Calculus PYQ — Page 9

If $\int_0^1 \frac{e^x}{1 + x} dx = m$, then the value of $\int_0^1 \frac{e^x}{(1 + x)^2} dx$ is:

$\int \sin x \sin 2x \sin 3x dx$ is equal to

The sum of order and degree of the differential equation $y = x\frac{dy}{dx} + 2\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ is

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$, $y(0) = 0$ is

If $x^2 - y^2 = 1$, then which of the following is correct? (A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$ (B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$ Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = \frac{ax + c}{by + d}$ represents a circle when

If $x = e^{\cos 2t}$, $y = e^{\sin 2t}$, then $\frac{dy}{dx}$ equals to

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$ | (I) 4 | | (B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$ | (II) 1 | | (C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$ | (III) Not defined | | (D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$ | (IV) 3 | Choose the correct answer from the options given below:

If $x = e^t$ and $y = e^{2t}$ then $\frac{d^2y}{dx^2} =$

The value of which of the following integrals is zero? (A) $\int_0^1 x dx$ (B) $\int_{-1}^1 x dx$ (C) $\int_{-1}^1 x^2 dx$ (D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$ Choose the correct answer from the options given below:

If $y = (x+1)(x^2+1)(x^4+1)(x^8+1)$ then $\frac{dy}{dx}$ at $x = -1$ is

The area (in square units) of the region bounded by the curve $x^2 = y$ and the straight line $y = 4$ in the first quadrant is equal to

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|) dx$ is

Let $e^y(x+1) = 1$. Then which of the following are TRUE? (A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$ (B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$ (C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$ (D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$ (E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$ Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by the parabola $y^2 = 4x$ and the line $x = 1$ is

Area of region bounded by the curves $x = y^3$, $x = 0$ between $y = -1$ and $y = 2$ is:

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is

If $g(x) = \begin{cases} \frac{αx}{|x|}, & \text{if } x < 0 \\ 5, & \text{if } x ≥ 0 \end{cases}$ is continuous at x = 0, then the value of α is

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3 - x} + \sqrt{x}} dx$ is equal to

For the function $f(x) = sinx + cosx, x \in [0, \pi]$, which one of the following is correct?

Which of the following statements are true? (A) The function $f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x$ has 3 critical points. (B) The function $f(x) = |x| + 3$ has no minimum value. (C) A local maximum value is always the absolute maximum value. (D) $f(x) = x^2$ has minima at $x=0$. Choose the **correct** answer from the options given below:

Solution of the differential equation $\frac{dy}{dx} = \sqrt{1 + x^2 + y^2 + x^2y^2}$ is : (Here $C$ is an arbitrary constant)