JEE MATH APEX

IAT 2018

Mathematics Questions

JEE MATH APEX
Let z be a given complex number with modulus |z| < 1. Then the set \{ \frac{z - w}{1 - \overline{z}w} : |w| = 1, w \in \mathbb{C} \} is a
(A) Straight line.
(B) Hyperbola.
(C) Circle.
(D) Parabola.
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JEE MATH APEX
Let f, g, h be functions from \mathbb{R} to \mathbb{R}, with f and g invertible. Which of the following is NOT always true?
(A) f \circ (g + h) = (f \circ g) + (f \circ h).
(B) f \circ (g \circ h) = (f \circ g) \circ h.
(C) (f \cdot g) \circ h = (f \circ h) \cdot (g \circ h).
(D) (f \circ g)^{-1} = g^{-1} \circ f^{-1}.
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JEE MATH APEX
Consider a matrix
\begin{bmatrix} 1 & 0 & 0 \\ 2 & x & y \\ 4 & 3 & 5 \end{bmatrix}
with integer entries and determinant -5. Then a possible value for y is
(A) 10.
(B) 6.
(C) 8.
(D) 1.
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JEE MATH APEX
The integral
\int_{-\pi/2}^{\pi/2} \left( \frac{(1 + \sqrt{|x|}) \sin^2(x) + \sin(x)}{1 + \sqrt{|x|}} \right) dx is equal to
(A) 0.
(B) \pi.
(C) \frac{\pi}{2}.
(D) 2\pi.
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JEE MATH APEX
Let \ell_0 be the line defined by the vector equation \hat{i} + 2\hat{j} + 3\hat{k} + \lambda (\hat{i} + \hat{j} + \hat{k}), with \lambda real. Which of the following vector equations, with \mu real, defines a line which intersects \ell_0?
(A) 2\hat{i} + 3\hat{j} + \mu (\hat{i} - \hat{j}).
(B) 3\hat{i} - 2\hat{j} + \hat{k} + \mu (-\hat{i} + \hat{j}).
(C) \hat{i} + 3\hat{j} + 5\hat{k} + \mu (2\hat{i} + 3\hat{j} + 4\hat{k}).
(D) -\hat{i} - 2\hat{j} - 3\hat{k} + \mu (-\hat{i} - \hat{j} - \hat{k}).
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JEE MATH APEX
For a given matrix, let R_i denote the sum of all entries in its i^{\text{th}} row and C_j denote the sum of all entries in its j^{\text{th}} column. How many 3 \times 3 matrices with nonnegative integer entries are there such that R_1 = R_2 = C_1 = C_2 = 2 and R_3 = C_3 = 1?
(A) 12.
(B) 11.
(C) 14.
(D) 13.
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JEE MATH APEX
Let P(n) be a statement for each natural number n. Assume that P(n+1) is a true statement whenever P(n) is a true statement. Suppose P(2018) is true. Then which one of the following statements is true?
(A) P(n) is true for exactly two values of n.
(B) P(n) is false for at most finitely many values of n.
(C) P(n) is false for infinitely many values of n.
(D) P(n) is true for all n.
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JEE MATH APEX
Let y = f(x) be the equation of the curve passing through the point (1,1) having slope \log_e x for positive values of x. Then the curve
(A) passes through the point (2,3 + \log_e 4).
(B) passes through the point (2, \log_e 4).
(C) passes through the point (2,3 - \log_e 4).
(D) does not pass through the point (2, -\log_e 4).
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JEE MATH APEX
How many functions f : \mathbb{R} \to \mathbb{R} satisfy f(1) = 10 and |f(x) - f(y)| = |x - y| for all x, y \in \mathbb{R}?
(A) 2.
(B) 3.
(C) 1.
(D) 4.
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JEE MATH APEX
For a real number a, let \tan^{-1}(a) denote the real number \theta, -\frac{\pi}{2} < \theta < \frac{\pi}{2} such that \tan(\theta) = a. The function f(x) = \tan^{-1}(b x^2 + 2 b x + c), where b and c are positive real numbers, is increasing on
(A) (-1,\infty).
(B) (-2,b).
(C) (-2,2).
(D) (-\infty,c).
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JEE MATH APEX
A number is picked uniformly randomly from the set of five digit natural numbers. What is the probability that at least one of the digits of the number thus picked is 0?
(A) 3987/10000
(B) 2601/10000
(C) 3439/10000
(D) 3095/10000
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JEE MATH APEX
How many functions f : \mathbb{N} \to \mathbb{N} satisfy lcm(f(n), n) - hcf(f(n), n) \leq 5? Here ‘lcm’ denotes the least common multiple and ‘hcf’ denotes the highest common factor.
(A) 0
(B) 1
(C) Infinitely many
(D) More than one but finitely many
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JEE MATH APEX
Let a, b be distinct positive real numbers, whose geometric mean equals \frac{a^{t - 99} + b^{t - 99}}{a^{t - 100} + b^{t - 100}}. Then t must equal
(A) \frac{199}{2}
(B) \frac{99}{2}
(C) 199
(D) 99
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JEE MATH APEX
Let f and g be two functions on \mathbb{R} defined by
f(x) = \sqrt{x^2 + 1} - x,
g(x) = \sin(\pi e^{1-x}).
Define a function h : \mathbb{R} \to \mathbb{R} by h(x) = \max\{f(x), g(x)\}. Then what can be said about \lim_{x \to \infty} h(x)?
(A) It does not exist.
(B) It is equal to 0.
(C) It is equal to -1.
(D) It is equal to 1.
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JEE MATH APEX
Let P denote a parabola in the plane and let a point A \in P be given. How many lines \ell in the plane satisfy \ell \cap P = \{A\}?
(A) 2
(B) 1
(C) Infinitely many
(D) 0
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