List of practice Questions

Let f : \(\R → \R\) be an infinitely differentiable function such that f" has exactly two distinct zeroes. Then

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

Let 𝑋₁,𝑋₂, … , 𝑋₅ be 𝑖. 𝑖. 𝑑. random variables, each having the 𝐵𝑖n (1, \(\frac{1}{2}\) ) distribution. Let 𝐾=𝑋₁+𝑋₂+ ⋯ + 𝑋₅ and
\(f(x) =\begin{cases}0,  & \quad \text{if }k=0,\\  x_1+x_2+...+x_k, & \quad if\,k=1,2,...,5. \end{cases}\)
Then, 𝐸(𝑈) equals ________

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

Let \(y : (\sqrt{\frac{2}{3}}, ∞) → \R\) be the solution of
(2x − y)y' + (2y − x) = 0,
y(1) = 3.
Then

Let 𝑋₁~Gamma(1,4), 𝑋₂~Gamma(2, 2) and 𝑋₃~Gamma(3, 4) be three independent random variables. If 𝑌=𝑋₁+2𝑋₂+𝑋₃, then 𝐸((\(\frac{y}{4}\))⁴ ) equals ______

Let 𝑋₁, 𝑋₂ be a random sample from a 𝑈(0, 𝜃) distribution, where 𝜃>0 is an unknown parameter. For testing the null hypothesis 𝐻₀ ∶ 𝜃∈(0,1]∪[2, ∞) against 𝐻₁: 𝜃∈(1, 2), consider the critical region
\(𝑅={(𝑥_1, 𝑥_2 )∈ℝ × ℝ∶\frac{5}{4}<max\,{𝑥_1, 𝑥_2 }<\frac{7}{4}}. \)
Then, the size of the critical region equals____.

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

Let f : \(\R → \R\) be a bijective function such that for all x ∈ R, f(x) = \(\sum\limits^{\infin}_{n=1}a_nx^n\) and \(f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n\), where f^-1 is the inverse function of f. If a₁ = 2 and a₂ = 4, then b₁ is equal to _________.

Let 𝑋₁,𝑋₂,𝑋₅ be a random sample from a 𝐵𝑖𝑛(1, 𝜃) distribution, where 𝜃∈(0,1) is an unknown parameter. For testing the null hypothesis 𝐻₀ ∶ 𝜃≤ 0.5 against 𝐻₁ :𝜃>0.5, consider the two tests 𝑇1 and 𝑇2 defined as:
𝑇₁: Reject 𝐻₀ if, and only if, \(∑^5_{i=1}\) 𝑋_𝑖=5. 
𝑇₂: Reject 𝐻₀ if, and only if, \(∑^5_{i=1}\) X_i≥3. 
Let 𝛽_𝑖 be the probability of making Type-II error, at 𝜃=\(\frac{2}{3}\), when the test 𝑇_𝑖 , 𝑖=1,2 , is used. Then, the value of 𝛽₁+𝛽₂ equals ________(round off to two decimal places)

Let \(y_c:\R \rightarrow(0,\infin)\) be the solution of the Bernoulli’s equation
\(\frac{dy}{dx}-y+y^3=0,\ \ \ \ \ \ \ y(0)=c \gt 0.\)
Then, for every 𝑐 > 0, which one of the following is true ?

For a twice continuously differentiable function 𝑔: ℝ → ℝ, define
\(u_g(x,y)=\frac{1}{y}\int^y_{-y}g(x+t)dt\ \ \ \text{for}(x,y)\in \R^2, \ \ \ y \gt0.\)
Which one of the following holds for all such 𝑔 ?

Let 𝑋₁~𝑁(2, 1), 𝑋₂~𝑁(−1,4) and 𝑋₃~𝑁(0, 1) be mutually independent random variables. Then, the probability that exactly two of these three random variables are less than 1, equals____(round off to two decimal places)

Let 𝑦(𝑥) be the solution of the differential equation
\(\frac{dy}{dx}=1+y\sec x\ \ \text{for}\ x \in(-\frac{\pi}{2},\frac{\pi}{2})\)
that satisfies 𝑦(0) = 0. Then, the value of \(y(\frac{\pi}{6})\) equals

Let F be the family of curves given by
x² + 2hxy + y² = 1, -1 < h < 1.
Then, the differential equation for the family of orthogonal trajectories to F is

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is 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

Which one of the following is TRUE for the symmetric group S₁₃ ?

Which of the following is a subspace of the real vector space \(\R^3\) ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

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

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?