List of practice Questions

An object of mass \( m \) is projected from the origin in a vertical \( xy \)-plane at an angle \( 45^\circ \) with the x-axis with an initial velocity \( v_0 \). The magnitude and direction of the angular momentum of the object with respect to the origin, when it reaches the maximum height, will be:

What is the time period of a simple pendulum of length \( L \) and acceleration due to gravity \( g \)?

Aman has been asked to synthesise the molecule:

Using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory.
He thought of performing ozonolysis reaction on the alkene to
produce a dicarbonyl compound followed by aldol reaction to prepare "x".
Predict the suitable alkene that can lead to the formation of "x".

A car of mass \( m \) moves on a banked road having radius \( r \) and banking angle \( \theta \). To avoid slipping from the banked road, the maximum permissible speed of the car is \( v_0 \). The coefficient of friction \( \mu \) between the wheels of the car and the banked road is:

If the equation of a circle is \( 4x^2 + 4y^2 - 12x + 8y = 0 \), what is the radius of the circle?

An alternating current is given by \[ I = I_A \sin \omega t + I_B \cos \omega t. \] The r.m.s. current will be:

For a reaction, \[ {N}_2{O}_5(g) \rightarrow 2{NO}_2(g) + \frac{1}{2} {O}_2(g) \] in a constant volume container, no products were present initially. The final pressure of the system when 50% of the reaction gets completed is:

Let in a \( \triangle ABC \), the length of the side AC is 6, the vertex B is \( (1, 2, 3) \) and the vertices A, C lie on the line \[ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. \] Then the area (in sq. units) of \( \triangle ABC \) is:

Let the foci of a hyperbola be \( (1, 14) \) and \( (1, -12) \). If it passes through the point \( (1, 6) \), then the length of its latus-rectum is:

In the above diagram, the standard electrode potentials are given in volts (over the arrow). The value of \( E^\circ_{\text{FeO}_4^{2-}/\text{Fe}^{2+}} \) is:

If \(\sum\)\(_{r=1}^n T_r\) = \(\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\) , then \( \lim_{n \to \infty} \sum_{r=1}^n \frac{1}{T_r} \) is equal to :

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is \(\frac{m}{n}\), where gcd(m, n) = 1, then m + n is equal to:

The interior angles of a polygon with \( n \) sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then \( n \) is equal to:

The major product of the following reaction is:

The incorrect statements among the following is:

Given below are two statements:
Statement I: Fructose does not contain an aldehydic group but still reduces Tollen’s reagent.
Statement II: In the presence of base, fructose undergoes rearrangement to give glucose.
In the light of the above statements, choose the correct answer from the options given below.

A closed organ and an open organ tube filled by two different gases having the same bulk modulus but different densities \( \rho_1 \) and \( \rho_2 \), respectively. The frequency of the 9th harmonic of the closed tube is identical with the 4th harmonic of the open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is \( \rho_1 : \rho_2 = 1 : 16 \), then the length of the open tube is:

Standard electrode potentials for a few half-cells are mentioned below: \[ \begin{array}{|c|c|} \hline \text{Half-cell} & \text{Standard Electrode Potential}
\hline \text{Cu}^{2+} / \text{Cu} & +0.34 \, \text{V}
\text{Zn}^{2+} / \text{Zn} & -0.76 \, \text{V}
\text{Ag}^+ / \text{Ag} & +0.80 \, \text{V}
\text{Mg}^{2+} / \text{Mg} & -2.37 \, \text{V}
\hline \end{array} \] Which one of the following cells gives the most negative value of \( \Delta G^\circ \)?

Let \[ f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{\tan \left( \frac{x}{2^{r+1}} \right) + \tan^3 \left( \frac{x}{2^{r+1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{r+1}} \right)} \right) \] Then, \( \lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)} \) is equal to:

Match the LIST-I with LIST-II

Match the LIST-I with LIST-II:

Choose the correct answer from the options given below :

Two soap bubbles of radius 2 cm and 4 cm, respectively, are in contact with each other. The radius of curvature of the common surface, in cm, is _______________.

A bob of mass \(m\) is suspended at a point \(O\) by a light string of length \(l\) and left to perform vertical motion (circular) as shown in the figure. Initially, by applying horizontal velocity \(v_0\) at the point ‘A’, the string becomes slack when the bob reaches at the point ‘D’. The ratio of the kinetic energy of the bob at the points B and C is:

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \( r \) be the radius of a circle that has centre at the point \( (2, 5) \) and intersects the circle C at exactly two points. If the set of all possible values of \( r \) is the interval \( (\alpha, \beta) \), then \( 3\beta - 2\alpha \) is equal to:

A parallel-plate capacitor of capacitance 40µF is connected to a 100 V power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant K = 2. Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are -