List of practice Questions

Let three real numbers a,b,c be in arithmetic progression and a + 1, b, c + 3 be in geometric progression. If a>10 and the arithmetic mean of a,b and c is 8, then the cube of the geometric mean of a,b and c is

Let the mean and the variance of 6 observation a,b, 68, 44, 48, 60 be 55 and 194, respectively if a > b, then a + 3b is

Let a relation \( R \) on \( \mathbb{N} \times \mathbb{N} \) be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$ 
Consider the two statements:
[(I)] \( R \) is reflexive but not symmetric. 
[(II)] \( R \) is transitive.
Then which one of the following is true:

The area of the region enclosed by the parabola \( y = 4x - x^2 \) and \( 3y = (x - 4)^2 \) is equal to

Let \( f : \mathbb{R} \rightarrow (0, \infty) \) be a strictly increasing function such that \(\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1.\)Then, the value of \(\lim_{x \to \infty} \left[ \frac{f(5x)}{f(x)} - 1 \right]\)is equal to

Let C be a circle with radius \( \sqrt{10} \) units and centre at the origin. Let the line \( x + y = 2 \) intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope \(-1\). Then, a distance (in units) between the chord PQ and the chord MN is

The temperature \( T(t) \) of a body at time \( t = 0 \) is \( 160^\circ \)F and it decreases continuously as per the differential equation \[ \frac{dT}{dt} = -K(T - 80), \] **where \( K \) is a positive constant. If \( T(15) = 120^\circ \)F, then \( T(45) \) is equal to

Let \( P \) be a parabola with vertex \( (2, 3) \) and directrix \( 2x + y = 6 \). Let an ellipse \( E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \), of eccentricity \( \frac{1}{\sqrt{2}} \) pass through the focus of the parabola \( P \). Then the square of the length of the latus rectum of \( E \) is

Let \( f, g : (0, \infty) \rightarrow \mathbb{R} \) be two functions defined by \(f(x) = \int_{-x}^{x} (|t| - t^2) e^{-t^2} \, dt \quad \text{and} \quad g(x) = \int_{0}^{x} t^{1/2} e^{-t} \, dt.\)Then the value of \( f \left( \sqrt{\log_e 9} \right) + g \left( \sqrt{\log_e 9} \right) \) is equal to

Let \( (\alpha, \beta, \gamma) \) be the mirror image of the point \( (2, 3, 5) \) in the line \(\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}\). Then \( 2\alpha + 3\beta + 4\gamma \) is equal to

Let a variable line passing through the centre of the circle x² + y² – 16x – 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \( z_1 + z_2 = 5 \) and \( z_1^3 + z_2^3 = 20 + 15i \). Then \( \left| z_1^4 + z_2^4 \right| \) equals

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

The following data were obtained during the first order thermal decomposition of a gas A at constant volume:\[\text{A(g)} \rightarrow 2\text{B(g)} + \text{C(g)}\]\[\begin{array}{|c|c|c|}\hline\text{S.No} & \text{Time/s} & \text{Total pressure/(atm)} \\\hline1 & 0 & 0.1 \\2 & 115 & 0.28 \\\hline\end{array}\]The rate constant of the reaction is _____ $\times 10^{-2}$ s$^{-1}$ (nearest integer).

Following Kjeldahl's method, 1g of organic compound released ammonia that neutralized 10 mL of 2M H\(_2\)SO\(_4\). The percentage of nitrogen in the compound is _________.%.

The number of tripeptides formed by three different amino acids using each amino acid once is ______.

Mass of ethylene glycol (antifreeze) to be added to 18.6 kg of water to protect the freezing point at $-24^\circ C$ is ____ kg (Molar mass in g mol$^{-1}$ for ethylene glycol = 62, $K_f$ of water = 1.86 K kg mol$^{-1}$).

Total number of isomeric compounds (including stereoisomers) formed by monochlorination of 2-methylbutane is________.

Solubility of calcium phosphate (molecular mass, \( M \)) in water is \( W_g \) per 100 mL at \( 25^\circ C \). Its solubility product at \( 25^\circ C \) will be approximately:

Given below are two statements :
Statement (I) : Dimethyl glyoxime forms a six-membered covalent chelate when treated with NiCl2 solution in presence of NH4OH.
Statement (II) : Prussian blue precipitate contains iron both in (+2) and (+3) oxidation states. In the light of the above statements,
choose the most appropriate answer from the options given below:

Acid D formed in above reaction is :

A light planet is revolving around a massive star in a circular orbit of radius $R$ with a period of revolution $T$. If the force of attraction between the planet and the star is proportional to $R^{-\frac{3}{2}}$, then choose the correct option:

\([ \text{Co(NH}_3)_6 ]^{3+}\) and \([ \text{CoF}_6 ]^{3-}\) are respectively known as:

A body of mass $4 \, \text{kg}$ experiences two forces \[\vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k} \quad \text{and} \quad \vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k}.\]The acceleration acting on the body is:

Given below are two statements:
Statement (I): \( \text{SiO}_2 \) and \( \text{GeO}_2 \) are acidic while \( \text{SnO} \) and \( \text{PbO} \) are amphoteric in nature.
Statement (II): Allotropic forms of carbon are due to property of catenation and \( p\pi-d\pi \) bond formation.
In the light of the above statements, choose the most appropriate answer from the options given below: