List of practice Questions

A student has been given a compound "x" of molecular formula- $C_6H_7N$. 'x' is sparingly soluble in water... On treatment with benzenesulphonyl chloride, 'x' gives a compound 'z' which is soluble in alkali. The number of different "H" atoms present in 'z' is:-

Consider the following reaction of benzene. the percentage of oxygen is _______ %. (Nearest integer)

A Wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances \[ (R_1=R_2=R_3=R_4). \] When \(R_3\) resistance is heated, its resistance value increases by \(10%\). The potential difference \((V_a-V_b)\) after \(R_3\) is heated is _______ V.

Let $y=y(x)$ be a differentiable function in the interval $(0,\infty)$ such that $y(1)=2$, and \[ \lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3 \text{ for each } x>0. \] Then $2y(2)$ is equal to
The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is $5$, is

A thin uniform rod (\(X\)) of mass \(M\) and length \(L\) is pivoted at a height \( \left(\dfrac{L}{3}\right) \) as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top is ________. (\(g\) = gravitational acceleration) 

A uniform solid cylinder of length \(L\) and radius \(R\) has moment of inertia about its axis equal to \(I_1\). A small co-centric cylinder of length \(L/2\) and radius \(R/3\) carved from it has moment of inertia about its axis equal to \(I_2\). The ratio \(I_1/I_2\) is ________.
A light wave described by E = 60[\(\sin(3 \times 10^{15})t + \sin(12 \times 10^{15})t\)] (in SI units) falls on a metal surface of work function 2.8 eV. The maximum kinetic energy of ejected photoelectron is (approximately)________ eV. (h=\(6.6 \times 10^{-34}\) J.s. and e=\(1.6 \times 10^{-19}\) C)
Name the painting done by the artist "Ustad Faqirullah Khan":
Given below are two statements:
Statement I: The number of paramagnetic species among $[\mathrm{CoF_6}]^{3-}$, $[\mathrm{TiF_6}]^{3-}$, $\mathrm{V_2O_5}$ and $[\mathrm{Fe(CN)_6}]^{3-}$ is 3.
Statement II: $ \mathrm{K_4[Fe(CN)_6]}<\mathrm{K_3[Fe(CN)_6]}<\mathrm{[Fe(H_2O)_6]SO_4 \cdot H_2O}<\mathrm{[Fe(H_2O)_6]Cl_3} $ is the correct order in terms of number of unpaired electrons.
Choose the correct answer from the options given below:

What are the charges stored in the \( 1\,\mu\text{F} \) and \( 2\,\mu\text{F} \) capacitors in the circuit once current becomes steady? 

If \[ \int (\cos x)^{-5/2}(\sin x)^{-11/2}\,dx = \frac{p_1}{q_1}(\cot x)^{9/2} + \frac{p_2}{q_2}(\cot x)^{5/2} + \frac{p_3}{q_3}(\cot x)^{1/2} - \frac{p_4}{q_4}(\cot x)^{-3/2} + C, \] where \(C\) is the constant of integration, then find the value of \[ \frac{15\,p_1p_2p_3p_4}{q_1q_2q_3q_4}. \]
If the system of linear equations \[ \begin{cases} 3x + y + \beta z = 3 \\ 2x + \alpha y - z = 1 \\ x + 2y + z = 4 \end{cases} \] has infinitely many solutions, then the value of \(22\beta - 9\alpha\) is:
Let \( A = \{2, 3, 5, 7, 11\} \) and a relation \( R \) is defined as \[ R = \{(x, y) : x, y \in A, 2x \leq 3y\}. \] Then the minimum number of elements to be added to relation \( R \) such that \( R \) becomes symmetric is:
The number of values of \( x \) satisfying \( \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6} \) and \( x<\left[ \frac{-1}{2\sqrt{6}} , \frac{1}{2\sqrt{6}} \right] \) is:
Find the value of \[ \frac{6}{3^{26}} + 10\cdot\frac{1}{3^{25}} + 10\cdot\frac{2}{3^{24}} + \cdots + 10\cdot\frac{2^{24}}{3^{1}} : \]
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{12(3+[x])\,dx}{3+[\sin x]+[\cos x]} \] (where \([\,]\) denotes the greatest integer function) is:
The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;\text{and}\; y^2 \le 4x\} \] is
If \[ I(x) = 3\int \frac{dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}, \quad I(0) = \frac{\sqrt{3}}{4}, \] then find \( I(1) \):
If $y = y(x)$ satisfies
$(1+x^2)\frac{dy}{dx} + (2 - \tan^{-1}x) = 0$
and $y(0) = 0$, then the value of $y(1)$ is:
Maximum value of $n$ for which $40^n$ divides $60!$ is
If the system of equations \[ \begin{cases} 2x + y + pz = -1 \\ 3x - 2y + z = q \\ 5x - 8y + 9z = 5 \end{cases} \] has more than one solution, then \( q - p \) is equal to:
The least value of $(\cos^2 \theta - 6\sin \theta \cos \theta + 3\sin^2 \theta + 2)$ is
The area of the region enclosed between the circles $x^2 + y^2 = 4$ and $x^2 + (y - 2)^2 = 4$ is:
A bag contains 'k' red balls and (10 - k) black balls. If 3 balls are drawn at random and they are found to be black then the probability that bag has 9 black balls & 1 red ball is