List of practice Questions

The number of points of discontinuity of the function $ f(x) = \left\lfloor \frac{x^2}{2} \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor, \quad x \in [0, 4], $ where $ \left\lfloor \cdot \right\rfloor $ denotes the greatest integer function, is:

For a diatomic gas, if $ \gamma_1 = \frac{C_P}{C_V} $ for rigid molecules and $ \gamma_2 = \frac{C_P}{C_V} $ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (where $ C_P $ and $ C_V $ are specific heats of the gas at constant pressure and volume)

Let $ \mathbf{a} = 2\hat{i} - \hat{j} + 3\hat{k} $, $ \mathbf{b} = 3\hat{i} - 5\hat{j} + \hat{k} $, and $ \mathbf{c} $ be a vector such that $ \mathbf{a} \times \mathbf{c} = \mathbf{a} \times \mathbf{b} $ and \[ (\mathbf{a} + \mathbf{c}) \cdot (\mathbf{b} + \mathbf{c}) = 168. \] Then the maximum value of $ | \mathbf{c} |^2 $ is:

If the equation of the hyperbola with foci $ (4, 2) $ and $ (8, 2) $ is $ 3x^2 - y^2 - \alpha x + \beta y + \gamma = 0 $, then $ \alpha + \beta + \gamma $ is equal to _______

Let $ f : [1, \infty) \to [2, \infty) $ be a differentiable function. If

$ 10 \int_{1}^{x} f(t) \, dt = 5x f(x) - x^5 - 9 $ for all $ x \ge 1 $, then the value of $ f(3) $ is ______.

What is the freezing point depression constant of a solvent, 50 g of which contain 1 g non-volatile solute (molar mass 256 g mol$^{-1}$) and the decrease in freezing point is 0.40 K?

For the reaction:

The correct order of set of reagents for the above conversion is :

If \[ \text{C(diamond)} \rightarrow \text{C(graphite)} + X \, \text{kj mol}^{-1} \] \[ \text{C(diamond)} + \text{O}_2(g) \rightarrow \text{CO}(g) + Y \, \text{kj mol}^{-1} \] \[ \text{C(graphite)} + \text{O}_2(g) \rightarrow \text{CO}(g) + Z \, \text{kj mol}^{-1} \] At constant temperature. Then:

Match the LIST-I with LIST-II

\[ \begin{array}{|l|l|} \hline \text{LIST-I} & \text{LIST-II} \\ \hline \text{A. Gravitational constant} & \text{I. } [LT^{-2}] \\ \hline \text{B. Gravitational potential energy} & \text{II. } [L^2T^{-2}] \\ \hline \text{C. Gravitational potential} & \text{III. } [ML^2T^{-2}] \\ \hline \text{D. Acceleration due to gravity} & \text{IV. } [M^{-1}L^3T^{-2}] \\ \hline \end{array} \]

Choose the correct answer from the options given below:

Twins are born to a family that lives next door to you. The twins are a boy and a girl. Which of the following must be true?

Let the product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

If the locus of $ z \in \mathbb{C} $, such that $ \text{Re} \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{ \bar{z} - 1}{2 \bar{z} - i} \right) = 2, $ is a circle of radius $ r $ and center $ (a, b) $, then $ \frac{15ab}{r^2} \text{ is equal to:} $

If $ A $ is a square matrix such that $ \text{adj}(\text{adj}(A)) = A $, then $ |A| $ is:

A bead of mass $ m $ slides without friction on the wall of a vertical circular hoop of radius $ R $ as shown in figure. The bead moves under the combined action of gravity and a massless spring $ k $ attached to the bottom of the hoop. The equilibrium length of the spring is $ R $. If the bead is released from the top of the hoop with (negligible) zero initial speed, the velocity of the bead, when the length of spring becomes $ R $, would be (spring constant is $ k $, $ g $ is acceleration due to gravity):

10 mL of 2 M NaOH solution is added to 20 mL of 1 M HCl solution kept in a beaker. Now, 10 mL of this mixture is poured into a volumetric flask of 100 mL containing 2 moles of HCl and made the volume upto the mark with distilled water. The solution in this flask is :

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw is:

Which of these is aromatic in nature?

The correct orders among the following are:
[A.] Atomic radius : $B<Al<Ga<In<Tl$
[B.] Electronegativity : $Al<Ga<In<Tl<B$
[C.] Density : $Tl<In<Ga<Al<B$
[D.] 1st Ionisation Energy :
In$<Al<Ga<Tl<B$
Choose the correct answer from the options given below :

Among SO₃, NF₃, NH₃, XeF₂, CIF$_3$, and SF₆, the hybridization of the molecule with non-zero dipole moment and one or more lone-pairs of electrons on the central atom is:

A body of mass $100 \;g$ is moving in a circular path of radius $2\; m$ on a vertical plane as shown in the figure. The velocity of the body at point A is $10 m/s.$ The ratio of its kinetic energies at point B and C is: (Take acceleration due to gravity as $10 m/s^2$)

A vessel with square cross-section and height of 6 m is vertically partitioned. A small window of $ 100 \, \text{cm}^2 $ with hinged door is fitted at a depth of 3 m in the partition wall. One part of the vessel is filled completely with water and the other side is filled with the liquid having density $ 1.5 \times 10^3 \, \text{kg/m}^3 $. What force one needs to apply on the hinged door so that it does not open?

Let the domain of the function $ f(x) = \cos^{-1} \left( \frac{4x + 5}{3x - 7} \right) $ be $ [\alpha, \beta] $ and the domain of $ g(x) = \log_2 \left( 2 - 6 \log_2 \left( 2x + 5 \right) \right) $ be $ (\gamma, \delta) $. Then $ |7(\alpha + \beta) + 4(\gamma + \delta)| $ is equal to:

Student to attempt either option-(A) or (B):
(A) Write the features a molecule should have to act as a genetic material. In the light of the above features, evaluate and justify the suitability of the molecule that is preferred as an ideal genetic material.
OR
(B) Differentiate between the following:

[(i)] Polygenic Inheritance and Pleiotropy
[(ii)] Dominance, Codominance and Incomplete dominance

A monoatomic gas is stored in a thermally insulated container. The gas is suddenly compressed to $ \frac{1}{8} $th of its initial volume. Find the ratio of final pressure to initial pressure.

Let $ C_{t-1} = 28, C_t = 56 $ and $ C_{t+1} = 70 $. Let $ A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \text{ and } C(3r - n_1, r^2 - n - 1) $ be the vertices of a triangle ABC, where $ t $ is a parameter. If $ (3x - 1)^2 + (3y)^2 = \alpha $ is the locus of the centroid of triangle ABC, then $ \alpha $ equals: