List of practice Questions

A biased die has six faces numbered as \( k = 1, 2, 3, 4, 5, 6 \). On rolling the die, the probability of the number \( k \) appearing is proportional to \( k^2 \). What is the probability that an even number will appear on rolling the die?

A flocculation tank used for water treatment has a velocity gradient (G) of 800 s\(^{-1}\). The volume of the tank is 40 m\(^3\). The dynamic viscosity of water is \(9 \times 10^{-4}\) N.s/m\(^2\). The theoretical power required to maintain the given velocity gradient is _______ kW (rounded off to the nearest integer).

In a sample, the growth of microbial cells started with an initial concentration of \(5 \times 10^4\) cells per millilitre (mL) of the sample. After a certain time period, the cell concentration was found to be \(1280 \times 10^4\) cells per mL of the sample. Assuming binary fission of cells and no cell death, the number of generations of cell growth occurred in this time period is _______ (answer in integer).

A water jet discharging from a 4 cm diameter orifice has a diameter of 3.5 cm at its vena contracta. The coefficient of velocity is defined as the ratio of the actual velocity of the jet at vena contracta to the theoretical velocity of the jet. If the coefficient of velocity is 0.98, the coefficient of discharge for the orifice will be _______ (rounded off to two decimal places).

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has \( y \) as the dependent variable and \( x \) as the independent variable. Which of the following classification(s) is/are applicable to the equation?

Consider the following equation: \[ x^3 - 10x^2 + 31x - 30 = 0 \] Which of the following is/are the root(s) of the above equation?

A wind rose is a representation of meteorological conditions. Which of the following is/are included in this representation?

The oxidation states of ‘N’ in \( \text{NH}_4^+ \), \( \text{NO}_2^- \), and \( \text{NO} \) are _______ , respectively.

What is the pH of a water sample having \( H^+ \) concentration of 10 mg/L? The atomic weight of H is 1 g/mol.

Which of the following pairing of nucleotide bases is present in double helix DNA?

Which of the following is/are both greenhouse gas(es) and ozone depleting substance(s)?

The United Nations Conference on Environment and Development was held in 1992 in Rio de Janeiro, Brazil. During this conference, several environmental management principles were adopted by many countries. Which one of the following principles allows the governments to take mitigation measures on the environmental issues having serious threats or irreversible damage, even if there is scientific uncertainty about such issues?

Choose the correct order of biodegradability (highest to lowest) of the following municipal solid waste components.

In proximate analysis, when a 10 kg moisture-free solid sample is heated in a furnace at 950 °C in the absence of air, its mass is reduced by 6 kg. If the same 10 kg moisture-free solid sample is heated in the furnace at the same temperature in the presence of air, its mass is reduced by 7 kg. The percentage of fixed carbon in the sample is _______.

Chlorine is most effective as a water disinfectant at a pH of _______.

A tangent is drawn on the curve of the function \( y = x^2 \) at the point \( (x, y) = (3, 9) \). The slope of the tangent is _______.

Consider the multiple linear regression model

\[ Y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \dots + \beta_{22} x_{22,i} + \epsilon_i, \quad i = 1, 2, \dots, 123, \]

where, for \( j = 0, 1, 2, \dots, 22 \), \( \beta_j \)'s are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed \( N(0, \sigma^2) \), \( \sigma>0 \), random variables. If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and \( R^2_{\text{adj}} \) denotes the value of adjusted \( R^2 \), then

\[ 100 R^2_{\text{adj}} = \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( X_1, X_2, \dots, X_{10} \) be a random sample from a probability density function

\[ f_\theta(x) = f(x - \theta), \quad -\infty<x<\infty, \]

where \( -\infty<\theta<\infty \) and \( f(-x) = f(x) \) for \( -\infty<x<\infty \). For testing

\[ H_0: \theta = 1.2 \quad \text{against} \quad H_1: \theta \neq 1.2, \]

let \( T^+ \) denote the Wilcoxon Signed-rank test statistic. If \( \eta \) denotes the probability of the event \( \{T^+<50\} \) under \( H_0 \), then \( 32\eta \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( \{0.90, 0.50, 0.01, 0.95\} \) be a realization of a random sample of size 4 from the probability density function

\[ f(x) = \begin{cases} \frac{\theta}{(1-\theta)} x^{(2\theta-1)/(1-\theta)}, & 0<x<1 \\ 0, & \text{otherwise} \end{cases} \]

where \( 0.5 \leq \theta<1 \). Then the maximum likelihood estimate of \( \theta \) based on the observed sample equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having probability density function

\[ f(x) = \begin{cases} e^{-x}, & x>0 \\ 0, & \text{otherwise} \end{cases} \]

Let \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \) for \( n \geq 1 \). If \( Z \) is the random variable to which

\[ \{ X_{(n)} - \log n \}_{n \geq 1} \]

converges in distribution, as \( n \to \infty \), then the median of \( Z \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( X \) be a random variable having the moment generating function

\[ M(t) = \frac{e^t - 1}{t(1 - t)}, \quad t<1. \]

Then

\[ P(X>1) = \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let

\[ A = \begin{bmatrix} a & u_1 & u_2 & u_3 \end{bmatrix}, \quad B = \begin{bmatrix} b & u_1 & u_2 & u_3 \end{bmatrix}, \quad C = \begin{bmatrix} u_2 & u_3 & u_1 & a + b \end{bmatrix}. \]

Let \( \det(A), \det(B), \det(C) \) denote the determinants of the matrices \( A \), \( B \), and \( C \), respectively. If

\[ \det(A) = 6, \quad \det(B) = 2, \quad \text{then } \det(A + B) - \det(C) = \underline{\hspace{2cm}}. \]