If a tangent of slope 2 to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) touches the circle \(x^2 + y^2 = 4\), then the maximum value of ab is:
If \[ f(x) = \begin{cases} x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \] Which of the following is true?
If the equation of the tangent at (2, 3) on y2 = ax3 + b is y = 4x - 5, then the value of a2 + b2 is:
If Rolle's theorem is applicable for the function \(f(x) = x(x+3)e^{-x/2}\) on \([-3, 0]\), then the value of \(c\) is:
For all x ∈ [0, 2024] assume that f (x) is differentiable. f (0) = −2 and f ′(x) ≥ 5. Then the least possible value of f (2024) is:
Let f(x) = \(\int \frac{x}{(x^2+1)(x^2+3)} dx\). If f(3) = \(\frac{1}{4} \log(\frac{5}{6})\), then f(0) =
\(\int\frac{2\cos 2x}{(1+\sin 2x)(1+\cos 2x)}dx=\)
If \(\lim_{n\rightarrow\infty}[(1+\frac{1}{n^{2}})(1+\frac{4}{n^{2}})(1+\frac{9}{n^{2}})\cdots(1+\frac{n^{2}}{n^{2}})]^{\frac{1}{n}}=ae^{b}\), then a+b=