List of practice Questions

The volume of the solid standing on the area common to the curves \( x^2 = y, y = x \) and cut off by the surface \( z = y - x^2 \) is:

In a submarine telegraph cable, the speed of signaling varies as \( x^2 \log(1/x) \), where \( x \) is the ratio of the radius of the core to that of the covering. To attain the greatest speed, the value of this ratio is:

The asymptote of the spiral \( r = \frac{\theta}{\sin \theta} \) is:

If the radius of curvature of the Folium \( x^3 + y^3 - 3xy = 0 \) at the point \( (3/2, 3/2) \) is 5, then the value of \( b^2 + 2a + 1 \) is:

Given below are two statements:
Statement (I): The nth derivative of the function \( e^x \cos x \cos 2x \) is:
\[\frac{e^x}{2} \big[ (10^n) \cos(3x + n \tan^{-1} 3) + (2^n) \cos(x + \frac{n \pi}{4}) \big]\]
Statement (II): The nth derivative of the function \( x \cos 2x \cos 3x \) is:
\[\frac{1}{2} \big[ (2^n) \cos(2x + \frac{n \pi}{2}) + (4^n) \cos(4x + \frac{n \pi}{2}) + (6^n) \cos(6x + \frac{n \pi}{2}) \big]\]
In light of the above statements, choose the most appropriate answer from the options given below:

If \[ \theta = r e^{-x^2}, \] then for what value of \( n \), the following holds: \[ \frac{1}{2} \left( \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial r^2} \right) = \frac{\partial \theta}{\partial x}. \]

The principal value of \( i^i \) is:

The general value of \( \log(1 + i) + \log(1 - i) \) is:

Statement (I): The determinant of a matrix A and its transpose A^T are equal.
Statement (II): The determinant of the product of two matrices A and B is the product of their determinants.
In light of the above statements, choose the most appropriate answer from the options given below:

If \( (\sqrt{3} + 1)^n + (\sqrt{3} - 1)^n = 4 \), then the value of \( n \) is:

In the system of linear equations AX = B, if A is a singular matrix and B is a null matrix, then which of the following is correct?

In the matrix equation: \[ \begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}, \] the values of \( x \) and \( y \) are:

If A is a skew-symmetric matrix of odd order, then the determinant of A is:

Let P and Q be two matrices such that P Q = 0 and P is non-singular, then
(a) Q is also non-singular
(b) Q = 0
(c) Q is singular
(d) P = Q
Choose the correct answer from the options given below:

	LIST I (Type of the Matrix)	LIST II (Property)
A.	Symmetric Matrix	I. a_ij = a_ji, for values of i and j
B.	Hermitian Matrix	II. a_ij = ā_ji, for values of i and j
C.	Skew-Hermitian matrix	III. a_ij = -ā_ji, for values of i and j
D.	Skew-Symmetric matrix	IV. a_ij = -a_ji, for values of i and j

Choose the correct answer from the options given below:

Specialized epidermal cells surrounding the guard cells are:

Which of the following is used as an enriching agent in wheat flour?

In a survey of IT professionals in Bangalore, those who regularly practiced yoga reported fewer instances of work-related stress than those who did not. Priyanka, an IT professional from Bangalore, rarely feels work-related stress. Which of the following can be inferred?

BELL : DATV :: MATE : ?

On the basis of the given information, calculate the values of the following:

Read the following text carefully from ‘The Economic Times’ dated 8th June, 2023:
{“The Reserve Bank of India’s (RBI’s) rate-setting panel unanimously decided to keep the benchmark lending rate unchanged at 6.5%. The committee voted to remain focused on the withdrawal of accommodative monetary policy.”}
On the basis of the given text and common understanding, answer the following questions: (a) Identify and discuss the economic issue indicated in the above text.

Arrange the following in chronological order: (A) Ashoka’s role in Magadha (B) Bimbisara’s rule in Magadha (C) Chandragupta ruled Mauryan Empire (D) Udayan’s reign in Magadha

The potential energy between two nucleons inside a nucleus is minimum at a distance of about:

Two statements are given below
Statement I: Benzanamine can be prepared from phthalimide.
Statement II: Benzanamine is less basic than phenyl methanamine.

What are X and Z in the following reaction sequence?