List of top Mathematics Questions on Calculus asked in CUET (PG)

For the function \( f(x, y) = x^3 + y^3 - 3x - 12y + 12 \), which of the following are correct:
A. minima at (1,2)
B. maxima at (-1,-2)
C. neither a maxima nor a minima at (1,-2) and (-1,2)
D. the saddle points are (-1,2) and (1,-2)

The value of \( \lim_{n \to \infty} (\sqrt{4n^2+n} - 2n) \) is:

The surface area of the solid generated by revolving the curve \(x = e^t \cos t, y = e^t \sin t\) about y-axis for \(0 \le t \le \pi/2\) is

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; \lim_{x \to 0} \frac{\ln(1+x)}{\sin x} & (I) \; 1 \\ (B) \; \lim_{x \to \infty} 2x \tan\left(\tfrac{1}{x}\right) & (II) \; 0 \\ (C) \; \lim_{x \to \infty} \frac{x^2}{e^x} & (III) \; 2 \\ (D) \; \lim_{x \to 1} x^{\tfrac{1}{x-1}} & (IV) \; e \\ \hline \end{array} \]
Choose the correct answer from the options given below:

If \( S = \lim_{n \to \infty} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right) ... \left(1-\frac{1}{n^2}\right) \), then S is equal to:

Let f be a continuous function on \(\mathbb{R}\) and \( F(x) = \int_{x-2^{x+2} f(t) dt \), then F'(x) is}

Consider the function \( f(x, y) = x^2 + xy^2 + y^4 \), then which of the following statement is correct:

The value of \( \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x} dx \) is:

Let \( f(x) = |x| + |x-1| + |x+1| \) be a function defined on \( \mathbb{R} \), then \( f(x) \) is:

Let \(f(x)\) be a real valued function defined for all \(x \in \mathbb{R}\), such that \(|f(x)-f(y)| \le (x-y)^2\), for all \(x,y \in \mathbb{R}\), then