IAT (IISER) 2022

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

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

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

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 $\angle P R Q$ is:

Let $\omega$ be a complex root of the quadratic polynomial $x^2+x+1$. The value of

\(\left(\omega+\frac{1}{\omega}\right) \cdot\left(\omega^2+\frac{1}{\omega^2}\right) \cdots\left(\omega^{100}+\frac{1}{\omega^{100}}\right)\)
is

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

For a natural number $n$, let $C_n$ be the curve in the $X Y$-plane given by $y=x^n$, 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

The value of the integral
\(\int_1^{100} \frac{[x]}{x} d x\)
where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is

For arbitrary constants $\alpha, \beta$, the differential equation representing the family of curves $y=$ $(\alpha x+\beta) e^x$ is

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

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

A lab reports that the global average temperature in the year 2020 was $14.9^{\circ} \mathrm{C}$ and predicts that the global average temperature will increase at the rate of $1 \%$ per year. What will be the global average temperature in t...

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

Consider the vectors $\vec{a}=\hat{\imath}+x \hat{\jmath}+2 \hat{k}, \vec{b}=\hat{\imath}+2 \hat{\jmath}+x \hat{k}, \vec{c}=2 \hat{\imath}+\hat{\jmath}+3 \hat{k}$. The values of $x$ for Which there is at least one nonzero vector perpendicul...

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