List of practice Questions

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 :
A free electron of 2.6 eV energy collides with a \(H^+\) ion. This results in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon. (\(h = 6.6 \times 10^{-34}\) J s)
Statement I : If three forces \(\vec{F_1}\), \(\vec{F_2}\) and \(\vec{F_3}\) are represented by three sides of a triangle and \(\vec{F_1} + \vec{F_2} = -\vec{F_3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II : A triangle made up of three forces \(\vec{F_1}\), \(\vec{F_2}\) and \(\vec{F_3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below :
Two thin metallic spherical shells of radii \(r_1\) and \(r_2\) (\(r_1<r_2\)) are placed with their centres coinciding. A material of thermal conductivity K is filled in the space between the shells. The inner shell is maintained at temperature \(\theta_1\) and the outer shell at temperature \(\theta_2\) (\(\theta_1<\theta_2\)). The rate at which heat flows radially through the material is :

Reactant A decomposes to products B and C in the presence of an enzyme in a well-stirred batch reactor. The kinetic rate expression is given by 
\[ -r_A = \frac{0.01 C_A}{0.05 + C_A} \, \text{(mol L}^{-1} \text{min}^{-1}) \] If the initial concentration of A is 0.02 mol L\(^{-1}\), the time taken to achieve 50% conversion of A is \(\underline{\hspace{1cm}}\) min (rounded to 2 decimal places).
 

The following homogeneous, irreversible reaction involving ideal gases: 
\[ A \longrightarrow B + C \left( -r_A = 0.5 C_A \, \text{(mol L}^{-1} \text{s}^{-1}) \right) \] is carried out in a steady state ideal plug flow reactor (PFR) operating at isothermal and isobaric conditions. The feed stream consists of pure A, entering at 2 m s\(^{-1}\). In order to achieve 50% conversion of A, the required length of the PFR is \(\underline{\hspace{1cm}}\) meter (rounded off to 2 decimal places).
 

The following isothermal autocatalytic reaction: 
\[ A + B \longrightarrow 2B \left( -r_A = 0.1 C_A C_B \, \text{(mol L}^{-1} \text{s}^{-1}) \right) \] is carried out in an ideal continuous stirred tank reactor (CSTR) operating at steady state. Pure A at 1 mol L\(^{-1}\) is fed, and 90% of A is converted in the CSTR. The space time of the CSTR is \(\underline{\hspace{1cm}}\) seconds.
 

As shown in the figure below, air flows in parallel to a freshly painted solid surface of width 10 m, along the z-direction. The equilibrium vapor concentration of the volatile component A in the paint, at the air-paint interface, is \( C_{A,i} \). The concentration \( C_A \) decreases linearly from this value to zero along the y-direction over a distance \( \delta \) of 0.1 m in the air phase. Over this distance, the average velocity of the air stream is 0.033 m s\(^{-1}\) and its velocity profile \( v_z(y) \) is given by \[ v_z(y) = 10 y^2 \] where \( y \) is in meter. Let \( C_{A,m} \) represent the flow averaged concentration. The ratio of \( C_{A,m} \) to \( C_{A,i} \) is \(\underline{\hspace{1cm}}\) (round off to 2 decimal places). 

Consider a tank filled with 3 immiscible liquids A, B, and C at static equilibrium. At 2 cm below the liquid A-liquid B interface, a tube is connected from the side of the tank. Both the tank and the tube are open to the atmosphere. At the operating temperature and pressure, the specific gravities of liquids A, B, and C are 1, 2, and 4, respectively. Neglect any surface tension effects in the calculations. The length of the tube \( L \) that is wetted by liquid B is \(\underline{\hspace{1cm}}\) cm. 

The probability distribution function of a random variable \( X \) is shown in the figure. From this distribution, random samples with sample size \( n = 68 \) are taken. If \( \bar{X} \) is the sample mean, the standard deviation of the probability distribution of \( \bar{X} \), i.e. \( \sigma_{\bar{X}} \), is \(\underline{\hspace{1cm}}\) (rounded off to 3 decimal places). 

To solve an algebraic equation \( f(x) = 0 \), an iterative scheme of the type \( x_{n+1} = g(x_n) \) is proposed, where \( g(x) = x - \frac{f(x)}{f'(x)} \). At the solution \( x = s \), \( g'(s) = 0 \) and \( g''(s) \neq 0 \). The order of convergence for this iterative scheme near the solution is \(\underline{\hspace{1cm}}\).
 

Match the reaction in Group – 1 with the reaction type in Group – 2. 

It is required to control the volume of the contents in the jacketed reactor shown in the figure. 
 

Which one of the following schemes can be used for feedback control? 
 

Which of these symbols can be found in piping and instrumentation diagrams? 

Water of density 1000 kg m\(^{-3}\) flows in a horizontal pipe of 10 cm diameter at an average velocity of 0.5 m s\(^{-1}\). The following plot shows the pressure measured at various distances from the pipe entrance. 
 

A ⇌ B ⇌ C reactions are at equilibrium. Given \(k_1 = 0.1\), \(k_{-1} = 0.2\), \(k_2 = 1\), \(k_{-2} = 10\), \(k_3 = 10\). Determine \(k_{-3}\) (round to 1 decimal). 

A, B, C and D are vectors of length 4. The rank of the matrix 
A = \(\begin{bmatrix} a_1 & a_2 & a_3 & a_4 \end{bmatrix}\), 
B = \(\begin{bmatrix} b_1 & b_2 & b_3 & b_4 \end{bmatrix}\), 
C = \(\begin{bmatrix} c_1 & c_2 & c_3 & c_4 \end{bmatrix}\), 
D = \(\begin{bmatrix} d_1 & d_2 & d_3 & d_4 \end{bmatrix}\) 
It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, \( D = 3A + 2B + C \) 

The rank of the matrix \( \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \\ b_1 & b_2 & b_3 & b_4 \\ c_1 & c_2 & c_3 & c_4 \\ d_1 & d_2 & d_3 & d_4 \end{bmatrix} \) is \(\underline{\hspace{1cm}}\)

The inherent characteristics of three control valves P, Q and R are shown in the figure. 

The CORRECT option(s) is/are 
 

Feed solution F is contacted with solvent B in an extraction process. Carrier liquid in the feed is A and the solute is C. The ternary diagram depicting a single ideal stage extraction is given below. The dashed lines represent the tie-lines.

The CORRECT option(s) is/are

For the function $f(x)=\begin{cases} -x, & x<0 \\ x^{2}, & x\ge 0 \end{cases}$ the CORRECT statement(s) is/are
 

Match the common name of chemicals in Group – 1 with their chemical formulae in Group – 2. 

The correct combination is: 
 

The mean of 10 numbers \(7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \dots\) is ___________