List of practice Questions

The area bounded by curves \( y = -\frac{2}{3}x + 2 \), \( x = -1 \), \( x = 2 \) and the x-axis is:

If \( y = \frac{x^2}{1 + x^{b-a} + x^{c-a}} + \frac{x^2}{1 + x^{a-b} + x^{c-b}} + \frac{x^2}{1 + x^{a-c} + x^{b-c}} \), then \( \frac{dy}{dx} \) is:

A firm anticipates an expenditure of Rs.5,000,000 for plant modernization at the end of 10 years from now, then the amount the company should deposit at the end of each year into a sinking fund earning interest 5% per annum is [use (1.05)¹⁰ = 1.629]:

Mr. Sanjay borrowed Rs.10,00,000 from a bank to purchase a car on reducing balance payment for a period of 10 years. If bank charges interest at 9% per annum compounded monthly and EMI is Rs.12,658 to be paid by him. Then principal outstanding after payment of 12th EMI is: (Use \( (1.0075)^{12 = 1.0938 \))}

A measurable characteristic of a sample is known as:

If a machine is correctly set up, it produces 80% acceptable items. If it is incorrectly set up, it produces only 30% acceptable items. From the past experience it was known that 90% of the setups are correctly done. If after a certain setup, the machine produces 2 acceptable items then the probability that the machine was correctly set up, is:

A line passes through (2, 1, 3) and (1, 2, -1), then
(A) Equation is \( \frac{x-2}{1} = \frac{y-1}{1} = \frac{z-3}{4} \)
(B) Equation is \( \frac{x+2}{-1} = \frac{y+1}{1} = \frac{z+3}{4} \)
(C) Equation is \( \vec{r} = 2\vec{i} + \vec{j} + 3\vec{k} + \lambda(\vec{i} - \vec{j} + 4\vec{k}) \)
(D) Equation is \( \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z+1}{4} \)
Choose the correct answer:

The probability of drawing a one-rupee coin from a purse with two compartments, one of which contains 3 fifty paise coins and 2 one-rupee coins and other contains 2 fifty paise coins and 3 one-rupee coins, is

The integral of \( \int_{-2}^{2} x^4 \, dx \) denominator \( (1+5x^2) \) is:

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \) then \( A^2 - (2 \cos \alpha) A \) is equal to: (Where I is identity matrix of order 2)

If \( A = \begin{bmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{bmatrix} \) then \( |adj(adj A)| \) is equal to

The solution set of the inequation \( 2x + 3y > 12 \) is

The area bounded by the lines \( y = 1 + |x + 1| \), \( x = -3 \), \( x = 3 \) and \( y = 0 \) is

If A and B are two independent events and P(A) = 1/2, P(B) = 1/3, then Match List-I with List-II

Solution of \( \frac{x^2 - 4x + 7}{x^2 - 7x + 12} \le \frac{2}{3} \) is/are:
Choose the correct answer:

If \( \int \frac{1 + \cos\theta}{\tan 2\theta - \cot 2\theta} d\theta = \lambda \cos\theta + c \), then \( \lambda \) is equal to (where c is constant of integration)

The maximum value of the linear programming problem, max. \( z = 3x + 4y \) subject to the constraints: \( x - y \le -1 \), \( x \ge y \), \( x, y \ge 0 \) is

General solution of the differential equation \( (x + 2y^3) dy = y dx \) is (Where C is an arbitrary constant)

If A, B and C are square matrices of order n x n, then which of the following are TRUE? [Where $A^T$ is transpose of matrix A]
(A) \( (A + B)^T = A^T + B^T \)
(B) \( (AB)^T = A^T B^T \)
(C) \( (ABC)^T = C^T B^T A^T \)
(D) \( (BA)^T = A^T B^T \)
Choose the correct answer:

The integral \( \int \frac{\cos 5x + \cos 4x}{1 - 2\cos 3x} \, dx \) is equal to (where C is an arbitrary constant):

Let $R = \{(1,1),(2,2),(3,3),(1,2)\}$ be a relation on $\{1,2,3\}$. The minimum number of elements to be added so that $R$ is an equivalence relation is:}

If $A_1, A_2, A_3$ are independent events such that $P(A_i)=\frac{1}{i+1}$, then probability that none occur is:}

Let $\mathbb{N}, \mathbb{Z}$ and $\mathbb{R}$ be the set of natural numbers, integers and real numbers respectively, $[\cdot]$ denotes the greatest integer function. Match List-I with List-II:}

The determinant \( \begin{vmatrix} \lambda & \sin\theta & \cos\theta \\ -\sin\theta & -\lambda & 1 \cos\theta & 1 & \lambda \end{vmatrix} \) is equal to:

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}+\vec{b}$ is also a unit vector. Then which of the following are TRUE?} (A) $|\vec{a}-\vec{b}|=0$
(B) $|\vec{a}-\vec{b}|=\sqrt{3}$
(C) Angle between $\vec{a}$ and $\vec{b}=\frac{2\pi}{3}$
(D) Angle between $\vec{a}$ and $\vec{b}=\frac{\pi}{3}$