List of practice Questions

Given the function:

\[ f(x) = \frac{2x - 3}{3x - 2} \]

and if \( f_n(x) = (f \circ f \circ \ldots \circ f)(x) \) is applied \( n \) times, find \( f_{32}(x) \).

If \(^nC_r = C_r\) and \[ 2 \frac{C_1}{C_0} + 4 \frac{C_2}{C_1} + 6 \frac{C_3}{C_2} + \dots + 2n \frac{C_n}{C_{n-1}} = 650 \] then \[ ^nC_2 = \ ? \]

When 2 dice are thrown, it is observed that the sum of the numbers appeared on the top faces of both the dice is a prime number. Then the probability of having a multiple of 3 among the pair of numbers thus obtained is:

A body thrown vertically upwards from the ground reaches a maximum height ‘h’. The ratio of the kinetic and potential energies of the body at a height 40\% of h from the ground is

A question paper has 3 parts A, B, C. Part A contains 7 questions, part B contains 5 questions and Part C contains 3 questions. If a candidate is allowed to answer not more than 4 questions from part A; not more than 3 questions from part B and not more than 2 questions from part C, then the number of ways in which a candidate can answer exactly 7 questions is:

If \( \sinh^{-1}(-\sqrt{3}) + \cosh^{-1}(2) = K \), then \( \cosh K = \):

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + 3x^2 - 10x - 24 = 0 \). If \( \alpha>\beta>\gamma \) and \( \alpha^3 + 3\beta^2 - 10\gamma - 24 = 11k \), then \( k = \):

A conductor of length 1.5 m and area of cross-section \( 3 \times 10^{-5} \, \text{m}^2 \) has electrical resistance of 15 \( \Omega \). The current density in the conductor for an electric field of 21 \( \text{V/m} \) is.

If \( \alpha, \beta \) are the roots of the equation \( x + \frac{4}{x} = 2\sqrt{3} \), then \( \frac{2}{\sqrt{3}}\left| \alpha^{2024} - \beta^{2024} \right| \) is:

If \( a \neq b \neq c \), then

\[ \Delta_1 = \begin{vmatrix} 1 & a^2 & bc \\ 1 & b^2 & ca \\ 1 & c^2 & ab \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

and

\[ \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \]

then what is \( 11(a + b + c) \)?

On the basis of the data given below, estimate the value of Gross National Product at Factor Cost (GNP_FC): \[ \begin{array}{|c|l|r|} \hline \textbf{S.No.} & \textbf{Items} & \textbf{Amount (in ₹ crore)} \\ \hline (i) & \text{Wages and Salaries} & 2,000 \\ \hline (ii) & \text{Rent and Interest} & 800 \\ \hline (iii) & \text{Corporate Tax} & 500 \\ \hline (iv) & \text{Undistributed Profit} & 300 \\ \hline (v) & \text{Dividend} & 200 \\ \hline (vi) & \text{Depreciation} & 150 \\ \hline (vii) & \text{Net Factor Income from Abroad} & -50 \\ \hline \end{array} \]

Solving the System of Linear Equations

If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations:

3x - 4y + 2z + 7 = 0, 2x + 3y - z = 10, x - 2y - 3z = 3,

then α = ?

If \(AX = D\) represents the system of linear equations \[ 3x - 4y + 7z + 6=0,\quad 5x + 2y - 4z + 9=0,\quad 8x - 6y - z + 5=0, \] then AX = D ?

Matrix Inverse Sum Calculation

Given the matrix:

A = | 1 2 2 | | 3 2 3 | | 1 1 2 |

The inverse matrix is represented as:

A^-1 = | a₁₁ a₁₂ a₁₃ | | a₂₁ a₂₂ a₂₃ | | a₃₁ a₃₂ a₃₃ |

The sum of all elements in A^-1 is:

∑ a_ij = 3

Match the following:

A metal target with atomic number \( Z = 46 \) is bombarded with a high-energy electron beam. The emission of X-rays from the target is analyzed. The ratio \( r \) of the wavelengths of the \( K_\alpha \)-line and the cutoff is found to be \( r = 2 \). If the same electron beam bombards another metal target with \( Z = 41 \), the value of \( r \) will be:

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all \( x \in (-\infty, a] \cup (b, \infty) \), then what is \( 3^a + b^2 \)?

An object is placed at a distance of 18 cm in front of a mirror. If the image is formed at a distance of 4 cm on the other side, then the focal length, nature of the mirror and nature of the image are respectively:

∫ √(2x² - 5x + 2) dx = ∫ (41/60) dx,

and

-1/2 > α > 0, then α = ?

The number of common roots among the 12^th and 30^th roots of unity is ?

Atomicity of Phosphorus is:

Isotones are the atoms of different elements having:

Atomicity is given by:

Atomicity of an element is:

Avogadro’s hypothesis relates volume of gases and:

List of practice Questions

∑ aij = 3

∑ a_ij = 3