Calculus PYQ — Page 4

The value of $\int_{-5}^{5} |x + 3| dx$ is

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

The area (in sq. units) bounded by the parabola $y^2 = 4ax$, its latus rectum and the $x$-axis in the first quadrant is:

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

The general solution of the differential equation $\frac{dy}{dx} = e^{ax+by}$ is: (Here C is an arbitrary constant)

The value of the definite integral $I = \int_0^2 x\sqrt{2-x} dx$ is:

If a revenue function is given by $R(x) = 2027x - 1013x^2 - 675x^3$, then the marginal revenue function (MR) is:

The value of $\int_{-1}^{1} |x^3 - x| dx$ is

The area of the region (in square units) bounded by $x=1, x=2$ and the curve $y^2 = 4x$ in the first quadrant is

The area (in sq. units) of the region bounded by the curve $y = x^5$, the x-axis and the ordinates $x = -1$ and $x = 1$ is equal to

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

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is

The point on the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$ at which the tangent to the curve is parallel to the x-axis is

Match List-I with List-II | List-I | List-II | |---|---| | (A) Maximum value of $f(x) = \sin^2 x - \cos^2 x$, $\forall x \in (\pi, 2\pi)$ is | (I) 0 | | (B) Minimum value of $f(x) = \sin x \cos x$ | (II) 1 | | (C) Point of Minima of $f(x) = x^x$ $(x > 0)$ | (III) $-\frac{1}{2}$ | | (D) Maximum value of $f(x) = -x^{2026}$ | (IV) $\frac{1}{e}$ | Choose the correct answer from the options given below:

The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The area (in sq. units) of the region bounded by the curve $y = \sin x, -2\pi \leq x \leq 2\pi$ and $x - axis$ is equal to

The area (in sq. units) of the region bounded by the parabola $y^2 = 8x$ and the line $x = 2$ is

Let the degree and order of the differential equation $2x^3\frac{dy}{dx}-5\left(\frac{d^2y}{dx^2}\right)^2=6\left(\frac{dy}{dx}\right)^3$ be $m$ and $n$ respectively. Then (A) m = 2 (B) n = 3 (C) m = 3 (D) mn = 4 Choose the correct answer from the options given below:

For $x \in \mathbb{R} - \{-1,0,1\}$, $\int \frac{1}{x - x^5}dx$ is equal to

$\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to

The function $f(x) = x + \frac{1}{x}$ has

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration (D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, where C is constant of integration (E) If $y(1) = 1$, then its particular solution is $y = x$ Choose the correct answer from the options given below: