Calculus PYQ — Page 4

Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0, x=1, y^{2}=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}$. Then $(f(0)+f(1))$ is equal to

Let $f$ be a twice differentiable function such that $f(x)=\int_{0}^{x}\tan(t-x)dt-\int_{0}^{x}f(t)\tan t\,dt$, $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$. Then $f''\left(\dfrac{\pi}{6}\right)+12f'\left(-\dfrac{\pi}{6}\right)+f\left(\dfrac{\pi}{6}\right)$ is equal to ______

Let $f(x)=x^{3}+x^{2} f^{\prime}(1)+2 x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then the value of $f^{\prime}(5)$ is:

Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0$, $x \in (0, 1)$, $\displaystyle\lim_{x\to 1^-} y(x) = 1$. Then $y\left(\dfrac{1}{2}\right)$ equals:

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \dfrac{dx}{(e^{2x} - e^{-2x})}$ is :

If $\displaystyle\lim_{x \to 2} \dfrac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1)\log_e(x-1)} = m$, then $a + b + m$ is equal to :

Let $f(x)=[x]^{2}-[x+3]-3, x \in \mathbf{R}$, where [] is the greatest integer funtion. Then

If $\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx$, then $\alpha^2$ is equal to _______.

The value of the integral $\int\limits_{-1}^{1} \left(\dfrac{x^3 + |x| + 1}{x^2 + 2|x| + 1}\right) dx$ is equal to :

Let the solution curve of the differential equation $x d y-y d x=\sqrt{x^{2}+y^{2}} d x, x>0$, $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to

The value of $\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{r} x|\sin \pi x| d x\right)}\right|\right)$ is $\_\_\_\_$

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbf{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^{2}\right) \mathrm{d} y+\left(y-\tan ^{-1} x\right) d x=0, y(0)=1$. Then the value of $y(1)$ is :

If the area of the region $\left\{(x, y): 1-2 x \leqslant y \leqslant 4-x^{2}, x \geqslant 0, y \geqslant 0\right\}$ is $\frac{\alpha}{\beta}, \alpha, \beta \in \mathbf{N}, \operatorname{gcd}(\alpha, \beta)=1$, then the value of $(\alpha+\beta)$ is :

The integral $\int_{0}^{1}\cot^{-1}(1+x+x^2)dx$ is equal to:

Let the line $x=-1$ divide the area of the region $\left\{(x, y): 1+x^{2} \leq y \leq 3-x\right\}$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$. Then $m+n$ is equal to

The area of the region bounded by the curves $x+3y^2=0$ and $x+4y^2=1$ is equal to:

For the function $f(x) = e^{\sin|x|} - |x|$, $x \in \mathbb{R}$, consider the following statements: Statement I: $f$ is differentiable for all $x \in \mathbb{R}$. Statement II: $f$ is increasing in $\left(-\pi, -\dfrac{\pi}{2}\right)$. In the light of the above statements, choose the correct answer from the options given below:

Let $f(x) = \lim_{y \to 0} \dfrac{(1 - \cos(xy)) \tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x) = \sin x$, $x \in \mathbf{R}$ is :

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x) f''(x)$, for all $x \in \mathbb{R}$. If $f(0) = 0$, then $36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)$ is equal to:

Let $f(x)=\begin{cases} e^{x-1}, & x<0 \\ x^2-5x+6, & x \geq 0 \end{cases}$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha+\beta$ is equal to ______

Let $f(x)=x^{2025}-x^{2000}, x \in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

Let $\displaystyle\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta$, where $[\cdot]$ is the greatest integer function. Then $\beta \sin\left(\dfrac{\beta}{2}\right)$ equals: