IAT 2021

Mathematics Questions

JEE MATH APEX

What is the maximum value of $\cos^4(x) + \sin^4(x) + \cos(x)$ , when $x \geq 0$ ?

(A) 2

(B) 1

(D) $2\sqrt{2}$

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JEE MATH APEX

Out of a pack of ten cards numbered 1 to 10, a boy draws a card at random and keeps it back. Then a girl draws a card at random from the same pack. If the boy’s card reads $m$ , and the girl’s card reads $n$ , then what is the probability that $m > n$ , given that $m$ is even?

(A) $\frac{1}{2}$

(B) $\frac{1}{3}$

(D) $\frac{1}{5}$

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JEE MATH APEX

Suppose $a, b \in \mathbb{R}$ are such that the points $(a, b)$ , $(a^2, b^3)$ , and $(a^3, b^3)$ in the coordinate plane are distinct, collinear, and the line passing through these points is not parallel to the $y$ -axis. Then, for all such choices of $(a, b)$ , the slope of the line passing through these three points can take

(A) exactly two values.

(B) infinitely many values.

(D) exactly one value.

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JEE MATH APEX

Define $f : [2, 22] \rightarrow \mathbb{R}$ by $f(x) = \max\{n(1 - |x - (2n + 1)|) : n = 1, 2, \ldots, 10\}$ . The area of the region $\{(x, y) : 0 \leq y \leq f(x), x \in [2, 22]\}$ is

(A) 55.

(B) 60.

(D) 10.

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JEE MATH APEX

The value of the limit
$\lim_{x \to 0^+} (\sin x)^{\sqrt{(e^x + x)^{\frac{1}{x}}}}$ is

(A) $e^2$ .

(B) 1.

(D) 0.

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JEE MATH APEX

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying
$f(x) = e^{x^2/2} + \int_0^x t f(t) \, dt \text{ for all } x$ .
Then which of the following is correct?

(A) $5 < f(\sqrt{2}) < 6$

(B) $2 < f(\sqrt{2}) < 3$

(D) $4 < f(\sqrt{2}) < 5$

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JEE MATH APEX

For each $a \in \mathbb{R}$ , define
$p_a(z) = z^2 + 2e^a z + e^{2a} - e^{-a}$ .
Then, which one of the followings is true?

(A) $p_a$ has only non-real complex roots for all $a \in \mathbb{R}$ .

(B) $p_a$ has a real root for all $a \in \mathbb{R}$ .

(D) $p_a$ has a real root if and only if $a \leq -1$ .

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JEE MATH APEX

The number of functions $f : \{1, 2, 3, 4, 5\} \rightarrow \{1, 2, 3, 4, 5\}$ such that $f(f(n)) = n$ for all $n \in \{1, 2, 3, 4, 5\}$ , is

(A) 41.

(B) 25.

(D) 120.

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JEE MATH APEX

Let $P_1 : x + y + z = 1$ , $P_2 : 2x + y + z = 3$ be two planes, and let $L$ denote the line of intersection of $P_1$ and $P_2$ . Let $P$ be the plane passing through the point (1, 2, 1), and normal to $L$ . Which of the following equations represents $P$ ?

(A) $y - z = 1$

(B) $x + z = 2$

(D) $x + y + 2z = 5$

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JEE MATH APEX

Let $f(x) = \ln(1 + x)$ for $x \geq 0$ . The value of
$\int_0^{\frac{\pi}{2}} \frac{f(\sqrt{3} \cos \theta)}{f(\sqrt{3} \sin \theta) + f(\sqrt{3} \cos \theta)} \, d\theta$
is

(A) $\frac{\pi}{4}$

(B) $\frac{\pi}{6}$

(D) $\frac{\pi}{2}$

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JEE MATH APEX

The area enclosed by the curves $y = 1 + |\sin x|$ , $y = -|\sin x|$ and the lines $x = 0$ , $x = 2\pi$ is

(A) $8 + 2\pi$.

(B) $8 + 4\pi$.

(D) $8 + 8\pi$.

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JEE MATH APEX

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by
$f(x) = x^2 |x|$ .
Then, which of the following statements is correct?

(A) $f'$ is differentiable but $f''$ is not differentiable.

(B) $f$ is continuous but not differentiable.

(D) $f''$ is differentiable.

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JEE MATH APEX

Let $A$ be a $2 \times 2$ matrix such that
$A^2 + A + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ .
Let $I$ denote the $2 \times 2$ identity matrix. Which of the following statements is correct?

(A) Both $A$ and $A + I$ are invertible.

(B) $A$ is invertible but $A + I$ may not be invertible.

(D) Neither $A + I$ nor $A$ may be invertible.

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JEE MATH APEX

If three real numbers $a$ , $b$ , $c$ are in arithmetic progression, the value of the determinant
$\begin{vmatrix} x^2 + 3 & x^2 + 4 & x^2 + 5 \\ x^2 + 4 & x^2 + 5 & x^2 + 6 \\ x^2 + a & x^2 + b & x^2 + c \end{vmatrix}$
is

(A) 0.

(B) 2a.

(D) $x^2 + 2b$ .

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JEE MATH APEX

Consider the two data sets
$S_1 = \{1, 2, 4, 8, 9, 11, 15, 20, 27, 29, 33\}$ ,
$S_2 = \{51, 52, 54, 58, 59, 61, 65, 70, 77, 79, 83\}$ .
Let $m_1$ , $m_2$ and $v_1$ , $v_2$ be the means and the variances of $S_1$ and $S_2$ , respectively. Then, which of the following relations is correct?