AP EAPCET (AP EAMCET) 2024 Question Paper May 23 Shift 1 (Available): Download MPC Question Paper with Answer Key PDF

If \( A \subseteq \mathbb{Z} \) and the function \( f: A \to \mathbb{R} \) is defined by \[ f(x) = \frac{1}{\sqrt{64 - (0.5)^{24+x- x^2} }} \]
then the sum of all absolute values of elements of \( A \) is:

• (1) \( 36 \)
• (2) \( 5 \)
• (3) \( 25 \)
• (4) \( 11 \)

Which of the following functions are odd?

• (1) II
• (2) I, II
• (3) III
• (4) I

The \( n^{th} \) term of the series \[ 1 + (3+5+7) + (9+11+13+15+17) + \dots \]
is:

• (1) \( (2n+1) \left[n^2 - (n-1)^2 \right] \)
• (2) \( (2n-1) \left[(n-1)^2 - n^2 \right] \)
• (3) \( (2n+1) \left[(n-1)^2 - n^2 \right] \)
• (4) \( (2n-1) \left[(n-1)^2 + n^2 \right] \)

If A = , then \(det (A - A^T) = \):

• (1) \( 0 \)
• (2) \( -7851 \)
• (3) \( 2442 \)
• (4) \( 1 \)

If the matrix is:


\(>\) 0, \text{ then abc \(>\) ?

• (1) \( 1 \)
• (2) \( -8 \)
• (3) \( 8 \)
• (4) \( 3 \)

If the system of equations: \[ a_1 x + b_1 y + c_1 z = 0, \quad a_2 x + b_2 y + c_2 z = 0, \quad a_3 x + b_3 y + c_3 z = 0 \]
has only the trivial solution, then the rank of the matrix:





is:

• (1) \( 2 \)
• (2) \( 1 \)
• (3) \( 3 \)
• (4) \( 0 \)

If \( \omega \) is a complex cube root of unity and if \( Z \) is a complex number satisfying \( |Z - 1| \leq 2 \) and \[ |\omega^2 Z - 1 - \omega| = a, \]
then the set of possible values of \( a \) is:

• (1) \( 0 \leq a \leq 2 \)
• (2) \( \frac{1}{2} \leq a \leq \frac{\sqrt{3}}{2} \)
• (3) \( |\omega| \leq a \leq \frac{\sqrt{3}}{2} + 2 \)
• (4) \( 0 \leq a \leq 4 \)

If the roots of the equation \[ Z^3 + iZ^2 + 2i = 0 \]
are the vertices of a triangle ABC, then that triangle ABC is:

• (1) A right-angled triangle
• (2) An equilateral triangle
• (3) An isosceles triangle
• (4) A right-angled isosceles triangle

If \( (r, \theta) \) denotes \( r (\cos \theta + i \sin \theta) \). If \[ x = (1, \alpha), \quad y = (1, \beta), \quad z = (1, \gamma) \]
and \( x + y + z = 0 \), then \[ \sum \cos (2\alpha - \beta - \gamma) = \]

• (1) \( 3 \)
• (2) \( 0 \)
• (3) \( 1 \)
• (4) \( -1 \)

The set of all real values of \(x\) satisfying the inequality \(\frac{7x^2 - 5x - 18}{2x^2 + x - 6} < 2\) is

• (1) \((-\infty, -\frac{2}{3}] \cup [3, \infty)\)
• (2) \((-2, -\frac{2}{3}) \cup (\frac{3}{2}, 3)\)
• (3) \((-\infty, -2) \cup (\frac{3}{2}, \infty)\)
• (4) \([-\frac{2}{3}, \frac{3}{2})\)

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

• (1) \((-\frac{13}{7}, 1)\)
• (2) \((-1, \frac{13}{7})\)
• (3) \((-\infty, -\frac{13}{7}) \cup (1, \infty)\)
• (4) \((-\infty, -1) \cup (\frac{13}{7}, \infty)\)

The cubic equation whose roots are the squares of the roots of the equation \( 12x^3 - 20x^2 + x + 3 = 0 \) is:

• (1) \( x^3 + 376x^2 - 121x - 9 = 0 \)
• (2) \( 144x^3 - 400x^2 + 121x + 98 = 0 \)
• (3) \( 144x^3 - 376x^2 + 121x - 9 = 0 \)
• (4) \( x^3 + 400x^2 - 121x - 98 = 0 \)

If \( \alpha, \beta, \gamma \) are the roots of the equation \[ x^3 + 3x^2 - 10x - 24 = 0. \]
If \( \alpha(\beta + \gamma), \beta(\gamma + \alpha) \), and \( \gamma(\alpha + \beta) \) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \]
then find the value of \( q \).

• (1) \( -44 \)
• (2) \( -28 \)
• (3) \( 44 \)
• (4) \( 28 \)

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:

• (1) \( 140 \)
• (2) \( 160 \)
• (3) \( 180 \)
• (4) \( 200 \)

The number of ways of arranging 2 red, 3 white, and 5 yellow roses of different sizes into a garland such that no two yellow roses come together is:

• (1) \( 2880 \)
• (2) \( 144 \)
• (3) \( 1440 \)
• (4) \( 288 \)

The number of ways of selecting 3 numbers that are in GP from the set \( \{1, 2, 3, \dots, 100\} \) is:

• (1) 18
• (2) 52
• (3) 14
• (4) 53

The independent term in the expansion of \( (1 + x + 2x^2) \left( \frac{3x^2}{2} - \frac{1}{3x} \right)^9 \) is:

• (1) \( \frac{18}{7} \)
• (2) \( \frac{7}{18} \)
• (3) \( \frac{-7}{18} \)
• (4) \( \frac{-18}{7} \)

For \(|x|<\frac{1}{\sqrt{2}}\) the coefficient of \(x\) in the expansion of \(\frac{(1-4x)^2(1-2x^2)^{1/2}}{(4-x)^{3/2}}\) is

• (1) \(\frac{61}{64}\)
• (2) \(-\frac{61}{64}\)
• (3) \(\frac{69}{64}\)
• (4) \(-\frac{69}{64}\)

Given the partial fraction decomposition: \[ \frac{4x^2 + 5}{(x - 2)^4} = \frac{A}{(x - 2)} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} + \frac{D}{(x - 2)^4} \]
then the value of \[ \sqrt{\frac{A}{C} + \frac{B}{C} + \frac{D}{C}} \]
is:

• (1) \( \frac{\sqrt{29}}{4} \)
• (2) \( \frac{\sqrt{23}}{4} \)
• (3) \( \frac{5}{4} \)
• (4) \( \frac{4}{5} \)

Evaluate the sum: \[ \tan^2 \frac{\pi}{16} + \tan^2 \frac{2\pi}{16} + \tan^2 \frac{3\pi}{16} + \tan^2 \frac{4\pi}{16} + \tan^2 \frac{5\pi}{16} + \tan^2 \frac{6\pi}{16} + \tan^2 \frac{7\pi}{16} \]

• (1) \( 35 \)
• (2) \( 41 \)
• (3) \( 37 \)
• (4) \( 33 \)

Evaluate the sum: \[ \sin^2 18^\circ + \sin^2 24^\circ + \sin^2 36^\circ + \sin^2 42^\circ + \sin^2 78^\circ + \sin^2 90^\circ + \sin^2 96^\circ + \sin^2 102^\circ + \sin^2 138^\circ + \sin^2 162^\circ. \]

• (1) \( \frac{11}{2} \)
• (2) \( \frac{9}{2} \)
• (3) \( 5 \)
• (4) \( 4 \)

If A, B, C are the angles of a triangle, then \(\frac{\sin A + \sin B + \sin C}{\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} - 1} =\)

• (1) \( -2 \tan \frac{B}{2} \)
• (2) \( -2 \cot \frac{B}{2} \)
• (3) \( 2 \tan \frac{B}{2} \)
• (4) \( 2 \cot \frac{B}{2} \)

The general solution of \( \cot \frac{x}{2} - \cot x = \csc \frac{x}{2} \) is:

• (1) \( \{ 2n\pi + \frac{\pi}{3} | n \in \mathbb{Z} \} \)
• (2) \( \{ 4n\pi + \frac{\pi}{3} | n \in \mathbb{Z} \} \)
• (3) \( \{ 2n\pi + \frac{2\pi}{3} | n \in \mathbb{Z} \} \)
• (4) \( \{ 4n\pi \pm \frac{2\pi}{3} | n \in \mathbb{Z} \} \)

If \( 0 < x < \frac{1}{2} \) and \( \alpha = \sin^{-1} x + \cos^{-1} \left(\frac{x}{2} +\frac{\sqrt{3} - 3x^2}{2} \right) \), then \( \tan \alpha + \cot \alpha = \):

• (1) \( \frac{4}{\sqrt{3}} \)
• (2) \( 4\sqrt{3} \)
• (3) \( \frac{4x}{1 - x^2} \)
• (4) \( x\sqrt{1 - x^2} \)

Evaluate \( \cosh (\log 4) \):

• (1) \( \frac{8}{17} \)
• (2) \( \frac{17}{8} \)
• (3) \( 0 \)
• (4) \( \frac{9}{8} \)

In \( \triangle ABC \), prove the identity: \[ a^2 \sin 2B + b^2 \sin 2A = \]

• (1) \( 2ab \cos A \)
• (2) \( 2ab \sin A \)
• (3) \( 2ab \sin C \)
• (4) \( 2ab \cos C \)

In \( \triangle ABC \), evaluate: \[ \frac{r_2 (r_1 + r_3)}{\sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}}. \]

• (1) \( a \)
• (2) \( b \)
• (3) \( c \)
• (4) \( s \)

In \( \triangle ABC \), \( (r_2 + r_3) \csc^2 \frac{A}{2} =\)

• (1) \( 4R \)
• (2) \( 4R \cot^2 \frac{A}{2} \)
• (3) \( 4R \tan^2 \frac{A}{2} \)
• (4) \( R \tan^2 \frac{A}{2} \)

If the vectors \(a\bar{i} + \bar{j} + \bar{k}\), \(\bar{i} + b\bar{j} + \bar{k}\), \(\bar{i} + \bar{j} + c\bar{k}\) (\(a \ne b \ne c \ne 1\)) are coplanar, then \(\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =\)

• (1) \(0\)
• (2) \(1\)
• (3) \(2\)
• (4) \(-1\)

If \( AB = 2i + 3j - 6k \), \( BC = 6i - 2j + 3k \) are the vectors along two sides of a triangle ABC, then the perimeter of triangle ABC is:

• (1) 21
• (2) \( \sqrt{74} + 14 \)
• (3) \( \sqrt{74} + 19 \)
• (4) \( \sqrt{74} + 3 \)

The orthogonal projection vector of \( \bar{a} = 2\bar{i} + 3\bar{j} + 3\bar{k} \) on \( \bar{b} = \bar{i} - 2\bar{j} + \bar{k} \) is:

• (1) \( -\frac{1}{6} (2\bar{i} + 3\bar{j} + 3\bar{k}) \)
• (2) \( \frac{1}{6} (-\bar{i} + 2\bar{j} - \bar{k}) \)
• (3) \( \bar{i} - 2\bar{j} + \bar{k} \)
• (4) \( -\bar{i} + 2\bar{j} - \bar{k} \)

If \( \bar{a} = -4\bar{i} + 2\bar{j} + 4\bar{k} \), \( \bar{b} = \sqrt{2} \bar{i} - \sqrt{2} \bar{j} \) are two vectors, then the angle between the vectors \( 2\bar{a} \) and \( \frac{\bar{b}}{2} \) is:

• (1) \( 30^\circ \)
• (2) \( 135^\circ \)
• (3) \( 90^\circ \)
• (4) \( 0^\circ \)

A unit vector perpendicular to the vectors \( \bar{a} = 2\bar{i} + 3\bar{j} + 4\bar{k} \) and \( \bar{b} = 3\bar{j} + 2\bar{k} \) is:

• (1) \( \frac{3\bar{i} + 2\bar{j} - 2\bar{k}}{\sqrt{22}} \)
• (2) \( \frac{3\bar{i} + 2\bar{j} - 3\bar{k}}{\sqrt{22}} \)
• (3) \( \frac{3\bar{i} - 2\bar{j} + 3\bar{k}}{\sqrt{22}} \)
• (4) \( \frac{3\bar{i} + 2\bar{j} + 3\bar{k}}{\sqrt{22}} \)

If the mean of the data 7, 8, 9, 7, 8, 7, \(\lambda\), 8 is 8, then the variance of the data:

• (1) 2
• (2) \( \frac{7}{8} \)
• (3) \( \frac{9}{8} \)
• (4) 1

When two dice are thrown, the probability of getting the sum of the values on them as 10 or 11 is:

• (1) \( \frac{7}{36} \)
• (2) \( \frac{5}{36} \)
• (3) \( \frac{5}{18} \)
• (4) \( \frac{7}{18} \)

It is given that in a random experiment, events A and B are such that \( P(A) = \frac{1}{4} ,P(A|B) = \frac{1}{2} \) and \( P(B|A) = \frac{2}{3} \). Then \( P(B) \) is:

• (1) \( \frac{1}{3} \)
• (2) \( \frac{2}{3} \)
• (3) \( \frac{1}{2} \)
• (4) \( \frac{1}{6} \)

The probability that A speaks truth is 75% and the probability that B speaks truth is 80%. The probability that they contradict each other when asked to speak on a fact is:

• (1) \( \frac{3}{20} \)
• (2) \( \frac{4}{20} \)
• (3) \( \frac{7}{20} \)
• (4) \( \frac{5}{20} \)

Bag A contains 2 white and 3 red balls, and Bag B contains 4 white and 5 red balls. If one ball is drawn at random from one of the bags and is found to be red, then the probability that it was drawn from Bag B is:

• (1) \( \frac{23}{54} \)
• (2) \( \frac{25}{51} \)
• (3) \( \frac{25}{52} \)
• (4) \( \frac{27}{55} \)

If the probability distribution of a random variable \( X \) is given as follows, then find \( k \):

• (1) \( \frac{1}{10} \)
• (2) \( \frac{2}{10} \)
• (3) \( \frac{3}{10} \)
• (4) \( \frac{4}{10} \)

In a Binomial distribution \( B(n,p) \), the sum and product of the mean and the variance are 5 and 6 respectively, then \( 6(n + p - q) = \):

• (1) 50
• (2) 53
• (3) 52
• (4) 51

The locus of the midpoint of the portion of the line \( x \cos \alpha + y \sin \alpha = p \) intercepted by the coordinate axes, where \( p \) is a constant, is:

• (1) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{3}{p^2} \)
• (2) \( \frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2} \)
• (3) \( x^2 + y^2 = 2p^2 \)
• (4) \( \frac{2}{x^2} + \frac{1}{p^2} = 1 \)

The origin is shifted to the point \( (2, 3) \) by translation of axes and then the coordinate axes are rotated about the origin through an angle \( \theta \) in the counter-clockwise sense. Due to this if the equation \( 3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0 \) is transformed to \( 4x^2 + 2y^2 - 1 = 0 \), then the angle \( \theta = \):

• (1) \( \frac{\pi}{4} \)
• (2) \( \frac{\pi}{3} \)
• (3) \( \frac{\pi}{6} \)
• (4) \( \frac{\pi}{2} \)

If the straight line passing through \( P(3,4) \) makes an angle \( \frac{\pi}{6} \) with the positive x-axis in the anticlockwise direction and meets the line \( 12x + 5y + 10 = 0 \) at \( Q \), then the length of the segment \( PQ \) is:

• (1) \( \frac{64}{12\sqrt{2} + 1} \)
• (2) \( \frac{96}{9\sqrt{2} - 1} \)
• (3) \( \frac{112}{10\sqrt{3} + 3} \)
• (4) \( \frac{132}{12\sqrt{3} + 5} \)

The equations of the perpendicular bisectors of the sides AB and AC of \( \triangle ABC \) are \( x - y + 5 = 0 \) and \( x + 2y = 0 \) respectively. If the coordinates of \( A \) are \( (1, -2) \), then the equation of the line BC is:

• (1) \( 14x + 23y - 40 = 0 \)
• (2) \( 13x - 9y - 14 = 0 \)
• (3) \( 9x - 14y - 25 = 0 \)
• (4) \( 8x + 15y - 30 = 0 \)

A pair of lines drawn through the origin forms a right-angled isosceles triangle with right angle at the origin with the line \( 2x + 3y = 6 \). The area (in square units) of the triangle thus formed is:

• (1) \( \frac{36}{13} \)
• (2) \( \frac{32}{13} \)
• (3) \( \frac{18}{5} \)
• (4) \( \frac{25}{9} \)

The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve \( x^2 + y^2 + xy + x + 3y + 1 = 0 \) and the line \( x + y + 2 = 0 \) is:

• (1) \( x^2 + 4xy - y^2 = 0 \)
• (2) \( x^2 - 4xy + y^2 = 0 \)
• (3) \( 2x^2 - 3xy + y^2 = 0 \)
• (4) \( x^2 + 2xy - 3y^2 = 0 \)

The circumference of a circle passing through the point \( (4, 6) \) with two normals represented by \( 2x - 3y + 4 = 0 \) and \( x + y - 3 = 0 \) is:

• (1) \( 5\pi \)
• (2) \( 10\pi \)
• (3) \( 25\pi \)
• (4) \( 8\pi \)

If the line through the point \( P(5,3) \) meets the circle \( x^2 + y^2 - 2x - 4y + \alpha = 0 \) at \( A(4, 2) \) and \( B(x_1, y_1) \), then \( PA \times PB = \):

• (1) 6
• (2) 12
• (3) 9
• (4) 8

Consider the point \( P(\alpha, \beta) \) on the line \( 2x + y = 1 \). If the points \( P \) and \( (3,2) \) are conjugate points with respect to the circle \( x^2 + y^2 = 4 \), then find \( \alpha + \beta \):

• (1) \( 3 \)
• (2) \( -1 \)
• (3) \( -5 \)
• (4) \( 7 \)

If \( (1,3) \) is the midpoint of a chord of the circle \( x^2 + y^2 - 4x - 8y + 16 = 0 \), then the area of the triangle formed by that chord with the coordinate axes is:

• (1) \( 16 \)
• (2) \( 8 \)
• (3) \( 4 \)
• (4) \( 8\sqrt{2} \)

If the circles \( x^2 + y^2 + 2ax + 2y - 8 = 0 \) and \( x^2 + y^2 - 2x + ay - 14 = 0 \) intersect orthogonally, then the distance between their centers is:

• (1) \( \sqrt{242} \)
• (2) \( \sqrt{970} \)
• (3) \( \sqrt{629} \)
• (4) \( \sqrt{541} \)

If \( P \) is a point which divides the line segment joining the focus of the parabola \( y^2 = 12x \) and a point on the parabola in the ratio 1:2, then the locus of \( P \) is:

• (1) \( y^2 = 2(x - 2) \)
• (2) \( y^2 = 4x \)
• (3) \( y^2 = 4(x - 2) \)
• (4) \( y^2 = 9(x - 3) \)

Let \( T_1 \) be the tangent drawn at a point \( P(\sqrt{2}, \sqrt{3}) \) on the ellipse \( \frac{x^2}{4} + \frac{y^2}{6} = 1 \). If \( (a, \beta) \) is the point where \( T_1 \) intersects another tangent \( T_2 \) to the ellipse perpendicularly, then \( a^2 + \beta^2 = \):

• (1) 10
• (2) 52
• (3) 26
• (4) \( \frac{5}{12} \)

If \( y = x + \sqrt{2} \) is a tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), then equations of its directrices are:

• (1) \( x = \pm \sqrt{3} \)
• (2) \( x = \pm \sqrt{\frac{8}{3}} \)
• (3) \( x = \pm \sqrt{\frac{2}{3}} \)
• (4) \( x = \pm \sqrt{\frac{4}{3}} \)

The area of the quadrilateral formed with the foci of the hyperbola \[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]
and its conjugate hyperbola is (in square units):

• (1) \( 24 \)
• (2) \( 16 \)
• (3) \( 25 \)
• (4) \( 50 \)

The length of the internal bisector of angle A in \( \triangle ABC \) with vertices \( A(4,7,8) \), \( B(2,3,4) \), and \( C(2,5,7) \) is:

• (1) \( \frac{1}{3} \sqrt{29} \)
• (2) \( \frac{2}{3} \sqrt{29} \)
• (3) \( \frac{2}{3} \sqrt{34} \)
• (4) \( \frac{4}{3} \sqrt{34} \)

If the direction cosines of two lines are given by \[ l + m + n = 0 \quad and \quad mn - 2lm - 2nl = 0, \]
then the acute angle between those lines is:

• (1) \( \frac{2\pi}{5} \)
• (2) \( \frac{\pi}{3} \)
• (3) \( \frac{\pi}{4} \)
• (4) \( \frac{\pi}{60} \)

If the angle \( \theta \) between the line \( \frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2} \) and the plane \( 2x - y + \sqrt{\lambda}z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( \lambda \) is:

• (1) \( \frac{3}{5} \)
• (2) \( \frac{5}{4} \)
• (3) \( \frac{5}{3} \)
• (4) \( \frac{4}{3} \)

Let \( f(x) = \begin{cases} 1 + \frac{2x}{a}, & 0 \le x \le 1
ax, & 1 < x \le 2 \end{cases} \). If \( \lim_{x \to 1} f(x) \) exists, then the sum of the cubes of the possible values of \( a \) is:

• (1) 1
• (2) 5
• (3) 7
• (4) 9

Let \( [P] \) denote the greatest integer \( \leq P \). If \( 0 \leq a \leq 2 \), then the number of integral values of \( a \) such that \( \lim_{x \to a} [x^2] - [x]^2 \) does not exist is:

• (1) 3
• (2) 2
• (3) 1
• (4) 0

If \( f(x) = \begin{cases} \frac{\sqrt{a^2 - ax - x^2} - \sqrt{x^2 + ax + a^2}}{\sqrt{a + x} - \sqrt{a - x}}, & x \ne 0
K, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then \( K = \)

• (1) \( -\sqrt{a} \)
• (2) \( \sqrt{a} \)
• (3) \( -1 \)
• (4) \( a + \sqrt{a} \)

If \( y = \sinh^{-1} \left(\frac{1 - x}{1 + x} \right) \), then \( \frac{dy}{dx} \) is given by:

• (1) \( \frac{-\sqrt{2}}{|1 + x| \sqrt{1 + x^2}} \)
• (2) \( \frac{-1}{(1 + x) \sqrt{x}} \)
• (3) \( \frac{1}{(1 + x^2) \sqrt{1 + x}} \)
• (4) \( \frac{-\sqrt{2}}{(1 + x) \sqrt{1 - x}} \)

If \[ y = (x - 1)(x + 2)(x^2 + 5)(x^4 + 8), \]
then \[ \lim\limits_{x \to -1} \left( \frac{dy}{dx} \right) = ? \]

• (1) \( -30 \)
• (2) \( 30 \)
• (3) \( 52 \)
• (4) \( -52 \)

If \( f(x) = \begin{cases} ax^2 + bx - \frac{13}{8}, & x \le 1
3x - 3, & 1 < x \le 2
bx^3 + 1, & x > 2 \end{cases} \) is differentiable \(\forall x \in \mathbb{R}\), then \( a - b = \)

• (1) \( \frac{9}{8} \)
• (2) \( \frac{5}{4} \)
• (3) \( \frac{11}{8} \)
• (4) \( \frac{1}{4} \)

A is a point on the circle with radius 8 and center at O. A particle P is moving on the circumference of the circle starting from A. M is the foot of the perpendicular from P on OA and \( \angle POM = \theta \). When \( OM = 4 \) and \( \frac{d\theta}{dt} = 6 \) radians/sec, then the rate of change of PM is (in units/sec):

• (1) \( 24\sqrt{3} \)
• (2) \( 24 \)
• (3) \( 15\sqrt{3} \)
• (4) \( 48\sqrt{3} \)

If the length of the sub-tangent at any point P on a curve is proportional to the abscissa of the point P, then the equation of that curve is (C is an arbitrary constant):

• (1) \( y^k + x^k = C \)
• (2) \( x^{1/k} C = y^k \)
• (3) \( (x + y)^k = C \)
• (4) \( y = x^{1/k} C \)

In each of the following options, a function and an interval are given. Choose the option containing the function and the interval for which Lagrange’s Mean Value Theorem is not applicable.

• (1) \( f(x) = |x| \), \( 1 \leq x \leq 5 \)
• (2) \( f(x) = [x] \), \( [\sqrt{2}, \sqrt{3}] \)
• (3) \( f(x) = \log (x^2 - 1) \), \( \left[\frac{1}{e}, e-2\right] \)
• (4) \( f(x) = e^x \), \( [-e, e] \)

The function \( f(x) \) is defined as: \[ f(x) = \begin{cases} \frac{x - |x|}{x}, & x \neq 0
2, & x = 0 \end{cases} \]
Which of the following is true for \( f(x) \)?

• (1) is continuous for all \( x \in \mathbb{R} \)
• (2) has maximum value 2
• (3) has neither minimum nor maximum
• (4) has minimum value 2

If \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] then \( \frac{2}{3} (A + B + C + D) = \)

• (1) \( \frac{2}{3} \)
• (2) \( -\frac{2}{3} \)
• (3) \( \frac{4}{3} \)
• (4) \( -\frac{4}{3} \)

Evaluate \( \int (\log x)^m x^n dx \).

• (1) \( \int t^m e^{nt} dt, \quad t = e^x \)
• (2) \( \int t^m e^{(n+1)t} dt, \quad t = e^x \)
• (3) \( \int t^m e^{(n+1)t} dt, \quad x = e^t \)
• (4) \( \int t^m e^{nt} dt, \quad x = e^t \)

Evaluate the integral: \[ \int \sin^{-1} \left( \sqrt{\frac{x - a}{x}} \right) dx \]

• (1) \( x \cos^{-1} \left(\sqrt{\frac{a}{x}}\right) - \sqrt{ax - a^2} + C \)
• (2) \( x \sec^{-1} \left(\sqrt{\frac{a}{x}}\right) + \sqrt{x^2 - ax} + C \)
• (3) \( x \sin^{-1} \left(\sqrt{\frac{x}{a}}\right) + \sqrt{x^2 + ax} + C \)
• (4) \( \frac{x}{a} \sin^{-1} \left(\frac{x}{a}\right) + \frac{x^2}{a} \sqrt{1 + a^2} + C \)

Find the domain of \( f(x) \) given: \[ \int \frac{\sin x \cos x}{\sqrt{\cos^4 x - \sin^4 x}} dx = -\frac{f(x)}{2} + C. \]
then domain of f{x) is

• (1) \( [2n\pi, (2n+1)\pi] \), \( n = 0, 1, 2, \dots \)
• (2) \( [(4n - 1) \frac{\pi}{2}, (4n+1) \frac{\pi}{2}] \), \( n = 0, 1, 2, \dots \)
• (3) \( [(4n - 1) \frac{\pi}{4}, (4n+1) \frac{\pi}{4}] \), \( n = 0, 1, 2, \dots \)
• (4) \( [(2n \frac{\pi}{4}, (2n+1) \frac{\pi}{4}] \), \( n = 0, 1, 2, \dots \)

Given the equation: \[ y = (\tan^{-1} 2x)^2 + (\cot^{-1} 2x)^2, \]
find the expression: \[ (1+4x^2)^2 y'' - 16. \]

• (1) \( 8x y' \)
• (2) \( -8x(1 + 4x^2) y' \)
• (3) \( 8x(1 + 4x^2) y' \)
• (4) \( -8x y' \)

If \( \int_0^{2\pi} (\sin^4 x + \cos^4 x) \, dx = K \int_0^\pi \sin^2 x \, dx + L \int_0^\frac{\pi}{2} \cos^2 x \, dx \) and \( K, L \in \mathbb{N} \), then the number of possible ordered pairs \( (K, L) \) is

• (1) \( 1 \)
• (2) \( 2 \)
• (3) \( 3 \)
• (4) \( 4 \)

Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{4 \cos^2 x + 3 \sin^2 x} dx \):

• (1) \( \frac{\pi^2}{6\sqrt{3}} \)
• (2) \( \frac{\pi}{3\sqrt{3}} \)
• (3) \( \frac{\pi^2}{3\sqrt{3}} \)
• (4) \( \sqrt{3}\pi^2 \)

If \( A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \) and \( B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx \), then:

• (1) \( 2A = B \)
• (2) \( A = B \)
• (3) \( 2B = A \)
• (4) \( 2B + A = 0 \)

If \( (a, b) \) is the stationary point of the curve \( y = 2x - x^2 \), then the area bounded by the curves \( y = 2x - x^2 \), \( y = x^2 - 2x \), and \( x = a \) is:

• (1) \( \frac{3 \log 2 + 4}{2} \)
• (2) \( \frac{3 + \log 4}{6} \)
• (3) \( \frac{3 - \log 4}{3} \)
• (4) \( \frac{1}{\log^2 2} + \frac{3}{4} \)

Among the options given below, from which option a differential equation of order two can be formed?

• (1) All circles passing through the origin
• (2) All parabolas passing through the origin and having focus on x-axis
• (3) All the lines passing through the origin
• (4) All hyperbolas of the form \( x^2 - y^2 = k^2 \)

The differential equation for which \( ax + by = 1 \) is the general solution is:

• (1) \( \frac{dy}{dx} = x + c \)
• (2) \( y \frac{d^2 y}{dx^2} + x = 1 \)
• (3) \( \frac{d^2 y}{dx^2} = 0 \)
• (4) \( \frac{d^3 y}{dx^3} = 0 \)

The solution of the differential equation \( e^x y dx + e^x dy + xdx = 0 \) is:

• (1) \( e^x + yx^2 = c \)
• (2) \( 2ye^x + x^2 = c \)
• (3) \( ye^x + x^2 e^y = c \)
• (4) \( e^x + x e^y = c \)

Which of the following is NOT a unit of permeability?

• (1) \( Henry meter^{-1} \)
• (2) \( Weber ampere^{-1} meter^{-1} \)
• (3) \( Ohm second meter^{-1} \)
• (4) \( Volt second meter^{-1} \)

A diving board is at a height of \( h \) from the water surface. A swimmer standing on this board throws a stone vertically upward with a velocity 16 ms\(^{-1}\). It reaches the water surface in a time of 5 s. In the next 0.2s, the diver can hear the sound from the water surface. The speed of sound is (acceleration due to gravity \( g = 10 \) ms\(^{-2} \))

• (1) \( 450 \) ms\(^{-1} \)
• (2) \( 225 \) ms\(^{-1} \)
• (3) \( 200 \) ms\(^{-1} \)
• (4) \( 275 \) ms\(^{-1} \)

Path of projectile is given by the equation \( Y = Px - Qx^2 \), match the following accordingly (acceleration due to gravity = \( g \))

• (1) a-i, b-iii, c-iv, d-ii
• (2) a-i, b-iii, c-ii, d-iv
• (3) a-iii, b-i, c-iv, d-ii
• (4) a-iv, b-ii, c-iii, d-i

A bowling machine placed at a height \( h \) above the earth surface releases different balls with different angles but with the same velocity \( 10 \sqrt{3} \, ms^{-1} \). All these balls landing velocities make angles 30° or more with horizontal. Then the height \( h \) (in meters) is:

• (1) 15
• (2) 12
• (3) 10
• (4) 5

A balloon carrying some sand of mass \( M \) is moving down with a constant acceleration \( a_0 \). The mass \( m \) of sand to be removed so that the balloon moves up with double the acceleration \( a_0 \) is:

• (1) m = \( \frac{2 M a_0}{a_0 + g} \)
• (2) m = \( \frac{2 M a_0}{a_0 - g} \)
• (3) m = \( \frac{3 M a_0}{g + 2 a_0} \)
• (4) m = \( \frac{3 M a_0}{g - 2 a_0} \)

A person walks up a stalled escalator in 90s. When standing on the same moving escalator, he reached in 60s. The time it would take him to walk up the moving escalator will be:

• (1) 36 s
• (2) 72 s
• (3) 18 s
• (4) 27 s

A particle of mass \( m \) at rest on a rough horizontal surface with a coefficient of friction \( \mu \) is given a velocity \( u \). The average power imparted by friction before it stops is:

• (1) Zero
• (2) \( \frac{1}{2} \mu m g u \)
• (3) \( \mu m g u \)
• (4) \( 2 \mu m g u \)

A soccer ball of mass 250 g is moving horizontally to the left with a speed of 22 m/s. This ball is kicked towards right with a velocity 30 m/s at an angle 53° with the horizontal in upward direction. Assuming that it took 0.01 s for the collision to take place, the average force acting is:

• (1) 1000 N
• (2) 986 N
• (3) 1166 N
• (4) 2000 N

The moment of inertia of a solid sphere about its diameter is 20 kg m². The moment of inertia of a thin spherical shell having the same mass and radius about its diameter is:

• (1) 16.6 kg m²
• (2) 30.3 kg m²
• (3) 33.3 kg m²
• (4) 66.6 kg m²

One ring, one solid sphere, and one solid cylinder are rolling down on the same inclined plane starting from rest. The radius of all three are equal. The object that reaches down with maximum velocity is

• (1) Solid cylinder
• (2) Solid sphere
• (3) Ring
• (4) Solid sphere and Ring

As shown in the figure, two blocks of masses \(m_1\) and \(m_2\) are connected to a spring of force constant \(k\). The blocks are slightly displaced in opposite directions to \(x_1, x_2\) distances and released. If the system executes simple harmonic motion, then the frequency of oscillation of the system (\(\omega\)) is:

• (1) \(\left( \frac{1}{m_1} + \frac{1}{m_2} \right) k^2\)
• (2) \(\sqrt{\left( \frac{1}{m_1} + \frac{1}{m_2} \right) k^2}\)
• (3) \(\sqrt{\left( \frac{1}{m_1} + \frac{1}{m_2} \right)}\)
• (4) \(\sqrt{\left( \frac{1}{m_1} + \frac{1}{m_2} \right) k}\)

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:

• (1) \( \sqrt{\frac{m^2 + M^2}{M^2}} \)
• (2) \( \sqrt{\frac{m+M}{M^2}} \)
• (3) \( \sqrt{\frac{m+M}{M}} \)
• (4) \( \frac{m+M}{M} \)

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:

• (1) \( 64 \) minutes
• (2) \( 640 \) minutes
• (3) \( 320 \) minutes
• (4) \( 240 \) minutes

The work done on a wire of volume \( 2 \) cm\(^3 \) is \( 16 \times 10^2 \) J. If the Young's modulus of the material of the wire is \( 4 \times 10^{12} \) Nm\(^{-2} \), then the strain produced in the wire is

• (1) \( 0.03 \) m
• (2) \( 0.04 \) m
• (3) \( 0.01 \) m
• (4) \( 0.02 \) m

Water flows from a tap of diameter 1.5 cm with velocity \( 7.5 \times 10^{-5} \, m^{3}s^{-1} \). The coefficient of viscosity of water is \( 10^{-3} \) Pas. The flow is:

• (1) Turbulent with Reynolds number less than 6000
• (2) Steady flow with Reynolds number less than 2000
• (3) Turbulent with Reynolds number greater than 6000
• (4) Steady flow with Reynolds number more than 6000

A uniform metal solid sphere is rotating with angular speed \( \omega_0 \) about its diameter. If the temperature is raised by 50°C, the angular speed will be: Given \( \alpha_{metal} = 20 \times 10^{-5} \, °C^{-1} \)

• (1) \( 0.95 \omega_0 \)
• (2) \( 0.96 \omega_0 \)
• (3) \( 0.98 \omega_0 \)
• (4) \( \omega_0 \) (Angular velocity is same)

When 2 moles of a monatomic gas expands adiabatically from a temperature of 80°C to 50°C, the work done is \( W \). The work done when 3 moles of a diatomic gas expands adiabatically from 50°C to 20°C is:

• (1) 7 W
• (2) 5 W
• (3) 2.5 W
• (4) 3.5 W

A gas absorbs 18 J of heat and work done on the gas is 12 J. Then the change in internal energy of the gas is:

• (1) \( 24 \) J
• (2) \( 36 \) J
• (3) \( 6 \) J
• (4) \( 30 \) J

If the ratio of the absolute temperature of the sink and source of a Carnot engine is changed from 2:3 to 3:4, the efficiency of the engine changes by:

• (1) \( 25% \)
• (2) \( 40% \)
• (3) \( 50% \)
• (4) \( 15% \)

The ratio of the molar specific heat capacities of monatomic and diatomic gases at constant pressure is:

• (1) \( 1:7 \)
• (2) \( 5:7 \)
• (3) \( 3:7 \)
• (4) \( 2:7 \)

The frequency of fifth harmonic of a closed pipe is equal to the frequency of third harmonic of an open pipe. If the length of the open pipe is 72 cm, then the length of the closed pipe is:

• (1) 60 cm
• (2) 45 cm
• (3) 30 cm
• (4) 75 cm

When a convex lens is immersed in two different liquids of refractive indices 1.25 and 1.5, the ratio of the focal lengths of the lens is 5:16. The refractive index of the material of the lens is:

• (1) 1.55
• (2) 1.5
• (3) 1.65
• (4) 1.6

Two light waves of intensities \( I \) and \( 2I \) superimpose on each other. If the path difference between the light waves reaching a point is 12.5% of the wavelength of the light, then the resultant intensity at the point is (Both the light waves have the same wavelength)

• (1) \( I \)
• (2) \( 9I \)
• (3) \( 3I \)
• (4) \( 5I \)

A particle of mass 0.5 g and charge 10 \(\mu\)C is subjected to a uniform electric field of 8 NC\(^{-1}\). If the particle is initially at rest, the velocity of the particle after a time of 5 seconds is

• (1) \( 5 \) ms\(^{-1} \)
• (2) \( 0.5 \) ms\(^{-1} \)
• (3) \( 8 \) ms\(^{-1} \)
• (4) \( 0.8 \) ms\(^{-1} \)

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:

• (1) \( 30 \) V
• (2) \( 15 \) V
• (3) \( 1.5 \) V
• (4) \( 3 \) V

Two particles of charges 4 nC and \( Q \) are kept in air with a separation of 10 cm between them. If the electrostatic potential energy of the system is 1.8 μJ, then \( Q \) is:

• (1) \( 12 \) nC
• (2) \( 9 \) nC
• (3) \( 5 \) nC
• (4) \( 7 \) nC

The emf of a cell of internal resistance 2 Ω is measured using a voltmeter of resistance 998 Ω. The error in the emf measured is:

• (1) \( 0.4% \)
• (2) \( 4% \)
• (3) \( 2% \)
• (4) \( 0.2% \)

In a meter bridge experiment, a resistance of 9 Ω is connected in the left gap and an unknown resistance greater than 9 Ω is connected in the right gap. If the resistance in the gaps are interchanged, the balancing point shifts by 10 cm. The unknown resistance is:

• (1) \( 18 \) Ω
• (2) \( 22 \) Ω
• (3) \( 11 \) Ω
• (4) \( 36 \) Ω

A charge ‘q’ is spread uniformly over an isolated ring of radius ‘R’. The ring is rotated about its natural axis with angular speed \( \omega \). The magnetic dipole moment of the ring is:

• (1) \( \frac{q \omega R}{2} \)
• (2) \( q \omega R^2 \)
• (3) \( \frac{q \omega R^2}{2} \)
• (4) \( \frac{q \omega}{2R} \)

Current sensitivities of two galvanometers \( G_1 \) and \( G_2 \) of resistances 100 Ω and 50 Ω are \( 10^8 \) div/A and \( 0.5 \times 10^5 \) div/A respectively. The galvanometer in which the voltage sensitivity is more is:

• (1) Same in both galvanometers
• (2) More in \( G_2 \)
• (3) Zero
• (4) More in \( G_1 \)

The relation between \(\mu\) and \(H\) for a specimen of iron is \(\mu = \left[ \frac{0.4}{H} + 12 \times 10^{-4} \right] Hm^{-1}\). The value of \(H\) which produces flux density of 1 T will be (\(\mu\) = magnetic permeability, \(H\) = magnetic intensity)

• (1) \( 250 Am^{-1} \)
• (2) \( 500 Am^{-1} \)
• (3) \( 750 Am^{-1} \)
• (4) \( 10^3 Am^{-1} \)

In a circuit, the current falls from 14 A to 4 A in a time 0.2 ms. If the induced emf is 150 V, then the self-inductance of the circuit is:

• (1) \( 6 \) H
• (2) \( 6 \) mH
• (3) \( 3 \) mH
• (4) \( 3 \) H

An alternating current is given by \( i = (3 \sin \omega t + 4 \cos \omega t) \) A. The rms current will be:

• (1) \( \frac{7}{\sqrt{2}} \) A
• (2) \( \frac{1}{\sqrt{2}} \) A
• (3) \( \frac{5}{\sqrt{2}} \) A
• (4) \( \frac{3}{\sqrt{2}} \) A

For plane electromagnetic waves propagating in the positive Z-direction, the combination which gives the correct possible direction for \( E \) and \( B \) fields respectively is:

• (1) \( (-2i - 3j) \) and \( (3i - 2j) \)
• (2) \( (3i + 4j) \) and \( (4i - 3j) \)
• (3) \( (i - 2j) \) and \( (-2i - j) \)
• (4) \( (-2i + 3j) \) and \( (i + 2j) \)

A photon incident on a metal of work function 2 eV produced a photoelectron of maximum kinetic energy of 2 eV. The wavelength associated with the photon is:

• (1) \( 6200 \) Å
• (2) \( 3100 \) Å
• (3) \( 9300 \) Å
• (4) \( 2000 \) Å

Energy levels A, B, and C of a certain atom correspond to increasing values of energy, i.e., \( E_A < E_B < E_C \). If \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) are the wavelengths of a photon corresponding to the transitions shown, then:

• (1) \( \lambda_3 = \lambda_1 + \lambda_2 \)
• (2) \( \lambda_3 = \frac{(\lambda_1 + \lambda_2)}{\lambda_1 \lambda_2} \)
• (3) \( \lambda_3^2 = \lambda_1^2 + \lambda_2^2 \)
• (4) \( \lambda_3 = \frac{\lambda_1 \lambda_2}{(\lambda_1 + \lambda_2)} \)

In a nuclear reactor, the fuel is consumed at the rate of \( 1 \times 10^{-3} \) gs\(^{-1}\). The power generated in kW is:

• (1) \( 9 \times 10^{14} \)
• (2) \( 9 \times 10^{7} \)
• (3) \( 9 \times 10^{8} \)
• (4) \( 9 \times 10^{12} \)

In the diodes shown in the diagrams, which one is reverse biased?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (3)
View Solution

The following configuration of gates is equivalent to:

• (1) NAND
• (2) XOR
• (3) AND
• (4) OR

Size of the antenna for a carrier wave of frequency 3 MHz is:

• (1) 75 m
• (2) 50 m
• (3) 2.5 m
• (4) 25 m

The sum of number of angular nodes and radial nodes for 4d orbital is:

• (1) 2
• (2) 3
• (3) 4
• (4) 5

If the position of the electron was measured with an accuracy of \( \pm 0.002 \) nm, the uncertainty in the momentum of it would be (in kg ms\(^{-1}\)) (\( h = 6.626 \times 10^{-34} \) Js):

• (1) \( 2.637 \times 10^{-23} \)
• (2) \( 2.637 \times 10^{-24} \)
• (3) \( 8.283 \times 10^{-23} \)
• (4) \( 8.283 \times 10^{-24} \)

Match the following:

• (1) A-III, B-IV, C-I, D-II
• (2) A-IV, B-III, C-I, D-II
• (3) A-III, B-I, C-IV, D-II
• (4) A-II, B-IV, C-I, D-III

Identify the set of molecules in which the central atom has only one lone pair of electrons in their valence shells:

• (1) BrF\(_5\), SF\(_4\), SnCl\(_2\)
• (2) BrF\(_5\), XeF\(_4\), SnCl\(_2\)
• (3) XeF\(_4\), NH\(_3\), ClF\(_3\)
• (4) XeF\(_6\), ClF\(_3\), SF\(_4\)

The bond order of which of the following two species is the same?

• (1) \( O_2, N_2 \)
• (2) \( C_2, O_2 \)
• (3) \( B_2, C_2 \)
• (4) \( F_2, C_2 \)

The rms velocity (\( u_{rms} \)), mean velocity (\( u_{av} \)), and most probable velocity (\( u_{mp} \)) of a gas differ from each other at a given temperature. Which of the following ratios regarding them is correct?

• (1) \( \frac{u_{rms}}{u_{av}} = 1.20 \)
• (2) \( \frac{u_{av}}{u_{mp}} = 1.12 \)
• (3) \( \frac{u_{rms}}{u_{mp}} = 1.15 \)
• (4) \( \frac{u_{av}}{u_{rms}} = 0.98 \)

60 cm\(^3\) of SO\(_2\) gas diffused through a porous membrane in \( x \) minutes. Under similar conditions, 360 cm\(^3\) of another gas (molar mass 4 g mol\(^{-1}\)) diffused in \( y \) minutes. The ratio of \( x \) and \( y \) is:

• (1) \( 3:2 \)
• (2) \( 2:3 \)
• (3) \( 1:3 \)
• (4) \( 3:1 \)

Observe the following reactions

(i) \( 2KClO_3(s) \xrightarrow{\Delta} 2KCl(s) + 3O_2(g) \)

(ii) \( 2H_2O_2(aq) \xrightarrow{\Delta} 2H_2O(l) + O_2(g) \)

(iii) \( AgNO_3(aq) + KCl(aq) \longrightarrow AgCl(s) + KNO_3(aq) \)

(iv) \( 2Na(s) + \frac{1}{2} O_2(g) \longrightarrow Na_2O(s) \)

The number of redox reactions in this list is

• (1) \( 3 \)
• (2) \( 4 \)
• (3) \( 2 \)
• (4) \( 1 \)

A 10 L vessel contains 1 mole of an ideal gas with pressure of P(atm) and temperature of T(K). The vessel is divided into two equal parts. The pressure (in atm) and temperature (in K) in each part is respectively.

• (1) \( \frac{P}{2} \), \( \frac{T}{2} \)
• (2) \( P \), \( T \)
• (3) \( \frac{P}{2} \), \( T \)
• (4) \( P \), \( \frac{T}{2} \)

Observe the following reactions:

I. \( CaCO_3(s) \rightarrow CaO(s) + CO_2(g) \)

II. \( Cl_2(g) \rightarrow 2 Cl(g) \)

III. \( H_2O(l) \rightarrow H_2O(s) \)

Identify the reactions in which entropy increases.

• (1) I, III
• (2) I, II Only
• (3) I, II, III
• (4) II, III Only

At 300 K, for the reaction, \[ A_2 B_2(g) \rightleftharpoons A_2(g) + B_2(g) \]
\text{is 100 mol L^{-1. \text{What is its K_p \text{(in atm) \text{at the same temperature?

\text{(R = 0.082 L atm mol^{-1 \text{K^{-1)

• (1) 100
• (2) 2460
• (3) 4.06
• (4) 246

At 27°C, the degree of dissociation of HA (weak acid) in 0.5 M of its solution is 1%. The concentrations of H\(_3\)O\(^+\), A\(^-\), and HA at equilibrium (in mol L\(^{-1}\)) are respectively:

• (1) \( 0.005, 0.005, 0.495 \)
• (2) \( 0.05, 0.05, 0.45 \)
• (3) \( 0.01, 0.01, 0.49 \)
• (4) \( 0.005, 0.495, 0.005 \)

Which of the following sets are correctly matched?

(i) \( B_2H_6 \) - electron deficient hydride

(ii) \( NH_3 \) - electron precise hydride

(iii) \( H_2O \) - electron rich hydride

• (1) \( i, iii \) only
• (2) \( i, ii, iii \)
• (3) \( ii, iii \) only
• (4) \( i, ii \) only

Which of the following, on thermal decomposition, form both acidic and basic oxides along with O\(_2\)?

(i) NaNO\(_3\)

(ii) Ca(NO\(_3\))\(_2\)

(iii) Be(NO\(_3\))\(_2\)

(iv) LiNO\(_3\)

• (1) \( ii, iii \) only
• (2) \( iii, iv \) only
• (3) \( ii, iv \) only
• (4) \( i, ii, iii \)

Identify the correct sets:

(i) Boron fibres - bulletproof vest

(ii) Metal borides - protective shields

(iii) Borax - glass wool

• (1) \( i, ii \) only
• (2) \( i, ii, iii \)
• (3) \( i, iii \) only
• (4) \( ii, iii \) only

Which of the following is/are ionic in nature?

(i) GeF\(_4\)

(ii) SnF\(_4\)

(iii) PbF\(_4\)

• (1) \( iii \) only
• (2) \( ii, iii \) only
• (3) \( i \) only
• (4) \( i, ii \) only

Which of the following is a lung irritant?

• (1) \( CO \)
• (2) \( NO_2 \)
• (3) \( CO_2 \)
• (4) \( CH_4 \)

Which of the following sequence of reagents converts 3-hexene to propane?

• (1) \( KMnO_4 | H^+; NaOH, CaO \)
• (2) \( (i) O_3, (ii) Zn, H_2O; NaBH_4 \)
• (3) \( (i) O_3, (ii) Zn, H_2O; Zn-Hg, HCl \)
• (4) \( KMnO_4 | H^+; LiAlH_4, H_2O \)

The number of alicyclic compounds from the following is:

Cyclohexene, Anisole, Pyridine, Tetrahydrofuran, Biphenyl

• (1) \( 2 \)
• (2) \( 3 \)
• (3) \( 1 \)
• (4) \( 4 \)

The molecular formula of a crystalline solid is X\( _3 \)Y\( _2 \). Atoms of Y form ccp lattice, and atoms of X occupy 50% octahedral voids and x% of tetrahedral voids. What is the percentage of unoccupied tetrahedral voids?

• (1) \( 66.6 \)
• (2) \( 25 \)
• (3) \( 50 \)
• (4) \( 33.3 \)

At 300 K, the vapour pressures of A and B liquids are 500 and 400 mm Hg respectively. Equal moles of A and B are mixed to form an ideal solution. The mole fraction of A and B in vapor state is respectively.

• (1) \( 0.5, 0.5 \)
• (2) \( 0.666, 0.333 \)
• (3) \( 0.444, 0.555 \)
• (4) \( 0.555, 0.444 \)

Two statements are given below:

Statement - I: Liquids A and B form a non-ideal solution with positive deviation. The interactions between A and B are weaker than A-A and B-B interactions.

Statement - II: For an ideal solution, \( \Delta H_{mix} = 0 \), \( \Delta V_{mix} = 0 \).

The correct answer is

• (1) Both Statements - I and Statement - II are correct
• (2) Both Statements - I and Statement - II are not correct
• (3) Statement - I is correct but Statement - II is not correct
• (4) Statement - I is not correct but Statement - II is correct

At 300 K, the \( E^\ominus_{cell} \) of \[ A(s) + B^{2+}(aq) \rightleftharpoons A^{2+}(aq) + B(s) \]
is 1.0 V. If \( \Delta_r S^\ominus \) of this reaction is 100 J K\(^{-1}\), what is \( \Delta_r H^\ominus \) (in kJ mol\(^{-1}\)) of this reaction?

(F = 96500 C mol\(^{-1}\))

• (1) -163
• (2) -223
• (3) -193
• (4) -163000

A → P is a first order reaction. The following graph is obtained for this reaction. (x-axis = time; y-axis = conc. of A). The instantaneous rate of the reaction at point C is

Two statements are given below

Statement I: Adsorption of a gas on the surface of charcoal is primarily an exothermic reaction

Statement II: A closed vessel contains O\(_2\), H\(_2\), Cl\(_2\), NH\(_3\) gases. Its pressure is P (atm). About 1 g of charcoal is added to this vessel and after some time its pressure was found to be less than P (atm)

The correct answer is

• (1) Both Statements - I and Statement - II are correct
• (2) Both Statements - I and Statement - II are not correct
• (3) Statement - I is correct but Statement - II is not correct
• (4) Statement - I is not correct but Statement - II is correct

The critical temperature of A, B, C, D gases are 190 K, 630 K, 261 K, 400 K respectively. The quantity of gas adsorbed per gram of charcoal at same pressure is least for the gas

• (1) D
• (2) C
• (3) B
• (4) A

In the extraction of iron, the reaction which occurs at 900-1500 K in blast furnace is

• (1) Fe\(_2\)O\(_3\) + 3C → 2Fe + 3CO
• (2) FeO + CO → Fe + CO\(_2\)
• (3) 3Fe\(_2\)O\(_3\) + CO → 2Fe\(_3\)O\(_4\) + CO\(_2\)
• (4) Fe\(_3\)O\(_4\) + 4CO → 3Fe + 4CO\(_2\)

Hydrolysis of XeF\(_4\) gives HF, O\(_2\), Xe and 'X'. The structure of 'X' is

• (1) pyramidal
• (2) Square pyramidal
• (3) octahedral
• (4) Square planar

Acidification of chromate gives 'Z'. The oxidation state of chromium in 'Z' is

• (1) +3
• (2) +6
• (3) +7
• (4) +2

Arrange the following in the increasing order of their magnetic moments



I. \( [Mn(CN)_6]^{3-} \)

II. \( [MnCl_6]^{3-} \)

III. \( [Fe(CN)_6]^{3-} \)

IV. \( [FeF_6]^{3-} \)

• (1) \(III < I < IV < II\)
• (2) \(III < IV < II < I\)
• (3) \(IV < II < I < III\)
• (4) \(III < I < II < IV\)

The X formed in the following reaction sequence and its structural type are respectively.

• (1) Novolac - linear polymer
• (2) Bakelite - cross linked polymer
• (3) Novolac - cross linked polymer
• (4) Bakelite - linear polymer

Which of the following acts as intracellular messengers?

• (1) Enzymes
• (2) Hormones
• (3) Receptors
• (4) Carrier proteins

The deficiency of vitamin \( x \) causes beri-beri and deficiency of vitamin \( y \) causes convulsions. What are \( x \) and \( y \) respectively?

• (1) B2, B12
• (2) B1, B12
• (3) B2, B6
• (4) B1, B6

Which of the following is incorrect?

• (1) Shaving soaps contain glycerol
• (2) Branched chain detergents are easily biodegradable
• (3) Dettol is a mixture of chloroxylenol and terpineol

What are X and Y respectively in the following reactions?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (3)
View Solution

The sequence of reagents required to convert ethyl bromide to propanal is:

• (1) \( CH_3COOAg, DIBAL-H, H_2O \)
• (2) \( CH_3COOAg, LiAlH_4, H_2O \)
• (3) \( Mg/ether, HCHO, H_2O \)
• (4) \( CH_3COOH, LiAlH_4, H_2O \)

What are X, Y, Z in the following reaction sequence respectively?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (2)
View Solution

Toluene on reaction with the reagent X gave Y, which dissolves in NaHCO\( _3 \) and when reacted with Br\( _2 \)/Fe gave Z. What are X and Z?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (4)
View Solution

A Grignard reagent (X) on reaction with carbonyl compound (Y) followed by hydrolysis gave Z. Z reacts with conc. HCl at room temperature. X and Y respectively are

• (1) \( CH_3MgBr, CH_3CH_2CHO \)
• (2) \( CH_3MgBr, CH_3COCH_3 \)
• (3) \( CH_3CH_2CH_2MgBr, HCHO \)
• (4) \( CH_3CH_2MgBr, CH_3CHO \)

p-Methyl benzene nitrile can be prepared from which of the following?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (3)
View Solution