List of practice Questions

The value of \[ \lim_{x\to 0}\frac{\log_e\!\big(\sec(ex)\cdot \sec(e^2x)\cdots \sec(e^{10}x)\big)} {e^2-e^{2\cos x}} \] is equal to:

The common difference of the A.P.: $ a_1, a_2, \ldots, a_m $ is 13 more than the common difference of the A.P.: $ b_1, b_2, \ldots, b_n $. If $ b_{31} = -277 $, $ b_{43} = -385 $ and $ a_{78} = 327 $, then $ a_1 $ is equal to:

Let $ y = y(x) $ be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If $ y(2) = 0 $, then $ \tan(y(1)) $ is equal to:

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are $ 2, 3, 5, 10, 11, 13, 15, 21 $, then the mean deviation about the median of all the 10 observations is:

Let $ S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} $ be a set of polynomials. Then the number of polynomials in $ S $, which are divisible by $ x^2 + 2 $, is:

Let $ y = x $ be the equation of a chord of the circle $ C_1 $ (in the closed half-plane $ x \ge 0 $) of diameter 10 passing through the origin. Let $ C_2 $ be another circle described on the given chord as diameter. If the equation of the chord of the circle $ C_2 $, which passes through the point $ (2, 3) $ and is farthest from the center of $ C_2 $, is $ x + ay + b = 0 $, then $ b $ is equal to:

If $ g(x) = 3x^2 + 2x - 3 $, $ f(0) = -3 $ and $ 4g(f(x)) = 3x^2 - 32x + 72 $, then $ f(g(2)) $ is equal to:

After release, the blocks moves 81 cm in 9 seconds. Find moment of inertia of the pulley :
(Given $m_{1} = 400$ gm, $m_{2} = 350$ gm, R = 2 cm, g = 10 m/s$^2$)

Density of water at 4$^\circ$C is 1000 kg/m$^3$ and at 20$^\circ$C it is 998 kg/m$^3$. If 4kg of water is heated from 4$^\circ$C to 20$^\circ$C, the change in internal energy of water is : (Given : specific heat capacity of water = 4200 J/kg K, Atmospheric pressure P = $10^5$ Pa).

From the following : (A) $[\text{Co}(\text{NH}_3)_6]^{3+}$ : Inner orbital complex, $d^2sp^3$ hybridization (B) $[\text{MnCl}_4]^{2-}$ : Outer orbital complex, $sp^3d^2$ hybridization (C) $[\text{CoF}_6]^{3-}$ : Outer orbital complex, $d^2sp^3$ hybridization (D) $[\text{FeF}_6]^{3-}$ : Outer orbital complex, $sp^3d^2$ hybridization (E) $[\text{Ni}(\text{CN})_4]^{2-}$ : Inner orbital complex, $sp^3$ hybridization Choose the correct answer from the given options.

For the following change: H$_2$O(l) $\rightarrow$ H$_2$O(g)
5$^\circ$C $\rightarrow$ 100$^\circ$C
Select the correct answer

It is noticed that $Pb^{2+}$ is more stable than $Pb^{4+}$ but $Sn^{2+}$ is less stable than $Sn^{4+}$. Observe the following reactions.
$PbO_2 + Pb \to 2PbO ; \Delta_rG^\circ(1)$
$SnO_2 + Sn \to 2SnO ; \Delta_rG^\circ(2)$
Identify the correct set from the following

The ratio of de Broglie wavelength of a deuteron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2E$ is $n:1$. (Assume mass of proton $=$ mass of neutron.) Find the value of $n$.

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is_______ cm. [Assume that the mass of the pendulum remains same]}

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Consider a ferromagnetic material :
Assertion (A): The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains.
Reason (R): At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature.
In the light of the above statements, choose the correct answer from the options given below :

Match List-I with List-II.

Choose the correct answer from the options given below :}

In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is_______ mm. [Assume least count is 0.01 mm]}

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of 60° with vertical, then maximum induced EMF between the point of suspension and point of oscillation is_______ mV. (Take g = 10 m/s²)}

Using a variable frequency ac voltage source the maximum current measured in the given LCR circuit is 50 mA for V = 5 sin (100t) The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is_______ µF.

The space between the plates of a parallel plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants $k_1 = 2$, $k_2 = 3$ and $k_3 = 3$ (as shown in figure). If new capacitance is n/3 C then the value of n is_______.

The equation of the electric field of an electromagnetic wave propagating through free space is given by: $E = \sqrt{377} \sin(6.27 \times 10^3 t - 2.09 \times 10^{-5} x)$ N/C. The average power of the electromagnetic wave is (1/a) W/m². The value of a is_______. (Take $\sqrt{\mu_0/\varepsilon_0} = 377$ in SI units)}

In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are 2:1 and 1:2 respectively. The corresponding ratio of the distances between the slits and the respective screens (D₁/D₂) is_______.}

The correct statements from the following are :
A. Ionic radii of trivalent cations of group 13 elements decreases down the group.
B. Electronegativity of group 13 elements decreases down the group.
C. Among the group 13 elements, Boron has highest first ionisation enthalpy.
D. The trichloride and triiodide of group 13 elements are covalent in nature.
Choose the correct answer from the options given below :

Given below are two statements :
Statement I: Sublimation is used for the separation and purification of compounds with low melting point.
Statement II: The boiling point of a liquid increases as the external pressure is reduced.
In the light of the above statements, choose the correct answer from the options given below :