List of top Mathematics Questions asked in CPET

For a function \(f: \mathbb{R}\to\mathbb{R}\), consider the following statements:
(I) If \(\lim_{h\to0}\{f(x+h)-f(x-h)\} = 0\), then f is continuous on \(\mathbb{R}\).
(II) If \(\lim_{h\to0}\dfrac{f(x+h)-f(x-h)}{2h} = 0\), then f is differentiable on \(\mathbb{R}\).
Choose the correct answer.

Which one of the following options is true for the function \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x) = \min\{|x|,\, x^2-1\}\), \(x\in\mathbb{R}\)?

If a function \(f:[0,1]\to\mathbb{R}\) is differentiable, \(f'(0) = -1\) and \(f'(1) = 5\), then which one of the following statements is true?

Consider the improper integrals \(I_\lambda = \displaystyle\int_0^1 \dfrac{dx}{(1-x)^\lambda}\) and \(J_\lambda = \displaystyle\int_1^\infty \dfrac{dx}{x^\lambda}\), \(\lambda \in \mathbb{R}\), and the statements:
(I) For \(\lambda = 1\), \(I_\lambda\) converges, but \(J_\lambda\) diverges.
(II) For \(\lambda = 2\), \(I_\lambda\) diverges, but \(J_\lambda\) converges.
Choose the correct option.

Consider the following statements:
(I) If \(a_n \ge 0\) for each \(n \in \mathbb{N}\) and the series \(\sum_{n=1}^\infty a_n\) converges, then the series \(\sum_{n=1}^\infty \dfrac{\sqrt{a_n}}{n^p}\) converges for \(p > 1/2\).
(II) The series \(\sum_{n=1}^\infty (-1)^n \sin(1/n)\) is conditionally convergent.
Which one of the options given below is correct?

Which one of the following options is always true?

Consider the sequences \(\{a_n\}\), \(\{b_n\}\), \(\{c_n\}\) defined for \(n \in \mathbb{N}\) by \[\text{(I) } a_n = n^2 \sin\left(\frac{1}{n}\right), \qquad \text{(II) } b_n = 1 + \frac{(-1)^n}{n}, \qquad \text{(III) } c_n = n\cos\left(\frac{1}{n}\right).\] Pick out the correct option.

Let the limits of the sequences \(\{x_n\}_{n\ge1}\) and \(\{y_n\}_{n\ge1}\) be \(\lambda\) and \(\lambda^3\) respectively. If the interleaved sequence \(x_1, y_1, x_2, y_2, x_3, y_3, \dots\) has a limit, then its value is ____.

Which of the following assertions is correct? \[\text{(I) } \lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}e^{-n} = 1 \qquad \text{(II) For } 0 < a < 1 < b,\ \lim_{n\to\infty}\left(a+\frac{b}{n}\right)^n = e^{b/a}\]

Which one of the following options is incorrect for the set \(S = \{x \in \mathbb{Q} : x^2 < 5\}\)?

Which one of the following statements is incorrect?

For a system \(AX = b\) of linear equations to have infinitely many solutions, which one of the following options must be true?

In a simple graph, which of the following statements is/are true?
(I) Adjacency matrix is symmetric.
(II) Trace of the adjacency matrix is 1.

What is the value of \(a_{32}\) from the recurrence relation: \(a_1 = 4,\ a_n = 5n + a_{n-1}\ (n \ge 2)\)?

Which one of the following statements is true for every square matrix with only real eigenvalues?

In how many ways can 12 distinct people be arranged around a circular table, if exactly two of them must always have at least one person between them?

Consider the following statements:
(I) The set \(\{\sin(1/x): x \in (0,1)\}\) is uncountable.
(II) The set \(\{(x,y) \in \mathbb{R}^2 : xy \in \mathbb{Z}\}\) is uncountable.
(III) The set of all \(2 \times 2\) real matrices with rational eigenvalues is uncountable.
Choose the correct answer:

For the propositions p and q, consider the following statements:
(I) \((p \vee q) \wedge (\neg p \vee \neg q)\) is a tautology.
(II) \((p \vee q) \wedge (\neg p \wedge \neg q)\) is a contradiction.
Choose the correct answer:

Which one of the following options is the general solution(s) of the linear congruences \(x \equiv 1 \pmod{3}\), \(x \equiv 2 \pmod{5}\) and \(x \equiv 3 \pmod{7}\)?

If a, b are integers and p is a prime number, then which of the following statements is/are false?
(I) If \((a,p) = (b,p) = 1\) and \(a^k \equiv b^k \pmod{p}\) (\(k \in \mathbb{N}\)), then \(a \equiv b \pmod{p}\).
(II) If \((a,p) = 1\) and \(ax \equiv 1 \pmod{p}\), then \(a \equiv a^{p-2} \pmod{p}\).

If \(\vec{a}, \vec{b}, \vec{c}\) are three vectors satisfying \(|\vec{a}|=1, |\vec{b}|=2, |\vec{c}|=1\) and \(\vec{a}\times(\vec{a}\times\vec{b}) + \vec{c} = 0\), then what is the angle between the vectors \(\vec{a}\) and \(\vec{b}\)?

Which one of the following equations is obtained from the equation \(xy=-8\) by rotating the axes counterclockwise through an angle of \(45^\circ\)? (Here \(X, Y\) denote the new coordinates.)

Which one of the following statements is incorrect?

What is the surface area of the solid obtained by revolving the curve \(y=\sqrt{9-x^2}\), \(-2\le x\le 2\), about the x-axis?

The value of \(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos^4(x)\,dx\) is ____.