AP EAPCET (AP EAMCET) 2025 Question Paper May 22 Shift 1 (Available): Download Solutions with Answer Key

Let \( f : \mathbb{N} \to \mathbb{N} \) be a function such that \( f(x + y) = f(x) + f(y) + xy \) for every \( x, y \in \mathbb{N} \). If \( f(1) = 2 \), then \( \sum\limits_{k=0}^{10} f(k) \) is equal to:

• (A) \(1650\)
• (B) \(275\)
• (C) \(550\)
• (D) \(1025\)

If a real-valued function \( f : [-1, 2] \to B \) is defined by: \[ f(x) = \begin{cases} 1 - x, & when -1 \leq x \leq 1
x - 1, & when 1 < x \leq 2 \end{cases} and f is a surjection, then B = ? \]

• (A) \( [-1, 2] \)
• (B) \( [-1, 1] \)
• (C) \( [0, 2] \)
• (D) \( [0, 1] \)

For all \( n \in \mathbb{N} \), which of the following is less than or equal to \( \frac{3^n - 1}{2} \)?

• (A) \( n^2 \left(2^{n/2} \right) \)
• (B) \( n^2 \left(\frac{n - 1}{3^2} \right) \)
• (C) \( n^3 \left(\frac{n - 1}{3^2} \right) \)
• (D) \( n \left(\frac{n - 1}{3^2} \right) \)

The values of \( p \) and \( q \) so that the system of equations \[ \begin{aligned} 2x + py + 6z &= 8,
x + 2y + qz &= 5,
x + y + 3z &= 4 \end{aligned} \]
may have no solution are:

• (A) \( p \neq 2, \, q = 3 \)
• (B) \( p \neq 2, \, q \neq 3 \)
• (C) \( p = 2, \, q = \frac{15}{4} \)
• (D) \( p = 2, \, q = 3 \)

A is the set of all matrices of order 3 with entries 0 or 1 only.
Let \( B \subset A \) be the subset consisting of all matrices with determinant value 1.
Let \( C \subset A \) be the subset consisting of all matrices with determinant value -1.
Then, which of the following is true?

• (A) \( A = B \cup C \)
• (B) \( C \) is empty
• (C) \( B \) and \( C \) contain the same number of elements
• (D) \( B \) has twice as many elements as \( C \)

Consider the matrices \[ A = \begin{bmatrix} x & y & 0
-3 & 1 & 2
1 & -2 & z \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 & -2
2 & 0 & 1
2 & 1 & 0 \end{bmatrix} \]
If the cofactors of the elements \( z, 1 \) in 3rd row and \( x \) of \( A \) are 9, 4, 3 respectively, then \( AB = \)

• (A) \( \begin{bmatrix} -7 & -4 & -8
  -1 & 8 & 7
  3 & -3 & -4 \end{bmatrix} \)
• (B) \( \begin{bmatrix} 7 & -6 & 8
  -5 & 4 & -5
  -5 & -3 & -4 \end{bmatrix} \)
• (C) \( \begin{bmatrix} 7 & -6 & -4
  3 & 8 & 7
  -5 & -3 & -4 \end{bmatrix} \)
• (D) \( \begin{bmatrix} 7 & -6 & 8
  -1 & 8 & -5
  3 & -3 & -4 \end{bmatrix} \)

The minimum value of \( |z - 1| + |z - 5| \) is:

• (A) \( 3 \)
• (B) \( 5 \)
• (C) \( 4 \)
• (D) \( 2 \)

If \( z = x + iy \) and if the point \( P \) in the Argand diagram represents \( z \), then the locus of the point \( P \) satisfying the equation \( 2|z - 2 - 3i| = 3|z + i - 2| \) is a circle with centre:

• (A) \( (10, -21) \)
• (B) \( (-10, 21) \)
• (C) \( \left(2, -\frac{21}{5}\right) \)
• (D) \( \left(-2, \frac{21}{5}\right) \)

If \( z \) is a non-real root of \( z^7 = 1 \), then the value of \[ 1 + 3z + 5z^2 + 7z^3 + 9z^4 + 11z^5 + 13z^6 \]
is:

• (A) \( \frac{14}{1 - z} \)
• (B) \( \frac{-14}{1 - z} \)
• (C) \( \frac{15}{1 - z} \)
• (D) \( \frac{-15}{1 - z} \)

If \( (2k - 1)x^2 - 2(3k - 2)x + 4k > 0 \) for every \( x \in \mathbb{R} \), then the sum of all possible integral values of \( k \) is:

• (A) \(21\)
• (B) \(27\)
• (C) \(36\)
• (D) \(28\)

The sum of the least positive integer and the greatest negative integer in the range of the function \( f(x) = \dfrac{x^2 - 5x + 7}{x^2 - 5x - 7} \) is:

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(-1\)

If \( \alpha \) is a repeated root of multiplicity 2 of the equation \( 18x^3 - 33x^2 + 20x - 4 = 0 \), then:

• (A) \( 3\alpha^2 - 8\alpha + 4 = 0 \)
• (B) \( 3\alpha^2 + 8\alpha + 4 = 0 \)
• (C) \( 3\alpha^2 - \alpha - 4 = 0 \)
• (D) \( 3\alpha^2 + 2\alpha - 4 = 0 \)

The equation \( 6x^4 - 5x^3 + 13x^2 - 5x + 6 = 0 \) will have:

• (A) only real roots
• (B) only complex roots
• (C) two real and two complex roots
• (D) two real and two purely imaginary roots

All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order. Then the rank of the word TETLER is:

• (A) \(171\)
• (B) \(138\)
• (C) \(141\)
• (D) \(168\)

5-digit numbers are formed by using the digits 0, 1, 2, 3, 5, 7 without repetition and all of them are arranged in ascending order. Then the rank of the number 70513 is:

• (A) \(500\)
• (B) \(499\)
• (C) \(497\)
• (D) \(503\)

The number of divisors of \(7!\) is:

• (A) \(72\)
• (B) \(24\)
• (C) \(64\)
• (D) \(60\)

If \(k\) is a positive integer and \(10^k\) is a divisor of the number \(9^{11} + 11^9\), then the greatest value of \(k\) is:

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

The number of all possible values of \( k \) for which the expansion \( (\sqrt{x} + k\sqrt{y})^{10} \) will have exactly nine irrational terms is:

• (A) 3
• (B) 4
• (C) 5
• (D) 6

If \[ \frac{x+1}{(x - 1)^2(x^2 + 1)} = \frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{Cx + D}{x^2 + 1}, \]
then \[ \sqrt{3A^2 + 4D^2 + 5C^2 + B^2} = ? \]

• (1) \dfrac{3}{2}
• (2) \dfrac{1}{2}
• (3) 1
• (4) 2

If \( A + B = \frac{\pi}{4} \), then \( \dfrac{\cos B - \sin B}{\cos B + \sin B} \) is equal to:

• (1) \( \sin A \)
• (2) \( \cos A \)
• (3) \( \tan A \)
• (4) \( \cot A \)

If \( 7\cos\theta - \sin\theta = 5 \) and \( \tan\theta > 0 \), then \( \tan\theta = \)

• (1) \( \dfrac{7}{12} \)
• (2) \( \dfrac{3}{4} \)
• (3) \( \dfrac{4}{3} \)
• (4) \( \dfrac{12}{7} \)

Evaluate: \[ \sin^3 10^\circ + \sin^3 50^\circ - \sin^3 70^\circ = ? \]

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

The number of solutions of \[ \sin 2x + \cos 4x = 2 \quad in the interval [-\pi, \pi] is \]

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

The range of the real-valued function \[ f(x) = \cos^{-1}(-x) + \sin^{-1}(-x) + \csc^{-1}(x) \]
is

• (1) \( \left\{ 0, \dfrac{\pi}{2} \right\} \)
• (2) \( \left[ 0, \dfrac{\pi}{2} \right) \cup \left( \dfrac{\pi}{2}, \pi \right] \)
• (3) \( \left( 0, \dfrac{\pi}{2} \right) \)
• (4) \( \left\{ 0, \pi \right\} \)

If \( \cosh 2x = 199 \), then \( \cot hx = \)

• (1) \( \dfrac{5}{3\sqrt{11}} \)
• (2) \( \dfrac{5}{6\sqrt{11}} \)
• (3) \( \dfrac{7}{3\sqrt{11}} \)
• (4) \( \dfrac{10}{3\sqrt{11}} \)

The horizontal distance between a tower and a building is \( 10\sqrt{3} \) units. If the angle of depression of the foot of the building from the top of the tower is \( 60^\circ \) and the angle of elevation of the top of the building from the foot of the tower is \( 30^\circ \), then the sum of the heights of the tower and the building is:

• (1) 60
• (2) 50
• (3) 40
• (4) 30

In a \( \triangle ABC \), \( A - B = 120^\circ \), \( R = 8r \), then \[ \frac{1 + \cos C}{1 - \cos C} =\ ? \]

• (1) 16
• (2) 14
• (3) 15
• (4) 10

In \( \triangle ABC \), if \[ \angle ABC = \delta,\quad \delta = \cos^{-1} \left( \sqrt{ \frac{r_2}{r_3 r_1} } \right), \]
then the expression \[ \angle ABC = \delta = \cos^{-1} \left( \sqrt{ \frac{r_2}{r_3 r_1} } \right) \]

• (1) \( (r_3 - r_2)(r_1 - r_2) \)
• (2) \( r_3 + r_1 \)
• (3) \( \dfrac{b}{r_3 - r_1} \)
• (4) \( \dfrac{b}{r_3 + r_1} \)

\( \vec{i} - 2\vec{j} \) is a point on the line parallel to the vector \( 2\vec{i} + \vec{k} \). \( \vec{i} + 2\vec{j} \) is a point on the plane parallel to the vectors \( 2\vec{j} - \vec{k} \) and \( \vec{i} + 2\vec{k} \).
Then, the point of intersection of the line and the plane is:

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

Points \( P \) and \( Q \) are given by \( \vec{OP} = \vec{i} - \vec{j} - \vec{k} \) and \( \vec{OQ} = -\vec{i} + \vec{j} + \vec{k} \).
A line along the vector \( \vec{a} = \vec{i} + \vec{j} \) passes through point \( P \), and another line along the vector \( \vec{b} = \vec{j} - \vec{k} \) passes through point \( Q \).
If a line along the vector \( \vec{c} = \vec{i} - \vec{j} + \vec{k} \) intersects both the lines along \( \vec{a} \) and \( \vec{b} \) at \( L \) and \( M \) respectively, then \( \vec{PM} = \) ?

• (1) \( \vec{i} - \vec{j} + 2\vec{k} \)
• (2) \( 4\vec{i} + 4\vec{j} \)
• (3) \( -2\vec{i} + 10\vec{j} - 6\vec{k} \)
• (4) \( 3\vec{i} - 2\vec{j} + \vec{k} \)

Angle between a diagonal of a cube and a diagonal of its face which are coterminus is

• (1) \( \dfrac{\pi}{2} \)
• (2) \( \cos^{-1}\left( \dfrac{2}{\sqrt{3}} \right) \)
• (3) \( \cos^{-1}\left( \dfrac{1}{\sqrt{3}} \right) \)
• (4) \( \cos^{-1}\left( \dfrac{\sqrt{3}}{2} \right) \)

For \( a \in \mathbb{R} \), if the vectors \( \vec{p} = (a+1)\hat{i} + a\hat{j} + a\hat{k} \), \( \vec{q} = a\hat{i} + (a+1)\hat{j} + a\hat{k} \) and \( \vec{r} = a\hat{i} + a\hat{j} + (a+1)\hat{k} \) are coplanar and \( 3\left(\vec{p} \cdot \vec{q} \right)^2 - \lambda \left|\vec{r} \times \vec{q} \right|^2 = 0 \), then the value of \( \lambda \) is

• (1) \( \dfrac{2}{3} \)
• (2) \( \dfrac{3}{2} \)
• (3) \( 2 \)
• (4) \( 1 \)

If \( \vec{a} = \hat{i} + 4\hat{j} - 4\hat{k}, \ \vec{b} = -2\hat{i} + 5\hat{j} - 2\hat{k} \), and \( \vec{c} = 3\hat{i} - 2\hat{j} - 4\hat{k} \) are three vectors such that \( (\vec{b} \times \vec{c}) \times \vec{a} = x\hat{i} + y\hat{j} + z\hat{k} \), then \( x + y - z = \)

• (1) \( 75 \)
• (2) \( -89 \)
• (3) \( 125 \)
• (4) \( -389 \)

The following data represents the frequency distribution of 20 observations:


Then its mean deviation about the mean is:

• (1) \( 3 \)
• (2) \( 2.4 \)
• (3) \( 2.7 \)
• (4) \( 2.9 \)

The probability that a person A completes a work in a given time is \( \dfrac{2}{3} \), and the probability that another person B completes the same work in the same time is \( \dfrac{3}{4} \). If both A and B start doing this work at the same time, then the probability that the work is completed in the given time is:

• (1) \( \dfrac{11}{12} \)
• (2) \( \dfrac{1}{2} \)
• (3) \( \dfrac{5}{12} \)
• (4) \( \dfrac{8}{9} \)

If \( l, m \) represent any two elements (identical or different) of the set \( \{1, 2, 3, 4, 5, 6, 7\} \), then the probability that \( lx^2 + mx + 1 > 0 \,\, \forall x \in \mathbb{R} \) is

• (1) \( \dfrac{12}{\binom{7}{2}} \)
• (2) \( \dfrac{22}{72} \)
• (3) \( \dfrac{10}{\binom{7}{2}} \)
• (4) \( \dfrac{36}{72} \)

A and B are playing chess game with each other. The probability that A wins the game is 0.6, the probability that he loses is 0.3 and the probability it is draw is 0.1. If they played three games, then the probability that A wins at least two games is

• (1) \( \dfrac{54}{125} \)
• (2) \( \dfrac{81}{125} \)
• (3) \( \dfrac{18}{25} \)
• (4) \( \dfrac{9}{25} \)

\(U_1\), \(U_2\), \(U_3\) are three urns. \(U_1\) contains 5 red, 3 white, 2 black balls; \(U_2\) contains 4 red, 4 white, 2 black balls; and \(U_3\) contains 3 red, 4 white, 3 black balls. If a ball is chosen at random from an urn chosen at random, then the probability of not getting a black ball is

• (1) \( \dfrac{7}{30} \)
• (2) \( \dfrac{23}{30} \)
• (3) \( \dfrac{2}{5} \)
• (4) \( \dfrac{11}{30} \)

If the probability distribution of a random variable \(X\) is as follows, then \(P(X \leq 2) =\)

• (1) \( \dfrac{14}{25} \)
• (2) \( \dfrac{23}{32} \)
• (3) \( \dfrac{41}{49} \)
• (4) \( \dfrac{83}{100} \)

If \(X\) follows Poisson distribution with variance 2, then \(P(X \geq 3) = \)

• (1) \( \dfrac{5}{e^2} \)
• (2) \( \dfrac{5 + 2}{e^2} \)
• (3) \( \dfrac{e^2 - 5}{e^2} \)
• (4) \( \dfrac{5 - e^2}{4} \)

A straight line passing through a fixed point (2, 3) intersects the coordinate axes at points \( P \) and \( Q \). If \( O \) is the origin and \( R \) is a variable point such that \( OPRQ \) is a rectangle, then the locus of \( R \) is

• (1) \( 3x + 2y = xy \)
• (2) \( 2x + 3y = xy \)
• (3) \( 3x + 2y = 6 \)
• (4) \( 3x + 2y = 6xy \)

By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation \( x^2 + 4xy + y^2 = 1 \) is transformed to \( \frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1 \), then \( \sqrt{\frac{a^2 + b^2}{a^2}} = \)

• (1) \( 2 \)
• (2) \( \frac{\sqrt{13}}{3} \)
• (3) \( \frac{3}{2} \)
• (4) \( \sqrt{10} \)

If the lines \( x + 2ay + a = 0 \), \( x + 3by + b = 0 \), \( x + 4cy + c = 0 \) are concurrent, then \( a, b, c \) are in

• (1) Arithmetic Progression
• (2) Geometric Progression
• (3) Harmonic Progression
• (4) Arithmetico-geometric Progression

If \( M \) is the foot of the perpendicular drawn from the origin to the line \( x - 2y + 3 = 0 \), which meets the X and Y-axes at \( A \) and \( B \) respectively, then \( AM = \)

• (1) \( \dfrac{6\sqrt{10}}{5} \)
• (2) \( 6\sqrt{5} \)
• (3) \( \dfrac{6\sqrt{5}}{5} \)
• (4) \( \sqrt{10} \)

One line of the pair of lines \( x^2 + xy - 2y^2 = 0 \) is perpendicular to one line of the pair of lines \( 3y^2 - 5xy - 2x^2 = 0 \). If the combined equation of the two lines other than those two perpendicular lines is \( ax^2 + 2hxy + by^2 = 0 \), then \( a + 2h + b = \)

• (1) \(-1\)
• (2) \(1\)
• (3) \(0\)
• (4) \(-5\)

If the angle between the lines joining the origin to the points of intersection of \( x + 2y + \lambda = 0 \) and \( 2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0 \) is \( \dfrac{\pi}{2} \), then a value of \( \lambda \) is:

• (1) \(1\)
• (2) \(\dfrac{1}{2}\)
• (3) \(2\)
• (4) \(\dfrac{3}{2}\)

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

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

If the circle passing through the points \( (3,5), (5,5), (3,-3) \) cuts the circle \( x^2 + y^2 + 2x + 2fy = 0 \) orthogonally, then the value of \( f \) is:

• (1) \(-12\)
• (2) \(-3\)
• (3) \(-15\)
• (4) \(-4\)

Length of the common chord of two circles of same radius is \( 2\sqrt{17} \). If one of the two circles is \( x^2 + y^2 + 6x + 4y - 12 = 0 \), then the acute angle between the two circles is:

• (1) \( \dfrac{\pi}{2} \)
• (2) \( \sin^{-1} \left( \dfrac{3}{5} \right) \)
• (3) \( \cos^{-1} \left( \dfrac{9}{25} \right) \)
• (4) \( \tan^{-1} \left( \dfrac{9}{17} \right) \)

A circle \( S = x^2 + y^2 - 16 = 0 \) intersects another circle \( S' = 0 \) of radius 5 units such that their common chord is of maximum length. If the slope of that chord is \( \dfrac{3}{4} \), then the centre of such a circle \( S' = 0 \) is:

• (1) \( \left( \dfrac{9}{5}, \dfrac{12}{5} \right) \)
• (2) \( \left( \dfrac{9}{5}, -\dfrac{12}{5} \right) \)
• (3) \( \left( -\dfrac{9}{5}, \dfrac{12}{5} \right) \)
• (4) \( \left( \dfrac{3}{5}, \dfrac{4}{5} \right) \)

Let \( \theta \) be the angle between the circles \( S = x^2 + y^2 + 2x - 2y + c = 0 \) and \( S' = x^2 + y^2 - 6x - 8y + 9 = 0 \). If \( c \) is an integer and \( \cos\theta = \dfrac{5}{16} \), then the radius of the circle \( S = 0 \) is:

• (1) 2
• (2) 4
• (3) 3
• (4) 1

PQ is a focal chord of the parabola \( y^2 = 4x \) with focus S. If \( P = (4,4) \), then SQ = ?

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

The angle between the tangents drawn from a point \( (-3, 2) \) to the ellipse \( 4x^2 + 9y^2 - 36 = 0 \) is:

• (1) \(45^\circ\)
• (2) \( \tan^{-1}\left(\frac{2}{3} \right) \)
• (3) \( \tan^{-1}\left(\frac{3}{2} \right) \)
• (4) \(90^\circ\)

If a tangent to the hyperbola \( xy = -1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of that tangent is:

• (1) \( 3y + x = 2 \)
• (2) \( y = 3x + 4 \)
• (3) \( y = x + 2 \)
• (4) \( y = 2x + 1 \)

The distance between the tangents of the hyperbola \( 2x^2 - 3y^2 = 6 \) which are perpendicular to the line \( x - 2y + 5 = 0 \) is:

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

If \( A(0,0,0),\ B(3,4,0),\ C(0,12,5) \) are the vertices of a triangle ABC, then the x-coordinate of its incenter is:

• (1) \( \dfrac{25}{18 + 7\sqrt{2}} \)
• (2) \( \dfrac{25}{26} \)
• (3) \( \dfrac{39}{18 + 7\sqrt{2}} \)
• (4) \( \dfrac{39}{26} \)

If \( A = (0, 4, -3),\ B = (5, 0, 12),\ C = (7, 24, 0) \), then \( \angle BAC = \)

• (1) \( 60^\circ \)
• (2) \( \cos^{-1}\left( \dfrac{16}{\sqrt{13}} \right) \)
• (3) \( \cos^{-1}\left( \dfrac{13}{38} \right) \)
• (4) \( 90^\circ \)

A plane \( \pi \) is passing through the points \( A(1, -2, 3) \) and \( B(6, 4, 5) \). If the plane \( \pi \) is perpendicular to the plane \( 3x - y + z = 2 \), then the perpendicular distance from \( (0, 0, 0) \) to the plane \( \pi \) is

• (1) \( \dfrac{63}{\sqrt{594}} \)
• (2) \( \dfrac{32}{\sqrt{594}} \)
• (3) \( \dfrac{72}{\sqrt{435}} \)
• (4) \( \dfrac{23}{\sqrt{135}} \)

\[ \lim_{y \to 0} \frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt{2}}{y^4} = \ ? \]

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

\[ If \lim_{x \to 0} \frac{\cos 2x - \cos 4x}{1 - \cos 2x} = k,\ then evaluate \lim_{x \to k} \frac{x^k - 27}{x^{k+1} - 81} \]

• (1) \( 0 \)
• (2) \( 1 \)
• (3) \( \dfrac{1}{2} \)
• (4) \( \dfrac{1}{4} \)

If the function  is continuous everywhere, then a^2 + b^2 =  ? 

• (1) \( 4 \)
• (2) \( 8 \)
• (3) \( 6 \)
• (4) \( 12 \)

\[ If x = 2 \cos^3 \theta and y = 3 \sin^2 \theta, then \frac{dy}{dx} =\ ? \]

• (1) \( -\sec \theta \)
• (2) \( \cos \theta \)
• (3) \( -\csc \theta \)
• (4) \( \sin \theta \)

\[ Assertion (A): If y = f(x) = (|x| - |x - 1|)^2, then \left.\frac{dy}{dx}\right|_{x = 1} = 1 \]
\[ Reason (R): If \lim_{x \to a} \frac{f(x) - f(a)}{x - a} exists, then it is called the derivative of f(x) at x = a. \]

Then:

• (1) (A) is true, (R) is true, (R) is correct explanation to (A)
• (2) (A) is true, (R) is true, (R) is not the correct explanation to (A)
• (3) (A) is true, (R) is false
• (4) (A) is false, (R) is true

\[ If y = |\cos x - \sin x| + |\tan x - \cot x|, then \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{3}} + \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{6}} = \]

• (1) 1
• (2) -1
• (3) 2
• (4) 0

If the tangent drawn at the point \((\alpha, \beta)\) on the curve \[ x^{2/3} + y^{2/3} = 4 \]
is parallel to the line \[ \sqrt{3}x + y = 1, \]
then \( \alpha^2 + \beta^2 =\)

• (1) 10
• (2) 9
• (3) 28
• (4) 19

The displacement \(S\) of a particle measured from a fixed point \(O\) on a line is given by \[ S = t^3 - 16t^2 + 64t - 16. \]
Then the time at which the displacement of the particle is maximum is

• (1) 8
• (2) 4
• (3) \(\dfrac{8}{3}\)
• (4) \(\dfrac{4}{3}\)

If the extreme value of the function \( f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x} \) in \(\left[0, \frac{\pi}{2}\right]\) is \(m\) and it exists at \(x = k\), then \(\cos k =\)

• (1) \(\dfrac{\sqrt{m}}{4}\)
• (2) \(\dfrac{\sqrt{m}+1}{\sqrt{2}}\)
• (3) \(\dfrac{\sqrt{5}}{\sqrt{m}}\)
• (4) \(\sin \theta\)

The interval in which the curve represented by \( f(x) = 2x + \log\left(\frac{x}{2 + x}\right) \) is increasing is

• (1) \((- \infty, 0)\)
• (2) \((-2, \infty)\)
• (3) \((- \infty, -2) \cup (0, \infty)\)
• (4) \((-2, 0)\)

Evaluate the integral: \[ \int \frac{1}{9\cos^2 x - 24 \sin x \cos x + 16 \sin^2 x} \, dx = \]

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

If \[ \int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} \, dx = A \log \left| \cos \frac{x}{2} \right| + B \log \left| \cos \frac{x}{3} \right| + C \log \left| \cos \frac{x}{6} \right| + k, \]
then \(A + B + C =\)

• (1) 7
• (2) 11
• (3) -7
• (4) 1

Evaluate the integral: \[ \int \frac{\sin x + \cos x}{\sin x - \cos x} \, dx =\ ? \]

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

Evaluate the integral: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx =\ ? \]

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

Evaluate the integral: \[ \int \frac{(3x - 2)\tan\left(\sqrt{9x^2 - 12x + 1}\right)}{\sqrt{9x^2 - 12x + 1}} \, dx =\ ?\]

• (1) \(\frac{1}{3} \sec^2 \left( \sqrt{9x^2 - 12x + 1} \right) + c\)
• (2) \(\frac{1}{3} \sec^2 x + c\)
• (3) \(\frac{1}{2} \log \left| \sec \left( \sqrt{9x^2 - 12x + 1} \right) \right| + c\)
• (4) \(\frac{1}{3} \log \left| \sec \left( \sqrt{9x^2 - 12x + 1} \right) \right| + c\)

Evaluate the integral: \[ \left| \int_{-\pi/4}^{\pi/3} \tan\left(x - \frac{\pi}{6}\right) dx \right| \]

• (1) \(\dfrac{\log(\sqrt{3} - 1)}{\sqrt{6}}\)
• (2) \(\log(2\sqrt{2}(\sqrt{3} + 1))\)
• (3) \(\dfrac{\log(\sqrt{3} + 1)}{\sqrt{6}}\)
• (4) \(\log(2\sqrt{2}(\sqrt{3} - 1))\)

Evaluate the integral: \[ \int_0^{\pi} \frac{x \sin x}{\sin^2 x + 2\cos^2 x} \, dx \]

• (1) \(\dfrac{\pi}{2}\)
• (2) \(\dfrac{\pi^2}{2}\)
• (3) \(\dfrac{\pi^2}{4}\)
• (4) \(\dfrac{\pi}{4}\)

The area of the region lying between the curves \( y = \sqrt{4 - x^2} \), \( y^2 = 3x \) and the Y-axis is:

• (1) \(\dfrac{\pi}{3} - \dfrac{1}{2\sqrt{3}}\)
• (2) \(\dfrac{\pi}{6} + \dfrac{1}{2\sqrt{3}}\)
• (3) \(\dfrac{\pi}{3} + \dfrac{1}{2\sqrt{3}}\)
• (4) \(\dfrac{\pi}{6} - \dfrac{1}{2\sqrt{3}}\)

Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{1^2 + n^2} + \frac{2}{2^2 + n^2} + \frac{3}{3^2 + n^2} + \cdots + \frac{n}{n^2 + n^2} \right) \]

• (1) \(1\)
• (2) \(\dfrac{1}{2} \log 2\)
• (3) \(2 \log 2\)
• (4) \(0\)

The general solution of the differential equation \[ \frac{dy}{dx} = \frac{2x^2 - xy - y^2}{x^2 - y^2} \]

• (1) \(\log \left| \frac{y^2 - 2x^2}{x^2} \right| + \sqrt{2} \log \left| \frac{y - \sqrt{2}x}{y + \sqrt{2}x} \right| + 2\sqrt{2} \log |x| = c\)
• (2) \(\sqrt{2} \log \left| \frac{y^2 - 2x^2}{x^2} \right| + \log \left| \frac{y - \sqrt{2}x}{y + \sqrt{2}x} \right| + 2\sqrt{2} \log |x| = c\)
• (3) \(\sqrt{2} \log \left| \frac{y^2 + 2x^2}{x^2} \right| + \log \left| \frac{y + \sqrt{2}x}{y - \sqrt{2}x} \right| + 2\sqrt{2} \log |x| = c\)
• (4) \(\log \left| \frac{2x^2 - y^2}{x^2} \right| + \sqrt{2} \log \left| \frac{y + \sqrt{2}x}{y - \sqrt{2}x} \right| + \log |x| = c\)

If the degree of the differential equation corresponding to the family of curves \[ y = ax + \frac{1}{a} \quad (where a \ne 0 is an arbitrary constant) \]
is \(r\) and its order is \(m\), then the solution of \[ \frac{dy}{dx} - \frac{y}{2x}, \quad y(1) = \sqrt{r + m} \]
is

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

The general solution of the differential equation \[ y + \cos x \left( \frac{dy}{dx} \right) - \cos^2 x = 0 \]
is

• (1) \((\sec x + \tan x) y = x + \cos x + c\)
• (2) \((1 + \cos x) y = (x + c) \cos x - \cos^2 x\)
• (3) \((1 + \sin x) y = (x + c) \cos x - \cos^2 x\)
• (4) \((\sec x + \tan x) y = x - \sin x + c\)

The dimensional formula of Planck’s constant is

• (1) [ML\textsuperscript{2}T\textsuperscript{-3}]
• (2) [ML\textsuperscript{2}T\textsuperscript{0}]
• (3) [ML\textsuperscript{2}T\textsuperscript{-1}]
• (4) [M\textsuperscript{0}L\textsuperscript{0}T\textsuperscript{0}]

The ratio of the displacements of a freely falling body during second and fifth seconds of its motion is

• (1) 1 : 1
• (2) 2 : 5
• (3) 4 : 25
• (4) 1 : 3

The magnitudes of two vectors are A and B (A > B). If the maximum resultant magnitude of the two vectors is ‘n’ times their minimum resultant magnitude, then \(\frac{A}{B} =\)

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

A particle crossing the origin at time \(t = 0\), moves in the xy-plane with a constant acceleration ‘a’ in y-direction. If the equation of motion of the particle is \(y = bx^2\) (where \(b\) is a constant), then its velocity component in the x-direction is

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

A train of mass \(10^6 \, kg\) is moving at a constant speed of \(108 \, kmph\). If the frictional force acting on it is \(0.5 \, N per 100 kg\), then the power of the train is

• (1) 300 kW
• (2) 150 kW
• (3) 75 kW
• (4) 225 kW

Two balls each of mass 250 g moving in opposite directions each with a speed 16 m/s collide and rebound with the same speeds. The impulse imparted to one ball due to the other is

• (1) 4 kg m s\textsuperscript{-1}
• (2) 16 kg m s\textsuperscript{-1}
• (3) 8 kg m s\textsuperscript{-1}
• (4) 2 kg m s\textsuperscript{-1}

A body is moving along a straight line under the influence of a constant power source. If the relation between the displacement (s) of the body and time (t) is \(s \propto t^x\), then \(x =\)

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

A body is projected at an angle of \(60^\circ\) with the horizontal. If the initial kinetic energy of the body is \(X\), then its kinetic energy at the highest point is

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

A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is

• (1) three times its rotational kinetic energy
• (2) three times its translational kinetic energy
• (3) one and half times its rotational kinetic energy
• (4) twice its translational kinetic energy

Three thin uniform rods each of mass \( M \) and length \( L \) are placed along the three axes of a Cartesian coordinate system with one end of all the rods at origin. The moment of inertia of the system of the rods about z-axis is

• (1) \(\dfrac{ML^2}{3}\)
• (2) \(\dfrac{2ML^2}{3}\)
• (3) \(\dfrac{ML^2}{2}\)
• (4) \(ML^2\)

For a particle executing simple harmonic motion, the ratio of kinetic and potential energies at a point where displacement is one half of the amplitude is

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

When the mass attached to a spring is increased from 4 kg to 9 kg, the time period of oscillation increases by \(0.2\pi\) s. Then the spring constant of the spring is

• (1) \(80~N m^{-1}\)
• (2) \(200~N m^{-1}\)
• (3) \(50~N m^{-1}\)
• (4) \(100~N m^{-1}\)

Two solid spheres each of radius \( R \) made of same material are placed in contact with each other. If the gravitational force acting between them is \( F \), then

• (1) \( F \propto R^4 \)
• (2) \( F \propto R^3 \)
• (3) \( F \propto R^2 \)
• (4) \( F \propto R \)

The force required to stretch a steel wire of area of cross-section \(1 \ mm^2\) to double its length is
(Young’s modulus of steel \(= 2 \times 10^{11} \ N/m^2\))

• (1) \(2 \times 10^3 \ N\)
• (2) \(2 \times 10^5 \ N\)
• (3) \(2 \times 10^2 \ N\)
• (4) \(2 \times 10^4 \ N\)

In a hydraulic lift, if the radius of the smaller piston is \(5 \ cm\), and the radius of the larger piston is \(50 \ cm\), then the weight that the larger piston can support when a force of \(250 \ N\) is applied to the smaller piston is

• (1) \(50 \ kN\)
• (2) \(100 \ kN\)
• (3) \(40 \ kN\)
• (4) \(25 \ kN\)

If the values of the temperature of a body in Fahrenheit and Celsius scales are in the ratio of 13 : 5, then the temperature of the body is

• (1) \(80^\circ F\)
• (2) \(104^\circ C\)
• (3) \(40^\circ C\)
• (4) \(40^\circ F\)

A Carnot heat engine absorbs 600 J of heat from a source at a temperature of 127°C and rejects 400 J of heat to a sink in each cycle. The temperature of the sink is

• (1) \(266.7\,K\)
• (2) \(166.7\,K\)
• (3) \(133.3\,K\)
• (4) \(333.3\,K\)

During adiabatic expansion, if the temperature of 3 moles of a diatomic gas decreases by \(50^\circC\), then the work done by the gas is

(R - Universal gas constant)

• (1) \(375R\)
• (2) \(750R\)
• (3) \(1500R\)
• (4) \(825R\)

The fundamental limitation to the coefficient of performance of a refrigerator is given by

• (1) first law of thermodynamics
• (2) Newton’s law of cooling
• (3) zeroth law of thermodynamics
• (4) second law of thermodynamics

If the ratio of specific heats of a gas at constant pressure and at constant volume is \(\gamma\), then the number of degrees of freedom of the rigid molecules of the gas is

• (1) \(\dfrac{3\gamma - 1}{2\gamma - 1}\)
• (2) \(\dfrac{2}{\gamma - 1}\)
• (3) \(\dfrac{9}{2}(\gamma - 1)\)
• (4) \(\dfrac{25}{2}(\gamma - 1)\)

A steel wire of length 81 cm has a mass of \(5 \times 10^{-3}\) kg. If the wire is under a tension of 50 N, then the speed of transverse waves on the wire is

• (1) 100 m s\(^{-1}\)
• (2) 105 m s\(^{-1}\)
• (3) 90 m s\(^{-1}\)
• (4) 60 m s\(^{-1}\)

A light ray incidents on an equilateral prism made of material of refractive index \(\sqrt{3}\). Inside the prism, if the light ray moves parallel to the base of the prism, then the angle of incidence of the light ray is

• (1) \(30^\circ\)
• (2) \(45^\circ\)
• (3) \(75^\circ\)
• (4) \(60^\circ\)

An unpolarised beam of light incidents on a group of three polarising sheets arranged such that the angle between the axes of any two adjacent sheets is \(30^\circ\). The ratio of the intensities of polarised light emerging from the second and third sheets is

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

In a region, the electric field is given by \(\vec{E} = (3\hat{i} + 5\hat{j} + 7\hat{k})\ N/C\). The electric flux through a surface of area \(3\ m^2\) in the \(yz\)-plane is (in SI units)

• (1) 21
• (2) 15
• (3) 12
• (4) 9

The energy stored in a capacitor is \(W\). To double the charge on the plates of the capacitor, the additional work to be done is

• (1) \(W\)
• (2) \(4W\)
• (3) \(\dfrac{4}{3}W\)
• (4) \(3W\)

The velocity acquired by an electron at rest when subjected to a uniform electric field of potential difference \(180~V\) is

(Mass of electron \(= 9 \times 10^{-31\) kg and charge of electron \(= 1.6 \times 10^{-19}\) C)

• (1) \(400~km s^{-1}\)
• (2) \(4000~km s^{-1}\)
• (3) \(800~km s^{-1}\)
• (4) \(8000~km s^{-1}\)

Charge 'Q' (in coulomb) flowing through a conductor in terms of time 't' (in seconds) is given by the equation \( Q = 3t^2 + t \). The current in the conductor at time \( t = 3 \, s \) is

• (1) 3 A
• (2) 7 A
• (3) 19 A
• (4) 21 A

In a metal, the charge carrier density is \( 9.1 \times 10^{28} \, m^{-3} \) and its electrical conductivity is \( 6.4 \times 10^7 \, S m^{-1} \). When an electric field of \( 10 \, N C^{-1} \) is applied to the metal, then the average time between two successive collisions of electrons in the metal is

(Mass of electron \( = 9.1 \times 10^{-31} \, kg \); charge of electron \( = 1.6 \times 10^{-19} \, C \))

• (1) \( 4.6 \times 10^{-14} \, s \)
• (2) \( 2.5 \times 10^{-13} \, s \)
• (3) \( 4.6 \times 10^{-13} \, s \)
• (4) \( 2.5 \times 10^{-14} \, s \)

The force per unit length on a straight wire carrying current of 8 A making an angle of \(30^\circ\) with a uniform magnetic field of 0.15 T is

• (1) \( 1.2 \, N m^{-1} \)
• (2) \( 1.02 \, N m^{-1} \)
• (3) \( 0.6 \, N m^{-1} \)
• (4) \( 2.4 \, N m^{-1} \)

A wire of length 10 m carrying current of 1 A is bent into a circular loop. If a magnetic field of \(2\pi \times 10^{-4} \, T\) is applied on the loop, then the maximum torque acting on it is

• (1) \(100 \times 10^{-4} \, Nm\)
• (2) \(50 \times 10^{-4} \, Nm\)
• (3) \(25 \times 10^{-4} \, Nm\)
• (4) \(75 \times 10^{-4} \, Nm\)

A short bar magnet has a magnetic moment of \(0.48 \, J T^{-1}\). The magnitude of magnetic field at a point at 10 cm distance from the centre of the magnet on its axis is

• (1) \(0.96 \, gauss\)
• (2) \(0.48 \, gauss\)
• (3) \(1.92 \, gauss\)
• (4) \(1.44 \, gauss\)

A coil of 45 turns and radius 4 cm is placed in a uniform magnetic field such that its plane is perpendicular to the direction of the field. If the magnetic field increases from 0 to 0.70 T at a constant rate in a time interval of 220 s, then the induced emf in the coil is

• (1) 0.32 mV
• (2) 0.50 mV
• (3) 0.72 mV
• (4) 0.96 mV

For better tuning of a series LCR circuit in a communication system, the preferred combination is

• (1) \( R = 20\,\Omega;\ L = 15\,H;\ C = 35\,\muF \)
• (2) \( R = 15\,\Omega;\ L = 40\,H;\ C = 20\,\muF \)
• (3) \( R = 25\,\Omega;\ L = 15\,H;\ C = 45\,\muF \)
• (4) \( R = 15\,\Omega;\ L = 20\,H;\ C = 45\,\muF \)

The magnitude of the electric field of a plane electromagnetic wave travelling in free space is \( E \). If \( \mu_0 \) and \( \varepsilon_0 \) are respectively permeability and permittivity of the free space, then the magnitude of magnetic field of the wave is

• (1) \( E \mu_0 \varepsilon_0 \)
• (2) \( \frac{E}{\mu_0 \varepsilon_0} \)
• (3) \( \frac{E}{\sqrt{\mu_0/\varepsilon_0}} \)
• (4) \( \frac{E}{\sqrt{\mu_0 \varepsilon_0}} \)

An alpha particle moves along a circular path of radius 0.5 mm in a magnetic field of \( 2 \times 10^{-2} \, T \). The de Broglie wavelength associated with the alpha particle is nearly

(Planck’s constant \( h = 6.63 \times 10^{-34} \, J\cdots \))

• (1) 3.1 \AA
• (2) 1.1 \AA
• (3) 0.1 \AA
• (4) 2.1 \AA

The difference between the frequencies of second and first Paschen lines of hydrogen atom is

(\( R \) - Rydberg constant and \( c \) - speed of light in vacuum)

• (1) \( \dfrac{9Rc}{16} \)
• (2) \( \dfrac{16Rc}{25} \)
• (3) \( \dfrac{9Rc}{400} \)
• (4) \( \dfrac{3Rc}{200} \)

If the time taken for a radioactive substance to decay 8% to 77% is 12 minutes, then the half life of the substance in minutes is

• (1) 24
• (2) 18
• (3) 12
• (4) 6

A transistor has 3 impurity regions of different doping levels. In the order of increasing doping level, the regions are

• (1) emitter, base, collector
• (2) collector, base, emitter
• (3) base, emitter, collector
• (4) base, collector, emitter

A cc camera is fabricated using a semiconducting material having a band gap of 3 eV. The wavelength of light it can detect is nearly

• (1) 210 nm
• (2) 546 nm
• (3) 413 nm
• (4) 345 nm

If in an amplitude modulated wave, the maximum amplitude is 14 V and the modulation index is 0.4, then the amplitude of the carrier wave is

• (1) 4 V
• (2) 8 V
• (3) 12 V
• (4) 10 V

The wavenumber of the first line (\(n_2 = 3\)) in the Balmer series of hydrogen is \( \overline{\nu}_1 \,cm^{-1} \). What is the wavenumber (in cm\(^{-1}\)) of the second line (\(n_2 = 4\)) in the Balmer series of He\(^{+}\)?

• (1) \( \frac{5\overline{\nu}_1}{27} \)
• (2) \( \frac{27\overline{\nu}_1}{5} \)
• (3) \( \frac{27\overline{\nu}_1}{20} \)
• (4) \( \frac{20\overline{\nu}_1}{27} \)

Which of the following sets of quantum numbers is not possible for the electron?

• (1) \( n = 3,\ l = 1,\ m = 0,\ s = \pm\frac{1}{2} \)
• (2) \( n = 4,\ l = 0,\ m = 0,\ s = -\frac{1}{2} \)
• (3) \( n = 3,\ l = 3,\ m = -3,\ s = +\frac{1}{2} \)
• (4) \( n = 1,\ l = 0,\ m = 0,\ s = -\frac{1}{2} \)

The correct order of atomic radii of C, Al and S is

• (1) \( C < Al < S \)
• (2) \( S < Al < C \)
• (3) \( Al < S < C \)
• (4) \( C < S < Al \)

How many of the following molecules / ions have trigonal planar structure?
\( BO_3^{3-}, NH_3, PCl_3, BCl_3, ClF_3, XeO_3 \)

• (1) 5
• (2) 2
• (3) 4
• (4) 3

Consider the following

Assertion (A): Dipole moment of NF\(_3\) is lesser than NH\(_3\)

Reason (R): In NF\(_3\), the orbital dipole due to lone pair of electrons is in the opposite direction to the resultant dipole moment of the three N-F bonds.

The correct answer is:

• (1) Both (A) and (R) are correct and (R) is the correct explanation of (A)
• (2) Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• (3) (A) is correct, but (R) is not correct
• (4) (A) is not correct, but (R) is correct

At T(K), a gaseous mixture contains H\(_2\) and O\(_2\). The total pressure of the mixture is 2 bar. The partial pressure of H\(_2\) is 1.778 bar. What is the weight (w/w) percentage of H\(_2\) in the mixture?

• (1) 66.67
• (2) 33.33
• (3) 80.00
• (4) 20.00

The most probable speed (\(u_{mp}\)) of 8 g of H\(_2\) is \(2 \times 10^2\) ms\(^{-1}\). The kinetic energy (in J) of the same amount of H\(_2\) gas is

• (1) 480
• (2) 240
• (3) 720
• (4) 120

1.84 g of a mixture of CaCO\(_3\) and MgCO\(_3\) is strongly heated to get a residue of 0.96 g. The percentage of CaCO\(_3\) in the mixture is

• (1) 50.34
• (2) 49.66
• (3) 54.34
• (4) 45.66

Identify the correct statements from the following:

I) Work is a path function.

II) Enthalpy is an extensive property.

III) Lattice enthalpy of ionic compounds can be obtained from Born–Haber cycle.

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

For which of the following processes entropy change (\(\Delta S\)) is negative?

I) Sublimation of dry ice

II) Freezing of water

III) Crystallisation of the dissolved substance

IV) Burning of rocket fuel

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

At 25\,\(^\circ\)C, the percentage of ionization of 'x' M acetic acid is 4.242. What is the value of x? \((K_a = 1.8 \times 10^{-5})\)

• (1) 0.05
• (2) 0.04
• (3) 0.02
• (4) 0.01

At T(K), \(K_c\) value for \(AO_2(g) + BO_2(g) \rightleftharpoons AO_3(g) + BO(g)\) is 16. In a closed 1 L flask, one mole each of \(AO_2\), \(BO_2\), \(AO_3\), and \(BO\) are taken and heated to T(K). What is the concentration (in mol L\(^{-1}\)) of \(AO_3\) at equilibrium?

• (1) 0.4
• (2) 0.6
• (3) 1.6
• (4) 1.4

The incorrect statement in the following is:

• (1) Ionic hydrides are crystalline in nature
• (2) Group 14 elements form electron precise hydrides
• (3) Covalent hydrides are non-volatile compounds
• (4) Generally, saline hydrides react violently with water

Which of the following statements are correct regarding lithium and magnesium?

I) They react slowly with water

II) Their bicarbonates are solids

III) Their chlorides are not soluble in ethanol

IV) Their nitrates decompose easily on heating

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

The incorrect statement from the following is

• (1) Aluminium dissolves in conc.\,HNO\(_3\) and liberates H\(_2\) gas
• (2) Borazole contains 12\(\sigma\) and 3\(\pi\) bonds
• (3) Gallium oxide is amphoteric in nature
• (4) BF\(_3\) is a Lewis acid

In Buckminster fullerene, the number of six-membered carbon rings is ‘x’ and five-membered carbon rings is ‘y’. What is the value of (x + y)?

• (1) 30
• (2) 31
• (3) 32
• (4) 33

Match the following:

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

Consider the following

Statement-I: Kolbe's electrolysis of sodium propionate gives \textbf{n}-hexane as product.

Statement-II: In Kolbe's process, CO\(_2\) is liberated at anode and H\(_2\) is liberated at cathode.

• (1) Both statement-I and statement-II are correct
• (2) Both statement-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 correct decreasing order of priority for the functional group of organic compounds in the IUPAC method of nomenclature is

• (1) –CHO \(>\) –OH \(>\) –CONH\(_2\) \(>\) –COCl
• (2) –CONH\(_2\) \(>\) –CHO \(>\) –COCl \(>\) –OH
• (3) –COCl \(>\) –CONH\(_2\) \(>\) –CHO \(>\) –OH
• (4) –CHO \(>\) –CONH\(_2\) \(>\) –COCl \(>\) –OH

A compound is formed by two elements A and B. Atoms of the element B (as anion) make ccp lattice and those of element A (as cation) occupy all tetrahedral voids. The formula of the compound is

• (1) A\(_4\)B\(_3\)
• (2) AB
• (3) AB\(_2\)
• (4) A\(_2\)B

The mole fractions of glucose and water in aqueous glucose solution are 0.0244 and 0.9756 respectively. What is the weight percentage (w/w) of glucose in this solution?

• (1) 40
• (2) 25
• (3) 20
• (4) 10

At T(K), the vapour pressure of an aqueous solution of a non-volatile solute,

whose mole fraction is 0.02 is found to be 34.65 mm Hg.

What is the vapour pressure (in mm Hg) of pure water at the same temperature?

• (1) 35.70
• (2) 35.36
• (3) 35.00
• (4) 34.30

If \(E^\circ_{Fe^{2+}/Fe} = -0.441 \, V\) and \(E^\circ_{Fe^{3+}/Fe^{2+}} = 0.771 \, V\),

the standard emf of the cell reaction \(Fe(s) + 2Fe^{3+}(aq) \rightarrow 3Fe^{2+}(aq)\) is

• (1) \(-1.212\) V
• (2) \(+1.212\) V
• (3) \(-2.424\) V
• (4) \(+2.424\) V

At T(K), the following equation is obtained for a first order reaction.
\[ \log \frac{k}{A} = -\frac{x}{T} \]
The activation energy for this reaction is equal to (R = gas constant)

• (1) \(2.303 \times R\)
• (2) \(\frac{2.303 \, R}{x}\)
• (3) \(\frac{x}{2.303 \, R}\)
• (4) \(\frac{1}{2.303 \times R}\)

Which one of the following is not the correct characteristic property of physical adsorption?

• (1) It is not specific in nature
• (2) Enthalpy of adsorption of this is low
• (3) It increases with increase of temperature
• (4) It is a multilayer adsorption under high pressure

In each of four separate beakers (I, II, III, IV), X mL of 1M Fe\(_2\)O\(_3\)·xH\(_2\)O colloidal solution is present.

Equal volume and equal concentration of KCl, K\(_4\)[Fe(CN)\(_6\)], K\(_3\)PO\(_4\) and K\(_2\)SO\(_4\) was added into I, II, III and IV respectively.

The efficiency of precipitations in these beakers follows the order

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

In the extraction of iron from haematite, the impurity (x) of the ore is removed in the form of 'y'.

What are x and y respectively?

• (1) SiO\(_2\), CaSiO\(_3\)
• (2) CaO, CaSiO\(_3\)
• (3) SiO\(_2\), FeSiO\(_3\)
• (4) P\(_2\)O\(_5\), Ca\(_3\)(PO\(_4\))\(_2\)

Which of the following is not correct?

• (1) Potassium permanganate on heating gives potassium manganate and manganese dioxide only
• (2) Phosphine is used in smoke screens
• (3) Bleaching action of chlorine is due to oxidation
• (4) Noble gases have very low boiling points

How many of the following lanthanide elements exhibit +4 oxidation state?

Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy

• (1) 5
• (2) 4
• (3) 3
• (4) 6

In which of the following, complex ions are not in correct order with respect to their magnitude of crystal field splitting?

• (1) [Fe(H\(_2\)O)\(_6\)]\(^{3+}\) \(>\) [FeF\(_6\)]\(^{3-}\)
• (2) [Fe(en)\(_3\)]\(^{3+}\) \(>\) [Fe(NCS)\(_6\)]\(^{3-}\)
• (3) [Fe(CN)\(_6\)]\(^{4-}\) \(>\) [Fe(H\(_2\)O)\(_6\)]\(^{2+}\)
• (4) [Fe(H\(_2\)O)\(_6\)]\(^{2+}\) \(>\) [Fe(NH\(_3\))\(_6\)]\(^{2+}\)

Novolac is formed by the polymerisation of monomer 'x' in the presence of OH\textsuperscript{–} ions. What is 'x'?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (1)
View Solution

Which of the following contain \(\alpha\)-D-glucose units?

a) cane sugar \quad b) milk sugar \quad c) cellulose \quad d) amylose

• (1) a, d
• (2) a, b
• (3) b, c
• (4) c, d

Identify the set containing purine and pyrimidine base of DNA respectively.

• (1) Adenine, Uracil
• (2) Cytosine, Guanine
• (3) Thymine, Uracil
• (4) Adenine, Cytosine

Bithionol is added to soaps to impart antiseptic properties. The number of –OH and –Cl groups in its structure are respectively

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

Which of the following is the product of Fittig reaction?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (3)
View Solution

Match the following:

List-I (Halide type) \hfill List-II (Example)

A) Vinyl \hfill I) 1-Bromo-1-phenylethane

B) Allyl \hfill II) 3-Bromotholene

C) Benzyl \hfill III) 1-Bromo-3-methylcyclohexene

D) Aryl \hfill IV) 3-Bromo-4-methylcyclohexene

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

Which of the following represents Etard reaction?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (2)
View Solution

The correct order of boiling points of the compounds given below is


[A)] Methoxy ethane
[B)] Propan-1-ol
[C)] Propanal
[D)] Propanone

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

The correct statement about the product of the following reaction is
\[ CH_3CHO \xrightarrow{(i)\ C_2H_5MgBr} \xrightarrow{(ii)\ H_2O} product \]

• (1) It undergoes dehydration with 20% H\(_3\)PO\(_4\) at 358 K
• (2) It gives ketone on oxidation with CrO\(_3\)
• (3) It does not give positive iodoform test
• (4) It is a vinylic alcohol

How many amines with molecular formula C\(_3\)H\(_9\)N can react with benzene sulphonlyl chloride?

• (1) 2
• (2) 3
• (3) 4
• (4) 1