List of practice Questions

The sum of the roots of the equation, \( x + 1 - 2\log_2(3 + 2^x) + 2\log_4(10 - 2^{-x}) = 0 \), is :

If \( \alpha + \beta + \gamma = 2\pi \), then the system of equations
\( x + (\cos \gamma)y + (\cos \beta)z = 0 \)
\( (\cos \gamma)x + y + (\cos \alpha)z = 0 \)
\( (\cos \beta)x + (\cos \alpha)y + z = 0 \)
has :

Let \( a_1, a_2, a_3, \dots \) be an A.P. If \( \frac{a_1 + a_2 + \dots + a_{10}}{a_1 + a_2 + \dots + a_p} = \frac{100}{p^2}, p \neq 10 \), then \( \frac{a_{11}}{a_{10}} \) is equal to :

An angle of intersection of the curves, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \( x^2 + y^2 = ab, a>b \), is :

The empirical formula for a compound with a cubic close packed arrangement of anions and with cations occupying all the octahedral sites is $A_x B$. The value of $x$ is ________. (Integer answer)

The value of magnetic quantum number of the outermost electron of $\text{Zn}^+$ ion is ________. (Integer answer)

1.22 g of an organic acid is separately dissolved in 100 g of benzene ($K_b = 2.6 \text{ K kg mol}^{-1}$) and 100 g of acetone ($K_b = 1.7 \text{ K kg mol}^{-1}$). The acid is known to dimerize in benzene but remain as a monomer in acetone. The boiling point of the solution in acetone increases by $0.17^\circ\text{C}$. The increase in boiling point of solution in benzene in $^\circ\text{C}$ is $x \times 10^{-2}$. The value of $x$ is ________. (Nearest integer) [Atomic mass : C = 12.0, H = 1.0, O = 16.0]

The pH of a solution obtained by mixing 50 mL of 1 M HCl and 30 mL of 1 M NaOH is $x \times 10^{-1}$. The value of $x$ is ________. (Nearest integer) [log 2.5 = 0.3979]

$\text{CH}_4$ is adsorbed on 1 g charcoal at $0^\circ\text{C}$ following the Freundlich adsorption isotherm. 10.0 mL of $\text{CH}_4$ is adsorbed at 100 mm of Hg, whereas 15.0 mL is adsorbed at 200 mm of Hg. The volume of $\text{CH}_4$ adsorbed at 300 mm of Hg is $10^x$ mL. The value of $x$ is ________ $\times 10^{-2}$. (Nearest integer) [Use $\log_{10} 2 = 0.3010, \log_{10} 3 = 0.4771$]

In the electrolytic refining of blister copper, the total number of main impurities, from the following, removed as anode mud is ________.
Pb, Sb, Se, Te, Ru, Ag, Au and Pt.

Which among the following is not a polyester ?

The deposition of X and Y on ground surfaces is referred as wet and dry depositions, respectively. X and Y are :

Which one of the following correctly represents the order of stability of oxides, \( X_2O \); (X = halogen) ?

Which one of the following statements is incorrect?

The number of S=O bonds present in sulphurous acid, peroxodisulphuric acid and pyrosulphuric acid, respectively are :

A bandwidth of 6 MHz is available for A.M. transmission. If the maximum audio signal frequency used for modulating the carrier wave is not to exceed 6 kHz. The number of stations that can be broadcasted within this band simultaneously without interfering with each other will be ________.

A particle is moving with constant acceleration 'a'. Following graph shows \(v^2\) versus x (displacement) plot. The acceleration of the particle is ________ m/s\(^2\).
\begin{center} \includegraphics[width=0.35\textwidth]{graph_v2_x} \end{center}

A long solenoid with 1000 turns/m has a core material with relative permeability 500 and volume \(10^3\) cm\(^3\). If the core material is replaced by another material having relative permeability of 750 with same volume maintaining same current of 0.75 A in the solenoid, the fractional change in the magnetic moment of the core would be approximately \(\left( \frac{x}{499} \right)\). Find the value of x.

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :

Four identical hollow cylindrical columns of mild steel support a big structure of mass $50 \times 10^3\,\text{kg}$. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local load distribution, calculate the compression strain of each column. [use $Y = 2.0 \times 10^{11}\,\text{Pa}$, $g = 9.8\,\text{m/s}^2$]

A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the ends of a light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ?

A current of 1.5 A is flowing through a triangle, of side 9 cm each. The magnetic field at the centroid of the triangle is : (Assume that the current is flowing in the clockwise direction.)

Statement I : Two forces \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\) where \(\vec{p} \perp \vec{q}\), when act at an angle \(\theta_1\) to each other, the magnitude of their resultant is \(\sqrt{3(p^2 + q^2)}\), when they act at an angle \(\theta_2\), the magnitude of their resultant becomes \(\sqrt{2(p^2 + q^2)}\). This is possible only when \(\theta_1<\theta_2\).
Statement II : In the situation given above, \(\theta_1 = 60^{\circ}\) and \(\theta_2 = 90^{\circ}\).
In the light of the above statements, choose the most appropriate answer from the options given below :

A mixture of hydrogen and oxygen has volume 500 \(cm^3\), temperature 300 K, pressure 400 kPa and mass 0.76 g. The ratio of masses of oxygen to hydrogen will be :

A block moving horizontally on a smooth surface with a speed of 40 m/s splits into two parts with masses in the ratio of \(1 : 2\). If the smaller part moves at 60 m/s in the same direction, then the fractional change in kinetic energy is :