) \( a = b = c \)

Question. The largest negative integer for which \( \frac{(x-4)(x-2)}{(x-1)(x-5)} > 0 \) is
(a) -2
(b) -1
(c) -3
(d) -4
Answer: (b) -1

Question. Let \( f(x) = x^2 + 2bx + 2c^2 \) and \( g(x) = -x^2 - 2cx + b^2 \), then \( \min\{f(x)\} > \max\{g(x)\} \) holds for
(a) \( 0 < c < b\sqrt{2} \)
(b) \( |c| > |b|\sqrt{2} \)
(c) \( |c| < |b|\sqrt{2} \)
(d) no real value of b and c
Answer: (b) \( |c| > |b|\sqrt{2} \)

Question. Number of rational roots of the equation \( |x^2 - 2x - 3| + 4x = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. The roots of the equation \( |x^2 - x - 6| = x + 2 \) are
(a) 1, -2, 2
(b) 1, 2, 4
(c) -2, 2, 4
(d) -2, 2, 3
Answer: (c) -2, 2, 4

Question. For a > 0, all the real roots of the equation \( x^2 - 3a|x-a| - 7a^2 = 0 \) are
(a) 4a, 5a
(b) -4a, 5a
(c) -4a, -5a
(d) 4a, -5a
Answer: (d) 4a, -5a