CUET UG Mathematics — Calculus previous year questions with solutions.
The value of which of the following integrals is zero? (A) $\int_0^1 x dx$ (B) $\int_{-1}^1 x dx$ (C) $\int_{-1}^1 x^2 dx$ (D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$ Choose the correct answer from the options given below:
The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at
The value of the integral $I = \int_0^1 \frac{1}{\sqrt{1+3\sqrt{x}}} dx$ is
Match List-I with List-II | List-I | List-II | |---|---| | (Function) | (Derivative with respect to 'x') | | (A) $f(x) = x^x$ | (I) $ax^{a-1}$ | | (B) $f(x) = a^x$ | (II) 0 | | (C) $f(x) = a^a$ | (III) $a^x log_e a$ | | (D) $f(x) = x^a$ | (IV) $x^x(1 + log_e x)$ | Choose the correct answer from the options given below:
Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Property** | | (A) $f(x) = \begin{cases} \frac{x}{\vert x \vert} & : x \neq 0 \\ 0 & : x = 0 \end{cases}$ | (I) continuous but not differentiable at $x= 0$ | | (B) $f(x) = \vert x \vert$ | (II) continuous but not differentiable at $x=1$ | | (C) $f(x) = \vert x^2 - 1 \vert$ | (III) discontinuous at $x = 0$ | | (D) $f(x) = \vert x - 1 \vert$ | (IV) continuous but not differentiable at $x = 1, -1$ | Choose the **correct** answer from the options given below:
$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{tanx}}$ is equal to
The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$ (A) 2.9 (B) 5 (C) -3 (D) 6.5 Choose the correct answer from the options given below:
The general solution of the differential equation $ydx - (x + 2y^2)dy = 0$
Function $f(x) = x^x, x > 0$ decreases on the interval
Match List-I with List-II | List-I | List-II | |------------|-------------| | (A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$ | (I) 1 | | (B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$ | (II) 4 | | (C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$ | (III) not defined | | (D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$ | (IV) 2 | Choose the correct answer from the options given below:
The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$, where m is arbitrary constant, is
The demand for a certain product is represented by the function $p = 150 + 10x - x^2$ (in Rs.) where $x$ is the number of units demanded and $p$ is the price per unit, then the value of marginal revenue, when 10 units are sold is
$\int \frac{\sqrt{16+(\log x)^2}}{x} dx$ is equal to (where C is an arbitrary constant)
The demand for a certain product is represented by the function $p = 300 + 25x - x^2$ (in rupees), where x is the number of units demanded and p is the price per unit, then the marginal revenue when 15 units are sold, is
$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx$ is equal to
The value of $\int \frac{x^5}{\sqrt{1 + x^3}} dx$ is
If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to
The value of k for which the function $f(x) = \begin{cases} \frac{1-\cos 8x}{16x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ is:
The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is
Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE? (A) $f'(x) = 3(x - 1)(x - 3)$ (B) The critical points of the function are $x = 1$ and $x = 3$ (C) $x = 1$ is the point of local minimum (D) The local maximum value is $-4$ Choose the correct answer from the options given below:
$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to
The area (in sq.units) of region bounded by $y^2 = 9x$, $x = 2$, $x = 4$ and the $x$-axis in the first quadrant is
If $\int \sqrt\frac{1-x}{{1+x}} dx = a\sqrt{1-x^2} + \beta \sin^{-1}x + C$, Where C is an arbitrary constant, then which of the following are TRUE? (A) $\alpha = 1$ (B) $\alpha = -1$ (C) $\beta = 1$ (D) $\beta = -1$ Choose the correct answer from the options given below:
$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$)