IAT 2019

Mathematics Questions

JEE MATH APEX

Let $1, \zeta_2, \zeta_3, \ldots, \zeta_n$ be the roots of the equation $x^n = 1$ , for $n \geq 3$ . Then, the sum
$\frac{1}{2 - \zeta_2} + \frac{1}{2 - \zeta_3} + \cdots + \frac{1}{2 - \zeta_n}$
equals to

(A) $\frac{1 + (n - 2)2^n}{2^{n-1}}$ .

(B) $\frac{1 + 2n^{n-1} - 2^n}{2^{n-1}}$ .

(D) $\frac{1 + (n-1)2^{n-2} - 2^{n-1}}{2^{n-1} + 1}$ .

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JEE MATH APEX

How many solutions does the equation
$\sin^2 x - 15 \sin x \cos x + 50 \cos^2 x = 0$
have in the interval $[0, 2\pi]$ ?

(A) 4

(B) 0

(D) 2

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JEE MATH APEX

Let $A$ be a $4 \times 4$ matrix with real entries. Consider the sets
$K = \left\{ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} : A \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \right\}$ ,
$J = \left\{ \begin{pmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \end{pmatrix} : A \begin{pmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \end{pmatrix} = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ y_4 \end{pmatrix}, \text{ for some } y_1, y_2, y_3, y_4 \in \mathbb{R} \right\}$ .
Suppose $K = J$ . Then, which one of the following statements is necessarily true?

(A) $A^2 = 0$ .

(B) $A$ is symmetric.

(D) $A^2 = A$ .

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JEE MATH APEX

Consider 5 straight lines in a plane such that no two of them are parallel and no three of them intersect at a point. Then, the number of disjoint regions into which the plane is divided by these lines equals to

(A) 17.

(B) 18.

(D) 20.

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JEE MATH APEX

Consider the circle $C$ that passes through the points $(1, 0)$ and $(0, 1)$ having the smallest area. Then, the equation of the tangent to the circle $C$ at $(0, 1)$ is

(A) $y = -x + 1$ .

(B) $y = x - 1$ .

(D) $y = x + 1$ .

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JEE MATH APEX

Let $f : [-1, 1] \rightarrow \mathbb{R}$ be a continuous function. Consider the region
$S = \{(x, y) : -1 \leq x \leq 1 \text{ and } 0 \leq y \leq f(x)\}$ .
For which one of the following functions $f$ , the area of the region $S$ is the largest?

(A) $f(x) = \pi^x |\sin \pi x|$ .

(B) $f(x) = \pi^x |\cos \pi x|$ .

(D) $f(x) = \frac{\pi^x}{|x| + 1}$ .

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JEE MATH APEX

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then, $f$ is surjective if

(A) $\lim_{x\to\infty} f(x) = \infty$ and $\lim_{x\to -\infty} f(x) = \infty$ .

(B) $\lim_{x\to\infty} f(x) = 0$ and $\lim_{x\to -\infty} f(x) = \infty$ .

(D) $\lim_{x\to\infty} f(x) = -\infty$ and $\lim_{x\to -\infty} f(x) = \infty$ .

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JEE MATH APEX

The limit $\lim_{n\to\infty} \frac{1}{n^{2020}} \sum_{k=1}^n k^{2019}$

(A) is $\frac{1}{2018}$ .

(B) is $\frac{1}{2020}$ .

(D) does not exist.

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JEE MATH APEX

The sum of $n$ consecutive terms of an arithmetic progression consisting of integers is 161. Then, a possible value of $n$ is

(A) 5.

(B) 7.

(D) 8.

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JEE MATH APEX

Let $f : \mathbb{R} \to \mathbb{R}$ be a non-zero even function such that $\int_{-1}^{1} f(x) \, dx = \alpha$ . Then, the value of the integral $\int_{-1}^{1} \frac{f(x)}{1 + e^{x}} \, dx$ is

(A) $\alpha$ .

(B) $\alpha e^{-\alpha}$ .

(D) $\frac{e^{-\alpha}}{2}$ .

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JEE MATH APEX

What is the mean deviation about the mean for the following data?

$x_i$	1	2	3	4
$f_i$	5	10	15	20

(A) $\frac{4}{5}$ .

(B) $\frac{3}{5}$ .

(D) 1.

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JEE MATH APEX

Let $S$ be a non-empty set such that the total number of subsets of $S$ containing at most two elements is equal to 16. Then, the number of elements in $S$ equals to

(A) 5.

(B) 6.

(D) 7.

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JEE MATH APEX

Consider the parallelogram $ABCD$ , where $\frac{AE}{AB} = \frac{CF}{CD} = \frac{1}{n}$ , for some positive integer $n$ .

(A) $\frac{a}{n}$ .

(B) $\frac{na}{n+1}$ .

(D) $\frac{(n-1)a}{n}$ .

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JEE MATH APEX

Let $\hat{i}, \hat{j}$ and $\hat{k}$ denote the standard unit vectors in $\mathbb{R}^3$ along the $x$-axis, $y$-axis and $z$-axis, respectively. Consider the sets
$X = \{ a\hat{i} + b\hat{j} + c\hat{k} : a, b, c \in \{-1, 0, 1\} \}$ and
$Y = \{ (\vec{v_1}, \vec{v_2}, \vec{v_3}) : \vec{v_1}, \vec{v_2}, \vec{v_3} \in X \text{ and } \vec{v_1}, \vec{v_2}, \vec{v_3} \text{ are mutually perpendicular unit vectors} \}$ .
Then, the number of elements in $Y$ is