List of practice Questions

Let A = [aij]n x n be a matrix. Then Match List-I with List-II
List-I
(A) AT = A
(B) AT = -A
(C) |A| = 0
(D) |A| $\neq$ 0
List-II
(I) A is a singular matrix
(II) A is a non-singular matrix
(III) A is a skew symmetric matrix
(IV) A is a symmetric matrix
Choose the correct answer from the options given below:
$\int_{1}^{4} |x - 2| dx$ is equal to
The integrating factor of the differential equation $(x \log_e x) \frac{dy}{dx} + y = 2\log_e x$ is

Consider the line \[ \vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k}) \]

Match List-I with List-II:

List-IList-II
(A) A point on the given line(I) \(\left(-\tfrac{1}{\sqrt{21}}, \tfrac{2}{\sqrt{21}}, -\tfrac{4}{\sqrt{21}}\right)\)
(B) Direction ratios of the line(II) (4, -2, -2)
(C) Direction cosines of the line(III) (1, -2, 4)
(D) Direction ratios of a line perpendicular to given line(IV) (-1, 2, -4)
Let F(Z) be the cumulative density function of the standard normal variate Z, then which of the following are correct?
(A) \(F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty < Z <\infty\)
(B) \(F(-Z) = 1 - F(Z)\)
(C) \(F(0) = 0\)
(D) \(F(\infty) = 1\)
Choose the correct answer from the options given below:
What is the mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on 1 face?
If A is a square matrix and I is the identity matrix of same order such that A2 = I, then (A - I)3 + (A + I)3 - 3A is equal to
Consider the LPP: Minimize Z = x + 2y subject to 2x + y $\ge$ 3, x + 2y $\ge$ 6, x, y $\ge$ 0. The optimal feasible solution occurs at
Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to
If \(A = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}\), \(X = \begin{bmatrix} \alpha \\ -2 \end{bmatrix}\), \(B = \begin{bmatrix} 7 \\ 32 \end{bmatrix}\) and \(AX = B\), then the value of the \(\alpha\) is

Match List-I with List-II

List-IList-II
(A) \( f(x) = |x| \)(I) Not differentiable at \( x = -2 \) only
(B) \( f(x) = |x + 2| \)(II) Not differentiable at \( x = 0 \) only
(C) \( f(x) = |x^2 - 4| \)(III) Not differentiable at \( x = 2 \) only
(D) \( f(x) = |x - 2| \)(IV) Not differentiable at \( x = 2, -2 \) only


Choose the correct answer from the options given below:

A person can row a boat in still water at the rate of 5 km/hr. It takes him 4 times as long to row upstream of a river as to row downstream to cover same distance in the same river. The speed of flow of the stream is
Which of the following inequalities holds true?
(A) \(\sqrt{5} + \sqrt{3}>\sqrt{6} + \sqrt{2}\)
(B) If \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\)
(C) \(\frac{1}{x^2}>\frac{1}{x}>1\), if \(0<x<1\)
(D) If a and b are positive integers and \(\frac{a-b}{6.25} = \frac{4}{2.5}\) then \(b>a\)
Choose the correct answer from the options given below:
If $ A $ is a $ 2 \times 2 $ matrix and $ |A| = 4 $, then $ |A^{-1}| $ is:
Corner points of a feasible bounded region are \((0, 10)\), \((4, 2)\), \((3, 7)\) and \((10, 6)\). Maximum value 50 of objective function \(z = ax + by\) occurs at two points \((0, 10)\) and \((10, 6)\). The value of \(a\) and \(b\) are:

Match List-I with List-II

List-I (Definite integral)List-II (Value)
(A) \( \int_{0}^{1} \frac{2x}{1+x^2}\, dx \)(I) 2
(B) \( \int_{-1}^{1} \sin^3x \cos^4x\, dx \)(II) \(\log_e\!\left(\tfrac{3}{2}\right)\)
(C) \( \int_{0}^{\pi} \sin x\, dx \)(III) \(\log_e 2\)
(D) \( \int_{2}^{3} \frac{2}{x^2 - 1}\, dx \)(IV) 0


Choose the correct answer from the options given below:

Consider the differential equation, $x \frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is $\log_e(\frac{y}{x}) = Cx$, where C is constant of integration
(D) Its general solution is $\log_e(\frac{x}{y}) = Cy$, where C is constant of integration
(E) If y(1) = 1, then its particular solution is y = x
Choose the correct answer from the options given below:
If \(e^y = \log x\), then which of the following is true?

Given: The least-squares straight line is \(y = a + b(x - 2022)\).
 

Year (x)20202021202220232024
Profit (Rs. '000) (y)23452
A person wishes to purchase a house for Rupess 39,65,000 with a down payment of Rupees 5,00,000 and balance in equal monthly installments (EMI) for 25 years. If bank charges 6% per annum compounded monthly, then EMI on reducing balance payment method is:
[Given \((1.005)^{300} = 4.465\)]
Let A = $\begin{bmatrix} 1 & 2 & 1 \\ 1 & 3 & 2 \\ 2 & 4 & 1 \end{bmatrix}$ and Mij, Aij respectively denote the minor, co-factor of an element aij of matrix A, then which of the following are true?
(A) M22 = -1
(B) A23 = 0
(C) A32 = 3
(D) M23 = 1
(E) M32 = -3
Choose the correct answer from the options given below:
If A and B are invertible matrices then which of the following statement is NOT correct?
Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
If P, Q and R are three singular matrices given by \(P = \begin{bmatrix} 2 & 3a \\ 4 & 3 \end{bmatrix}\), \(Q = \begin{bmatrix} b & 5 \\ 2a & 6 \end{bmatrix}\) and \(R = \begin{bmatrix} a^2 + b^2 - c & 1-c \\ c+1 & c \end{bmatrix}\), then the value of \((2a + 6b + 17c)\) is