List of practice Questions

Which of the following statements is incorrect?

A person wants to invest at least ₹20,000 in plan A and ₹30,000 in plan B. The return rates are 9% and 10% respectively. He wants the total investment to be ₹80,000 and investment in A should not exceed investment in B. Which of the following is the correct LPP model (maximize return $ Z $)?

The vertices of a closed convex polygon representing the feasible region of the LPP with objective function $z = 5x + 3y$ are $(0,0)$, $(3,1)$, $(1,3)$ and $(0,2)$. The maximum value of $z$ is:

The corner points of the feasible region associated with the LPP: Maximise Z = px + qy, p, q > 0 subject to 2x + y $\le$ 10, x + 3y $\le$ 15, x,y $\ge$ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). If optimum value occurs at both (3, 4) and (0, 5), then

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of x-axis, y-axis and z-axis respectively, then $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ is equal to

Let A = $\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$ and I = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If AT + A = I, then

The area (in sq. units) of the region bounded by y = $2\sqrt{1-x^2}$, x $\in$ [0,1] and x-axis is equal to

If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true?
(A) $\hat{i} \times \hat{i} = \vec{0}$
(B) $\hat{i} \times \hat{k} = \hat{j}$
(C) $\hat{i} . \hat{i} = 1$
(D) $\hat{i} . \hat{j} = 0$
Choose the correct answer from the options given below:

If a 95% confidence interval for a population mean was reported to be 132 to 160 and sample standard deviation s = 50, then the size of the sample in the study is:
(Given $Z_{0.025}$ = 1.96)

A sofa set costing Rupees 36000 has a useful life of 10 years. If the annual depreciation is Rupees 3000, then the scrap value by linear method is:

The interval, on which the function f(x) = x2e-x is increasing, is equal to

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

Which of the following are linear first order differential equations?
(A) $\frac{dy}{dx} + P(x)y = Q(x)$
(B) $\frac{dx}{dy} + P(y)x = Q(y)$
(C) $(x - y)\frac{dy}{dx} = x + 2y$
(D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$
Choose the correct answer from the options given below:

If the function f(x) = $\begin{cases}\frac{k\cos x}{\pi - 2x} & ; x \neq \frac{\pi}{2} \\ 3 & ; x = \frac{\pi}{2} \end{cases}$ is continuous at x = $\frac{\pi}{2}$, then k is equal to

The integrating factor of the differential equation $(x \log_e x) \frac{dy}{dx} + y = 2\log_e x$ is

Which of the following statements are correct in reference to the linear programming problem (LPP):
Maximize Z = 5x + 2y
subject to the following constraints
3x + 5y $\le$ 15,
5x + 2y $\le$ 10,
x $\ge$ 0, y $\ge$ 0.
(A) The LPP has a unique optimal solution at (2, 0) only.
(B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3).
(C) The optimal value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is unbounded.
Choose the correct answer from the options given below:

Match List-I with List-II

List-I (Matrix)	List-II (Inverse of the Matrix)
(A) $\begin{bmatrix} 1 & 7 \\ 4 & -2 \end{bmatrix}$	(I) $\begin{bmatrix} \tfrac{2}{15} & \tfrac{1}{10} \\[6pt] -\tfrac{1}{15} & \tfrac{1}{5} \end{bmatrix}$
(B) $\begin{bmatrix} 6 & -3 \\ 2 & 4 \end{bmatrix}$	(II) $\begin{bmatrix} \tfrac{1}{5} & -\tfrac{2}{15} \\[6pt] -\tfrac{1}{10} & \tfrac{7}{30} \end{bmatrix}$
(C) $\begin{bmatrix} 5 & 2 \\ -5 & 4 \end{bmatrix}$	(III) $\begin{bmatrix} \tfrac{1}{15} & \tfrac{7}{30} \\[6pt] \tfrac{2}{15} & -\tfrac{1}{30} \end{bmatrix}$
(D) $\begin{bmatrix} 7 & 4 \\ 3 & 6 \end{bmatrix}$	(IV) $\begin{bmatrix} \tfrac{2}{15} & -\tfrac{1}{15} \\[6pt] \tfrac{1}{6} & \tfrac{1}{6} \end{bmatrix}$

The least non-negative remainder when $3^{128}$ is divided by 7 is:

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 $(1.005)^{300} = 4.465$]

If A = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ then the matrix AB is equal to

If y = 3e2x + 2e3x, then $\frac{d^2y}{dx^2} + 6y$ is equal to

Which of the following statements is incorrect?

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?

Which one of the following set of constraints does the given shaded region represent?

Given: $y = a + b(x - 2022)$ is the least-squares straight-line trend.

Year (x)	2020	2021	2022	2023	2024
Profit (Rs. '000) (y)	2	3	4	5	2

List-I (Matrix)	List-II (Inverse of the Matrix)
(A) \(\begin{bmatrix} 1 & 7 \\ 4 & -2 \end{bmatrix}\)	(I) \(\begin{bmatrix} \tfrac{2}{15} & \tfrac{1}{10} \\[6pt] -\tfrac{1}{15} & \tfrac{1}{5} \end{bmatrix}\)
(B) \(\begin{bmatrix} 6 & -3 \\ 2 & 4 \end{bmatrix}\)	(II) \(\begin{bmatrix} \tfrac{1}{5} & -\tfrac{2}{15} \\[6pt] -\tfrac{1}{10} & \tfrac{7}{30} \end{bmatrix}\)
(C) \(\begin{bmatrix} 5 & 2 \\ -5 & 4 \end{bmatrix}\)	(III) \(\begin{bmatrix} \tfrac{1}{15} & \tfrac{7}{30} \\[6pt] \tfrac{2}{15} & -\tfrac{1}{30} \end{bmatrix}\)
(D) \(\begin{bmatrix} 7 & 4 \\ 3 & 6 \end{bmatrix}\)	(IV) \(\begin{bmatrix} \tfrac{2}{15} & -\tfrac{1}{15} \\[6pt] \tfrac{1}{6} & \tfrac{1}{6} \end{bmatrix}\)