List of practice Questions

If \[ \int_{0}^{x} t^2 \sin(x - t)\,dt = x^2, \] then the sum of values of \( x \), where \( x \in [0,100] \), is:
Let \[ f(x)= \begin{cases} \dfrac{|a|x + 2x^2 - 2\sin|x|\cos|x|}{x}, & x \neq 0 \\ b, & x = 0 \end{cases} \] If \( f(x) \) is continuous, then the value of \( a + b \) is:
Let the sets \[ A = \{x : |x - 3| - 3 \le 1,\; x \in \mathbb{Z}\} \] \[ B = \left\{ x : x \in \mathbb{R},\; x \ne 1,2,\; \frac{(x-2)(x-4)}{(x-1)} \log_e |x-2| = 0 \right\} \] Then the number of onto functions from \( A \) to \( B \) is:
Find the area bounded by the curves \[ x^2 + y^2 = 4 \quad \text{and} \quad x^2 + (y-2)^2 = 4. \]
The following table gives a frequency distribution:
The mean and variance of the above data are \( \mu \) and 19 respectively, where \( \mu \) is an integer. Find the value of \( (\lambda + \mu) \):
On two metal surfaces, a monochromatic light of $6 \text{ eV}$ was incident. They have ratio of their work function and maximum KE as $\frac{\Phi_1}{\Phi_2} = \frac{1}{2}$ and $\frac{(KE_{\max})_1}{(KE_{\max})_2} = \frac{2.62}{1}$. Then $\Phi_1$ and $\Phi_2$ values are respectively (in $\text{eV}$).
For given reaction $X_2(g) + Y_2(g) \rightleftharpoons 2Z (g)$. Moles of $X_2, Y_2 & Z$ are at equilibrium are $3 \text{ mole}, 3 \text{ mole} & 9 \text{ mole}$ respectively. If $10 \text{ moles}$ of $Z$ are added at constant $T$ then find moles of $Z$ at re-established equilibrium.
$K_2Cr_2O_7 + KI \xrightarrow{H^+} I_2 + Cr^{3+}$
Statement-I : Size of $O^{2-}$ is smaller than $F^-$.
Statement-II: Second ionization energy of $\text{Na}$ is greater than second ionization energy of $\text{Mg}$.
Which of the following are isobars?
Organic compound (P) $\xrightarrow{(i) \text{ excess of} HI, (ii) \text{ Aq. } NaOH} Q + R$. $Q$ and $R$ both gives Iodoform test, Which among the following is (P) from the given organic compound?
Identify (A)
$0.245 \text{ gm}$ of an unknown organic compound gave $0.5453 \text{ gm}$ of $\text{AgCl}$ through Carious method. Calculate $%$ of $\text{Cl}$ in unknown compound.
. Number of different $\text{H}$-atoms in $(\text{Y})$
Consider the following carbocations: (A) $\text{CH}_3\text{C}^+\text{H}_2 \rightarrow \text{B} \rightarrow \text{CH}_3-\text{C}^+\text{H}-\text{CH}_3 \rightarrow \text{C} \rightarrow (\text{CH}_3)_3\text{C}^+$. Statement-I : B is more stable due to $9\alpha H$. Statement-II: A is less stable due to $3\alpha H$.
Correct statement is : (A) $\text{NaOCl}$ when reacted with $\text{KI}$ gives $\text{KOI}$ (B) $\text{KOI}$ is best reducing agent (C) Methanoic acid gives iodoform test (D) Isopropyl alcohol gives iodoform test (E) $\text{H}_3\text{C}–\text{CH}=\text{CH}–\text{CO}–\text{CH}_3$ gives iodoform test
Two beads P & Q move along two wires straight and semi circle. At some instant both are shown in figure having same horizontal component of velocity V. Find relation of time taken to reach A by beads :
A metallic sphere of diameter 2mm and density 10.5 g/cc is dropped in glycerine having viscosity 10 poise and density 1.5 g/cc. The terminal velocity attained by sphere is ______ cm/s. [\(\pi = \frac{22}{7}, g = 10 \text{ m/s}^2\)]
A particle is projected from bottom of inclined plane with speed u find distance covered along plane before coming to rest :
Energy released per fission of U-235 is 190 MeV, then total energy released by 47 g of U-235 is \(x \times 10^{23}\) MeV. Find the value of x.
A parachutist jumps from a helicopter. It falls freely for 2 sec. Then he opens parachute which produces retardation of 3 m/s\(^2\). When his height from ground is 10 m his velocity is 5 m/s. Find his initial height from ground.
A bomb at rest explodes into three pieces in the ratio of masses 2 : 2 : 3. The identical masses fly off perpendicular to each other with 18 m/s. Find speed of the third piece.
A disc of mass (M, R) is given. Two discs of radius \(\frac{R}{4}\) are cut from this, whose centers are at 135° angle. Their peripheries touch the larger disc as shown. If moment of inertia of remaining disc about the center is \(\frac{\alpha}{256}MR^2\), find \(\alpha\):
Which of the following is an isobaric pair :
In phosphorus estimation, \(0.60\,\text{g}\) of an organic compound gives \(0.93\,\text{g}\) of \( \mathrm{Mg_2P_2O_7} \). Calculate the percentage of phosphorus (nearest integer).
Determine the largest value of \( n \) for which \[ 2^{n} \mid (75)! \]