CUET UG Mathematics — Calculus previous year questions with solutions.
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:
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 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:
The maximum height (in meters) achieved in the first jump is
Evaluate $\int_0^{\frac{1}{3}} y \, dx$
The area (in square units) bounded by the curve $y^2 = 4x$ and the line $x=1$ is :
If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.
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 :
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 :
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 : If area of the flower bed is maximum, then area (in $m^2$) of the garden, which is outside the flower bed is :
If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :
$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :
The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$, is :
For $a, b \in R$ and $a < b$, $\int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} \, dx =$
$f(x) = [x]$, 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 :
If curve represented by differential equation $x \frac{dy}{dx} + y = e^x$ passes through (1, 1), then $y(-1)$ is :
The region bounded by curve $y = x|x|$, x-axis and lines $x = \pm 1$ is best represented graphically by :
The slope of the tangent drawn at the point whose x coordinates is 2 on the curve $y = x|x|$.
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 (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:
$\int \frac{x^2 + 1}{x^4 + 1} dx =$
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 :
$\int \sqrt{1 - 49x^2} \, dx$ is equal to