CUET UG 2022 — Calculus

A function $f(x)$ is defined by : $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$ Which of the following is true ?

An energy DRONE is flying along the curve $y = x^2 + 7$. A soldier is placed at $(3, 7)$. The nearest distance of the DRONE from soldier's position is

Based on above information answer the following question : $x$ and $y$ will satisfy :

Based on above information answer the following question : The area of the flower bed $(A(x))$ is given by :

Based on above information answer the following question : If area of the flower bed is maximum, then area (in $m^2$) of the garden, which is outside the flower bed is :

Based on above information answer the following question : The maximum area (in $m^2$) of the flower bed is :

Based on above information answer the following question : $\frac{dA(x)}{dx} =$

Choose the correct statements: A. The order and degree (if defined) of a differential equation are always positive integrals B. The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation C. If $\frac{dy}{dx} + P(x)y = Q(x)$ then Integrating factor $= e^{\int P(x)dx}$ D. The sum of order and degree of differential equation $1 + (y'')^5 = (y''')^3$ is $8$ E. If the solution of a differential equation of order $n$, contains $n$ arbitrary constant, then it is called a general solution Choose the correct answer from the options given below:

Consider the differential equation $\frac{dy}{dx} = \frac{y+1}{x+1}$, and $y=0$ when $x=2$. The value of $y$ at $x=3$ is :

Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct : (A) $f(x) = x|x|$ is differentiable at $x = 0$. (B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$. (C) $f(x) = x|x|$ has point of infection at $x = 0$. (D) $f(x) = x|x|$ is symmetrical about y-axis. Choose the correct answer from the options given below :

Consider the function $f(x) = x^{\frac{1}{x}}$. Its

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

Distance between Sumit and Amit in terms of $x$ is :

Evaluate $\int_0^{\frac{1}{3}} y \, dx$

For $a, b \in R$ and $a < b$, $\int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} \, dx =$

General solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} = 0$ is A. $\tan^{-1} x + \tan^{-1} y = C$ B. $\sin^{-1} x - \cos^{-1} y = C$ C. $x\sqrt{1 - y^2} - y\sqrt{1 - x^2} = C$ D. $\sin^{-1} x + \sin^{-1} y = C$ E. $\cos^{-1} x + \cos^{-1} y = C$ (where C is arbitrary constant) Choose the correct answer from the options given below

If $x = \int_0^y \frac{dt}{\sqrt{1+9t^2}}$ and $\frac{d^2y}{dx^2} = \lambda y$, then, $\lambda$ is equal to

If $f'(x) = 4x^5 - 6x$ and $f(0) = 3$, then $f(3)$ is equal to

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is ( where t is a parameter )

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

If curve represented by differential equation $x \frac{dy}{dx} + y = e^x$ passes through (1, 1), then $y(-1)$ is :

If D is the distance between Sumit and Amit, then the value of $x$ for which D is minimum, is :

If $f$ is a function defined by $f(x) = \begin{cases} 5x^2 - x + 3, & x < 1 \\ 3x + 4, & x \geq 1 \end{cases}$, then, at $x = 1$, $f$ is

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$. Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

If the order and the degree of the differential equation $\left(\frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{5}}$ are O and S respectively, then $S - O$ is equal to

If the solution curve of the differentiable equation $\frac{dy}{dx} + 2y = e^{3x}$, passes through the point $\left(0, \frac{6}{5}\right)$, then the value of $y(\log_e 2)$ is:

If $y = \frac{\log_e x}{x}$, then $\frac{d^2y}{dx^2} =$

If $x^y = e^{x-y}$, then $\frac{dy}{dx} =$

If $y = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$, then $(x^2-1)^{3/2} \frac{d^2y}{dx^2} =$

If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$, then $\phi(x)$ is equal to :

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

If $g(x) = \int \frac{dx}{x^{1/2} + x^{1/6}}$, then $g(1) - g(0)$ is :

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$, then $\int_{-1}^{1} f(x) \, dx =$

If $U(x) = x + \sqrt{1 + x^2}$, then solution of the differential equation $\frac{dy}{dx} + \sqrt{\frac{1 + y^2}{1 + x^2}} = 0$, is :

If $2x + y = 6$ then the maximum value of $x^2 y$ is :

If $\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\lambda}{\sqrt{2}}$, then the value of $\lambda$ is

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :

If $y = x^{\sin x}$, then value of $x^{-\sin x} \frac{dy}{dx} - \cos x \log x$ is :

If $\int(\sqrt{x+1} + \sqrt{x-1})^2 dx = \alpha x^2 + \beta x\sqrt{x^2-1} + \gamma \log|x+\sqrt{x^2-1}| + C$, then value of $\alpha + \beta - 2\gamma$ is :

If $\int (x + \sqrt{x^2 - 1})^2 \, dx = \alpha \cdot x + \beta x^3 + \gamma (x^2 - 1)^{\frac{3}{2}} + C$, where $C$ is arbitrary constant, then the value of $3(\alpha + \beta + \gamma)$ is

If x is real, then minimum value of $x^2 - 8x + 17$ is :

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

$\int \sqrt{1 - 49x^2} \, dx$ is equal to

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :

$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

$\int x\sqrt{x + 2} \, dx$ is equal to :

$y(x)$ is strictly increasing in the interval

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ : (A) $a = 1 + x^2$ (B) $b = 0$ (C) $c = 0$ (D) $b = x$ (E) $c = 2$ Choose the correct answer from the options given below :

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

Let $\int \frac{dx}{\left(\sqrt{x} - \sqrt{x-1}\right)^2} = \alpha u(x) + \beta v(x) + C$, where $u(x) = x^2 - x + \left(x - \frac{1}{2}\right)\sqrt{x^2 - x}$ and $v(x) = \log\left|x - \frac{1}{2} + \sqrt{x^2 - x}\right|$. The value of $\alpha + \beta$ is :

Match List - I with List - II | List - I (Differential Equation) | List - II (Degree) | |---|---| | A. $\left[1 + (y')^2\right]^2 = y''$ | I. $2$ | | B. $\left[1 + (y'')^3\right]^{\frac{1}{2}} = (y')^3$ | II. $4$ | | C. $(y''')^2 + y'' + 3y' + 5y = e^x$ | III. $1$ | | D. $\left[1 + (y')^3\right]^{\frac{1}{2}} = (y'')^2$ | IV. $3$ | Choose the correct answer from the option given below:

Match List I with List - II | List - I | List - II | |---|---| | A. An even function | I. $x^2 + \cos x$ | | B. For an even function, $\int_{-a}^{a} f(x)dx =$ | II. 0 | | C. If $f(2a-x) = -f(x)$, then $\int_{0}^{2a} f(x)dx =$ | III. $2\int_{0}^{a} f(x)dx$ | | D. An odd function | IV. $x^3 + \sin x$ | Choose the correct answer from the options given below:

Match List I with List II | List - I | List - II | |---|---| | A. $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$ | I. $\frac{\pi}{2}$ | | B. $\int_{-\pi/2}^{\pi/2} \sin^2 x \, dx$ | II. $\frac{\pi}{4}$ | | C. $\int_0^{\pi/2} \frac{1}{1 + \tan x} \, dx$ | III. 0 | | D. $\int_0^{\pi} \lvert \cos x \rvert \, dx$ | IV. 2 |

Match List I with List II | List - I | List - II | |---|---| | A. $\int \frac{dx}{x^2 - a^2} =$ | I. $\log_e\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{a^2 - x^2} =$ | II. $\frac{1}{2a}\log_e\left\lvert\frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | III. $\frac{1}{2a}\log_e\left\lvert\frac{a+x}{a-x}\right\rvert + C$ | | D. $\int \frac{dx}{\sqrt{x^2 + a^2}} =$ | IV. $\log_e\left\lvert x + \sqrt{x^2 - a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

Match list I with list II | List - I | List - II | |---|---| | A. Slope of tangent to the curve $y = x^3 - x$ at $x = 2$ | I. $-2$ | | B. Slope of tangent to the curve $y = 3x^3 - 4x$ at $x = 0$ | II. $11$ | | C. Slope of normal to the curve $y = \sin\theta$ at $\theta = \frac{\pi}{3}$ | III. $2$ | | D. Slope of normal to the curve $y = \cos\theta$ at $\theta = \frac{\pi}{6}$ | IV. $-4$ | Choose the correct answer from the option given below :

Match List I with List II | List I | List II | |---|---| | A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then $\frac{g(x)}{f(x)}$ is | I. discontinuous at exactly three points. | | B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is | II. continuous everywhere | | C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is | III. discontinuous at $x = 0$. | | D. The function $f(x) = \lvert \sin x \rvert$ is | IV. continuous at $x = 0$ and $x = 1$ | Choose the correct answer from the options given below:

Match List I with List II | List I | List II : Order and degree respectively | |---|---| | A. $\frac{dy}{dx} - (x^2+3) = 0$ | I. 2 and 1 | | B. $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$ | II. 2 and 3 | | C. $y''' + y^2 + e^{y'} = 0$ | III. 1 and 1 | | D. $\left(\frac{ds}{dt}\right)^4 + 3s\left(\frac{d^2s}{dt^2}\right)^3 = 0$ | IV. 3 and not defined | Choose the correct answer from the options given below:

Match List I with List II | List I | List II | |---|---| | A. The number of arbitrary constants in the particular solution of differential equation of order 2 | I. 1 | | B. The number of arbitrary constants in the general solution of differential equation of order 2 | II. 0 | | C. The integrating factor of differential equation $\frac{dy}{dx} + \frac{1}{x}y = 3, x > 0$, is | III. 2 | | D. For differential equation, $x^2 \frac{dy}{dx} + x = xy, x > 0, \lim_{x \to 0^+} y(x)$ is equal to | IV. $x$ | Choose the correct answer from the options given below:

Match List I with List II | List I | List II | |---|---| | A. $\int \frac{dx}{x^2 - a^2}$ is equal to | I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to | II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to | III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$ | | D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to | IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (a) If $x = t^2$ and $y = t^3$, then $\frac{d^2y}{dx^2}$ at $t = 1$ | (i) $-2$ | | (b) If $f(x) = \sqrt{x} + 1$, then $f''(1)$ | (ii) $-1$ | | (c) The minimum value of $f(x) = 9x^2 + 12x + 2$ is | (iii) $\frac{3}{4}$ | | (d) The point of inflexion of the function $f(x) = (x-2)^4 (x+1)^3$ is | (iv) $-\frac{1}{4}$ | Choose the correct answer from the options given below

Match List I with List II. | List I | List II | |---|---| | A. $\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ | I. order 3, degree 1 | | B. $\left(\frac{d^2y}{dx^2}\right)^2 = 0$ | II. order 2, degree 2 | | C. $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y = 0$ | III. order 2, degree 1 | | D. $\sin\left(\frac{dy}{dx}\right) + 5y = 0$ | IV. order 1, degree is not defined | Choose the correct answer from the options given below:

Minimum value of D is :

Order and degree of the differential equation $y\frac{dy}{dx} + \frac{4}{\frac{dy}{dx}} = 5$ are

$\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx =$

$\int \frac{1}{\cos^2 x (1 + \tan x)^3} \, dx =$

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$

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

$\int \frac{x}{(x^2+3)(x^2+4)} dx =$

$\int_{1}^{5} |x - 2| \, dx =$

$\int \frac{xe^x}{(x+1)^2} dx =$

$\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx =$

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

$\int \frac{x^2 + 1}{x^4 + 1} dx =$

$\int \frac{dx}{(e^x - 1)} =$

$\int \frac{x^2 - 4}{(x+2)(x-1)(x-3)} dx =$

$\int_{\frac{1}{3}}^{1} \frac{(x - x^3)^{\frac{1}{3}}}{x^4} dx =$

$\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx =$

$\int \frac{dx}{\sin^2 x \cos^2 x} =$

$\int \sqrt{x^2 - 4x + 5} \, dx =$

$\int_0^1 \frac{dx}{x^2 + x + 1}$

$\int \frac{\sin(\tan^{-1} x)}{1 + x^2} dx =$

Solution of the differential equation $(x + xy)dy - y(1 - x^2)dx = 0$ is

Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is:

Sumit's position, when $x = 10$ is :

The absolute maximum value of $y = x^3 - 3x + 2$, $0 \leq x \leq 2$, is

The area bounded by $x = \sqrt{9 - y^2}$, $x - y + 3 = 0$ and x-axis is :

The area bounded by the curve $x^2 = 4y$ and the line $x = 4y - 2$ is :

The area bounded by the parabola $y^2 = 4ax$ and $x^2 = 4ay$ is :

The area enclosed by the curve $y^2 = 4ax$ and its latus-rectum is

The area (in square units) bounded by the curve $y^2 = 4x$ and the line $x=1$ is :

The area (in square units) of minor segment of the circle $x^2 + y^2 = 25$ cut off by the line $x = \frac{5}{2}$ is

The area of the region bounded by $f(x) = x|x|$ and x-axis from $x = 0$ to $x = 4$ is :

The area of the region bounded by the curve $x^2 = 4y$ and the straight line $x = 4y - 2$ is:

The area of the region bounded by the curve $x = y^2$ and the line $x = 4$ is equal to :

The area of the region bounded by $f(x) = x|x|$, x-axis and from $x = -1$ to $x = 1$ is :

The area of the smaller region bounded by the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the line $\frac{x}{3} + \frac{y}{2} = 1$ is :

The curve passing through the point $(-1, 1)$, given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$ also passes through the point $\left( k, -\frac{1}{2} \right)$, then

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$, where $a$ and $b$ are arbitrary constants, is

The differential equation representing family of curves $y = e^{-2x}(a \cos x + b \sin x)$, where a and b are arbitrary constant, is :

The equation of the normal to the curve $y = x - \frac{1}{x}$ at $(1,0)$ is:

The function $f(x) = |x - 1|$ is

The function $f(x) = e^{|x|}$ is (a) continuous everywhere on $R$ (b) not continuous at $x = 0$ (c) Differentiable everywhere on $R$ (d) not differentiable at $x = 0$ (e) continuous and differentiable on $R$ Choose the most appropriate answer from the options given below :

The function $f(x) = 6(2x^4 - x^2)$ is strictly increasing in

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$, is :

The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$; has/have (a) local maxima at $x = 1$ (b) local maximum value is 0 (c) local minimum at $x = 3$ (d) local minimum value is $-28$ (e) The point of inflexion is $x = 1$ Choose the correct answer from the options given below

The integrating factor of differential equation $x\frac{dy}{dx} + 2y = x^2 \log x$ is

The integrating factor of the differential equation $x\frac{dy}{dx} - y = 2x^2$ is :

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

The interval in which $f(x) = \frac{x}{2} + \frac{2}{x}$ is a decreasing function of $x$ is :

The interval in which the function given by $f(x) = x^2 e^{-x}$ is strictly increasing is:

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing is -

The interval in which the function, $f(x) = 7 - 4x - x^2$ is strictly increasing is

The line $ax + by = 7$ is a tangent to the curve $y = \frac{x-7}{(x-2)(x-3)}$ at the point where it cuts the x-axis A. The y-intercept of the line is $-0.7$ B. $b = -7$ C. $a = 1$ D. the line passes through the point $(-13, -1)$ E. $b = -20$ Choose the correct answer from the options given below:

The line $y = x$, partition the area of the circle $(x-1)^2 + y^2 = 1$, into two segments. The area of the major segment is

The line $y = mx$ ($m > 0$) partitions the area of the circle $x^2 + y^2 = a^2$ ($a > 0$) in the ratio:

The maximum height (in meters) achieved in the first jump is

The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is

The maximum value of $x^{-x}$ is

The minimum value of $x^2 - 8x + 17$ on the set $\mathbb{R}$ of all real numbers is:

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

The order and degree of the differential equation $\left[\left(\frac{d^2y}{dx^2}\right)^2 - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}$ are

The order and the degree of the differential equation $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ are respectively:

The point(s) on the curve $\frac{x^2}{9} + \frac{y^2}{64} = 1$, at which the tangents are parallel to x-axis are

The portion of the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, that lies in the first quadrant is

The position of Sumit when Amit will hit the paper ball is :

The region bounded by curve $y = x|x|$, x-axis and lines $x = \pm 1$ is best represented graphically by :

The slope of normal to the curve $y = kx^2 - 3x + 2$ at $x = \frac{1}{2}$ is 5. The value of 'k' is

The slope of normal to the curve $y = 3x^2 + 3 \sin 3x$, at $x = 0$ is:

The slope of the tangent drawn at the point whose x coordinates is 2 on the curve $y = x|x|$.

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$. The value of $k$ is :

The smaller area enclosed by the curve $y = |x|$ and the circle $(x-a)^2 + y^2 = a^2$ is:

The smaller of the areas enclosed by the circle $x^2 + y^2 = 4$ and the line $x + y = 2$ is

The solution of differential equation $\sqrt{x+1} - \sqrt{x-1}\frac{dy}{dx} = 0$ is

The solution of differential equation $y(1 - x^2)\frac{dy}{dx} = x(1 + y^2)$ is :

The solution of the differential equation $\frac{dy}{dx} = \frac{\lambda^2}{(x+y)^2}$ ($\lambda$ is constant) is:

The solution of the differential equation $(x+1)\frac{dy}{dx} = 1 + y$ is

The sum of order and degree of differential equation $2x^2 \cdot \left(\frac{d^2y}{dx^2}\right) - 3 \cdot \left(\frac{dy}{dx}\right)^3 + y = 0$ is

The tangent to the curve $y^2 + 2x - 5 = 0$ at the point (h, k) is parallel to the line $x + 2y = 4$, then the value of 'k' is:

The tangent to the curve $y = e^{3x}$ at the point $(0, 1)$, meets the x-axis at :

The value of $x$ for which $\frac{dy}{dx} = 0$, is

The value of $\int_{\pi/2}^{\pi} \frac{1}{1 + \cot x} dx$ is equal to:

The value of $\int_{-4}^{4} \log_e \left( \frac{1-x}{1+x} \right) dx$ is equal to

The value of $\int_0^1 e^x (x + 1) \, dx$ is equal to

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ is

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

The value of $\frac{dy}{dx}$ when $x = \frac{1}{6}$, is

$\int \frac{x}{(x-1)(x-2)} dx =$ (where c is an arbitrary constant)

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$, where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

$f(x) = [x]$, where $[\,]$ represents greatest integer function. (A) For $2 \leq x < 3$, $[x] = 3$ (B) For $2 \leq x < 3$, $[x] = 2$ (C) Right hand derivative of f at $x = 2$ is not defined (D) Left hand derivative of f at $x = 2$ is zero (E) $f(x)$ is not differentiable at $x = 2$ Choose the correct answer from the options given below :