List of practice Questions

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

In a uniform magnetic field of \(0.049 T\), a magnetic needle performs \(20\) complete oscillations in \(5\) seconds as shown. The moment of inertia of the needle is \(9.8 \times 10 kg m^2\). If the magnitude of magnetic moment of the needle is \(x \times 10^{-5} Am^2\); then the value of '\(x\)' 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 \( k = \):

Two ambitious friends, Sana and Mihir, having a passion for creating innovative software applications, established a tech startup named ‘Quick Solutions'. Their software was developed in such a way that it caters to the needs of all sections. Since its inception, 'Quick Solutions' was earning enough revenue to cover the costs. They faced countless challenges from competing with large and well-established companies. But they learned from their mistakes and continuously improved their product. Slowly, their software gained recognition for its quality and uniqueness, and the company started making profits. It was a big incentive for Sana and Mihir for the continued successful operation of the enterprise. Within two years, the customer base increased manifold. Now, Sana and Mihir decided to make additional capital investment and hired more employees. They invested in research and development and expanded their product line to meet emerging market demands. ‘Quick Solutions' ultimately became the industry leader because of the tireless efforts of Sana and Mihir. Quoting lines, identify and state the objectives of management discussed in the above case which 'Quick Solutions' seeks to achieve.

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 real roots of the equation \( 12x^\frac{1}{3} - 25x^\frac{1}{6} + 12 = 0 \), if \( \alpha>\beta \), then \( 6\sqrt{\frac{\alpha}{\beta}} = \):

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 \( 11(a + b + c) \)?

If \( A = \begin{pmatrix} 9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2 \end{pmatrix} \) and

\[ A^T A^{-2} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

then

\[ \sum_{1 \leq i \leq 3} \sum_{1 \leq j \leq 3} a_{ij} \]

is:

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

The temperature at which the rms speed of oxygen molecules is 75\% of the rms speed of nitrogen molecules at a temperature of \( 287^\circ C \) is:

Match the following: