List of practice Questions

The range of the real-valued function \( f(x) = \sin^{-1} \left( \frac{1 + x^2}{2x} \right) + \cos^{-1} \left( \frac{2x}{1 + x^2} \right) \) is:
In a triangle ABC, if \( r : R = 1 : 3 : 7 \), then \( \sin(A + C) + \sin B = \)
2.644 g of metal (M) was deposited when 8040 coulombs of electricity was passed through molten MF\(_2\) salt. What is the atomic mass of M? (F = 96500 C mol\(^{-1}\))
The set of all values of \(k\) for which the inequality \(x^2 - (3k+1)x + 4k^2 + 3k - 3>0\) is true for all real values of \(x\) is
If \(\sec \theta + \tan \theta = \frac{1}{3}\), then the quadrant in which \(2\theta\) lies is:
In a triangle ABC, if \( r_1 = 2r_2 = 3r_3 \), then \(\sin A\): \(\sin B\): \(\sin C\) = Options:
Among the 4-digit numbers formed using the digits \( 0, 1, 2, 3, 4 \) when repetition of digits is allowed, the number of numbers which are divisible by 4 is:

जिन स्वरों का उच्चारण केवल मुख से होता है, उन्हें कहते हैं-

At 300 K, \( \Delta_r G^\circ \) for the reaction \( A(g) \rightleftharpoons B(g) \) is \( -11.5 \) kJ mol\(^{-1}\). The equilibrium constant at 300 K is approximately: \((R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1})\)
A slab consists of two identical plates of copper and brass. The free face of the brass is at \( 0^\circ C \) and that of copper at \( 100^\circ C \). If the thermal conductivities of brass and copper are in the ratio \( 1:4 \), then the temperature of the interface is:

If

 
and \( AA^T = I \), then \( \frac{a}{b} + \frac{b}{a} = \):

125 identical charged small spheres coalesce to form a big charged sphere. If the electric potential on each small sphere is 60 mV, then the electric potential on the bigger sphere formed is:
Which among the following compounds show metal excess defect due to anionic vacancy?
In a region, the electric field is \( (30\hat{i} + 40\hat{j}) \) NC\(^{-1}\). If the electric potential at the origin is zero, the electric potential at the point (1 m, 2 m) is
If the roots of the quadratic equation \( x^2 - 35x + c = 0 \) are in the ratio 2:3 and \( c = 6K \), then \( K \) is:
If a direct common tangent is drawn to the circles \( x^2 + y^2 - 6x + 4y + 9 = 0 \) and \( x^2 + y^2 + 2x - 2y + 1 = 0 \) that touches the circles at points \( A \) and \( B \), then \( AB = \):
Number of moles of ions produced by the complete dissociation of one mole of Mohr’s salt in water is:
Eight drops of mercury of equal radii and possessing equal charge combine to form a big drop. If the capacity of each drop is \( C \), then capacity of the big drop is:
The time period of revolution of a satellite close to the planet’s surface is 80 minutes. The time period of another satellite which is at a height of 3 times the radius of the planet from the surface is:
Choose the appropriate word pair to complete the sentence: I didn’t know Rahul was in the hospital. If I __________, I would __________ him.
A mass \( M \), attached to a horizontal spring executes simple harmonic motion with amplitude \( A_1 \). When mass \( M \) passes mean position, then a smaller mass \( m \) is attached to it, and both of them together execute simple harmonic motion with amplitude \( A_2 \). Then the value of \( \frac{A_1}{A_2} \) is:
If \[ \frac{13x+43}{2x^2 + 17x + 30} = \frac{A}{2x+5} + \frac{B}{x+6} \text{ then } A + B = \]
When 80 J of heat is absorbed by a monotonic gas, its volume increases by \( 16 \times 10^5 \) \( m^3 \). The pressure of the gas is:

A person climbs up a conveyor belt with a constant acceleration. The speed of the belt is \( \sqrt{\frac{g h}{6}} \) and the coefficient of friction is \( \frac{5}{3\sqrt{3}} \). The time taken by the person to reach from A to B with maximum possible acceleration is: 

A light body of momentum \( P_L \) and a heavy body of momentum \( P_H \), both have the same kinetic energy, then: