JEE Mathematics Equation and Expression MCQs Set 03

Question. If \( px^2 + qx + r = 0 \) has no real roots and \( p, q, r \) are real such that \( p + r > 0 \) then
(a) \( p - q + r < 0 \)
(b) \( p - q + r > 0 \)
(c) \( p + r = 0 \)
(d) All of the options
Answer: (b) \( p - q + r > 0 \)

Question. If \( \alpha, \beta \) are the roots of \( x^2 - px + q = 0 \) then the product of the roots of the quadratic equation whose roots are \( \alpha^2 - \beta^2 \) and \( \alpha^3 - \beta^3 \) is
(a) \( p(p^2 - q)^2 \)
(b) \( p(p^2 - q)(p^2 - 4q) \)
(c) \( p(p^2 - 4q)(p^2 + q) \)
(d) None of the options
Answer: (b) \( p(p^2 - q)(p^2 - 4q) \)

Question. If the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is equal to the sum of the squares of their reciprocals then \( \frac{b^2}{ac} + \frac{bc}{a^2} \) is equal to
(a) 2
(b) -2
(c) 1
(d) -1
Answer: (a) 2

Question. If the absolute value of the difference of roots of the equation \( x^2 + px + 1 = 0 \) exceed \( \sqrt{3p} \) then
(a) \( p < -1 \) or \( p > 4 \)
(b) \( p > 4 \)
(c) \( -1 < p < 4 \)
(d) \( 0 \le p < 4 \)
Answer: (b) \( p > 4 \)

Question. If \( \alpha, \beta \) are roots of \( x^2 + px + q = 0 \) and \( \gamma, \delta \) are the roots of \( x^2 + px - r = 0 \) then \( (\alpha - \gamma)(\alpha - \delta) \) is equal to
(a) \( q + r \)
(b) \( q - r \)
(c) \( -(q + r) \)
(d) \( -(p + q + r) \)
Answer: (c) \( -(q + r) \)

Question. If \( \alpha, \beta \) are roots of \( 375x^2 - 25x - 2 = 0 \) and \( s_n = \alpha^n + \beta^n \) then \( \lim_{n \to \infty} \sum_{r=1}^n s_r \) is
(a) \( \frac{7}{116} \)
(b) \( \frac{1}{12} \)
(c) \( \frac{29}{358} \)
(d) None of the options
Answer: (b) \( \frac{1}{12} \)

Question. The quadratic equation whose roots are the AM and HM of the roots of the equation \( x^2 + 7x - 1 = 0 \) is
(a) \( 14x^2 + 14x - 45 = 0 \)
(b) \( 45x^2 - 14x + 14 = 0 \)
(c) \( 14x^2 + 45x - 14 = 0 \)
(d) None of the options
Answer: (c) \( 14x^2 + 45x - 14 = 0 \)

Question. Let \( \alpha \neq \beta \) and \( \alpha^2 + 3 = 5\alpha \) while \( \beta^2 = 5\beta - 3 \). The quadratic equation whose roots are \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) is
(a) \( 3x^2 - 31x + 3 = 0 \)
(b) \( 3x^2 - 19x + 3 = 0 \)
(c) \( 3x^2 + 19x + 3 = 0 \)
(d) None of the options
Answer: (b) \( 3x^2 - 19x + 3 = 0 \)

Question. If \( a \) and \( b \) are rational and \( b \) is not a perfect square then the quadratic equation with rational coefficients whose one root is \( \frac{1}{a + \sqrt{b}} \) is
(a) \( x^2 - 2ax + (a^2 - b) = 0 \)
(b) \( (a^2 - b)x^2 - 2ax + 1 = 0 \)
(c) \( (a^2 - b)x^2 - 2bx + 1 = 0 \)
(d) None of the options
Answer: (b) \( (a^2 - b)x^2 - 2ax + 1 = 0 \)

Question. If \( \frac{1}{4 - 3i} \) is a root of \( ax^2 + bx + 1 = 0 \), where \( a, b \) are real, then
(a) \( a = 25, b = -8 \)
(b) \( a = 25, b = 8 \)
(c) \( a = 5, b = 4 \)
(d) None of the options
Answer: (a) \( a = 25, b = -8 \)

Question. If \( \alpha, \beta, \gamma \) be the roots of the equation \( x(1 + x^2) + x^2(6 + x) + 2 = 0 \) then the value of \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \) is
(a) -3
(b) \( \frac{1}{2} \)
(c) \( -\frac{1}{2} \)
(d) None of the options
Answer: (c) \( -\frac{1}{2} \)

Question. If the roots of \( x^3 - 12x^2 + 39x - 28 = 0 \) are in AP then their common difference is
(a) \( \pm 1 \)
(b) \( \pm 2 \)
(c) \( \pm 3 \)
(d) \( \pm 4 \)
Answer: (c) \( \pm 3 \)

Question. The roots of the equation \( x^3 + 14x^2 - 84x - 216 = 0 \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If \( z_0 = \alpha + i\beta, i = \sqrt{-1} \), then the roots of the cubic equation \( x^3 - 2(1 + \alpha)x^2 + (4\alpha + \alpha^2 + \beta^2)x - 2(\alpha^2 + \beta^2) = 0 \) are
(a) \( 2, z_0, \bar{z}_0 \)
(b) \( 1, z_0, -z_0 \)
(c) \( 2, z_0, -\bar{z}_0 \)
(d) \( 2, -\bar{z}_0, \bar{z}_0 \)
Answer: (a) \( 2, z_0, \bar{z}_0 \)

Question. If 3 and \( 1 + \sqrt{2} \) are two roots of a cubic equation with rational coefficients then the equation is
(a) \( x^3 - 5x^2 + 9x - 9 = 0 \)
(b) \( x^3 - 3x^2 - 4x + 12 = 0 \)
(c) \( x^3 - 5x^2 + 7x + 3 = 0 \)
(d) None of the options
Answer: (d) None of the options

Question. Let \( a, b, c \) be real numbers and \( a \neq 0 \). If \( \alpha \) is a root of \( a^2x^2 + bx + c = 0 \), \( \beta \) is a root of \( a^2x^2 - bx - c = 0 \), and \( 0 < \alpha < \beta \) then the equation \( a^2x^2 + 2bx + 2c = 0 \) has a root \( \gamma \) that always satisfies
(a) \( \gamma = \frac{1}{2}(\alpha + \beta) \)
(b) \( \gamma = \alpha + \frac{\beta}{2} \)
(c) \( \gamma = \alpha \)
(d) \( \alpha < \gamma < \beta \)
Answer: (d) \( \alpha < \gamma < \beta \)

Question. Let \( a, b, c \) three real number such that \( 2a + 3b + 6c = 0 \). Then the quadratic equation \( ax^2 + bx + c = 0 \) has
(a) imaginary roots
(b) at least one root in (0, 1)
(c) at least one root in (-1, 0)
(d) both roots in (1, 2)
Answer: (b) at least one root in (0, 1)

Question. If the equations \( 2x^2 - 7x + 1 = 0 \) and \( ax^2 + bx + 2 = 0 \) have a common root then
(a) \( a = 2, b = -7 \)
(b) \( a = -7/2, b = 1 \)
(c) \( a = 4, b = -14 \)
(d) None of the options
Answer: (c) \( a = 4, b = -14 \)

Question. The quadratic equations \( x^2 + (a^2 - 2)x - 2a^2 = 0 \) and \( x^2 - 3x + 2 = 0 \) have
(a) no common root for all \( a \in R \)
(b) exactly one common root for all \( a \in R \)
(c) two common roots for some \( a \in R \)
(d) None of the options
Answer: (b) exactly one common root for all \( a \in R \)

Question. If the equation \( ax^2 + bx + c = 0 \) and \( cx^2 + bx + a = 0, a \neq c \) have a negative common root then the value of \( a - b + c \) is
(a) 0
(b) 2
(c) 1
(d) None of the options
Answer: (a) 0

Question. If the equations \( x^2 + ix + a = 0 \), \( x^2 - 2x + ia = 0, a \neq 0 \) have a common root then
(a) \( a \) is real
(b) \( a = \frac{1}{2} + i \)
(c) \( a = \frac{1}{2} - i \)
(d) the other root is also common
Answer: (c) \( a = \frac{1}{2} - i \)

Question. If \( x^2 - 2^r \cdot p_r x + r = 0 \); \( r = 1, 2, 3 \) are three quadratic equations of which each pair has exactly one root common then the number of solutions of the triplet \( (p_1, p_2, p_3) \) is
(a) 2
(b) 1
(c) 9
(d) 27
Answer: (a) 2

Question. If \( (\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x < 1 \) for all \( x \in R \) then \( \lambda \) belongs to the interval
(a) (-2, 1)
(b) \( (-2, \frac{2}{5}) \)
(c) \( (\frac{2}{5}, 1) \)
(d) None of the options
Answer: (b) \( (-2, \frac{2}{5}) \)

Question. The least integral value of \( k \) for which \( (k - 2)x^2 + 8x + k + 4 > 0 \) for all \( x \in R \), is
(a) 5
(b) 4
(c) 3
(d) None of the options
Answer: (a) 5

Question. The set of possible values of \( x \) such that \( 5^x + (2\sqrt{3})^{2x} - 169 \) is always positive is
(a) [3, +∞)
(b) [2, +∞)
(c) (2, +∞)
(d) None of the options
Answer: (c) (2, +∞)

Question. If all real values of \( x \) obtained from the equation \( 4^x - (a - 3)2^x + a - 4 = 0 \) are nonpositive then
(a) \( a \in (4, 5] \)
(b) \( a \in (0, 4) \)
(c) \( a \in (4, +∞) \)
(d) None of the options
Answer: (a) \( a \in (4, 5] \)

Question. The set of possible values of \( \lambda \) for which \( x^2 - (\lambda^2 - 5\lambda + 5)x + (2\lambda^2 - 3\lambda - 4) = 0 \) has roots whose sum and product are both less than 1 is
(a) \( (1, \frac{5}{2}) \)
(b) (1, 4)
(c) \( [1, \frac{5}{2}] \)
(d) None of the options
Answer: (d) None of the options

Question. If \( \log_{10} x + \log_{10} y \ge 2 \) then the smallest possible value of \( x + y \) is
(a) 10
(b) 30
(c) 20
(d) None of the options
Answer: (c) 20

Question. If \( f(x) = \frac{x^2 - 1}{x^2 + 1} \) for every real number \( x \) then the minimum value of \( f \)
(a) does not exist because f is unbounded
(b) is not attained even though f is bounded
(c) is equal to 1
(d) is equal to -1
Answer: (d) is equal to -1

Question. If \( ax^2 + bx + 6 = 0 \) does not have two distinct real roots, where \( a \in R, b \in R \), then the least value of \( 3a + b \) is
(a) 4
(b) -1
(c) 1
(d) -2
Answer: (d) -2

Question. If \( ab = 2a + 3b, a > 0, b > 0 \) then the minimum value of \( ab \) is
(a) 12
(b) 24
(c) \( \frac{1}{4} \)
(d) None of the options
Answer: (b) 24

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

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

Question. The number of values of \( k \) for which \( \{x^2 - (k - 2)x + k^2\}\{x^2 + kx + (2k - 1)\} \) is a perfect square is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (a) 1

Question. If \( x + \lambda y - 2 \) and \( x - \mu y + 1 \) are factors of the expression \( 6x^2 - xy - y^2 - 6x + 8y - 12 \) then
(a) \( \lambda = \frac{1}{3}, \mu = \frac{1}{2} \)
(b) \( \lambda = 2, \mu = 3 \)
(c) \( \lambda = \frac{1}{3}, \mu = -\frac{1}{2} \)
(d) None of the options
Answer: (a) \( \lambda = \frac{1}{3}, \mu = \frac{1}{2} \)

Question. If \( x - y \) and \( y - 2x \) are two factors of the expression \( x^3 - 3x^2y + \lambda xy^2 + \mu y^3 \) then
(a) \( \lambda = 11, \mu = -3 \)
(b) \( \lambda = 3, \mu = -11 \)
(c) \( \lambda = \frac{11}{4}, \mu = -\frac{3}{4} \)
(d) None of the options
Answer: (c) \( \lambda = \frac{11}{4}, \mu = -\frac{3}{4} \)