IAT 2022

Mathematics Questions

JEE MATH APEX

A randomly chosen card from a deck of 52 cards is given to be a black card (i.e., Spade or Club). What is the probability that it is either a face card (i.e., King, Queen or Jack) or a Spade?

(A) 8/13

(B) 9/13

(D) 7/13

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JEE MATH APEX

Let $\omega$ be a complex root of the quadratic polynomial $x^{2} + x + 1$ . The value of
$\left( \omega + \frac{1}{\omega} \right) \left( \omega^{2} + \frac{1}{\omega^{2}} \right) \cdots \left( \omega^{100} + \frac{1}{\omega^{100}} \right)$
is

(A) $-2^{33}$

(B) $-2^{31}$

(D) $2^{31}$

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JEE MATH APEX

Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ be a polynomial. Suppose that
$f(0) = 0, \left. \frac{df}{dx} \right|_{x=0} = 1, \left. \frac{d^2 f}{dx^2} \right|_{x=0} = 4 \text{ and } \frac{d^3 f}{dx^3} = \frac{d^5 f}{dx^5}$ .
Then $f(5) =$

(A) 55

(B) 25

(D) 105

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JEE MATH APEX

Let $S$ be the set of all unit vectors in the XY-plane. Then the set $S$ has

(A) Infinitely many elements

(B) 2 elements

(D) 8 elements

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JEE MATH APEX

A lab reports that the global average temperature in the year 2020 was 14.9°C and predicts that the global average temperature will increase at the rate of 1% per year. What will be the global average temperature in the year 2035?

(A) 17.298°C

(B) 15.049°C

(D) 17.471°C

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JEE MATH APEX

Let $X$ be the set of all $2 \times 2$ matrices with real entries and $R \subseteq X \times X$ be the relation
$R = \{(A, B) : AB = BA\}$ .
Which of the following statements is true?

(A) $R$ is reflexive and symmetric but not transitive

(B) $R$ is reflexive and transitive but not symmetric

(D) $R$ is an equivalence relation

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JEE MATH APEX

For a natural number $n$ , let $C_n$ be the curve in the XY-plane given by $y = x^{2n}$ , where $0 \leq x \leq 1$ . Let $A_n$ denote the area of the region bounded between $C_n$ and $C_{n+1}$ . Then the largest value of $A_n$ is

(A) $\frac{1}{6}$

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

(D) $\frac{1}{12}$

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JEE MATH APEX

Let $f$ be a continuous function on $[0, 1]$ and $F$ be its antiderivative. If $F(0) = 1$ and $\int_0^1 f(x) \, dx = 1$ , then $F(1)$ is

(A) 2

(B) 0

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

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JEE MATH APEX

Let $a$ be a nonzero real number and $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f'(x) > 0$ for all $x \in \mathbb{R}$ . Consider $g(x) = f(2a^2 x - ax^2)$ . Then $g$ has

(A) Local maxima at $x = a$ if $a > 0$

(B) Local maxima at $x = a$ if $a < 0$

(D) A point of inflection at $x = a$

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JEE MATH APEX

Let $A$ be the matrix
$\begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 1 & 1 & 1 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$ .
For any natural number $k$ , the determinant of $A^k$ is

(A) 1

(B) -1

(D) 0

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JEE MATH APEX

Consider the vectors
$\vec{a} = \hat{i} + x\hat{j} + 2\hat{k}$ ,
$\vec{b} = \hat{i} + 2\hat{j} + x\hat{k}$ ,
$\vec{c} = 2\hat{i} + \hat{j} + 3\hat{k}$ .
The values of $x$ for which there is at least one nonzero vector perpendicular to the vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ are

(A) 0, 2

(B) -2, 2

(D) 4, -2

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JEE MATH APEX

Consider the tangent lines to the circle $x^2 + y^2 = 1$ at points $P = (1, 0)$ and $Q = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ . If $R$ is the point of intersection of these two tangent lines, then $\angle PRQ$ is:

(A) $\frac{3\pi}{4}$

(B) $\pi$

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

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JEE MATH APEX

The function given by $f(x) = 2x^3 - 15x^2 + 36x - 5$ is

(A) Increasing on the interval (0, 2)

(B) Decreasing on the interval (-3, 0)

(D) Decreasing on the interval (3, $\infty$ )

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JEE MATH APEX

The value of the integral
$\int_1^{100} \frac{\lfloor x \rfloor}{x} \, dx$ ,
where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ for any real number $x$ , is

(A) $\log \left( \frac{100^{99}}{99!} \right)$