IAT 2020

Mathematics Questions

JEE MATH APEX

If $S$ is the set of all points $(x, y)$ satisfying the equation $\sin x = \cos y$ , then which of the following statements is correct?

(A) $S$ contains all points of a parabola.

(B) $S$ contains all points of a circle.

(D) $S$ contains all points of a straight line.

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JEE MATH APEX

If $A = \begin{bmatrix} 1 & a & 0 \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}$ , then the determinant of $I - A + A^2 - A^3 + A^4 - \cdots + A^{2020}$ is

(A) 2020

(B) $a^{2020} - b a^{2019} + \cdots - b^{2019} a + b^{2020}$

(D) 1

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JEE MATH APEX

Let $S = \{(a, b) : a, b \in \mathbb{Q}\}$ and $\ell$ be the line $y = mx + c$ . Which of the following statements is not correct?

(A) If $\ell$ passes through two points of $S$ , then $m$ must be rational.

(B) If $m$ is rational and $c$ is irrational, then $\ell$ passes through at least one point of $S$ .

(D) If $m$ is irrational and $c$ is rational, then $\ell$ passes through exactly one point of $S$ .

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JEE MATH APEX

If $p(t) = \frac{t(t-1)\cdots(t-2019)}{2019!}$ , then the value of $\int_0^1 \left(\frac{1}{t+1} + \frac{1}{t+2} + \cdots + \frac{1}{t+2020}\right) p(-t - 1) \, dt$ is:

(A) $2019^2$

(B) 2019

(D) 2020

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JEE MATH APEX

Let $a, b$ be nonzero real numbers. If $p(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_0$ , where $c_{n-1}, \ldots, c_0$ are integers and $n \geq 3$ , then which of the following statements is correct?

(A) If $a + ib$ is a root of $p(x)$ , then $n$ must be even.

(B) If $a$ is a root of $p(x)$ , then $n$ must be odd.

(D) If $a$ is a root of $p(x)$ , then $a$ must be an integer.

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JEE MATH APEX

The number of skew-symmetric matrices $A = [a_{ij}]_{3 \times 3}$ , where $a_{ij} \in \{-3, -2, -1, 0, 1, 2, 3\}$ is:

(A) $7^3$

(B) $3^7$

(D) $7^6$

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JEE MATH APEX

Define binary operation $*$ on $\mathbb{Q}$ by $a * b = a + b - 3$ . The inverse of 2 with respect to $*$ is:

(A) 4

(B) 2

(D) 5

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JEE MATH APEX

Let $[x]$ denote the greatest integer less than or equal to $x$ . The number of positive integer solutions of the equation $\left[\frac{x}{19}\right] = \left[\frac{x}{20}\right]$ are:

(A) 190

(B) 188

(D) $187 \sum_{n=0}^{\infty}$

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JEE MATH APEX

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two functions such that $g \circ f(x) = \begin{cases} x^2 & ; x \geq 0 \\ e^x - 1 & ; x < 0 \end{cases}$ .
Then which of the following statements is correct?

(A) $g$ is onto.

(B) $f$ is onto.

(D) $f$ is one-one.

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JEE MATH APEX

If $F(x) = \int_0^x (t^3 + 2t^2 - t - 2) \, dt$ , then for how many real numbers $x$ does $F'(x) = 0$ ?

(A) 0

(B) 3

(D) 1

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JEE MATH APEX

Let $C$ be a circle which passes through $(-2, 1)$ and $(0, 3)$ and whose centre lies on the line $y - x = 1$ . Then which of the following points lies on $C$ ?

(A) $(\sqrt{2} + 1, 1)$

(B) $(1, \sqrt{2} + 1)$

(D) $(1, \sqrt{3} - 1)$

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JEE MATH APEX

For what value of $t$ does $f(x) = x^3 + t x^2 + (2 - t) x - 3$ have only real roots?

(A) $t = 0$

(B) $t = 4$

(D) $t = -1$

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JEE MATH APEX

If $f(x) = ax^2 + bx + c$ and $f\left(\frac{1}{n}\right) = \frac{n+1}{n^2}$ for all $n \in \mathbb{N}$ , then what is the value of $\lim_{x \to 0} f(x)$ ?

(A) 2

(B) 0

(D) -1

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JEE MATH APEX

A bag contains 8 balls, of which 5 are green and 3 are red. The probability that two balls chosen at random are of the same colour is:

(A) 12/28

(B) 14/28

(D) 13/28

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JEE MATH APEX

Consider the differential equation $f'' + p f' + q f = 0$ where $p$ and $q$ are constants, $p^2 = 4q$ and $q \neq 0$ . If $y$ is a nonzero solution of the equation, then which of the following statements is correct?