Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 14

Question. If \( n \) is a positive integer and \( n \in [5, 100] \) then the number of positive integral roots of the equation \( x^2 + 2x - n = 0 \) is
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (c) 8

Question. The number of solutions of the system of equations given below is \( |x| + |y| = 1 ; x^2 + y^2 = a^2 ; \left( \frac{1}{\sqrt{2}} < a < 1 \right) \)
(a) infinite
(b) 2
(c) 4
(d) 8
Answer: (d) 8

Question. If \( \alpha, \beta \) are the roots of \( x^2 + x + 1 = 0 \) and \( S_n = \alpha^n + \beta^n \) then \( \begin{vmatrix} 3 & 1 + S_1 & 1 + S_2 \\ 1 + S_1 & 1 + S_2 & 1 + S_3 \\ 1 + S_2 & 1 + S_3 & 1 + S_4 \end{vmatrix} = \)
(a) 27
(b) -27
(c) -3
(d) 9
Answer: (b) -27

Question. If \( \alpha, \beta \) are the roots of the equation \( \lambda(x^2 - x) + x + 5 = 0 \) and if \( \lambda_1 \) and \( \lambda_2 \) are the two values of \( \lambda \) for which the roots \( \alpha, \beta \) are connected by the relation \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{5} \), then the value of \( \frac{\lambda_1}{\lambda_2} + \frac{\lambda_2}{\lambda_1} = \)
(a) 250
(b) 1022
(c) 254
(d) 0
Answer: (c) 254

Question. If \( 0 < x < 1000 \) and \( \left[\frac{x}{2}\right] + \left[\frac{x}{3}\right] + \left[\frac{x}{5}\right] = \frac{31}{30}x \) where [x] is the greatest Integer less than or equal to x, then the number of possible values of x is
(a) 34
(b) 32
(c) 33
(d) 35
Answer: (c) 33

Question. If \( \alpha, \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \) then the value of \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \) is
(a) positive
(b) negative
(c) non-negative
(d) non positive
Answer: (c) non-negative

Question. The equation \( (x - 3)^9 + (x - 3^2)^9 + \dots + (x - 3^9)^9 = 0 \) has
(a) all the roots are real
(b) one real and 8 imaginary roots
(c) real roots namely \( x = 3, 3^2, \dots, 3^9 \)
(d) five real and 4 imaginary roots
Answer: (b) one real and 8 imaginary roots

Question. The root of the equation \( 2(1 + i)x^2 - 4(2 - i)x - 5 - 3i = 0 \), where \( i = \sqrt{-1} \), which has greater modulus is
(a) \( (3 - 5i) / 2 \)
(b) \( (5 - 3i) / 2 \)
(c) \( (3 + i) / 2 \)
(d) \( (1 + 3i) / 2 \)
Answer: (a) \( (3 - 5i) / 2 \)

Question. If \( xy = 2(x + y), x \leq y \) and \( x, y \in N \), then number of solutions of the equaiton
(a) two
(b) three
(c) no solution
(d) infinitely many solution
Answer: (a) two

Question. If \( c > 0 \) and the equation \( 3ax^2 + 4bx + c = 0 \) has no real root, then
(a) \( 2a + c > b \)
(b) \( a + 2c > b \)
(c) \( 3a + c > 4b \)
(d) \( a + 3c < b \)
Answer: (c) \( 3a + c > 4b \)

Question. The polynomial \( (ax^2 + bx + c)(ax^2 - dx - c), ac \neq 0, \) has
(a) four real zeroes
(b) at least two real zeroes
(c) at most two real zeros
(d) All of the options
Answer: (b) at least two real zeroes

Question. The range of a for which the equation \( x^2 + ax - 4 = 0 \) has its smaller root in the interval \( (-1, 2) \) is
(a) \( (-\infty, -3) \)
(b) \( (0, 3) \)
(c) \( (0, \infty) \)
(d) \( (-\infty, -3) \cup (0, \infty) \)
Answer: (a) \( (-\infty, -3) \)

Question. If x, y, z are real and distinct, then x² + 4y² + 9z² - 6yz - 3zx - 2xy is always
(a) positive
(b) non negative
(c) negative
(d) zero
Answer: (b) non negative

Question. If \( \sqrt{9x^2 + 6x + 1} < 2 - x \) then
(a) \( x \in \left( -\frac{3}{2}, \frac{1}{4} \right) \)
(b) \( x \in \left( -\frac{3}{2}, \frac{1}{4} \right] \)
(c) \( x \in \left[ -\frac{3}{2}, \frac{1}{4} \right) \)
(d) \( x < \frac{1}{4} \)
Answer: (a) \( x \in \left( -\frac{3}{2}, \frac{1}{4} \right) \)

Question. For the quadratic equation \( 4x^2 - 2(a + c - 1)x + ac - b = 0 \ (a > b > c) \)
(a) both roots are greater than a
(b) both roots are less than c
(c) both roots lie between c/2 and a/2
(d) exactly one of the roots lies between c/2 and a/2
Answer: (d) exactly one of the roots lies between c/2 and a/2

Question. If \( \frac{2x}{2x^2 + 5x + 2} > \frac{1}{x + 1} \), then
(a) \( x < 2 \)
(b) \( -1 < x < 1 \)
(c) \( -2 < x < 0 \)
(d) \( -\frac{2}{3} < x < -\frac{1}{2} \)
Answer: (d) \( -\frac{2}{3} < x < -\frac{1}{2} \)

Question. If \( x, y, z \) are real such that \( x + y + z = 4 \), \( x^2 + y^2 + z^2 = 6 \), then \( x \in \)
(a) \( (-1, 1) \)
(b) \( [0, 2] \)
(c) \( [2, 3] \)
(d) \( \left[ \frac{2}{3}, 2 \right] \)
Answer: (d) \( \left[ \frac{2}{3}, 2 \right] \)

Question. The equation \( 2^{2x} + (a - 1) 2^{x+1} + a = 0 \) has roots of opposite signs then exhaustive set of values of a is
(a) \( a \in (-1, 0) \)
(b) \( a < 0 \)
(c) \( a \in (-\infty, 1/3) \)
(d) \( a \in (0, 1/3) \)
Answer: (c) \( a \in (-\infty, 1/3) \)

Question. If the equation \( |x^2 + bx + c| = k \) has four real roots, then
(a) \( b^2 - 4c > 0 \) and \( 0 < k < \frac{b^2 - 4c}{4} \)
(b) \( b^2 - 4c < 0 \) and \( 0 < k < \frac{b^2 - 4c}{4} \)
(c) \( b^2 - 4c > 0 \) and \( k > \frac{b^2 - 4c}{4} \)
(d) \( b^2 - 4c < 0 \) and \( k > \frac{b^2 - 4c}{4} \)
Answer: (a) \( b^2 - 4c > 0 \) and \( 0 < k < \frac{b^2 - 4c}{4} \)

Question. The set of values of a for which \( (a - 1)x^2 - (a + 1)x + a - 1 \geq 0 \) is true for all \( x \geq 2 \)
(a) \( (-\infty, 1) \)
(b) \( \left( 1, \frac{7}{3} \right] \)
(c) \( \left( \frac{7}{3}, \infty \right) \)
(d) \( (1, \infty) \)
Answer: (b) \( \left( 1, \frac{7}{3} \right] \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 + px + q = 0 \) and \( x^{2n} + p^n x^n + q^n = 0 \) and if \( \left(\frac{\alpha}{\beta}\right), \left(\frac{\beta}{\alpha}\right) \) are the roots of \( x^n + 1 + (x + 1)^n = 0, \ (n \in N) \) then
(a) must be an odd integer
(b) may be any integer
(c) must be an even integer
(d) cannot say anything
Answer: (c) must be an even integer

Question. If \( (x^2 + x + 2)^2 - (a - 3)(x^2 + x + 1)(x^2 + x + 2) + (a - 4)(x^2 + x + 1)^2 = 0 \) has at least one real root, then the complete set of values of a is.
(a) \( \left( 5, \frac{19}{3} \right] \)
(b) \( \left( \frac{19}{3}, \infty \right) \)
(c) \( (-\infty, 5) \)
(d) \( (1, 10) \)
Answer: (a) \( \left( 5, \frac{19}{3} \right] \)

Question. Let \( \alpha, \beta \) be the roots of the equation \( x^2 - px + r = 0 \) and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 - qx + r = 0 \). Then, the value of \( r \) is ( AIE - 2007 )
(a) \( \frac{2}{9}(p - q)(2q - p) \)
(b) \( \frac{2}{9}(q - p)(2p - q) \)
(c) \( \frac{2}{9}(q - 2p)(2q - p) \)
(d) \( \frac{2}{9}(2p - q)(2q - p) \)
Answer: (d) \( \frac{2}{9}(2p - q)(2q - p) \)

Question. Let \( p \) and \( q \) be real numbers such that \( p \neq 0 \), \( p^3 \neq q \) and \( p^3 \neq -q \). If \( \alpha \) and \( \beta \) are non-zero complex numbers satisfying \( \alpha + \beta = -p \) and \( \alpha^3 + \beta^3 = q \), then a quadratic equation having \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) as its roots is ( AIE - 2010 )
(a) \( (p^3 + q)x^2 - (p^3 + 2q)x + (p^3 + q) = 0 \)
(b) \( (p^3 + q)x^2 - (p^3 - 2q)x + (p^3 + q) = 0 \)
(c) \( (p^3 - q)x^2 - (5p^3 - 2q)x + (p^3 - q) = 0 \)
(d) \( (p^3 - q)x^2 - (5p^3 + 2q)x + (p^3 - q) = 0 \)
Answer: (b) \( (p^3 + q)x^2 - (p^3 - 2q)x + (p^3 + q) = 0 \)

Question. Let \( \alpha \) and \( \beta \) be the roots of \( x^2 - 6x - 2 = 0 \), with \( \alpha > \beta \). If \( a_n = \alpha^n - \beta^n \) for \( n \geq 1 \), then the value of \( \frac{a_{10} - 2a_8}{2a_9} \) is ( ( AIE - 2011 )
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. A value of b for which the equations \( x^2 + bx - 1 = 0 \), \( x^2 + x + b = 0 \) have one root in common is
(a) \( -\sqrt{2} \)
(b) \( -i\sqrt{3} \)
(c) \( i\sqrt{5} \)
(d) \( \sqrt{2} \)
Answer: (b) \( -i\sqrt{3} \)

Question. The real number k for which the equation \( 2x^3 + 3x + k = 0 \) has two distinct real roots in \( [0, 1] \) ( JEE MAIN - 2013 )
(a) lies between 1 and 2
(b) lies between 2 and 3
(c) lies between -1 and 0
(d) does not exist
Answer: (d) does not exist

Question. If the equations \( x^2 + 2x + 3 = 0 \) and \( ax^2 + bx + c = 0 \), a, b, c \( \in R \), have a common root then \( a : b : c \) is ( JEE MAIN- 2013 )
(a) 1 : 2 : 3
(b) 3 : 2 : 1
(c) 1 : 3 : 2
(d) 3 : 1 : 2
Answer: (a) 1 : 2 : 3

Question. If \( a \in R \) and the equation \( -3(x - [x])^2 + 2(x - [x]) + a^2 = 0 \) ( where, \( [x] \) denotes the greatest integer \( \leq x \) ) has no integral solution, then all possible values of a lie in the interval ( JEE - 2014 )
(a) \( (-1, 0) \cup (0, 1) \)
(b) \( (1, 2) \)
(c) \( (-2, -1) \)
(d) \( (-\infty, -2) \cup (2, \infty) \)
Answer: (a) \( (-1, 0) \cup (0, 1) \)

Question. Let \( \alpha \) and \( \beta \) be the roots of the equation \( px^2 + qx + r = 0 \), \( p \neq 0 \). If \( p, q \) and \( r \) are in AP and \( \frac{1}{\alpha} + \frac{1}{\beta} = 4 \), then the value of \( |\alpha - \beta| \) is ( ( JEE MAIN - 2014 )
(a) \( \frac{\sqrt{61}}{9} \)
(b) \( \frac{2\sqrt{17}}{9} \)
(c) \( \frac{\sqrt{34}}{9} \)
(d) \( \frac{2\sqrt{13}}{9} \)
Answer: (d) \( \frac{2\sqrt{13}}{9} \)

Question. \( \left\{ x \in R : \frac{14x}{x + 1} - \frac{9x - 30}{x - 4} < 0 \right\} \) is equal to ( EAM - 2010 )
(a) \( (-1, 4) \)
(b) \( (1, 4) \cup (5, 7) \)
(c) \( (1, 7) \)
(d) \( (-1, 1) \cup (4, 6) \)
Answer: (d) \( (-1, 1) \cup (4, 6) \)