CUET UG Mathematics — Calculus previous year questions with solutions.
If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$, then $\phi(x)$ is equal to :
$\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx =$
Sumit's position, when $x = 10$ is :
If D is the distance between Sumit and Amit, then the value of $x$ for which D is minimum, is :
The position of Sumit when Amit will hit the paper ball is :
Minimum value of D is :
The minimum value of $x^2 - 8x + 17$ on the set $\mathbb{R}$ of all real numbers is:
$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to
Match List I with List II. | List I | List II | |---|---| | A. $\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ | I. order 3, degree 1 | | B. $\left(\frac{d^2y}{dx^2}\right)^2 = 0$ | II. order 2, degree 2 | | C. $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y = 0$ | III. order 2, degree 1 | | D. $\sin\left(\frac{dy}{dx}\right) + 5y = 0$ | IV. order 1, degree is not defined | Choose the correct answer from the options given below:
Match List I with List II | List I | List II | |---|---| | A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then $\frac{g(x)}{f(x)}$ is | I. discontinuous at exactly three points. | | B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is | II. continuous everywhere | | C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is | III. discontinuous at $x = 0$. | | D. The function $f(x) = \lvert \sin x \rvert$ is | IV. continuous at $x = 0$ and $x = 1$ | Choose the correct answer from the options given below:
The function $f(x) = |x - 1|$ is
The line $y = mx$ ($m > 0$) partitions the area of the circle $x^2 + y^2 = a^2$ ($a > 0$) in the ratio:
Match List I with List II | List I | List II | |---|---| | A. $\int \frac{dx}{x^2 - a^2}$ is equal to | I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to | II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to | III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$ | | D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to | IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | Choose the correct answer from the options given below:
The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$; has/have (a) local maxima at $x = 1$ (b) local maximum value is 0 (c) local minimum at $x = 3$ (d) local minimum value is $-28$ (e) The point of inflexion is $x = 1$ Choose the correct answer from the options given below
The area enclosed by the curve $y^2 = 4ax$ and its latus-rectum is
The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is
The order and degree of the differential equation $\left[\left(\frac{d^2y}{dx^2}\right)^2 - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}$ are
$\int x\sqrt{x + 2} \, dx$ is equal to :
$\int \frac{x}{(x^2+3)(x^2+4)} dx =$
The area of the region bounded by the curve $x^2 = 4y$ and the straight line $x = 4y - 2$ is:
Match List I with List II | List I | List II : Order and degree respectively | |---|---| | A. $\frac{dy}{dx} - (x^2+3) = 0$ | I. 2 and 1 | | B. $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$ | II. 2 and 3 | | C. $y''' + y^2 + e^{y'} = 0$ | III. 1 and 1 | | D. $\left(\frac{ds}{dt}\right)^4 + 3s\left(\frac{d^2s}{dt^2}\right)^3 = 0$ | IV. 3 and not defined | Choose the correct answer from the options given below:
The slope of normal to the curve $y = kx^2 - 3x + 2$ at $x = \frac{1}{2}$ is 5. The value of 'k' is
Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is:
The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is