IAT 2023

Mathematics Questions

JEE MATH APEX

Let $f : \mathbb{R} \rightarrow (0, \infty)$ be a continuous decreasing function. Suppose $f(0), f(1), \ldots, f(10)$ are in a geometric progression with common ratio $\frac{1}{5}$ . In which of the following intervals does the value of $\int_0^{10} f(x) \, dx$ lie?

(A) $(0, 2f(0))$

(B) $(4f(0), 6f(0))$

(D) $(12f(0), 14f(0))$

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JEE MATH APEX

Let $M$ be a $3 \times 3$ matrix with real entries such that
$\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} : M \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} x_2 \\ x_3 \\ x_1 + x_2 = 0 = x_2 + x_3 \end{pmatrix}$ .
What is the value of the determinant of $M$ ?

(A) 0

(B) 1

(D) 3

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JEE MATH APEX

What is the locus of the center of circles passing through the origin $(0, 0)$ with fixed radius?

(A) Circle

(B) Hyperbola

(D) Line

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JEE MATH APEX

Let $\alpha$ be a real number. What is the total number of distinct point(s) of intersection between the parabola $y = x^2 + 4x \sin \alpha + 6$ and the pair of lines $y^2 = 1$ ?

(A) Zero

(B) One

(D) Four

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JEE MATH APEX

Let $L$ be a straight line passing through the origin, and it makes angles $\alpha$ , $\beta$ and $\gamma$ with the positive $X$ , $Y$ and $Z$ -axes, respectively. What is the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ ?

(A) $-1$

(B) 3

(D) $-3$

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JEE MATH APEX

What is the total number of distinct divisors of $2^9 \times 3^{19}$ ?

(A) 200

(B) 30

(D) 100

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JEE MATH APEX

Suppose the mean and the standard deviation of the data $\{x_1, x_2, \ldots, x_9\}$ are $\mu$ and $\sigma (\neq 0)$ , respectively. After including one more data value $x_{10}$ , the mean of the data $\{x_1, x_2, \ldots, x_9, x_{10}\}$ remains $\mu$ . What is the standard deviation of the data $\{x_1, x_2, \ldots, x_9, x_{10}\}$ ?

(A) $\frac{3}{\sqrt{10}} \sigma$

(B) $\frac{\sqrt{10}}{3} \sigma$

(D) $\frac{9}{10} \sigma$

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JEE MATH APEX

Consider three biased coins. Let the probability of getting head be $\frac{1}{3}$ and the probability of getting tail be $\frac{2}{3}$ in each of the coins. Consider the experiment of tossing the three coins one by one, and the following events:
$E$ : “at least two heads show up”
$F$ : “first coin shows tail”.
What is the conditional probability of $E$ given that $F$ has already occurred?

(A) $\frac{1}{9}$

(B) $\frac{2}{9}$

(D) $\frac{4}{9}$

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JEE MATH APEX

Which of the following differential equations has $y = e^x$ as one of its particular solutions?

(A) $y \frac{d^2 y}{dx^2} - e^x \frac{dy}{dx} + y^2 = e^{2x}$

(B) $y \frac{d^2 y}{dx^2} + e^x \frac{dy}{dx} + y^2 = e^{2x}$

(D) $y \frac{d^2 y}{dx^2} + e^x \frac{dy}{dx} + y^2 = e^x$

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JEE MATH APEX

What is the area of the region $\{(x, y) : 0 \leq y \leq x e^{x^2}, 0 \leq x \leq 1\}$ ?

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

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

(D) $e - 2$

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JEE MATH APEX

Consider the objective function
$Z = x - y$
subject to the constraints:
$x + 2y \leq 10;$
$x + y \geq 2;$
$x \geq 0; y \geq 0$ .
What is the minimum value of $Z$ subject to the above constraints?

(A) -5

(B) -2

(D) -10

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JEE MATH APEX

Let $p(x) = x^2 + bx + c$ be a quadratic polynomial with real coefficients $b$ and $c$ . Suppose $p(1) = 5$ and $p(-1) = 3$ . What is the product of the roots of $p(x) = 0$ ?

(A) 3

(B) -1

(D) 1

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JEE MATH APEX

Let $f : (-1, 2) \rightarrow \mathbb{R}$ be a differentiable function such that
$f'(x) = \frac{2}{x^2 - 5}$ and $f(0) = 0$ .
Then in which of the following intervals does $f(1)$ lie?