JEE Mathematics Equation and Expression MCQs Set D

Question. If \( x + y \) and \( y + 3x \) are two factors of the expression \( \lambda x^3 - \mu x^2y + xy^2 + y^3 \) then the third factor is
(a) \( y + 3x \)
(b) \( y - 3x \)
(c) \( y - x \)
(d) None of the options
Answer: (b) \( y - 3x \)

Question. If \( x, y, z \) are real and distinct then \( f(x, y) = x^2 + 4y^2 + 9z^2 - 6yz - 3zx - 2xy \) is always
(a) non-negative
(b) nonpositive
(c) zero
(d) None of the options
Answer: (a) non-negative

Question. If \( x^2 + y^2 + z^2 = 1 \) then the value of \( xy + yz + zx \) lies in the interval
(a) \( [\frac{1}{2}, 2] \)
(b) [-1, 2]
(c) \( [-\frac{1}{2}, 1] \)
(d) \( [-1, \frac{1}{2}] \)
Answer: (c) \( [-\frac{1}{2}, 1] \)

Question. If \( a \in R, b \in R \) then the factors of the expression \( a(x^2 - y^2) - bxy \) are
(a) real and different
(b) real and identical
(c) complex
(d) None of the options
Answer: (a) real and different

Question. If \( a, b, c \) are in HP then the expression \( a(b - c)x^2 + b(c - a)x + c(a - b) \)
(a) has real and distinct factors
(b) is a perfect square
(c) has no real factor
(d) None of the options
Answer: (b) is a perfect square

Question. The number of positive integral values of \( k \) for which \( (16x^2 + 12x + 39) + k(9x^2 - 2x + 11) \) is a perfect square is
(a) two
(b) zero
(c) one
(d) None of the options
Answer: (c) one

Question. If \( (x - 1)^3 \) is a factor of \( x^4 + ax^3 + bx^2 + cx - 1 \) then the other factor is
(a) \( x - 3 \)
(b) \( x + 1 \)
(c) \( x + 2 \)
(d) None of the options
Answer: (b) \( x + 1 \)

Question. Let \( x^2 - px + q = 0 \), where \( p \in R, q \in R \), have the roots \( \alpha, \beta \) such that \( \alpha + 2\beta = 0 \) then
(a) \( 2p^2 + q = 0 \)
(b) \( 2q^2 + p = 0 \)
(c) \( q < 0 \)
(d) None of the options
Answer: (a) \( 2p^2 + q = 0 \)
(c) \( q < 0 \)

Question. The cubic equation whose roots are the AM, GM and HM of the roots of \( x^2 - 2px + q^2 = 0 \) is
(a) \( (x - p)(x - q)(x - p - q) = 0 \)
(b) \( (x - p)(x - |q|)(px - q^2) = 0 \)
(c) \( x^3 - \left( p + |q| + \frac{q^2}{p} \right)x^2 + \left( p|q| + q^2 + \frac{|q|^3}{p} \right)x - |q|^3 = 0 \)
(d) None of the options
Answer: (b) \( (x - p)(x - |q|)(px - q^2) = 0 \)
(c) \( x^3 - \left( p + |q| + \frac{q^2}{p} \right)x^2 + \left( p|q| + q^2 + \frac{|q|^3}{p} \right)x - |q|^3 = 0 \)

Question. If \( x^2 + ax + b = 0 \) and \( x^2 + bx + a = 0, a \neq b \), have a common root \( \alpha \) then
(a) \( a + b = 1 \)
(b) \( \alpha + 1 = 0 \)
(c) \( \alpha = 1 \)
(d) \( a + b + 1 = 0 \)
Answer: (c) \( \alpha = 1 \)
(d) \( a + b + 1 = 0 \)

Question. The line \( y + 14 = 0 \) cuts the curve whose equation is \( x(x^2 + x + 1) + y = 0 \) at
(a) three real points
(b) one real point
(c) at least one real point
(d) no real point
Answer: (b) one real point

Question. If \( a, b, c \) are in GP, where \( a, c \) are positive, then the equation \( ax^2 + bc + c = 0 \) has
(a) real roots
(b) imaginary roots
(c) ratio of roots = 1 : w where w is a nonreal cube root of unity
(d) ratio of roots = b : ac
Answer: (b) imaginary roots
(c) ratio of roots = 1 : w where w is a nonreal cube root of unity

Question. If \( \alpha, \beta \) are the roots of the equation \( x^2 + x + 3 = 0 \) then the equation \( 3x^2 + 5x + 3 = 0 \) has a root
(a) \( \frac{\alpha}{\beta} \)
(b) \( \frac{\beta}{\alpha} \)
(c) \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \)
(d) None of the options
Answer: (a) \( \frac{\alpha}{\beta} \)
(b) \( \frac{\beta}{\alpha} \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 - 2ax + b^2 = 0 \) and \( \gamma, \delta \) are the roots of \( x^2 - 2bx + a^2 = 0 \) then
(a) AM of \( \alpha, \beta \) = GM of \( \gamma, \delta \)
(b) GM of \( \alpha, \beta \) = AM of \( \gamma, \delta \)
(c) \( \alpha, \beta, \gamma, \delta \) are in AP
(d) \( \alpha, \beta, \gamma, \delta \) are in GP
Answer: (a) AM of \( \alpha, \beta \) = GM of \( \gamma, \delta \)
(b) GM of \( \alpha, \beta \) = AM of \( \gamma, \delta \)

Question. If the roots of the equation \( ax^2 - 4x + a^2 = 0 \) are imaginary and the sum of the roots is equal to their product then \( a \) is
(a) -2
(b) 4
(c) 2
(d) None of the options
Answer: (c) 2

Question. If \( x, y, z \) are three consecutive terms of a GP, where \( x > 0 \) and the common ratio is \( r \), then the inequality \( z + 3x > 4y \) holds for
(a) \( r \in (-\infty, 1) \)
(b) \( r = \frac{24}{5} \)
(c) \( r \in (3, +\infty) \)
(d) \( r = \frac{1}{2} \)
Answer: (a) \( r \in (-\infty, 1) \)
(b) \( r = \frac{24}{5} \)
(c) \( r \in (3, +\infty) \)
(d) \( r = \frac{1}{2} \)

Question. The equation \( ||x - 1| + a| = 4 \) can have real solutions for \( x \) if \( a \) belongs to the interval
(a) \( (-\infty, 4] \)
(b) \( (-\infty, -4] \)
(c) \( (4, +\infty) \)
(d) \( [-4, 4] \)
Answer: (a) \( (-\infty, 4] \)
(b) \( (-\infty, -4] \)

Question. The equation \( |x + 1| |x - 1| = a^2 - 2a - 3 \) can have real solutions for \( x \) if \( a \) belongs to
(a) \( (-\infty, -1] \cup [3, +\infty) \)
(b) \( [1 - \sqrt{5}, 1 + \sqrt{5}] \)
(c) \( [1 - \sqrt{5}, -1] \cup [3, 1 + \sqrt{5}] \)
(d) None of the options
Answer: (a) \( (-\infty, -1] \cup [3, +\infty) \)
(c) \( [1 - \sqrt{5}, -1] \cup [3, 1 + \sqrt{5}] \)

Question. The common roots of the equations \( x^3 + 2x^2 + 2x + 1 = 0 \) and \( 1 + x^{130} + x^{1988} = 0 \) are (where \( \omega \) is a nonreal cube root of unity)
(a) \( \omega \)
(b) \( \omega^2 \)
(c) -1
(d) \( \omega - \omega^2 \)
Answer: (a) \( \omega \)
(b) \( \omega^2 \)

Question. If \( \alpha \) is a root of the equation \( 2x(2x + 1) = 1 \) then the other root is
(a) \( 3\alpha^3 - 4\alpha \)
(b) \( -2\alpha(\alpha + 1) \)
(c) \( 4\alpha^3 - 3\alpha \)
(d) None of the options
Answer: (b) \( -2\alpha(\alpha + 1) \)
(c) \( 4\alpha^3 - 3\alpha \)

Question. For the equation \( 2x^2 + 6\sqrt{2}x + 1 = 0 \)
(a) roots are rational
(b) if one root is \( p + \sqrt{q} \) then the other is \( -p + \sqrt{q} \)
(c) roots are irrational
(d) if one root is \( P + \sqrt{q} \) then the other is \( p - \sqrt{q} \)
Answer: (b) if one root is \( p + \sqrt{q} \) then the other is \( -p + \sqrt{q} \)
(c) roots are irrational

Question. If \( \alpha, \beta \) are the real roots of \( x^2 + px + q = 0 \) and \( \alpha^4, \beta^4 \) are the roots of \( x^2 - rx + s = 0 \) then the equation \( x^2 - 4qx + 2q^2 - r = 0 \) has always
(a) two real roots
(b) two negative roots
(c) two positive roots
(d) one positive root and one negative root
Answer: (a) two real roots
(d) one positive root and one negative root

Question. The equation \( x^{3/4(\log_2 x)^2 + \log_2 x - 5/4} = \sqrt{2} \) has
(a) at least one negative solution
(b) exactly one irrational solution
(c) exactly three real solutions
(d) two nonreal complex roots
Answer: (b) exactly one irrational solution
(c) exactly three real solutions

Question. If \( a, b, c \) are rational and no two of them are equal then the equations \( (b - c)x^2 + (c - a)x + a - b = 0 \) and \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \)
(a) have rational roots
(b) will be such that at least one has rational roots
(c) have exactly one root common
(d) have at least one root common
Answer: (a) have rational roots
(c) have exactly one root common

Question. The equations \( x^2 + b^2 = 1 - 2bx \) and \( x^2 + a^2 = 1 - 2ax \) have one and only one root common. Then
(a) \( a - b = 2 \)
(b) \( a - b + 2 = 0 \)
(c) \( |a - b| = 2 \)
(d) None of the options
Answer: (a) \( a - b = 2 \)
(b) \( a - b + 2 = 0 \)
(c) \( |a - b| = 2 \)

Question. Let \( p \) and \( q \) be roots of the equation \( x^2 - 2x + A = 0 \), and let \( r \) and \( s \) be the roots of the equation \( x^2 - 18x + B = 0 \). If \( p < q < r < s \) are in arithmetic progression then
(a) \( A = -83, B = -3 \)
(b) \( A = -3, B = 77 \)
(c) \( q = 3, r = 7 \)
(d) \( p + q + r + s = 20 \)
Answer: (b) \( A = -3, B = 77 \)
(c) \( q = 3, r = 7 \)
(d) \( p + q + r + s = 20 \)

Question. The quadratic equation \( x^2 - 2x - \lambda = 0, \lambda \neq 0 \)
(a) cannot have a real root if \( \lambda < -1 \)
(b) can have a rational root if \( \lambda \) is a perfect square
(c) cannot have an integral root if \( n^2 - 1 < \lambda < n^2 + 2n \) where \( n = 0, 1, 2, 3, ..... \)
(d) None of the options
Answer: (a) cannot have a real root if \( \lambda < -1 \)
(c) cannot have an integral root if \( n^2 - 1 < \lambda < n^2 + 2n \) where \( n = 0, 1, 2, 3, ..... \)

Question. A quadratic equation whose roots are \( \left( \frac{\gamma}{\alpha} \right)^2 \) and \( \left( \frac{\beta}{\alpha} \right)^2 \), where \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 27 = 0 \), is
(a) \( x^2 - x + 1 = 0 \)
(b) \( x^2 + 3x + 9 = 0 \)
(c) \( x^2 + x + 1 = 0 \)
(d) \( x^2 - 3x + 9 = 0 \)
Answer: (c) \( x^2 + x + 1 = 0 \)

Question. The graph of the curve \( x^2 = 3x - y - 2 \) is
(a) between the lines \( x = 1 \) and \( x = \frac{3}{2} \)
(b) between the lines \( x = 1 \) and \( x = 2 \)
(c) strictly below the line \( 4y = 1 \)
(d) None of the options
Answer: (c) strictly below the line \( 4y = 1 \)

Question. \( a(x^2 - y^2) + \lambda\{x(y + 1) + 1\} \) can be resolved into linear rational factors. Then
(a) \( \lambda = 1 \)
(b) \( \lambda = \frac{4a^2}{a - 1}, a \neq 1 \)
(c) \( \lambda = 0, a = 1 \)
(d) None of the options
Answer: (b) \( \lambda = \frac{4a^2}{a - 1}, a \neq 1 \)
(c) \( \lambda = 0, a = 1 \)

Question. \( x^2 - 4 \) is a factor of \( f(x) = (a_1x^2 + b_1x + c_1) \cdot (a_2x^2 + b_2x + c_2) \) if
(a) \( b_1 = 0, c_1 + 4a_1 = 0 \)
(b) \( b_2 = 0, c_2 + 4a_2 = 0 \)
(c) \( 4a_1 + 2b_1 + c_1 = 0, 4a_2 + c_2 = 2b_2 \)
(d) \( 4a_1 + c_1 = 2b_1, 4a_2 + 2b_2 + c_2 = 0 \)
Answer: (a) \( b_1 = 0, c_1 + 4a_1 = 0 \)
(b) \( b_2 = 0, c_2 + 4a_2 = 0 \)
(c) \( 4a_1 + 2b_1 + c_1 = 0, 4a_2 + c_2 = 2b_2 \)
(d) \( 4a_1 + c_1 = 2b_1, 4a_2 + 2b_2 + c_2 = 0 \)

Question. \( ax^2 + by^2 + cz^2 + 2ayz + 2bzx + 2cxy \) can be resolved into linear factors if \( a, b, c \) are such that
(a) \( a = b = c \)
(b) \( ab + bc + ca = 1 \)
(c) \( a + b + c = 0 \)
(d) None of the options
Answer: (a) \( a = b = c \)
(c) \( a + b + c = 0 \)

Question. If \( a, b \) are the real roots of \( x^2 + px + 1 = 0 \) and \( c, d \) are the real roots of \( x^2 + qx + 1 = 0 \) then \( (a - c)(b - c)(a + d)(b + d) \) is divisible by
(a) \( a + b + c + d \)
(b) \( a + b - c - d \)
(c) \( a - b + c - d \)
(d) \( a - b - c - d \)
Answer: (a) \( a + b + c + d \)
(b) \( a + b - c - d \)

Question. If \( x \in [2, 4] \) then for the expression \( x^2 - 6x + 5 \)
(a) the least value = -4
(b) the greatest value = 4
(c) the least value = 3
(d) the greatest value = -4
Answer: (a) the least value = -4
(d) the greatest value = -4

Question. If \( 0 < a < 5 \), \( 0 < b < 5 \) and \( \frac{x^2 + 5}{2} = x - 2 \cos(a + bx) \) is satisfied for at least one real \( x \) then the greatest value of \( a + b \) is
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( 3\pi \)
(d) \( 4\pi \)
Answer: (c) \( 3\pi \)

Question. Let \( f(x) = x^2(x + 2) + x + 3 \). Then
(a) \( f(-3 - k) < 0 \) and \( f(-2 + k) > 0 \) for all \( k > 0 \)
(b) \( f(-3 - k) > 0 \) and \( f(-2 + k) < 0 \) for all \( k > 0 \)
(c) \( f(x) = 0 \) has a root \( \alpha \) such that \( [\alpha] + 3 = 0 \), where \( [\alpha] \) is the greatest integer less than or equal to \( \alpha \)
(d) \( f(x) = 0 \) has exactly one root \( \alpha \) such that \( (\alpha) + 2 = 0 \), where \( (\alpha) \) is the smallest integer greater than or equal to \( \alpha \)
Answer: (a) \( f(-3 - k) < 0 \) and \( f(-2 + k) > 0 \) for all \( k > 0 \)
(c) \( f(x) = 0 \) has a root \( \alpha \) such that \( [\alpha] + 3 = 0 \), where \( [\alpha] \) is the greatest integer less than or equal to \( \alpha \)
(d) \( f(x) = 0 \) has exactly one root \( \alpha \) such that \( (\alpha) + 2 = 0 \), where \( (\alpha) \) is the smallest integer greater than or equal to \( \alpha \)