CUET UG Mathematics — Calculus previous year questions with solutions.
Which of the following differential equation represents the family of circles touching the x-axis at the origin ?
The general solution of $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$ is: (given that $C$ is the constant of integration)
Match List - I with List - II. Match the integrating factors : | List - I (Differential Equation) | List - II (Integrating factor) | |---|---| | (A) $\frac{dy}{dx} + 3y = e^{-2x}$ | (I) $\frac{1}{x}$ | | (B) $x\frac{dy}{dx} + y = 3x^2$ | (II) $e^{-x}$ | | (C) $x\frac{dy}{dx} - y = 3x^2$ | (III) $x$ | | (D) $\frac{dy}{dx} - y = x$ | (IV) $e^{3x}$ | Choose the correct answer from the options given below :
The two curves $x^3 - 3xy^2 + 15 = 0$ and $3x^2 y - y^3 + 17 = 0$ :
The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :
$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$
Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :
If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} = $
The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :
The area enclosed between the curve $x^2 + y^2 = 16$ and the coordinate axes in the first quadrant is :
Match List I with List II | LIST I | LIST II | |---|---| | A. Maximum value of $f(x) = -\lvert x+1 \rvert + 3$ | I. 6 | | B. Minimum value of $f(x) = (2x-1)^2 + 5$ | II. 5 | | C. Maximum value of $f(x) = 6 - x^2$ | III. no maximum value | | D. Maximum value of $f(x) = x^3 + 1$ | IV. 3 | Choose the correct answer from the options given below:
Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :
The equation of the tangent, to the curve $y = x^2 - 2x - 3$ which is perpendicular to the line $x + 2y + 3 = 0$, is
If $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $\frac{dy}{dx} =$
The solution of $y' - y'' = 2x$ is: A. $y = x^2 + 2x + 2$ B. $y = x^2 + 2x + 1$ C. $y = x + 2$ D. $y = x^2 - 2x + 1$ Choose the correct answer from the options given below:
The differential equation $\frac{dy}{dx} + \frac{x}{y} = 0$, represents the family of curves:
Area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is :
The degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2$ is :
If the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its local maximum value at $x = 1$, then a is equal to :
The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :
In the context of differential equation Match List I with List II | LIST I | LIST II | |---|---| | A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ | I. Not a differential equation | | B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$ | II. Linear first order | | C. $\sin x + y = \cos(x+y)$ | III. Variable separable | | D. $(x+y)\frac{dy}{dx} = 1$ | IV. Homogenous | Choose the correct answer from the options given below:
The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :
The differential equation of the family of curves $y = a \sin(bx + c)$, a and c are parameters, is :
The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :