JEE MATH APEX

IAT 2017

Mathematics Questions

JEE MATH APEX
The number of solutions of the equation 2 \sin^2 x + 1 = 3 \sin x in the interval (0, \pi) is
(A) 0
(B) 2
(C) 1
(D) 3
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JEE MATH APEX
For n \geq 2, the number of onto functions from the set \{1, 2, \ldots, n\} to the set \{1, 2\} is
(A) 2^n
(B) 2^n - 2
(C) n!
(D) 2^n - 1
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JEE MATH APEX
Let f : \mathbb{R} \to \mathbb{R} be a differentiable function such that f'(0) = 1 and f(x + y) = f(x) f(y) for all x, y \in \mathbb{R}. Which of the following is true?
(A) f is a decreasing function but f' is an increasing function
(B) Both f and f' are increasing functions
(C) f is an increasing function but f' is a decreasing function
(D) Both f and f' are decreasing functions
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JEE MATH APEX
The number of points of intersection of the curves x^2 + 8y^2 = 4 and x^2 + y^2 = 1 is
(A) 0
(B) 4
(C) 1
(D) 2
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JEE MATH APEX
The coefficient of x^9 in \left(x^2 - \frac{1}{3x}\right)^9 is
(A) -3
(B) -\frac{56}{9}
(C) -\frac{28}{9}
(D) \frac{28}{9}
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JEE MATH APEX
The roots of the polynomial x^3 - 39x^2 + 471x - 1729 are in an arithmetic progression. Which of the following is a common difference of the progression?
(A) 6
(B) 19
(C) 13
(D) 7
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JEE MATH APEX
Let f : \mathbb{R} \to \mathbb{R} be defined as
f(x) = \begin{cases} \frac{e^x - e^{-x} - e^{-2x} + 1}{x^2} & \text{if } x > 0, \\ a & \text{if } x = 0, \\ \frac{1 - \cos(2x)}{x^2} & \text{if } x < 0. \end{cases}
The value of a for which f is continuous at 0 is
(A) 1
(B) 2
(C) 0
(D) 3
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JEE MATH APEX
Let S be a set with 3 elements. What is the probability of choosing an ordered pair (A, B) of subsets of S such that A and B are disjoint?
(A) \frac{1}{2}
(B) \frac{26}{64}
(C) \frac{27}{64}
(D) \frac{1}{8}
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JEE MATH APEX
Consider the functions y_1 and y_2 satisfying
\frac{dy_1}{dx} = -y_2, \quad \frac{dy_2}{dx} = y_1, \quad y_1(0) = 1, \quad y_2(0) = 0.
The set S = \{(y_1(x), y_2(x)) : x \in \mathbb{R}\} lies on a
(A) hyperbola
(B) straight line
(C) circle
(D) parabola
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JEE MATH APEX
A function f satisfying f \circ f(x) = x for all x \in \mathbb{R} is
(A) onto but not one-one
(B) neither one-one nor onto
(C) one-one and onto
(D) one-one but not onto
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JEE MATH APEX
The function f(x) = \sin x + \frac{1}{\sin x} in the interval (0, \pi) has a
(A) local minima at \pi/2
(B) local maxima at \pi/3
(C) local minima at \pi/6
(D) local maxima at \pi/4
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JEE MATH APEX
Let p be a polynomial with real coefficients such that \int_0^1 p(t) \, dt = 0. Which of the following statements is always true?
(A) p has no roots in the interval [0, 1]
(B) All roots of p lie in the interval [0, 1]
(C) p has exactly one root in the interval [0, 1]
(D) p has a root in the interval [0, 1]
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JEE MATH APEX
For a complex number z, let (1 + z)^{15} = a_0 + a_1 z + \cdots + a_{15} z^{15}. The value of
(a_0 - 4a_2 + 16a_4 + \cdots - 2^{14} a_{14})^2 + (2a_1 - 8a_3 + 32a_5 \cdots - 2^{15} a_{15})^2
is
(A) 2^{30}
(B) 2^{15}
(C) 1
(D) 5^{15}
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JEE MATH APEX
Let A be a 3 \times 3 matrix which has determinant 3 and satisfies the equation A^2 - 7A + 4I = 0. The value of \det(A - 2I) is
(A) 5
(B) 1
(C) 3
(D) 9
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JEE MATH APEX
Consider the functions
g(x) = \begin{cases} 1 & \text{if } x \in [-1, 1], \\ 0 & \text{otherwise}, \end{cases}
and
f(x) = \lim_{h \to 0} \frac{1}{h} \int_{x-h}^{x+h} g(y) \, dy.
The value of f(1) is
(A) 1/2
(B) -1
(C) 1
(D) 0
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