List of practice Questions

If $\int_{-2}^2 ([\sin x] + |x \sin x|) dx = 2 \sin 2 - 4 \cos 2 - \beta$, then the value of $|\beta|$ where $[\cdot]$ is GIF is
Consider the differential equation $\frac{dy}{dx} = (x^2 + x + 1)(y^2 - y + 1)$. If $y(0) = \frac{1}{2}$, then the value of $2(y(1)) - 1$ is
The domain of $f(x) = \cos^{-1} \left( \frac{4x + 2[x]}{3} \right)$ (where $[x]$ denotes greatest integer function) is
If coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}$ is $\binom{100}{3} \left( \frac{101}{4} - 43n \right)$, then the value of $(k^3 + 43n)$ is
Let $y = \tan^{-1}\left( \frac{3\cos x - 4\sin x}{4\cos x + 3\sin x} \right) + \tan^{-1}\left( \frac{x}{1 + \sqrt{1 + x^2}} \right)$. Then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$ is :
Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{a, b, c\}$ then total number of functions from $A$ to $B$ which are not onto are
If Latus rectum of parabola $y^2 = 4kx$ and ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ coincide then the value of $e^2 + 2\sqrt{2}$ is, where $e$ is eccentricity of ellipse
In an A.P. first term is $\frac{10}{3}$ and first 30 terms are non-negative such that sum of first 30 terms = $(T_{30})^3$, then d is equal to (where $T_n$ is $n^{\text{th}}$ term of A.P., and d is the common difference of A.P.)
If $f(x)$ is a non-constant polynomial satisfying $f(x) = f'(x)f''(x)$ and $f(0) = 0$. Then the value of $\int_0^2 f(x) dx + f'(2) + f''(2)$ is :
Let in a $\triangle ABC$, given that $A \equiv (1, 2)$, mid-point of $AB$ is $(-5, -1)$ and centroid is $(3, 4)$ then circumcentre is $(\alpha, \beta)$, then the value of $21(\alpha + \beta)$ is :
Let mean and median of 9 observations 8, 13, a, 17, 21, 51, 103, b, 67 are 40 and 21 respectively where a > b. If mean deviation about median is 26 then 2a is :-
A circle $x^2 + y^2 + x - 3y = 0$ passes through $P(1, 2)$. If 2 chords (PS \& PR) drawn from P are bisected by $y$-axis, then mid point of RS is $(\alpha, \beta)$, find $6(\alpha + \beta)$
Find number of ways of arranging 4 Boys \& 3 Girls such that all girls are not together :
If $S = \{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, A^2 - 4A + 3I = \text{Null matrix}, a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } \text{Tr}(A) = 4 \}$. Then the number of elements of set $S$ is/are
IUPAC Name of the formed compound:
Consider the following reaction:
What is the colour of the final compound B?
Let $A$ is a matrix of order 3 such that $|A| = -4$, then the value of $|\text{adj}(\text{adj}(2\text{adj} A)^{-1})|$ is
Choose the correct match.
Match the column:
} Match correct reagents with given reactions:
Arrange the following groups according to their decreasing order of electron withdrawing nature – \( \text{COOH} \), \( \text{CN} \), \( I \), and \( \text{NO}_2 \).
Statement-1: Maltose is a non-reducing sugar.
Statement-2: Lactose is a reducing sugar.
For the reaction
\[ \text{A} + \text{B} \rightleftharpoons \text{C}, \quad K_p = 2.7 \times 10^{-5} \] Calculate \( K_p \) for the reaction
\[ \frac{1}{3} \text{A} + \frac{1}{3} \text{B} \rightleftharpoons \frac{1}{3} \text{C} \]
Calculate the number of molecules and moles of \( \text{SO_2 \) in its 1.479 liters at STP.}
An ideal gas is placed in a container at \( (P_1, V_1, T_1) \) and another ideal gas is placed in a different container at \( (P_2, V_2, T_2) \) are mixed at final pressure of \( P \) and final volume of \( V \). Calculate the final temperature.
Statement-1: Heat capacity at constant volume is always greater than heat capacity at constant pressure.
Statement-2: At constant volume as work done is zero, heat given to the chaotic motion is reflected by increase in temperature.