List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

For n ∈ \(\N\), if
\(a_n=\frac{1}{n^3+1}+\frac{2^2}{n^3+2}+...+\frac{n^2}{n^3+n}\)
then the sequence \(\left\{a_n\right\}_{n=1}^{\infin}\) converges to ____________ (rounded off to two decimal places)

For a ∈ (−2π, 0), consider the differential equation
\(x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+y=0\) for x > 0.
Let D be the set of all a ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0⁺. Then, the number of elements in D ∩ \(\Z\) equals ___________

Define the function \(f:(-1,1)\rightarrow (-\frac{\pi}{2},\frac{\pi}{2})\) by
\(f(x)=\sin^{-1}x\)
Let a₆ denote the coefficient of x⁶ in the Taylor series of (f(x))² about x = 0. Then, the value of 9a₆ equals ____________ (rounded off to two decimal places).

Consider the following two statements.
P : There exist functions f: ℝ → ℝ, g: ℝ → ℝ such that f is continuous at x = 1 and g is discontinuous at x = 1 but g ∘ f is continuous at x = 1.
Q : There exist functions f: ℝ → ℝ, g: ℝ → ℝ such that both f and g are discontinuous at x = 1 but g ∘ f is continuous at x = 1.
Which one of the following holds ?

Define the function \(f: \R^2 → \R\) by
f(x, y) = 12xy e^-(2x+3y-2).
If (a, b) is the point of local maximum of f, then f(a, b) equals

The area of the region
\(R=\left\{(x,y) \in \R^2\ : 0\le x \le1,0 \le y \le 1\ \text{and}\ \frac{1}{4} \le xy \le \frac{1}{2} \right\}\)
is __________ (rounded off to two decimal places).

Define the sequences \(\left\{a_n\right\}^{\infin}_{n=3}\) and \(\left\{b_n\right\}^{\infin}_{n=3}\) as
\(a_n=(\log n+\log\ \log n)^{\log n}\) and \(b_n=n^{(1+\frac{1}{\log n})}\).
Which one of the following is TRUE ?

For 0 < a < 4, define the sequence \(\left\{x_n\right\}^{\infin}_{n=1}\) of real numbers as follows :
x₁ = a and x_n+1 + 2 = −x_n(x_n - 4) for n ∈ \(\N\).
Which of the following is/are TRUE ?

Consider
\(𝐺 = \left\{𝑚 + 𝑛\sqrt2\ ∶\ 𝑚, 𝑛 ∈ \Z\right\}\)
as a subgroup of the additive group ℝ.
Which of the following statements is/are TRUE ?

Let f: ℝ → ℝ be defined by
\(f(x)=\frac{(x^2+1)^2}{x^4+x^2+1}\) for x ∈ \(\R\).
Then, which one of the following is TRUE ?

For the differential equation
y(8x − 9y)dx + 2x(x − 3y)dy = 0 ,
which one of the following statements is TRUE ?

For a > 0, let y_a(x) be the solution to the differential equation
\(2\frac{d^2y}{dx^2}-\frac{dy}{dx}-y=0\)
satisfying the conditions
y(0) = 1, y'(0) = a.
Then, the smallest value of a for which ya(x) has no critical points in ℝ equals ___________ (rounded off to the nearest integer).

Let P₇(x) be the real vector space of polynomials in x with degree at most 7, together with the zero polynomial. For r = 1, 2, … , 7, define
s_r(x) = x(x - 1) ⋯ (x − (r - 1)) and s₀(x) = 1.
Consider the fact that B = {s₀(x), s₁(x), … , s₇(x)} is a basis of P₇(𝑥).
If
\(x^5=\sum\limits_{k=0}^7a_{5,k}s_k(x),\)
where a_5,k ∈ \(\R\), then a_5,2 equals ___________ (rounded off to two decimal places)

Let
\(M=\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 2 & 0 & 0 & 0 & -4 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 & 2 \end{pmatrix}\)
If p(x) is the characteristic polynomial of M, then p(2) - 1 equals _________

The center Z(G) of a group G is defined as
z(G) = {x ∈ G ∶ xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G ?

For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M10(\(\R\)) satisfying A = BC ?

Define f: ℝ → ℝ and g: ℝ → ℝ as follows
\(f(x)=\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m!)^2}\) and \(g(x)=\frac{x}{2}\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m+1)!m!}\) for x ∈ \(\R\).
Let x₁, x₂, x₃, x₄ ∈ ℝ be such that 0 < x₁ < x₂ , 0 < x₃ < x₄,
f(x₁) = f(x₂) = 0,    f(x) ≠ 0 when x₁ < x < x₂,
g(x₃) = g(x₄) = 0 and g(x) ≠ 0 when x₃ < x < x₄.
Then, which of the following statements is/are TRUE ?

Let f: (1, ∞) → (0, ∞) be a continuous function such that for every n ∈ \(\N\), f(n) is the smallest prime factor of n. Then, which of the following options is/are CORRECT ?

Which of the following statements is/are TRUE ?

Which of the following statements is/are TRUE ?

Let
\(𝑆 = x\left\{(𝑥, 𝑦) ∈ ℝ^2 : 𝑥 > 0, 𝑦 > 0\right\} ,\)
and f: S → ℝ be given by
\(f(x,y)=2x^2+3y^2-\log x-\frac{1}{6}\log y.\)
Then, which of the following statements is/are TRUE ?

Let \(\left\{a_n\right\}^{\infin}_{n=1}\) be a sequence of real numbers.
Then, which of the following statements is/are always TRUE ?

Consider the group G = {A ∈ M₂ (ℝ): AA^T = I₂} with respect to matrix multiplication. Let
Z(G) = {A ∈ G : AB = BA, for all B ∈ G}.
Then, the cardinality of Z(G) is

The number of permutations in S₄ that have exactly two cycles in their cycle decompositions is equal to ___________.

How many group homomorphisms are there from \(\Z_2\) to S₅ ?