List of practice Questions

A periodic function f(x) = x² for -π < x < π is expanded in a Fourier series. Which of the following statement(s) is/are correct ?

The state of a harmonic oscillator is given as Ψ =\(\frac{1}{ √3}\) 𝜓₀ -\(\frac{1}{ √6}\) 𝜓₁+\(\frac{1}{ √2}\) 𝜓₂ , where 𝜓₀, 𝜓₁ and 𝜓₂ are the normalized wave functions of ground, first excited, and second excited states, respectively. Which of the following statement(s) is/are true?

Which of the following schematic plots correctly represent(s) a first order phase transition occurring at temperature 𝑇=𝑇_c? Here 𝑔, 𝑠, 𝑣 are specific Gibbs free energy, entropy and volume, respectively

A wave travelling along the x-axis with y representing its displacement is described by (𝑣 is the speed of the wave)

An objective lens with half angular aperture 𝛼 is illuminated with light of wavelength 𝜆. The refractive index of the medium between the sample and the objective is 𝑛. The lateral resolving power of the optical system can be increased by

Which of the following statement(s) is/are true for a LC circuit with 𝐿=25 mH and 𝐶=4 µF?

For a particle moving in a general central force field, which of the following statement(s) is/are true?

Let G be a finite group. Then G is necessarily a cyclic group if the order of G is

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

Consider the initial value problem
\(\frac{dy}{dx}+αy=0, \\ y(0)=1,\)
where α ∈ \(\R\). Then

Let p(x) = x⁵⁷ + 3x¹⁰ - 21x³ + x² + 21 and
\(q(x)=p(x)+\sum\limits_{j=1}^{57}p^{(j)}(x) \ \text{for all }x \in \R,\)
where p^(j)(x) denotes the j^th derivative of p(x). Then the function q admits

The limit
\(\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)\)
is

The value of
\(\int^1_0\int_0^{1-x}\cos(x^3+y^2)dy\ dx-\int^1_0\int_0^{1-x}\cos(x^3+y^2)dx\ dy\)
is

Let \(a_n=\left(1+\frac{1}{n}\right)^n\) and \(b_n=n\cos(\frac{n!\pi}{2^{10}})\) for n ∈ \(\N\). Then

Let (an) be a sequence of real numbers defined by
\(a_n=\begin{cases} 1 & \text{if } n \text{ is prime}\\    -1 & \text{if } n \text{ is not prime} \end{cases}\)
Let \(b_n=\frac{a_n}{n}\) for n ∈ \(\N\). Then

Let S and T be non-empty subsets of \(\R^2\), and W be a non-zero proper subspace of \(\R^2\). Consider the following statements :
I. If span(S) = \(\R^2\) , then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then

Consider the following statements :
I. Every infinite group has infinitely many subgroups.
II. There are only finitely many non-isomorphic groups of a given finite order.
Then

Let
\(f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.\)
Then \(\lim\limits_{n \rightarrow \infin}f(n,n^2)\) is

For each t ∈ (0, 1), the surface P_t in \(\R^3\) is defined by
\(P_t = \left\{(x, y, z) : (x^2 + y^2 )z = 1, t^2 ≤ x^2 + y^2 ≤ 1\right\}.\)
Let a_t ∈ R be the surface area of P_t. Then

Which of the following functions is/are Riemann integrable on [0, 1] ?

A subset S ⊆ \(\R^2\) is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of \(\R^2\) is/are bounded ?

Let f : \(\R^2 → \R\) be defined as follows :
\(f(x,y)=\begin{cases}    \frac{x^4y^3}{x^6+y^6} & \text{if }(x,y) \ne (0,0)\\     0  & \text{if } (x,y)=(0,0) \end{cases}\)
Then

Which of the following is/are true ?

Which of the following is/are linear transformations ?

Let R₁ and R₂ be the radii of convergence of the power series \(\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}\) and \(\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}\), respectively. Then