AP EAPCET (AP EAMCET) 2024 Question Paper May 21 Shift 1 (Available): Download MPC Question Paper with Answer Key PDF

Question 1:

The domain of the real-valued function \( f(x) = \log_2 \log_3 \log_5 (x^2 - 5x + 11) \) is:

• (1) \( (2, \infty) \)
• (2) \( (-\infty, 3) \)
• (3) \( (2, 3) \)
• (4) \( (-\infty, 2) \cup (3, \infty) \)

The range of the real valued function \( f(x) = \frac{x^2 + 2x - 15}{2x^2 + 13x + 15} \) is:

• (1) \( R = \left\{ -5, -\frac{3}{2} \right\} \)
• (2) \( R = \left\{ -5, -\frac{1}{2} \right\} \)
• (3) \( R = \left\{ -\frac{8}{7}, \frac{2}{7} \right\} \)
• (4) \( R = \left\{ -\frac{3}{2}, \frac{3}{7} \right\} \)

The sum of the series \( \frac{1}{1.5} + \frac{1}{5.9} + \frac{1}{9.13} + \cdots \) up to \( n \) terms is:

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

If \( A = \begin{bmatrix} 2 & 3
1 & k \end{bmatrix} \) is a singular matrix, then the quadratic equation having the roots \( k \) and \( \frac{1}{k} \) is:

• (1) \( 6x^2 + 13x + 6 = 0 \)
• (2) \( 12x^2 - 25x + 12 = 0 \)
• (3) \( 6x^2 - 13x + 6 = 0 \)
• (4) \( 2x^2 - 5x + 2 = 0 \)

Let \( A \) be a \( 4 \times 4 \) matrix and \( P \) be its adjoint matrix. If \( |P| = \left| \frac{A}{2} \right| \), then \( |A^{-1}| = ? \)

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

The system \( x + 2y + 3z = 4, \, 4x + 5y + 3z = 5, \, 3x + 4y + 3z = \lambda \) is consistent and \( 3\lambda = n + 100 \), then \( n = ? \)

• (1) \( -42 \)
• (2) \( -86 \)
• (3) \( 16 \)
• (4) \( -24 \)

The complex conjugate of \( (4 - 3i)(2 + 3i)(1 + 4i) \) is:

• (1) \( 7 + 74i \)
• (2) \( -7 + 74i \)
• (3) \( -7 - 74i \)
• (4) \( 7 - 74i \)

If the amplitude of \( (Z - 2) \) is \( \frac{\pi}{2} \), then the locus of \( Z \) is:

• (1) \( x = 0, \, y > 0 \)
• (2) \( x = 2, \, y > 0 \)
• (3) \( x > 0, \, y = 2 \)
• (4) \( x > 0, \, y = 0 \)

If \( \omega \) is the cube root of unity, then:
\[ \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} = \frac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} \]

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

Roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are:

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

If \( (3 + i) \) is a root of \( x^2 + ax + b = 0 \), then \( a = ? \)

• (1) \( 3 \)
• (2) \( \)
• (3) \( 6 \)
• (4) \( -6 \)

The algebraic equation of degree 4 whose roots are the translates of the roots of the equation \( x^4 + 5x^3 + 6x^2 + 7x + 9 = 0 \) by \( -1 \) is:

• (1) \( x^4 + 3x^3 - 3x^2 + 6x + 4 = 0 \)
• (2) \( x^4 + 9x^3 + 27x^2 + 38x + 28 = 0 \)
• (3) \( x^4 + 5x^3 + 6x^2 + 7x + 9 = 0 \)
• (4) \( x^4 + 5x^3 + 6x^2 - 7x + 9 = 0 \)

If the roots of the equation \( 4x^3 - 12x^2 + 11x + m = 0 \) are in arithmetic progression, then \( m = ? \)

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

The number of 5-digit odd numbers greater than 40,000 that can be formed by using 3, 4, 5, 6, 7, 0 such that at least one of its digits must be repeated is:

• (1) 2592
• (2) 240
• (3) 3032
• (4) 2352

The number of ways in which 3 men and 3 women can be arranged in a row of 6 seats, such that the first and last seats must be filled by men is:

• (1) 720
• (2) 36
• (3) 144
• (4) 72

If a committee of 10 members is to be formed from 8 men and 6 women, then the number of different possible committees in which the men are in majority is:

• (1) 931
• (2) 175
• (3) 48
• (4) 595

If the eleventh term in the binomial expansion of \( (x + a)^n \) is the geometric mean of the eighth and twelfth terms, then the greatest term in the expansion is:

• (1) 7th term
• (2) 8th term
• (3) 9th term
• (4) 10th term

The sum of the rational terms in the binomial expansion of \( \left( \sqrt{2} + 3^{1/5} \right)^{10} \) is:

• (1) 41
• (2) 39
• (3) 32
• (4) 30

If \[ \frac{1}{(3x+1)(x-2)} = \frac{A}{3x+1} + \frac{B}{x-2} \quad and \quad \frac{x+1}{(3x+1)(x-2)} = \frac{C}{3x+1} + \frac{D}{x-2}, \]
then \[ \frac{1}{(3x+1)(x-2)} = \frac{A}{3x+1} + \frac{B}{x-2}, find A + 3B = 0, A:C = 1:3, B:D = 2:3. \]

• (1) \( A + 3B = 0, A:C = 1:3, B:D = 2:3 \)
• (2) \( A + 3B = 0, A:C = 3:1, B:D = 3:2 \)
• (3) \( A + 3B = 0, A:C = 3:2, B:D = 1:3 \)
• (4) \( A + 3B = 0, A:C = 3:2, B:D = 1:3 \)

If the period of the function \( f(x) = \frac{\tan 5x \cos 3x}{\sin 6x} \) is \( \alpha \), then find \( f \left( \frac{\alpha}{8} \right) \):

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

If \( \sin x + \sin y = \alpha \), \( \cos x + \cos y = \beta \), then \( \csc(x + y) \) =

• (1) \( \frac{\beta^2 - \alpha^2}{\beta^2 + \alpha^2} \)
• (2) \( \frac{2 \alpha \beta}{\beta^2 - \alpha^2} \)
• (3) \( \frac{\beta^2 + \alpha^2}{2 \alpha \beta} \)
• (4) \( \frac{2 \alpha \beta}{\beta^2 + \alpha^2} \)

If \( P + Q + R = \frac{\pi}{4} \), then \[ \cos \left( \frac{\pi}{8} - P \right) + \cos \left( \frac{\pi}{8} - Q \right) + \cos \left( \frac{\pi}{8} - R \right) = P + Q + R = \frac{\pi}{4}. \]

• (1) \( 4 \cos \frac{P}{2} \cos \frac{Q}{2} - \cos \frac{\pi}{8} \)
• (2) \( 4 \cos \frac{P}{2} \sin \frac{R}{2} + \cos \frac{\pi}{8} \)
• (3) \( 4 \sin \frac{P}{2} \sin \frac{R}{2} - \cos \frac{\pi}{8} \)
• (4) \( 4 \sin \frac{P}{2} \cos \frac{R}{2} + \cos \frac{\pi}{8} \)

For \( a \in \mathbb{R} \setminus \{0\} \), if \( a \cos x + a \sin x + a = 2K + 1 \) has a solution, then \( K \) lies in the interval:

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

If the general solution set of \( \sin x + 3 \sin 3x + \sin 5x = 0 \) is \( S \), then \[ \sin a \quad for \quad a \in S \quad is \quad \{ \sin a \mid a \in S \} = \]

• (1) \( \{1, -1, 0\} \)
• (2) \( \left\{ \frac{1}{2}, -\frac{1}{2}, 0, 1, -1 \right\} \)
• (3) \( \left\{ \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2} \right\} \)
• (4) \( \left\{ 1, -1, \frac{\sqrt{3}}{2}, 0, -\frac{\sqrt{3}}{2} \right\} \)

If \( \theta \) is an acute angle, \( \cosh x = K \) and \( \sinh x = \tan \theta \), then \( \sin \theta = \dots \)

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

In a triangle, if the angles are in the ratio \( 3:2:1 \), then the ratio of its sides is:

• (1) \( 1 : 2 : 3 \)
• (2) \( 2 : \sqrt{3} : 1 \)
• (3) \( 3 : \sqrt{2} : 1 \)
• (4) \( 1 : \sqrt{3} : 3 \)

In a triangle ABC, if \( BC = 5 \), \( CA = 6 \), \( AB = 7 \), then the length of the median drawn from \( B \) onto \( AC \) is:

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

In \( \triangle ABC \), if \( AB:BC:CA = 6:4.5 \), then \( R : r = \)

• (1) \( 16 : 9 \)
• (2) \( 16 : 7 \)
• (3) \( 12 : 7 \)
• (4) \( 12 : 9 \)

If \( \vec{a} = i\hat{i} + j\hat{j} + 3k\hat{k} \), \( \vec{b} = i\hat{i} + 2k\hat{k} \), \( \vec{c} = -3i\hat{i} + 2j\hat{j} + k\hat{k} \) are linearly dependent vectors and the magnitude of \( \vec{a} \) is \( \sqrt{14} \), then if \( \alpha, \beta \) are integers, find \( \alpha + \beta \):

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

If \( \vec{c} \) is a vector along the bisector of the internal angle between the vectors \( \vec{a} = 4\hat{i} + 7\hat{j} - 4\hat{k} \) and \( \vec{b} = 12\hat{i} - 3\hat{j} + 4\hat{k} \), and the magnitude of \( \vec{c} \) is \( 3\sqrt{13} \), then \( \vec{c} = \):

• (1) \( 5\hat{i} - 8\hat{j} + 2\sqrt{2} \hat{k} \)
• (2) \( 10\hat{i} + 4\hat{j} - \hat{k} \)
• (3) \( 7\hat{i} - 10\hat{j} + 4\hat{k} \)
• (4) \( 2\sqrt{2} \hat{i} + 5\hat{j} - 8\hat{k} \)

If \( \vec{a} = \hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} + \hat{j} + \hat{k} \) are two vectors and \( \vec{c} \) is a unit vector lying in the plane of \( \vec{a} \) and \( \vec{b} \), and if \( \vec{c} \) is perpendicular to \( \vec{b} \), then \( \vec{c} \cdot (\hat{i} + 2\hat{k}) = \):

• (1) \( 0 \)
• (2) \( 5 \)
• (3) \( \frac{1}{\sqrt{21}} \)
• (4) \( \frac{2}{\sqrt{21}} \)

A(1, 2, 1), B(2, 3, 2), C(3, 1, 3) and D(2, 1, 3) are the vertices of a tetrahedron. If \( \theta \) is the angle between the faces ABC and ABD then \( \cos \theta \) is:

• (1) \( \frac{5}{\sqrt{14}} \)
• (2) \( \frac{15}{8\sqrt{7}} \)
• (3) \( \frac{3}{\sqrt{14}} \)
• (4) \( \frac{5}{2\sqrt{7}} \)

If \( \vec{a} = \hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} + \hat{j} - 2\hat{k} \), \( \vec{c} = 2\hat{i} - 3\hat{j} - 3\hat{k} \), and \( \vec{d} = 2\hat{i} + \hat{j} + \hat{k} \) are four vectors, then \( (\vec{a} \times \vec{c}) \times (\vec{b} \times \vec{d}) = \):

• (1) \( 2\hat{i} + 19\hat{j} - 11\hat{k} \)
• (2) \( -8\hat{i} + 19\hat{j} - 29\hat{k} \)
• (3) \( 2\hat{i} + \hat{j} - 11\hat{k} \)
• (4) \( -8\hat{i} + \hat{j} - 29\hat{k} \)

Mean deviation about the mean for the following data is:
\[ \begin{array}{|c|c|} \hline Class Interval & Frequency
\hline 0-6 & 1
6-12 & 2
12-18 & 3
18-24 & 2
24-30 & 1
\hline \end{array} \]

• (1) \( 5 \)
• (2) \( \frac{16}{3} \)
• (3) \( 6 \)
• (4) \( \frac{19}{3} \)

If 12 dice are thrown at a time, then the probability that a multiple of 3 does not appear on any die is:

• (1) \( \left( \frac{1}{2} \right)^{12} \)
• (2) \( \left( \frac{1}{3} \right)^{12} \)
• (3) \( \left( \frac{2}{3} \right)^{12} \)
• (4) \( \left( \frac{5}{6} \right)^{12} \)

If a number is drawn at random from the set \( \{1, 3, 5, 7, \dots, 59\} \), then the probability that it lies in the interval in which the function \( f(x) = x^3 - 16x^2 + 20x - 5 \) is strictly decreasing is:

• (1) \( \frac{1}{5} \)
• (2) \( \frac{1}{3} \)
• (3) \( \frac{1}{2} \)
• (4) \( \frac{1}{6} \)

In a class consisting of 40 boys and 30 girls, 30% of the boys and 40% of the girls are good at Mathematics. If a student selected at random from that class is found to be a girl, then the probability that she is not good at Mathematics is:

• (1) \( \frac{3}{5} \)
• (2) \( \frac{2}{5} \)
• (3) \( \frac{3}{10} \)
• (4) \( \frac{7}{10} \)

A basket contains 12 apples in which 3 are rotten. If 3 apples are drawn at random simultaneously from it, then the probability of getting at most one rotten apple is:

• (1) \( \frac{34}{55} \)
• (2) \( \frac{48}{55} \)
• (3) \( \frac{21}{55} \)
• (4) \( \frac{42}{55} \)

7 coins are tossed simultaneously and the number of heads turned up is denoted by the random variable \( X \). If \( \mu \) is the mean and \( \sigma^2 \) is the variance of \( X \), then \( \frac{\mu^2}{P(X = 3)} \) is:

• (1) \( \frac{56}{5} \)
• (2) \( \frac{84}{5} \)
• (3) \( \frac{112}{5} \)
• (4) \( \frac{224}{5} \)

A manufacturing company noticed that 1% of its products are defective. If a dealer orders 300 items from this company, then the probability that the number of defective items is at most one is:

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

P is a variable point such that the distance of P from A(4,0) is twice the distance of P from B(-4,0). If the line \( 3y - 3x - 20 = 0 \) intersects the locus of P at the points C and D, then the distance between C and D is:

• (1) \( 8 \)
• (2) \( 8\sqrt{2} \)
• (3) \( \frac{32}{3} \)
• (4) \( 32 \)

When the origin is shifted to \( (h, k) \) by translation of axes, the transformed equation of \( x^2 + 2x + 2y - 7 = 0 \) does not contain \( x \) and constant terms. Then \( (2h + k) = \):

• (1) \( \frac{7}{2} \)
• (2) 1
• (3) 0
• (4) \( \frac{1}{2} \)

Let \( \alpha \in \mathbb{R} \). If the line \( (a + 1)x + \alpha y + \alpha = 1 \) passes through a fixed point \( (h, k) \) for all \( a \), then \( h^2 + k^2 = \):

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

If \( (a, \beta) \) is the orthocenter of the triangle with the vertices \( A(2, 5), B(1, 5), C(1, 4) \), then \( a + \beta = \):

• (1) 6
• (2) 5
• (3) \( \frac{7}{2} \)
• (4) \( \frac{5}{2} \)

The area of the triangle formed by the lines represented by \( 3x + y + 15 = 0 \) and \( 3x^2 + 12xy - 13y^2 = 0 \) is:

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

If all chords of the curve \( 2x^2 - y^2 + 3x + 2y = 0 \), which subtend a right angle at the origin, always pass through the point \( (a, \beta) \), then \( (a, \beta) = \):

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

The equations \( 2x - 3y + 1 = 0 \) and \( 4x - 5y - 1 = 0 \) are the equations of two diameters of the circle \( S = x^2 + y^2 + 2gx + 2fy - 11 = 0 \) \text{ and \( R \) are the points of contact of the tangents drawn
from the point \( P(-2, -2) \) to this circle. If \( C \) is the centre of the circle, \( S = 0 \) is the equation
of the circle, then the area (in square units) of the quadrilateral \( PQCR \) is:

\flushleft

• (1) 25
• (2) 30
• (3) 24
• (4) 36

If the inverse point of the point \( (-1, 1) \) with respect to the circle \( x^2 + y^2 - 2x + 2y - 1 = 0 \) is \( (p, q) \), then \( p^2 + q^2 = \):

\flushleft

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

If \( (a, b) \) is the midpoint of the chord \( 2x - y + 3 = 0 \) of the circle \( x^2 + y^2 + 6x - 4y + 4 = 0 \), then \( 2a + 3b = \):

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

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 \( AB = \):

• (1) 9
• (2) 16
• (3) \( \sqrt{6} \)
• (4) \( 2\sqrt{6} \)

The radius of the circle which cuts the circles \( x^2 + y^2 - 4x - 4y + 7 = 0 \), \( x^2 + y^2 + 4x + 6 = 0 \), and \( x^2 + y^2 + 4x + 4y + 5 = 0 \) orthogonally is:

• (1) \( \frac{\sqrt{193}}{4\sqrt{2}} \)
• (2) \( \frac{\sqrt{193}}{8} \)
• (3) \( \frac{\sqrt{193}}{4} \)
• (4) \( \frac{\sqrt{193}}{2\sqrt{2}} \)

The equation of the normal drawn to the parabola \( y^2 = 6x \) at the point \( (24,12) \) is:

• (1) \( 3x - y = 60 \)
• (2) \( 4x + y = 108 \)
• (3) \( 2x + y = 60 \)
• (4) \( x - 2y = 0 \)

If \( A_1, A_2, A_3 \) are the areas of the ellipse \( x^2 + 4y^2 = 4 \), its director circle, and auxiliary circle respectively, then \( A_2 + A_3 - A_1 \) is:

• (1) \( 11\pi \)
• (2) \( 3\pi \)
• (3) \( 7\pi \)
• (4) \( 9\pi \)

The equation of the pair of asymptotes of the hyperbola \( 4x^2 - 9y^2 - 24x - 36y - 36 = 0 \) is:

• (1) \( 2x^2 - xy - 3y^2 - 14x - 9y - 12 = 0 \)
• (2) \( 2x^2 - xy - 3y^2 - 2x + 3y = 0 \)
• (3) \( 2x^2 - 5xy + 3y^2 - 22x + 27y + 60 = 0 \)
• (4) \( 4x^2 - 9y^2 - 24x - 36y = 0 \)

The equation of one of the tangents drawn from the point \( (0,1) \) to the hyperbola \( 45x^2 - 4y^2 = 5 \) is:

• (1) \( 4y + 5 = 0 \)
• (2) \( 3x + 4y - 4 = 0 \)
• (3) \( 5x - 6y + 6 = 0 \)
• (4) \( 9x - 2y + 2 = 0 \)

Consider the tetrahedron with the vertices \( A(3,2,4) \), \( B(x_1,y_1,0) \), \( C(x_2,y_2,0) \), and \( D(x_3,y_3,0) \). If the triangle \( BCD \) is formed by the lines \( y = x \), \( x + y = 6 \), and \( y = 1 \), then the centroid of the tetrahedron is:

• (1) \( \left(\frac{9}{4}, \frac{7}{4}, 1 \right) \)
• (2) \( \left(\frac{11}{4}, \frac{5}{4}, 1 \right) \)
• (3) \( \left(3, \frac{7}{4}, 1 \right) \)
• (4) \( (3,2,1) \)

If \( P(2, \beta, \alpha) \) lies on the plane \( x + 2y - z - 2 = 0 \) and \( Q (\alpha, -1, \beta) \) lies on the plane \( 2x - y + 3z + 6 = 0 \), then the direction cosines of the line \( PQ \) are:

• (1) \( \left( \frac{-4}{\sqrt{17}}, 0, \frac{-1}{\sqrt{17}} \right) \)
• (2) \( \left( \frac{4}{\sqrt{17}}, 0, \frac{1}{\sqrt{17}} \right) \)
• (3) \( \left( \frac{1}{\sqrt{17}}, 0, \frac{-4}{\sqrt{17}} \right) \)
• (4) \( \left( \frac{-1}{\sqrt{17}}, 0, \frac{4}{\sqrt{17}} \right) \)

Let \( \pi \) be the plane that passes through the point \( (-2,1,-1) \) and is parallel to the plane \( 2x - y + 2z = 0 \). Then the foot of the perpendicular drawn from the point \( (1,2,1) \) to the plane \( \pi \) is:

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

If \( f(x) = \frac{5x \csc(\sqrt{x}) - 1}{(x-2) \csc(\sqrt{x})} \), then \( \lim\limits_{x \to \infty} f(x^2) \) is:

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

Evaluate the limit: \[ \lim\limits_{x \to 2} \frac{\sqrt{1 + 4x} - \sqrt{3} + 3x}{x^3 - 8}. \]

• (1) \( \frac{1}{72} \)
• (2) \( \frac{1}{36} \)
• (3) \( \frac{1}{24} \)
• (4) \( \frac{1}{12} \)

If \[ \lim\limits_{x \to \infty} \frac{\left(\sqrt{2x+1} + \sqrt{2x-1}\right) + \left(\sqrt{2x+1} - \sqrt{2x-1}\right) P x^4 - 16} {(x+\sqrt{x^2 - 2}) + (x - \sqrt{x^2 - 2})} = 1, \]
then P = ?

• (1) \( 16 \)
• (2) \( 64 \)
• (3) \( \frac{1}{64} \)
• (4) \( \frac{1}{16} \)

The rate of change of \( x^{\sin x} \) with respect to \( (\sin x)^x \) is:

• (1) \( \frac{x^{\sin x} \left(\frac{\sin x}{x} + \cos x \log x \right)}{(\sin x)^x (x \cot x + \log \sin x)} \)
• (2) \( \frac{(\sin x)^x (x \cot x + \log \sin x)}{\sin x \left(\frac{\sin x}{x} + \cos x \log x \right)} \)
• (3) \( y \left( \frac{\sin x}{x} + \cos x \log x \right) \)
• (4) \( (\sin x)^x (x \cot x + \log \sin x) \)

If \( y = \frac{ax + \beta}{\gamma x + \delta} \), then \( 2y_1 y_3 = \) ?

• (1) \( 2y_2^3 \)
• (2) \( 3y_2^2 \)
• (3) \( y_2^2 \)
• (4) \( 3y_3^2 \)

Which one of the following is false?

• (1) \( \frac{d}{dx} \left[ Sec^{-1} (\cosh x) \right] = sech x \)
• (2) \( \frac{d}{dx} \left[ Cos^{-1} (sech x) \right] = sech x \)
• (3) \( \frac{d}{dx} \left[ Tan^{-1} (\sinh x) \right] = sech x \)
• (4) \( \frac{d}{dx} \left[ Tan^{-1} (\tan \frac{x}{2}) \right] = sech x \)

The point which lies on the tangent drawn to the curve \( x^4 e^y + 2 \sqrt{y} + 1 = 3 \) at the point \( (1,0) \) is:

• (1) \( (2,6) \)
• (2) \( (2,-6) \)
• (3) \( (-2,-6) \)
• (4) \( (-2,6) \)

If \( f(x) = x^x \), then the interval in which \( f(x) \) decreases is:

• (1) \( \left[ 0, \frac{1}{e} \right] \)
• (2) \( \left[ 0, e \right] \)
• (3) \( \left[ \frac{1}{e}, \infty \right] \)
• (4) \( \left[ 0, e^e \right] \)

If Rolle’s theorem is applicable for the function \( f(x) \) defined by \( f(x) = x^3 + Px - 12 \) on \( [0,1] \), then the value of \( C \) of the Rolle's theorem is:

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

The number of all the values of \( x \) for which the function \[ f(x) = \sin x + \frac{1 - \tan^2 x}{1 + \tan^2 x} \]
attains its maximum value on \( [0, 2\pi] \).

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

If \( x \notin \left[ 2n\pi - \frac{\pi}{4}, 2n\pi + \frac{3\pi}{4} \right] \) and \( n \in \mathbb{Z} \), then \[ \int \sqrt{1 - \sin 2x} \, dx = \]

• (1) \( -\cos x + \sin x + c \)
• (2) \( \cos x + \sin x + c \)
• (3) \( \cos x - \sin x + c \)
• (4) \( -\cos x - \sin x + c \)

Evaluate the integral: \[ \int e^x \left( \frac{x + 2}{(x+4)} \right)^2 dx. \]

• (1) \( \frac{-x e^x}{(x+4)^2} + c \)
• (2) \( \frac{-x e^x}{(x+4)} + c \)
• (3) \( \frac{x e^x}{(x+4)} + c \)
• (4) \( \frac{2x e^x}{(x+4)} + c \)

If \[ \int \frac{1}{1 - \cos x} \, dx = \tan \left( \frac{x}{4} + \beta \right) + c, \]
then one of the values of \( \frac{\pi}{4} - \beta \) is:

• (1) \( -\frac{\pi}{2} \)
• (2) \( \frac{\pi}{2} \)
• (3) \( 0 \)
• (4) \( \frac{\pi}{4} \)

If \[ 729 \int_1^3 \frac{1}{x^3 (x^2 + 9)^2} \, dx = a + \log b, \]
then \( a - b = \) ?

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

If \( n \geq 2 \) is a natural number and \( 0 < \theta < \frac{\pi}{2} \), then \[ \int \frac{(\cos^n \theta - \cos \theta)^{1/n}}{\cos^{n+1} \theta} \sin \theta \, d\theta = \]

• (1) \( \frac{n}{n-1} (\cos^{(1-n)}\theta -1)^2 + c \)
• (2) \( \frac{n}{(n+1)(1-n)} (\cos^{(1-n)}\theta -1)^{\frac{1}{n+1}} + c \)
• (3) \( \frac{1}{n-1} (\cos^{(n-\theta)} -1)^2 + c \)
• (4) \( \frac{n}{1 - n^2} \left( 1 - \cos^{(1-n)}\theta \right)^{(n+1)/n} \)

Evaluate the limit: \[ \lim_{n \to \infty} \frac{17^7 + 27^7 + \dots + n^{77}}{n^{78}}. \]

• (1) \( \frac{1}{77} \)
• (2) \( 1 \)
• (3) \( 76 \)
• (4) \( \frac{1}{78} \)

If \[ f(x) = \begin{cases} \frac{6x^2 + 1}{4x^3 + 2x + 3}, & 0 < x < 1
x^2 + 1, & 1 \leq x < 2 \end{cases} \]
then \[ \int_{0}^{2} f(x) \,dx = ? \]

• (1) \( \frac{1}{2} \log 3 + \frac{10}{3} \)
• (2) \( \frac{1}{2} \log 3 - \frac{10}{3} \)
• (3) \( \frac{1}{2} \log 3 + \frac{13}{3} \)
• (4) \( \frac{1}{2} \log 3 + \frac{20}{3} \)

If \[ \int_{1}^{n} f(x) \,dx = 120, \]
then \( n \) is:

• (1) \( 15 \)
• (2) \( 16 \)
• (3) \( 14 \)
• (4) \( 12 \)

The area of the region under the curve \( y = |\sin x - \cos x| \) in the interval \( 0 \leq x \leq \frac{\pi}{2} \), above the x-axis, is (in square units):

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

The differential equation formed by eliminating \( a \) and \( b \) from the equation \[ y = ae^{2x} + bxe^{2x} \]
is:

• (1) \( y''' - 4y'' - 4y' = 0 \)
• (2) \( y''' + 4y'' = 0 \)
• (3) \( y''' - 4y' = 0 \)
• (4) \( y''' - 4y'' + 4y' = 0 \)

If \[ y = a e^{bx} + c e^{dx} + x e^{bx} \]
is the general solution of a differential equation, where \( a \) and \( c \) are arbitrary constants and \( b \) is a fixed constant, then the order of the differential equation is:

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

The solution of the differential equation \[ (x + 2y^3) \frac{dy}{dx} = y \]
is:

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

The time period of revolution of a satellite (\( T \)) around the earth depends on the radius of the circular orbit (\( R \)), mass of the earth (\( M \)) and universal gravitational constant (\( G \)). The expression for \( T \), using dimensional analysis, is (where \( K \) is a constant of proportionality):

• (1) \( K \sqrt{\frac{R^2}{GM}} \)
• (2) \( K \sqrt{\frac{R}{GM}} \)
• (3) \( K \sqrt{\frac{R^3}{GM}} \)
• (4) \( K \sqrt{\frac{R^3}{GM^2}} \)

An object is projected upwards from the foot of a tower. The object crosses the top of the tower twice with an interval of 8 s and the object reaches the foot after 16 s. The height of the tower is (Given \( g = 10 \, m/s^2 \))

• (1) \( 220 \) m
• (2) \( 240 \) m
• (3) \( 640 \) m
• (4) \( 80 \) m

The centripetal acceleration of a particle in uniform circular motion is \( 18 \, ms^{-2} \). If the radius of the circular path is \( 50 \) cm, the change in velocity of the particle in a time of \( \frac{T}{18} \) is:

• (1) \( 9 \, ms^{-1} \)
• (2) \( 2 \, ms^{-1} \)
• (3) \( 3 \, ms^{-1} \)
• (4) \( 6 \, ms^{-1} \)

The horizontal range of a projectile projected at an angle of \( 45^\circ \) with the horizontal is 50 m. The height of the projectile when its horizontal displacement is 20 m is:

• (1) \( 18 \) m
• (2) \( 36 \) m
• (3) \( 12 \) m
• (4) \( 24 \) m

A body of mass 1.5 kg is moving towards south with a uniform velocity of \( 8 ms^{-1} \). A force of \( 6 \) N is applied to the body towards east. The displacement of the body 3 seconds after the application of the force is:

• (1) \( 24 \) m
• (2) \( 30 \) m
• (3) \( 18 \) m
• (4) \( 42 \) m

The upper \( \left(\frac{1}{n} \right)^{th} \) of an inclined plane is smooth, and the remaining lower part is rough with a coefficient of friction \( \mu_k \). If a body starting from rest at the top of the inclined plane will again come to rest at the bottom of the plane, then the angle of inclination of the inclined plane is:

• (1) \( \sin^{-1} \left[ \left( \frac{n}{n-1} \right) \mu_k \right] \)
• (2) \( \sin^{-1} \left[ \left( \frac{n-1}{n} \right) \mu_k \right] \)
• (3) \( \tan^{-1} \left[ \left( \frac{n}{n-1} \right) \mu_k \right] \)
• (4) \( \tan^{-1} \left[ \left( \frac{n-1}{n} \right) \mu_k \right] \)

A spring of spring constant \( 200 \, N/m \) is initially stretched by \( 10 \) cm from the unstretched position. The work to be done to stretch the spring further by another \( 10 \) cm is:

• (1) \( 3 \) J
• (2) \( 6 \) J
• (3) \( 9 \) J
• (4) \( 12 \) J

A ball falls freely from rest from a height of 6.25 m onto a hard horizontal surface. If the ball reaches a height of 81 cm after the second bounce from the surface, the coefficient of restitution is:

• (1) \( 0.3 \)
• (2) \( 0.45 \)
• (3) \( 0.75 \)
• (4) \( 0.6 \)

The masses of a solid cylinder and a hollow cylinder are 3.2 kg and 1.6 kg respectively. Both the solid cylinder and hollow cylinder start from rest from the top of an inclined plane and roll down without slipping. If both the cylinders have equal radius and the acceleration of the solid cylinder is \( 4 ms^{-2} \), the acceleration of the hollow cylinder is:

• (1) \( 2 ms^{-2} \)
• (2) \( 9 ms^{-2} \)
• (3) \( 6 ms^{-2} \)
• (4) \( 3 ms^{-2} \)

A solid sphere of mass 50 kg and radius 20 cm is rotating about its diameter with an angular velocity of 420 rpm. The angular momentum of the sphere is:

• (1) \( 8.8 \) Js
• (2) \( 70.4 \) Js
• (3) \( 17.6 \) Js
• (4) \( 35.2 \) Js

The mass of a particle is \( 1 \) kg and it is moving along the \( x \)-axis. The period of its oscillation is \( \frac{\pi}{2} \). Its potential energy at a displacement of \( 0.2 \) m is:

• (1) \( 0.24 \) J
• (2) \( 0.48 \) J
• (3) \( 0.32 \) J
• (4) \( 0.16 \) J

The potential energy of a particle of mass 10 g as a function of displacement \( x \) is \( (50 \, x^2 + 100) \). The frequency of oscillation is:

• (1) \( \frac{10}{\pi} \) s\(^{-1}\)
• (2) \( \frac{5}{\pi} \) s\(^{-1}\)
• (3) \( \frac{100}{\pi} \) s\(^{-1}\)
• (4) \( \frac{50}{\pi} \) s\(^{-1}\)

If the time period of revolution of a satellite is \( T \), then its kinetic energy is proportional to:

• (1) \( T^{-1} \)
• (2) \( T^{-2} \)
• (3) \( T^{-3} \)
• (4) \( T^{-2/3} \)

The elastic energy stored per unit volume in terms of longitudinal strain \( e \) and Young’s modulus \( Y \) is:

• (1) \( \frac{Y e^2}{2} \)
• (2) \( \frac{1}{2} Y e \)
• (3) \( 2 Y e^2 \)
• (4) \( 2 Y e \)

A large tank filled with water to a height \( h \) is to be emptied through a small hole at the bottom. The ratio of the time taken for the level to fall from \( h \) to \( \frac{h}{2} \) and that taken for the level to fall from \( \frac{h}{2} \) to \( 0 \) is:

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

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:

• (1) \( 20^\circ C \)
• (2) \( 40^\circ C \)
• (3) \( 60^\circ C \)
• (4) \( 80^\circ C \)

A monoatomic gas of \( n \)-moles is heated from temperature \( T_1 \) to \( T_2 \) under two different conditions:

At constant volume
At constant pressure

The change in internal energy of the gas is:

• (1) More when heated at constant volume
• (2) More when heated at constant pressure
• (3) Same in both the cases
• (4) Zero in both the cases

In a Carnot engine, when the temperatures are \( T_2 = 0^\circ C \) and \( T_1 = 200^\circ C \), its efficiency is \( \eta_1 \), and when the temperatures are \( T_1 = 0^\circ C \) and \( T_2 = -200^\circ C \), its efficiency is \( \eta_2 \). Then the value of \( \frac{\eta_1}{\eta_2} \) is:

• (1) \( 0.58 \)
• (2) \( 0.73 \)
• (3) \( 0.64 \)
• (4) \( 0.42 \)

Heat energy absorbed by a system going through the cyclic process shown in the figure is:

• (1) \( 10^7 \pi \) J
• (2) \( 10^4 \pi \) J
• (3) \( 10^2 \pi \) J
• (4) \( 10^{-3} \pi \) J

A polyatomic gas with \( n \) degrees of freedom has a mean kinetic energy per molecule given by (if \( N \) is Avogadro’s number):

• (1) \( \frac{n k T}{N} \)
• (2) \( \frac{n k T}{2N} \)
• (3) \( \frac{n k T}{2} \)
• (4) \( \frac{3 k T}{2} \)

A car sounding a horn of frequency 1000 Hz passes a stationary observer. The ratio of frequencies of the horn noted by the observer before and after passing of the car is 11:9. The speed of the car is (Speed of sound \( v = 340 \, ms^{-1} \)):

• (1) \( 34 \, ms^{-1} \)
• (2) \( 17 \, ms^{-1} \)
• (3) \( 170 \, ms^{-1} \)
• (4) \( 340 \, ms^{-1} \)

A ray of light travels from an optically denser to rarer medium. The critical angle for the two media is \( C \). The maximum possible deviation of the ray will be:

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

The angle of polarisation for a medium with respect to air is \( 60^\circ \). The critical angle of this medium with respect to air is:

• (1) \( \sin^{-1} \sqrt{3} \)
• (2) \( \tan^{-1} \sqrt{3} \)
• (3) \( \cos^{-1} \sqrt{3} \)
• (4) \( \sin^{-1} \frac{1}{\sqrt{3}} \)

A point charge \( q \) coulomb is placed at the centre of a cube of side length \( L \). Then the electric flux linked with each face of the cube is:

• (1) \( \frac{q}{\epsilon_0} \)
• (2) \( \frac{q}{L^2 \epsilon_0} \)
• (3) \( \frac{q}{6L^2 \epsilon_0} \)
• (4) \( \frac{q}{6 \epsilon_0} \)

Three equal electric charges of each charge \( q \) are placed at the vertices of an equilateral triangle of side length \( L \), then potential energy of the system is:

• (1) \( \frac{1}{4\pi\epsilon_0} \frac{3q^2}{L} \)
• (2) \( \frac{1}{4\pi\epsilon_0} \frac{q^2}{3L} \)
• (3) \( \frac{1}{4\pi\epsilon_0} \frac{2q^2}{3L} \)
• (4) \( \frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \)

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:

• (1) \( 4C \)
• (2) \( 2C \)
• (3) \( 8C \)
• (4) \( 16C \)

Five equal resistances each \( 2R \) are connected as shown in figure. A battery of \( V \) volts connected between A and B. Then current through FC is:

• (1) \( \frac{V}{4R} \)
• (2) \( \frac{V}{8R} \)
• (3) \( \frac{V}{R} \)
• (4) \( \frac{V}{2R} \)

A lamp is rated at 240V, 60W. When in use the resistance of the filament of the lamp is 20 times that of the cold filament. The resistance of the lamp when not in use is:

• (1) \( 54 \, \Omega \)
• (2) \( 60 \, \Omega \)
• (3) \( 50 \, \Omega \)
• (4) \( 48 \, \Omega \)

When an electron placed in a uniform magnetic field is accelerated from rest through a potential difference \( V_1 \), it experiences a force \( F \). If the potential difference is changed to \( V_2 \), the force experienced by the electron in the same magnetic field is \( 2F \), then the ratio of potential differences \( \frac{V_2}{V_1} \) is:

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

A rectangular loop of sides 25 cm and 10 cm carrying a current of 10 A is placed with its longer side parallel to a long straight conductor 10 cm apart carrying current 25 A. The net force on the loop is:

• (1) \( 6.25 \times 10^{-5} \, N \)
• (2) \( 5.5 \times 10^{-5} \, N \)
• (3) \( 3.75 \times 10^{-5} \, N \)
• (4) \( 8.75 \times 10^{-11} \, N \)

If the vertical component of the earth's magnetic field is 0.45 G at a location, and angle of dip is \( 60^\circ \), then magnetic field of earth at that location is:

• (1) \( 0.26 \, G \)
• (2) \( 0.52 \, G \)
• (3) \( 0.3 \, G \)
• (4) \( 0.7 \, G \)

X and Y are two circuits having coefficient of mutual inductance 3 mH and resistances 10 \( \Omega \) and 4 \( \Omega \) respectively. To have induced current 60 \( \times 10^{-4} \, A \) in circuit Y, the amount of current to be changed in circuit X in 0.02 sec is:

• (1) \( 1.6 \, A \)
• (2) \( 0.16 \, A \)
• (3) \( 0.32 \, A \)
• (4) \( 3.2 \, A \)

Two figures are shown as Fig. A and Fig. B. The time constant of Fig. A is \( \tau_A \), and time constant of Fig. B is \( \tau_B \). Then:

• (1) \( \tau_A = \frac{1}{4} \, s \) and \( \tau_B = 5 \, s \)
• (2) \( \tau_A = \frac{1}{2} \, s \) and \( \tau_B = \frac{1}{5} \, s \)
• (3) \( \tau_A = 4 \, s \) and \( \tau_B = 5 \, s \)
• (4) \( \tau_A = \frac{1}{3} \, s \) and \( \tau_B = 10 \, s \)

Which of the following produces electromagnetic waves?

• (1) Stationary charges
• (2) Charges in uniform motion
• (3) Accelerating charges
• (4) Stationary magnet

A blue lamp emits light of mean wavelength 4500\(Å\). The lamp is rated at 150 W and 8% efficiency. Then the number of photons are emitted by the lamp per second.

• (1) \( 27.17 \times 10^{18} \)
• (2) \( 17.17 \times 10^{18} \)
• (3) \( 27.17 \times 10^{15} \)
• (4) \( 54 \times 10^{16} \)

The ground state energy of hydrogen atom is -13.6 eV. The potential energy of the electron in this state is:

• (1) \( 27.2 \, eV \)
• (2) \( -27.2 \, eV \)
• (3) \( -13.6 \, eV \)
• (4) \( 13.6 \, eV \)

If the energy released per fission of a \(^{235}U\) nucleus is 200 MeV, the energy released in the fission of 0.1 kg of \(^{235}U\) in kilowatt-hour is:

• (1) \( 22.8 \times 10^5 \, kWh \)
• (2) \( 22.8 \times 10^7 \, kWh \)
• (3) \( 11.4 \times 10^5 \, kWh \)
• (4) \( 820 \times 10^{10} \, kWh \)

The semiconductor used for fabrication of visible LEDs must at least have a band gap of:

• (1) \( 0.6 \, eV \)
• (2) \( 1.2 \, eV \)
• (3) \( 1.8 \, eV \)
• (4) \( 0.9 \, eV \)

In a common emitter amplifier, a.c. current gain is 40 and input resistance is 1 kΩ. The load resistance is given as 10 kΩ. Then the voltage gain is:

• (1) 52
• (2) 125
• (3) 178
• (4) 200

An information signal of frequency 10 kHz is modulated with a carrier wave of frequency 3.61 MHz. The upper side and lower side frequencies are:

• (1) 3650 kHz and 3590 kHz
• (2) 3620 kHz and 3600 kHz
• (3) 3610 kHz and 3580 kHz
• (4) 3600 kHz and 3620 kHz

The energy of the third orbit of \( Li^{2+} \) ion (in J) is:

• (1) \( -2.18 \times 10^{-18} \, J \)
• (2) \( -6.54 \times 10^{-18} \, J \)
• (3) \( -7.3 \times 10^{-19} \, J \)
• (4) \( +2.18 \times 10^{-18} \, J \)

The number of \( d \) electrons in Fe is equal to which of the following?

• (i) Total number of \( s \)-electrons of Mg

The correct order of atomic radii of given elements is

• (A) \( B < Be < Mg \)
• (B) \( Mg < Be < B \)
• (C) \( Be < B < Mg \)
• (D) \( B < Mg < Be \)

Which of the following orders are correct regarding their covalent character?



(i) \( KF < KI \)

Observe the following sets:





Which of the above sets are correctly matched?

• (A) \( i, ii, iv only \)
• (B) \( i, iii only \)
• (C) \( ii, iii, iv only \)
• (D) \( i, iii, iv only \)

The RMS velocity (\(v_{rms}\)) of one mole of an ideal gas was measured at different temperatures and the following graph is obtained. What is the slope (m) of the straight line?

• (A) \( \frac{3R}{M} \)
• (B) \( \frac{M}{3R} \)
• (C) \( \frac{M}{3R} \)
• (D) \( \frac{3R}{M} \)

Two statements are given below:

Statement I: Viscosity of liquid decreases with increase in temperature.

Statement II: The units of viscosity coefficient are Pascal sec.

The correct answer is:

• (A) Both statement-I and statement-II are correct.
• (B) Both statement-I and statement-II are not correct.
• (C) Statement-I is correct but statement-II is not correct.
• (D) Statement-I is not correct but statement-II is correct.

0.1 mole of potassium permanganate was heated at 300°C. What is the weight (in g) of the residue?

• (1) 14.2 g
• (2) 1.6 g
• (3) 15.8 g
• (4) 7.1 g

Identify the correct statements from the following:


(i) \(\Delta G\) is zero for a \( A \rightarrow B \) reaction.

(ii) The entropy of pure crystalline solids approaches zero as the temperature approaches absolute zero.

(iii) \(\Delta U\) of a reaction can be determined using a bomb calorimeter.

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

Observe the following reactions:

\( AB(g) + 25 H_2O(l) \rightarrow AB(H_2S{O_4}) \quad \Delta H = x \, kJ/mol^{-1} \)
\( AB(g) + 50 H_2O(l) \rightarrow AB(H_2SO_4) \quad \Delta H = y \, kJ/mol^{-1} \)


The enthalpy of dilution, \( \Delta H_{dil} \) in kJ/mol\(^{-1}\), is:

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

K_c for the reaction \[ A(g) \rightleftharpoons T(K) + B(g) \]
is 39.0. In a closed one-litre flask, one mole of \( A(g) \) was heated to \( T(K) \). What are the concentrations of \( A(g) \) and \( B(g) \) (in mol L\(^{-1}\)) respectively at equilibrium?

• (1) \( 0.025, 0.975 \)
• (2) \( 0.975, 0.025 \)
• (3) \( 0.05, 0.95 \)
• (4) \( 0.02, 0.98 \)

At \( T(K) \), the solubility product of \( AX \) is \( 10^{-10} \). What is the molar solubility of \( AX \) in 0.1 M HX solution?

• (1) \( 10^{-5} \)
• (2) \( 10^{-10} \)
• (3) \( 10^{-9} \)
• (4) \( 10^{-8} \)

The equation that represents ‘coal gasification’ is:

• (1) \( CO(g) + H_2O(g) \xrightarrow{673 \, K} CO_2(g) + H_2(g) \, (catalyst) \)
• (2) \( C(s) + H_2O(g) \xrightarrow{1270 \, K} CO(g) + H_2(g) \)
• (3) \( 2C(s) + O_2(g) \xrightarrow{1273 \, K} 2CO(g) + 4N_2(g) \)
• (4) \( CH_4(g) \xrightarrow{1270 \, K} CO(g) + 3H_2(g) \, (Ni) \)

As per standard reduction potential values, which is the strongest reducing agent among the given elements?

• (1) Rb
• (2) Sr
• (3) Na
• (4) Mg

A Lewis acid 'X' reacts with LiAlH\(_4\) in either medium to give a highly toxic gas, 'Y'. 'Y' when heated with NH\(_3\) gives a compound known as inorganic benzene. 'Y' burns in oxygen and gives H\(_2\)O and 'Z'. 'Z' is:

• (1) Basic oxide
• (2) Acidic oxide
• (3) Amphoteric acid
• (4) Neutral oxide

The method for preparation of water gas is:

• (1) Burning coke in excess of air
• (2) Oxidation of C in limited supply of oxygen
• (3) Passing steam over hot coke
• (4) Passing air over hot coke

The BOD values for pure water and highly polluted water are respectively:

• (1) \( > 5 \, ppm, \leq 17 \, ppm \)
• (2) \( > 5 \, ppm, \geq 17 \, ppm \)
• (3) \( < 5 \, ppm, > 17 \, ppm \)
• (4) \( < 5 \, ppm, \leq 17 \, ppm \)

A mixture of ethyl iodide and n-propyl iodide is subjected to Wurtz reaction. The hydrocarbon which will not be formed is:

• (1) Butane
• (2) Propane
• (3) Pentane
• (4) Hexane

Which of the following alkenes does not undergo anti-Markovnikov addition of HBr?

• (1) Propene
• (2) 1-Butene
• (3) 2-Butene
• (4) 3-Methyl-2-Pentene

What are the variables in the graph of powder diffraction pattern of a crystalline solid?

• (1) \( x-axis = 2\theta; \, y-axis = intensity \)
• (2) \( x-axis = \theta; \, y-axis = 2\theta \)
• (3) \( x-axis = \theta; \, y-axis = intensity \)
• (4) \( x-axis = intensity; \, y-axis = \theta \)

100 mL of \( M \times 10^{-1} \) Ca(NO₃)₂ and 200 mL of \( M \times 10^{-1} \) KNO₃ solutions are mixed. What is the normality of the resulted solution with respect to NO₃⁻?

• (1) 0.1 N
• (2) 0.2 N
• (3) 0.13 N
• (4) 0.066 N

A solution was prepared by dissolving 0.1 mole of a non-volatile solute in 0.9 moles of water. What is the relative lowering of vapor pressure of the solution?

• (1) 0.9
• (2) 0.1
• (3) 0.05
• (4) 0.066

The standard free energy change (\( \Delta G^\circ \)) for the following reaction (in kJ) at 25°C is
\[ 3Ca(s) + 2 Au^{+}(aq, 1M) \rightleftharpoons 3Ca^{2+}(aq, 1M) + 2Au(s) \]
(given: \( E^\circ_{Au^{3+/2+}} = +1.50 \, V \), \( E^\circ_{Ca^{2+/Ca}} = -2.87 \, V \), \( F = 96500 \, C mol^{-1} \))

• (1) \(-2.53 \times 10^3 \)
• (2) \( +2.53 \times 10^3 \)
• (3) \(-2.53 \times 10^4 \)
• (4) \( +2.53 \times 10^4 \)

The rate constant of a first-order reaction is \( 3.46 \times 10^3 \, s^{-1} \) at 298K. What is the rate constant of the reaction at 350 K if its activation energy is 50.1 kJ mol\(^{-1}\) (R = 8.314 J K\(^{-1}\) mol\(^{-1}\))?

• (1) 0.592 s\(^{-1}\)
• (2) 0.692 s\(^{-1}\)
• (3) 0.792 s\(^{-1}\)
• (4) 0.892 s\(^{-1}\)

The correct statement regarding chemisorption is

• (1) It is a multilayered adsorption
• (2) The process is reversible in nature
• (3) The process is not specific in nature
• (4) Enthalpy of adsorption is in the range of 80-240 kJ mol\(^{-1}\)

Which of the following is incorrectly matched?

• (1) Multi molecular colloid - S\(_8\)
• (2) Macro molecular colloid - enzyme
• (3) As\(_2\)S\(_3\) sol - positively charged sol
• (4) Starch sol - lyophilic sol

Impure silver ore + CN⁻ + H₂O → [X]⁻ + OH⁻

[X]⁻ + Zn → [Y]²⁺ + Ag (pure)

The co-ordination numbers of the metals in [X], [Y] are respectively:

• (1) 3, 4
• (2) 1, 4
• (3) 2, 4
• (4) 2, 2

In the reaction of sulfur with concentrated sulfuric acid, the oxidised product is X and reduced product is Y. X and Y are respectively:

• (1) \( SO_3, SO_2 \)
• (2) \( SO_2, SO_2 \)
• (3) \( SO_2, H_2S \)
• (4) \( SO_2, H_2O \)

Which of the following lanthanides have [Xe] 4f⁵ 5d¹ 6s² configuration in their ground state?

• (1) Pr, Tb, Yb
• (2) Ce, Yb, Lu
• (3) Ce, Gd, Lu
• (4) Gd, Tb, Lu

How many of the following ligands are stronger than H₂O?
\[ S^{2-}, \, Br^-, \, C_2O_4^{2-}, \, CN^-, \, en, \, NH_3, \, CO, \, OH^- \]

• (1) 5
• (2) 3
• (3) 4
• (4) 6

The common monomer for both Terylene and Glyptal is

• (1)
• (2)
• (3)
• (4)

Which of the following structure of proteins represents its constitution?

• (1) Secondary structure
• (2) Quaternary structure
• (3) Primary structure
• (4) Tertiary structure

Carrot and curd are sources for the vitamins respectively.

• (1) A, B12
• (2) A, B1
• (3) E, Pyridoxine
• (4) E, Riboflavin

Match the following




Correct answer is

The major products X and Y respectively from the following reactions are

• (1)
• (2)
• (3)
• (4)

An isomer of C₆H₁₂ on reaction with Br₂/ light gave only one isomer C₆H₁₁Br (X). Reaction of X with AgNO₃ gave Y as the major product. What is Y?

• (1)
• (2)
• (3)
• (4)

What are the major products X and Y respectively in the following reactions?
\[ (CH_3)_3COONa + CH_2 CH Br \rightarrow X \] \[ (CH_3)_3COCH_2 CH_3 + CH_3COONa \rightarrow Y \]

• (1) \( CH_2 = CH_2, \, (CH_3)_3 COCH_2 CH_3 \)
• (2) \( (CH_3)_3 COCH_2 CH_3, \, (CH_3)_3 COCH_2 CH_3 \)
• (3) \( (CH_3)_3 COCH_2 CH_3, \, CH_2 = CH_2 \)
• (4) \( (CH_3)_3 COCH_2 CH_3, \, CH_2 = CH_2 \)

Match the following reagents with the products obtained when they react with a ketone:




Correct Answer is

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

What are X and Y respectively in the following reactions?

• (1)
• (2)
• (3)
• (4)

Arrange the following in decreasing order of their basicity:

• (1) \( B > C > A \)
• (2) \( B > A > C \)
• (3) \( A > B > C \)
• (4) \( A > C > B \)