Calculus PYQ — Page 6

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true? (A) Integrating factor = tan⁻¹x (B) Integrating factor = tan⁻¹y (C) Integrating factor = $e^{tan⁻¹y}$ (D) Degree = 1 Choose the correct answer from the options given below:

The greatest integer function $f(x) = [x]$ is differentiable for all values of

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

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Points of Non-Differentiability | | --- | --- | | (A) $f(x) = \vert x\vert + 1$ | (I) Not differentiable at $x = 3$ only | | (B) $f(x) = \vert x - 3\vert $ | (II) Not differentiable at $x = -3$ only | | (C) $f(x) = \vert x + 3\vert $ | (III) Not differentiable at $x = 3, -3$ only | | (D) $f(x) = \vert x^2 - 9\vert $ | (IV) Not differentiable at $x = 0$ only | Choose the correct answer from the options given below:

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

The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

A integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}$, $(x > 0)$ is equal to

The area of the region bounded by the curves $y = x^2 + 2$ and $x$-axis, between $x = 0$ and $x = 3$ in the first quadrant is:

Match List-I with List-II | List-I | List-II | | --- | --- | | Functions | Integrals | | --- | --- | | (A) $\int \frac{dx}{x^2 - 4}, x \neq \pm 2$ | (I) $\log \vert x + \sqrt{4 + x^2}\vert + C$, where $C$ is an arbitrary constant | | (B) $\int \frac{1}{\sqrt{16 - x^2}} dx; \vert x\vert < 4$ | (II) $\sin^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | | (C) $\int \frac{1}{16 + x^2} dx$ | (III) $\frac{1}{4} \log \left\vert \frac{x-2}{x+2} \right\vert + C$, where $C$ is an arbitrary constant | | (D) $\int \frac{1}{\sqrt{4 + x^2}} dx$ | (IV) $\frac{1}{4} \tan^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | Choose the correct answer from the options given below:

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

$\int_{-\frac{5}{2}}^{\frac{5}{2}} |x| dx$ is equal to

The particular solution of the differential equation $\frac{dy}{dx} + \frac{3y}{x} = 0$, $y(1) = 1$ is

$\int \left(\frac{1}{log_e t} - \frac{1}{(log_e t)^2}\right) dt$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$ is equal to

The area of the region bounded by the parabola $y^2 = x$ and the straight line $2y = x$ is

If $y = t - \frac{1}{t}$ and $x = t + \frac{1}{t}$, then $\frac{dy}{dx}$ is equal to

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:

The area bounded by the curve $y = 4 + 3x - x^2$ and $x$-axis is equal to

Match List-I with List-II: | List-I | List-II | | --- | --- | | **Differential Equations** | **Degree/Order** | | (A) Degree of the differential equation $\frac{d^3y}{dx^3} + 2 \log x.y = 0$ | (I) 3 | | (B) Order of the differential equation $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^4 + xy = 0$ | (II) 2 | | (C) Degree of the differential equation $\left(\frac{d^4y}{dx^4}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^2y = 0$ | (III) 1 | | (D) Order of the differential equation $\frac{d^3y}{dx^3} + y\left(\frac{dy}{dx}\right)^3 = 0$ | (IV) 4 |