JEE Mathematics Equation and Expression MCQs Set 01

Question. If \( x \) is a real number such that \( x(x^2 + 1) \), \( (-1/2)x^2 \), 6 are three consecutive terms of an AP then the next two consecutive term of the AP are
(a) 14, 6
(b) -2, -10
(c) 14, 22
(d) None of the options
Answer: (c) 14, 22

Question. The number of real solutions of \( x - \frac{1}{x^2 - 4} = 2 - \frac{1}{x^2 - 4} \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. The number of values of \( a \) for which \( (a^2 - 3a + 2)x^2 + (a^2 - 5a + 6)x + a^2 - 4 = 0 \) is an identity in \( x \) if
(a) 0
(b) 2
(c) 1
(d) 3
Answer: (c) 1

Question. The number of values of the pair \( (a, b) \) for which \( a(x + 1)^2 + b(x^2 - 3x - 2) + x + 1 = 0 \) is an identity in \( x \) is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (a) 0

Question. The number of values of the triplet \( (a, b, c) \) for which \( a \cos 2x + b \sin^2 x + c = 0 \) is satisfied by all real \( x \) is
(a) 0
(b) 2
(c) 3
(d) infinite
Answer: (d) infinite

Question. The polynomial \( (ax^2 + bx + c)(ax^2 - dx - c) \), \( ac \neq 0 \), has
(a) four real zeros
(b) at least two real zeros
(c) at most two real zeros
(d) no real zeros
Answer: (b) at least two real zeros

Question. Let \( f(x) = ax^3 + 5x^2 - bx + 1 \). If \( f(x) \) when divided by \( 2x + 1 \) leaves 5 as remainder, and \( f'(x) \) is divisible by \( 3x - 1 \) then
(a) \( a = 26, b = 10 \)
(b) \( a = 24, b = 12 \)
(c) \( a = 26, b = 12 \)
(d) None of the options
Answer: (c) \( a = 26, b = 12 \)

Question. \( x^{3^n} + y^{3^n} \) is divisible by \( x + y \) if
(a) \( n \) is any integer \( \ge 0 \)
(b) \( n \) is an odd positive integer
(c) \( n \) is an even positive integer
(d) \( n \) is a rational number
Answer: (a) \( n \) is any integer \( \ge 0 \)

Question. If \( x, y \) are rational numbers such that \( x + y + (x - 2y)\sqrt{2} = 2x - y + (x - y - 1)\sqrt{6} \) then
(a) \( x \) and \( y \) cannot be determined
(b) \( x = 2, y = 1 \)
(c) \( x = 5, y = 1 \)
(d) None of the options
Answer: (b) \( x = 2, y = 1 \)

Question. The number of real solutions of the equation \( 2^{x/2} + (\sqrt{2} + 1)^x = (5 + 2\sqrt{2})^{x/2} \) is
(a) one
(b) two
(c) four
(d) infinite
Answer: (a) one

Question. The number of real solutions of the equation \( e^x = x \) is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (c) 0

Question. The sum of the real roots of the equation \( x^2 + |x| - 6 = 0 \) is
(a) 4
(b) 0
(c) -1
(d) None of the options
Answer: (b) 0

Question. The solutions of the equation \( 2x - 2[x] = 1 \), where \( [x] = \) the greatest integer less than or equal to \( x \), are
(a) \( x = n + \frac{1}{2}, n \in N \)
(b) \( x = n - \frac{1}{2}, n \in N \)
(c) \( x = n + \frac{1}{2}, n \in Z \)
(d) \( n < x < n + 1, n \in Z \)
Answer: (c) \( x = n + \frac{1}{2}, n \in Z \)

Question. The number of real solutions of the equation \( \sin (e^x) = 5^x + 5^{-x} \) is
(a) 0
(b) 1
(c) 2
(d) infinitely many
Answer: (a) 0

Question. The number of real solution of \( 1 + |e^x - 1| = e^x(e^x - 2) \) is
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (b) 1

Question. The equation \( 2 \sin^2 \frac{x}{2} \cdot \cos^2 x = x + \frac{1}{x}, 0 < x \le \frac{\pi}{2} \) has
(a) one real solution
(b) no real solution
(c) infinitely many real solutions
(d) None of the options
Answer: (b) no real solution

Question. If \( y \neq 0 \) then the number of values of the pair \( (x, y) \) such that \( x + y + \frac{x}{y} = \frac{1}{2} \) and \( (x + y) \frac{x}{y} = -\frac{1}{2} \), is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (b) 2

Question. The number of real solutions of the equation \( \log_{0.5} x = |x| \) is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (a) 1

Question. The equation \( \sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1} \) has
(a) no solution
(b) one solution
(c) two solutions
(d) more than two solutions
Answer: (a) no solution

Question. The number of solutions of the equation \( |x| = \cos x \) is
(a) one
(b) two
(c) three
(d) zero
Answer: (b) two

Question. The product of all the solutions of the equation \( (x - 2)^2 - 3 |x - 2| + 2 = 0 \) is
(a) 2
(b) -4
(c) 0
(d) None of the options
Answer: (c) 0

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 \( x \), the number of possible values of \( x \) is
(a) 34
(b) 32
(c) 33
(d) None of the options
Answer: (c) 33

Question. The solution set of \( (x)^2 + (x + 1)^2= 25 \), where \( (x) \) is the least integer greater than or equal to \( x \), is
(a) (2, 4)
(b) (-5, -4] \( \cup \) (2, 3]
(c) [-4, -3) \( \cup \) [3, 4)
(d) None of the options
Answer: (b) (-5, -4] \( \cup \) (2, 3]

Question. If \( 3^{x+1} = 6^{\log_2 3} \) then \( x \) is
(a) 3
(b) 2
(c) \( \log_3 2 \)
(d) \( \log_2 3 \)
Answer: (d) \( \log_2 3 \)

Question. If \( (\sqrt{2})^x + (\sqrt{3})^x = (\sqrt{13})^{x/2} \) then the number of values of \( x \) is
(a) 2
(b) 4
(c) 1
(d) None of the options
Answer: (c) 1

Question. The number of real solutions of the equation \( \frac{6 - x}{x^2 - 4} = 2 + \frac{x}{x + 2} \) is
(a) two
(b) one
(c) zero
(d) None of the options
Answer: (b) one

Question. The number of real solutions of \( \sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6} \) is
(a) one
(b) two
(c) three
(d) None of the options
Answer: (a) one

Question. If \( [x] = \) the greatest integer less than or equal to \( x \), and \( (x) = \) the least integer greatest than or equal to \( x \), and \( [x]^2 + (x)^2 > 25 \) then \( x \) belongs to
(a) [3, 4]
(b) (-∞, -4]
(c) [4, +∞)
(d) (-∞, -4] \( \cup \) [4, +∞)
Answer: (d) (-∞, -4] \( \cup \) [4, +∞)

Question. Let \( R = \) the set of real numbers, \( Z = \) the set of integers, \( N = \) the set of natural numbers. If \( S \) be the solution set of the equation \( (x)^2 + [x]^2 = (x - 1)^2 + [x + 1]^2 \), where \( (x) = \) the least integer greater than or equal to \( x \) and \( [x] = \) the greatest integer less than or equal to \( x \), then
(a) \( S = R \)
(b) \( S = R - Z \)
(c) \( S = R - N \)
(d) None of the options
Answer: (b) \( S = R - Z \)

Question. If \( [x]^2 = [x + 2] \), where \( [x] = \) the greatest integer less than or equal to \( x \), then \( x \) must be such that
(a) \( x = 2, -1 \)
(b) \( x \in [2, 3) \)
(c) \( x \in [-1, 0) \)
(d) None of the options
Answer: (d) None of the options

Question. The solution set of \( \frac{x + 1}{x} + |x + 1| = \frac{(x + 1)^2}{|x|} \) is
(a) \( \{x | x \ge 0\} \)
(b) \( \{x | x > 0\} \cup \{-1\} \)
(c) \( \{-1, 1\} \)
(d) \( \{x | x \ge 1 \text{ or } x \le -1\} \)
Answer: (b) \( \{x | x > 0\} \cup \{-1\} \)

Question. The number of solutions of \( |[x] - 2x| = 4 \), where \( [x] \) is the greatest integer \( \le x \), is
(a) 2
(b) 4
(c) 1
(d) infinite
Answer: (b) 4

Question. The set of real values of \( x \) satisfying \( |x - 1| \le 3 \) and \( |x - 1| \ge 1 \) is
(a) [2, 4]
(b) (-∞, 2] \( \cup \) [4, +∞)
(c) [-2, 0] \( \cup \) [2, 4]
(d) None of the options
Answer: (c) [-2, 0] \( \cup \) [2, 4]

Question. The set of real values of \( x \) satisfying \( ||x - 1| - 1| \le 1 \) is
(a) [-1, 3]
(b) [0, 2]
(c) [-1, 1]
(d) None of the options
Answer: (a) [-1, 3]

Question. If \( x \in Z \) (the set of integers) such that \( x^2 - 3x < 4 \) then the number of possible values of \( x \) is
(a) 3
(b) 4
(c) 6
(d) None of the options
Answer: (b) 4

Question. If \( x \) is an integer satisfying \( x^2 - 6x + 5 \le 0 \) and \( x^2 - 2x > 0 \) then the number of possible values of \( x \) is
(a) 3
(b) 4
(c) 2
(d) infinite
Answer: (a) 3