IAT (IISER) 2023

Let $f(x)=\sin (3 x), x \in\left[0, \frac{\pi}{2}\right]$. Which of the following statements is true

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

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

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

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

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

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

Which one of the following functions is differentiable at $x=0$ ?

Consider the objective function $Z=x-y$ subject to the constraints
$$ \begin{aligned} & x+2 y \leq 10 \\ & x+y \geq 2 \\ & x \geq 0, y \geq 0 \end{aligned} $$
What is the minimum value of $Z$ subject to the above constraints?

Let $M$ be a $3 \times 3$ matrix with real entries such that
$$ \left\{\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]: M\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\...

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

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 threee coins one by one, and the following e...

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

Suppose the mean and the standard deviation of the data $\left\{x_1, x_2, \cdots, x_9\right\}$ are $\mu$ and $\sigma(\neq 0)$, respectively. After including one more data value $x_{10}$, the mean of the data $\left\{x_1, x_2, \cdots, x_9, x...

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 $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma$ ?