CUET UG 2024 — Calculus

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$ (A) $f^{\prime}(x)=\cos x-\sin 2 x$ (B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$ (C) The minimum value of the function is $2$ (D) The maximum value of the function is $\frac{3}{4}$ Choose the correct answer from the options given below :

Choose the **correct** answer from the options given below :

Choose the correct answer from the options given below: | List-I | List-II | | --- | --- | | (A) Integrating factor of $ x \, dy - (y + 2x^2) \, dx = 0 $ | (I) $ \frac{1}{x} $ | | (B) Integrating factor of $ (2x^2 - 3y) \, dx = x \, dy $ | (II) $ x $ | | (C) Integrating factor of $ (2y + 3x^2) \, dx + x \, dy = 0 $ | (III) $ x^2 $ | | (D) Integrating factor of $ 2x \, dy + (3x^3 + 2y) \, dx = 0 $ | (IV) $ x^3 $ |

The value of lim (x→0) (sin x)/x is

For the differential equation $\left(x \log _{e} x\right) d y=\left(\log _{e} x-y\right) d x$ (A) Degree of the given differential equation is $1$. (B) It is a homogeneous differential equation. (C) Solution is $2y \log _{\mathrm{e}} \mathrm{x}+A=\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}$, where $A$ is an arbitrary constant (D) Solution is $2 y \log _{e} x+A=\log _{e}\left(\log _{e} x\right)$, where $A$ is an arbitrary constant Choose the correct answer from the options given below :

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$, $x \in [0, 3]$, match List-I with List-II : | List-I | List-II | | --- | --- | | A. Absolute maximum value | (I) $ 3 $ | | B. Absolute minimum value | (II) $ 0 $ | | C. Point of maxima | (III) $ -5 $ | | D. Point of minima | (IV) $ 4 $ |

If a function $f(x)=x^{2}+b x+1$ is increasing in the interval $[1,2]$, then the least value of $b$ is :

If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

If the lengths of the three sides of a trapezium other than the base are $10 \mathrm{~cm}$ each, then the maximum area of the trapezium is:

If $f(x)=2\left(\tan ^{-1}\left(e^{x}\right)-\frac{\pi}{4}\right)$, then $f(x)$ is :

If $\sin y=x \sin (a+y)$, then $\frac{d y}{d x}$ is :

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$, then which of the following is true ?

Let $[x]$ denote the greatest integer function. Then match List-I with List-II: | List-I | List-II | | --- | --- | | (A) $ \vert x - 1\vert + \vert x - 2\vert $ | (I) is differentiable everywhere except at $ x = 0 $ | | (B) $ x - \vert x\vert $ | (II) is continuous everywhere | | (C) $ x - [x] $ | (III) is not differentiable at $ x = 1 $ | | (D) $ x \, \vert x\vert $ | (IV) is differentiable at $ x = 1 $ |

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

$\int \mathrm{e}^{\mathrm{x}}\left(\frac{2 \mathrm{x}+1}{2 \sqrt{\mathrm{x}}}\right) \mathrm{dx}$

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

The area of the region bounded by the lines $\frac{x}{7 \sqrt{3} a}+\frac{y}{b}=4, x=0$ and $y=0$ is :

The area of the region bounded by the lines $x+2 y=12, x=2, x=6$ and $x$-axis is :

The area of the region enclosed between the curves $4 x^{2}=y$ and $y=4$ is :

The degree of the differential equation $\left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}=k \frac{d^{2} y}{d x^{2}}$ is :

The equation of the tangent to the curve $\mathrm{x}^{\frac{5}{2}}+\mathrm{y}^{\frac{5}{2}}=33$ at the point $(1,4)$ is :

The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :

The second order derivative of which of the following functions is $5^{\mathrm{x}}$ ?

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :

The value of the integral $\int_{\log _{e} 2}^{\log _{e} 3} \frac{e^{2 x}-1}{e^{2 x}+1} d x$ is :