Calculus PYQ — Page 34

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

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

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

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

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

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

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

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 $y = 3x^2 + 3 \sin 3x$, at $x = 0$ is:

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

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

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

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

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

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