AP EAPCET (AP EAMCET) 2024 Question Paper May 22 Shift 1 (Available): Download MPC Question Paper with Answer Key PDF

The domain of the real valued function \( f(x) = \sqrt{9 - \sqrt{x^2 - 144}} \) is

• (1) \([-15, -12] \cup [12, 15]\)
• (2) \((-\infty, -12] \cup [12, \infty)\)
• (3) \([-15, 15]\)
• (4) \([-12, 12]\)

If set A has 5 elements, set B has 7 elements, then the number of one-one functions that can be defined from A to B is

• (1) \( 7^5 - 7 \)
• (2) \( 5^7 - 5 \)
• (3) \( 5^7 - 7P_5 \)
• (4) \( 7^5 - 7P_5 \)

Find the sum of the sequence: \( 2 + 3 + 5 + 6 + 8 + 9 + \dots + 2n \) terms.

• (1) \( 3n^2 + 2n \)
• (2) \( 4n^2 + 2n \)
• (3) \( 4n^2 \)
• (4) \( 5n^2 + 2n \)

If the system of equations has a unique solution, find the values of \( a \) and \( b \).

• (1) \( a = 8, b = 15 \)
• (2) \( a \neq 8, b \in \mathbb{R} \)
• (3) \( a = 8, b \neq 15 \)
• (4) \( a \neq 15, b = 8 \)

If \( P \) and \( Q \) are two \( 3 \times 3 \) matrices such that \( |PQ| = 1 \) and \( |P| = 9 \), then the determinant of adjoint of the matrix \( P . Adj \ 3Q \) is:

• (A) \(9^4\)
• (B) \(\frac{1}{9^4} \)
• (C) \(9^2\)
• (D) \(\frac{1}{9^2} \)

If \( A = \begin{bmatrix} a & 1 & 2\\
1 & b & 3\\
c & 1 & 3 \end{bmatrix} \) and \( Adj A = \begin{bmatrix} 7 & -1 & -5\\
-3 & 9 & 5\\
1 & -3 & 5 \end{bmatrix} \), then \( a^2 + b^2 + c^2 = \) ?

• (A) \(10\)
• (B) \(14\)
• (C) \(11\)
• (D) \(29\)

If \( Z \) is a complex number such that \( |Z| \leq 3 \) and \( -\frac{\pi}{2} \leq amp Z \leq \frac{\pi}{2} \), then the area of the region formed by the locus of \( Z \) is:

• (A) \( 9\pi \)
• (B) \( \frac{9\pi}{2} \)
• (C) \( 3\pi \)
• (D) \( \frac{9\pi}{4} \)

The locus of the complex number \( Z \) such that \( \arg \left( \frac{Z - 1}{Z + 1} \right) = \frac{\pi}{4} \) is:

• (A) A straight line
• (B) A circle
• (C) A parabola
• (D) An ellipse

All the values of \( (8i)^{\frac{1}{3}} \) are:

• (A) \( \pm (\sqrt{3} + i), -2i \)
• (B) \( \pm \sqrt{3} + i, -2i \)
• (C) \( \pm (\sqrt{3} - i), 2i \)
• (D) \( \pm (2 + i), i \)

If \( \alpha, \beta \) are the roots of the equation \( x^2 - 6x - 2 = 0 \), \( \alpha \(>\) \beta \), and \( a_n = \alpha^n - \beta^n, n \geq 1 \), then the value of \( \frac{a_{10} - 2 a_8}{2 a_9} \) is:

• (A) \(6 \)
• (B) \(4 \)
• (C) \(3 \)
• (D) \(2 \)

If both the roots of the equation \( x^2 - 6ax + 2 - 2a + 9a^2 = 0 \) exceed 3, then:

• (A) \( a \(<\) \frac{3}{2} \)
• (B) \( a \(>\) \frac{3}{2} \)
• (C) \( a \(<\) \frac{5}{2} \)
• (D) \( a \(>\) \frac{11}{9} \)

If \( \alpha \) and \( \beta \) are two distinct negative roots of the equation \( x^5 - 5x^3 + 5x^2 - 1 = 0 \), then the equation of least degree with integer coefficients having \( \sqrt{-\alpha} \) and \( \sqrt{-\beta} \) as its roots is:

• (A) \( x^2 - 3x + 1 = 0 \)
• (B) \( -x^4 + 5x^2 - 5x + 1 = 0 \)
• (C) \( -x^4 - 5x^2 + 5x + 1 = 0 \)
• (D) \( x^4 - 3x^2 + 1 = 0 \)

If the number of real roots of \( x^9 - x^5 + x^4 - 1 = 0 \) is \( n \), the number of complex roots having argument on imaginary axis is \( m \), and the number of complex roots having argument in the second quadrant is \( k \), then \( m.n.k \) is:

• (A) \( 6 \)
• (B) \( 9 \)
• (C) \( 12 \)
• (D) \( 24 \)

The rank of the word "TABLE" counted from the rank of the word "BLATE" in dictionary order is:

• (A) \( 50 \)
• (B) \( 97 \)
• (C) \( 61 \)
• (D) \( 37 \)

5 boys and 6 girls are arranged in all possible ways. Let \(X\) denote the number of linear arrangements in which no two boys sit together, and \(Y\) denote the number of linear arrangements in which no two girls sit together. If \(Z\) denotes the number of ways of arranging all of them around a circular table such that no two boys sit together, then \(X:Y:Z\) = ?

• (A) \(1:1:21\)
• (B) \(21:1:1\)
• (C) \(7:5:5\)
• (D) \(4:3:3\)

The number of ways of distributing 15 apples to three persons A, B, C such that A and C each get at least 2 apples and B gets at most 5 apples is:

• (A) \( 57 \)
• (B) \( 131 \)
• (C) \( 156 \)
• (D) \( 251 \)

If the \(2^{nd}\), \(3^{rd}\), and \(4^{th}\) terms in the expansion of \( (x + a)^n \) are 96, 216, and 216 respectively, and \( n \) is a positive integer, then \( a + x \) is:

• (A) \( n + 1 \)
• (B) \( n \)
• (C) \( n - 1 \)
• (D) \( \frac{n}{2} \)

If \( |x| \(<\) 1 \), then the number of terms in the expansion of \( \left[ \frac{1}{2} (1.2 + 2.3x + 3.4x^2 + \dots) \right]^{-25} \) is:

• (A) Infinite
• (B) 101
• (C) 76
• (D) 51

If \( |x| \(<\) 1 \), the coefficient of \( x^2 \) in the power series expansion of \( \frac{x^4}{(x+1)(x-2)} \) is:

• (A) 3
• (B) 0
• (C) -1
• (D) -3

If \( M_1 \) and \( M_2 \) are the maximum values of \( \frac{1}{11 \cos 2x + 60 \sin 2x + 69} \) and \( 3 \cos^2 5x + 4\sin^2 5x \) respectively, then \( \frac{M_1}{M_2} = \):

• (A) \( \frac{65}{2} \)
• (B) \( \frac{1}{32} \)
• (C) \( \frac{8}{3} \)
• (D) \( 2 \)

Evaluate the given trigonometric expression:
\[ 4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2\pi}{7} \cos \frac{2\pi}{5} \cos \frac{4\pi}{7} = \]

• (A) \( -\frac{1}{8} \)
• (B) \( \frac{1}{32} \)
• (C) \( -\frac{1}{32} \)
• (D) \( \frac{1}{8} \)

In a triangle \( ABC \), if \( A, B, C \) are in arithmetic progression and
\[ \cos A + \cos B + \cos C = \frac{1 + \sqrt{2} +\sqrt{3}}{2\sqrt{2}}, \]

then \( \tan A \) is:

• (A) \( \sqrt{3} \)
• (B) \( 2 + \sqrt{3} \)
• (C) \( 1 \)
• (D) \( 2 - \sqrt{3} \)

The general solution of the equation \( \tan x + \tan 2x - \tan 3x = 0 \) is:

• (A) \( \{ x | x = n\pi \pm \frac{\pi}{3} or \frac{n\pi}{2}, n \in \mathbb{Z} \} \)
• (B) \( \{ x | x = n\pi \pm \frac{\pi}{3} or n\pi, n \in \mathbb{Z} \} \)
• (C) \( \{ x | x = n\pi \pm \frac{\pi}{3} or \frac{n\pi}{2}, n \in \mathbb{Z} \} \)
• (D) \( \{ x | x = n\pi \pm \frac{\pi}{6} or \frac{n\pi}{2}, n \in \mathbb{Z} \} \)

The value of \( x \) such that \( \sin \left( 2 \tan^{-1} \frac{3}{4} \right) = \cos \left( 2 \tan^{-1} x \right) \) is:

• (A) \( 7 \)
• (B) \( \) (Blank)
• (C) \( \frac{1}{7} \)
• (D) \( \frac{4}{7} \)

If \( \tanh x = sech y = \frac{3}{5} \) and \( e^{x+y} \) is an integer, then \( e^{x+y} \) is:

• (A) 2
• (B) 8
• (C) 3
• (D) 6

In \( \triangle ABC \), if \( b + c : c + a : a + b = 7:8:9 \), then the smallest angle (in radians) of that triangle is:

• (A) \( \cos^{-1} \left( \frac{4}{5} \right) \)
• (B) \( \frac{\pi}{3} \)
• (C) \( \cos^{-1} \left( \frac{3}{5} \right) \)
• (D) \( \frac{\pi}{4} \)

In \( \triangle ABC \), if \( (a+c)^2 = b^2 + 3ca \), then \( \frac{a+c}{2R} \) is:

• (A) \( \frac{\sqrt{3}}{2} \)
• (B) \( \sqrt{3} \cos \left(\frac{A - C}{2} \right) \)
• (C) \( \cos \left(\frac{A - C}{2} \right) \)
• (D) \( \sin \left(\frac{A - C}{2} \right) \)

In \( \triangle ABC \), if \( A, B, C \) are in arithmetic progression, \( \Delta = \frac{\sqrt{3}}{2} \) and \( r_1 r_2 = r_3 r \), then \( R \) is:

• (A) \( \sqrt{3} \)
• (B) \( 2 \)
• (C) \( 1 \)
• (D) \( \sqrt{2} \)

Let \( \mathbf{a} = 3\hat{i} + 4\hat{j} - 5\hat{k} \), \( \mathbf{b} = 2\hat{i} + \hat{j} - 2\hat{k} \). The projection of the sum of the vectors \( \mathbf{a}, \mathbf{b} \) on the vector perpendicular to the plane of \( \mathbf{a}, \mathbf{b} \) is:

• (A) \( 0 \)
• (B) \( 4\sqrt{2} \)
• (C) \( 7\sqrt{2} \)
• (D) \( \frac{1}{\sqrt{2}} \)

In \( \triangle PQR \), \((4\overline{i} + 3\overline{j} + 6\overline{k} )\) and \((3\overline{i} + \overline{j} + 3\overline{k} )\) are the position vectors of the vertices P, Q, R respectively. Then the position vector of the point of intersection of the angle bisector of \( P \) with \( QR \).

• (A) \( 6\overline{i} + 5\overline{j} + 9\overline{k} \)
• (B) \( 2\overline{i} - \overline{j} + 3\overline{k} \)
• (C) \( (5\overline{i} + 3\overline{j} - 2\overline{k}) \)
• (D) \( \frac{5}{2} \overline{i} + \frac{3}{2} \overline{j} + 3\overline{k} \)

If \( \vec{f} = i + j + k \) and \( \vec{g} = 2i - j + 3k \), then the projection vector of \( \vec{f} \) on \( \vec{g} \) is:

• (A) \( \frac{2}{7} (i + j + k) \)
• (B) \( \frac{2}{7} (2i - j + 3k) \)
• (C) \( \frac{1}{3} (i + j + k) \)
• (D) \( \frac{1}{14} (2i - j + 3k) \)

If \( \theta \) is the angle between \( \vec{f} = i + 2j - 3k \) and \( \vec{g} = 2i - 3j + ak \) and \( \sin \theta = \frac{\sqrt{24}}{28} \), then \( 7a^2 + 24a = \) ?

• (A) \( 10 \)
• (B) \( 12 \)
• (C) \( 36 \)
• (D) \( 15 \)

The distance of a point \( (2,3,-5) \) from the plane \( \vec{r} \cdot (4i - 3j + 2k) = 4 \) is:

• (A) \( \frac{11}{2} \)
• (B) \( \frac{11}{\sqrt{29}} \)
• (C) \( \frac{15}{\sqrt{29}} \)
• (D) \( \frac{11}{\sqrt{38}} \)

If \( x_1, x_2, x_3, \dots, x_n \) are \( n \) observations such that \( \sum (x_i + 2)^2 = 28n \) and \( \sum (x_i - 2)^2 = 12n \), then the variance is:

• (A) \( 12 \)
• (B) \( 14 \)
• (C) \( 16 \)
• (D) \( 20 \)

Three numbers are chosen at random from 1 to 20. The probability that their sum is divisible by 3 is:

• (A) \( \frac{1}{114} \)
• (B) \( \frac{147}{342} \)
• (C) \( \frac{16}{47} \)
• (D) \( \frac{32}{85} \)

Two persons A and B throw three unbiased dice one after the other. If A gets the sum 13, then the probability that B gets a higher sum is:

• (A) \( \frac{5}{216} \)
• (B) \( \frac{4}{27} \)
• (C) \( \frac{35}{216} \)
• (D) \( \frac{20}{216} \)

8 teachers and 4 students are sitting around a circular table at random. The probability that no two students sit together is:

• (A) \( \frac{7}{88} \)
• (B) \( \frac{14}{33} \)
• (C) \( \frac{8}{33} \)
• (D) \( \frac{7}{33} \)

A bag contains 6 balls. If three balls are drawn at a time and all of them are found to be green, then the probability that exactly 5 of the balls in the bag are green is:

• (A) \( \frac{4}{35} \)
• (B) \( \frac{5}{35} \)
• (C) \( \frac{2}{7} \)
• (D) \( \frac{1}{7} \)

In a Binomial distribution, the difference between the mean and standard deviation is 3, and the difference between their squares is 21. Then, the ratio \( P(x = 1) : P(x = 2) \) is:

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

When an unfair dice is thrown, the probability of getting a number \( k \) on it is \( P(X = k) = k^2 P \), where \( k = 1, 2, 3, 4, 5, 6 \) and \( X \) is the random variable denoting a number on the dice. Then, the mean of \( X \) is:

• (A) \( 25 \)
• (B) \( 5 \)
• (C) \( \frac{441}{9} \)
• (D) \( \frac{441}{91} \)

The equation of the locus of points which are equidistant from the points \( (2,3) \) and \( (4,5) \) is:

• (A) \( x + y = 0 \)
• (B) \( x + y = 7 \)
• (C) \( 4x + 4y = 38 \)
• (D) \( x + y = 1 \)

The transformed equation of \( x^2 - y^2 + 2x + 4y = 0 \) when the origin is shifted to the point \( (-1,2) \) is:

• (A) \( x^2 + y^2 = 1 \)
• (B) \( x^2 + 3y^2 = 1 \)
• (C) \( x^2 - y^2 + 3 = 0 \)
• (D) \( 4x^2 + 9y^2 = 36 \)

The equation of the side of an equilateral triangle is \( x + y = 2 \) and one vertex is \( (2,-1) \). The length of the side is:

• (A) \( \frac{\sqrt{2}}{\sqrt{3}} \)
• (B) \( \frac{1}{2\sqrt{3}} \)
• (C) \( \frac{\sqrt{3}}{\sqrt{2}} \)
• (D) \( \frac{2}{\sqrt{3}} \)

The orthocentre of the triangle formed by lines \( x + y + 1 = 0 \), \( x - y - 1 = 0 \) and \( 3x + 4y + 5 = 0 \) is:

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

If the slope of one of the pair of lines represented by \( 2x^2 + 3xy + Ky^2 = 0 \) is 2, then the angle between the pair of lines is:

• (A) \( \frac{\pi}{2} \)
• (B) \( \frac{\pi}{3} \)
• (C) \( \frac{\pi}{6} \)
• (D) \( \frac{\pi}{4} \)

The length of x-intercept made by the pair of lines \( 2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0 \) is:

• (A) \( 2 \)
• (B) \( 10 \)
• (C) \( 5 \)
• (D) \( 20 \)

From a point \( (1,0) \) on the circle \( x^2 + y^2 - 2x + 2y + 1 = 0 \), if chords are drawn to this circle, then locus of the poles of these chords with respect to the circle \( x^2 + y^2 = 4 \) is:

• (A) \( x = 4 \)
• (B) \( x + 2y = 5 \)
• (C) \( x^2 + y^2 - x - y = 0 \)
• (D) \( 2y^2 = (x + 1) \)

If A and B are the centres of similitude with respect to the circles \( x^2 + y^2 - 14x + 6y + 33 = 0 \) and \( x^2 + y^2 + 30x - 2y + 1 = 0 \), then midpoint of \( AB \) is:

• (A) \( \left( \frac{7}{3}, \frac{4}{5} \right) \)
• (B) \( \left( \frac{3}{2}, \frac{1}{5} \right) \)
• (C) \( \left( \frac{39}{2}, \frac{-7}{4} \right) \)
• (D) \( \left( \frac{39}{4}, \frac{-7}{2} \right) \)

\( C_1 \) is the circle with centre at \( (0,0) \) and radius 4, \( C_2 \) is a variable circle with centre at \( (\alpha, \beta) \) and radius 5. If the common chord of \( C_1 \) and \( C_2 \) has slope \( \frac{3}{4} \) and of maximum length, then one of the possible values of \( \alpha + \beta \) is:

• (A) \( \frac{21}{5} \)
• (B) \( \frac{3}{5} \)
• (C) \( \frac{1}{5} \)
• (D) \( \frac{19}{5} \)

If the pair of tangents drawn to the circle \( x^2 + y^2 = a^2 \) from the point \( (10, 4) \) are perpendicular, then \( a \) is:

• (A) \( \sqrt{58} \)
• (B) \( 58 \)
• (C) \( 2\sqrt{63} \)
• (D) \( 2\sqrt{45} \)

If \( x - 4 = 0 \) is the radical axis of two orthogonal circles out of which one is \( x^2 + y^2 = 36 \), then the centre of the other circle is:

• (A) \( (8,0) \)
• (B) \( (9,0) \)
• (C) \( (6,0) \)
• (D) \( (12,0) \)

If the normal chord drawn at \( (2a,2a\sqrt{2}) \) on the parabola \( y^2 = 4ax \) subtends an angle \( \theta \) at its vertex, then \( \theta \) is:

• (A) \( 45^\circ \)
• (B) \( 90^\circ \)
• (C) \( 135^\circ \)
• (D) \( 60^\circ \)

If the ellipse \(4x^2 + 9y^2 = 36\) is confocal with a hyperbola whose length of the transverse axis is 2, then the points of intersection of the ellipse and hyperbola lie on the circle:

• (A) \( x^2 + y^2 = 81 \)
• (B) \( x^2 + y^2 = 16 \)
• (C) \( x^2 + y^2 = 25 \)
• (D) \( x^2 + y^2 = 5 \)

If the eccentricity of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \( \sec \alpha \), then the area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is:

• (A) \( a^2b^2 \sec^2\alpha \)
• (B) \( \frac{b^2}{|\tan \alpha|} \)
• (C) \( a^2\tan^2\alpha \)
• (D) \( (a^2+b^2)\tan^2\alpha \)

If \( e_1 \) and \( e_2 \) are respectively the eccentricities of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and its conjugate hyperbola, then the line \( \frac{x}{2e_1} + \frac{y}{2e_2} = 1 \) touches the circle having center at the origin, then its radius is:

• (A) \( 2 \)
• (B) \( e_1 + e_2 \)
• (C) \( e_1 e_2 \)
• (D) \( 4 \)

The orthocentre of triangle formed by points: \( (2,1,5) \), \( (3,2,3) \) and \( (4,0,4) \) is:

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

If \( P = (0,1,2) \), \( Q = (4,-2,-1) \) and \( O = (0,0,0) \), then \( \angle POQ \) is:

• (A) \( \frac{\pi}{6} \)
• (B) \( \frac{\pi}{4} \)
• (C) \( \frac{\pi}{3} \)
• (D) \( \frac{\pi}{2} \)

If the perpendicular distance from \( (1,2,4) \) to the plane \( 2x + 2y - z + k = 0 \) is 3, then \( k \) is:

• (A) \( 4 \)
• (B) \( 7 \)
• (C) \( 9 \)
• (D) \( 19 \)

Evaluate: \[ \lim_{x \to 0} \left[ \frac{1}{x} - \frac{1}{e^x - 1} \right] \]

• (A) \( 0 \)
• (B) \( 1 \)
• (C) \( 2 \)
• (D) \( \frac{1}{2} \)

Let \( f(x) \) be defined as: \[ f(x) = \begin{cases} 0, & x = 0
2 - x, & 0 \(<\) x \(<\) 1
2, & x = 1
1 - x, & 1 \(<\) x \(<\) 2
-\frac{3}{2}, & x \geq 2 \end{cases} \]

Then which of the following is true?

• (A) \( f \) is right continuous at \( x = 0 \)
• (B) \( f \) is left continuous at \( x = 1 \)
• (C) \( f \) is right continuous at \( x = 1 \)
• (D) \( f \) is continuous at \( x = 2 \)

If \( f(x) = \left(\frac{1+x}{1-x}\right)^{\frac{1}{x}} \) is continuous at \( x = 0 \), then \( f(0) \) is:

• (A) \( e^{\frac{1}{2}} \)
• (B) \( e^2 \)
• (C) \( e^{-2} \)
• (D) \( e^{-\frac{1}{2}} \)

The function \( f(x) = |x - 24| \) is:

• (A) Differentiable on \( [0,25] \)
• (B) Not continuous at \( x = 24 \)
• (C) Neither continuous nor differentiable on \( [0,25] \)
• (D) Continuous on \( [0,25] \), but not differentiable on \( [0,25] \)

If \( y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots \infty}}} \), then the value of \( \frac{d^2y}{dx^2} \) at the point \( (\pi,1) \) is:

• (A) \( 2 \)
• (B) \( -2 \)
• (C) \( -\frac{1}{2} \)
• (D) \( \frac{1}{2} \)

If \( f(0) = 0 \), \( f'(0) = 3 \), then the derivative of \( y = f(f(f(f(f(x))))) \) at \( x = 0 \) is:

• (A) \( 16 \)
• (B) \( 32 \)
• (C) \( 81 \)
• (D) \( 243 \)

The value \( c \) of Lagrange’s Mean Value Theorem for \( f(x) = e^x + 24 \) in \( [0,1] \) is:

• (1) \( \log(e - 1) \)
• (2) \( \log(e + 1) \)
• (3) \( \log(e + 24) \)
• (4) \( \log(e - 24) \)

Equation of the normal to the curve \( y = x^2 + x \) at the point \( (1,2) \) is:

• (1) \( x - 3y + 5 = 0 \)
• (2) \( x + 3y + 7 = 0 \)
• (3) \( x + 3y + 5 = 0 \)
• (4) \( x + 3y - 7 = 0 \)

Displacement \( s \) of a particle at time \( t \) is expressed as \( s = 2t^3 - 9t \). Find the acceleration at the time when the velocity vanishes.

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

If a running track of 500 ft. is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular portion is maximum is (in feet).

• (1) \( 100 \)
• (2) \( 125 \)
• (3) \( 150 \)
• (4) \( 200 \)

Evaluate the integral: \[ \int \frac{x^2 - 1}{x^3\sqrt{2x^4 - 2x^2 + 1}} \,dx. \]

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

Evaluate the integral: \[ \int \frac{x^3 \tan^{-1}(x^4)}{1 + x^8} \,dx. \]

• (1) \( \frac{(\tan^{-1}(x^4))^2}{8} + c \)
• (2) \( \frac{(\tan^{-1}(x^4))^3}{3} + c \)
• (3) \( \frac{(\tan^{-1}(x^4))^2}{4} + c \)
• (4) \( \frac{(\tan^{-1}(x^4))^2}{2} + c \)

Evaluate the integral: \[ I = \int \frac{2}{1 + x + x^2} \,dx. \]

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

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

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

Evaluate the integral: \[ I = \int \frac{\sin 7x}{\sin 2x \sin 5x} \,dx. \]

• (1) \( \log (\sin 5x \sin 2x) + c \)
• (2) \( \log \sin 5x + \log \sin 2x + c \)
• (3) \( \frac{1}{5} \log \sin 5x + \frac{1}{2} \log \sin 2x + c \)
• (4) \( \frac{1}{5} \log \sin x + \frac{1}{2} \log \sin x + c \)

Evaluate the integral: \[ I = \int_0^{\frac{\pi}{4}} \log(1 + \tan x) \,dx. \]

• (1) \( \pi \log 2 + 1 \)
• (2) \( \frac{\pi}{2} \log 2 + 1 \)
• (3) \( \frac{\pi}{4} \log 2 \)
• (4) \( \frac{\pi}{8} \log 2 \)

Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2 - 1}} + \dots + \frac{1}{\sqrt{n^2 - (n-1)^2}} \right). \]

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

The area (in square units) bounded by the curves \( x = y^2 \) and \( x = 3 - 2y^2 \) is:

• (1) \( 8 \)
• (2) \( \frac{8}{3} \)
• (3) \( 4 \)
• (4) \( 6 \)

Evaluate the integral: \[ I = \int_{-\pi}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \,dx. \]

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

The general solution of the differential equation: \[ (1 + \tan y) (dx - dy) + 2x \, dy = 0. \]

• (1) \( e^x(y \cos x + \sin x) + \sin x = c \)
• (2) \( e^x(y \cos x + y \sin x - \sin x) + \cos x = 0 \)
• (3) \( e^y(x \cos y + x \sin y - \sin y) = c \)
• (4) \( e^y(x \cos y + x \sin y + \sin y) = c \)

The general solution of the differential equation: \[ x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx. \]

• (1) \( y + \sqrt{x^2 + y^2} = c x^2 \)
• (2) \( y + \sqrt{x^2 + y^2} = cx \)
• (3) \( x + \sqrt{x^2 + y^2} = cy \)
• (4) \( x - \sqrt{x^2 + y^2} = c y^2 \)

The sum of the order and degree of the differential equation: \[ x \left( \frac{d^2 y}{dx^2} \right)^{1/2} = \left( 1 + \frac{dy}{dx} \right)^{4/3} \]
is:

• (1) \( 5 \)
• (2) \( 8 \)
• (3) \( 12 \)
• (4) \( 10 \)

The potential difference across the ends of a conductor is \( (30 \pm 0.3) V \) and the current through the conductor is \( (5 \pm 0.1) A \). The error in the determination of the resistance of the conductor is:

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

A body thrown vertically upwards reaches a maximum height \( H \). The ratio of the velocities of the body at heights \( \frac{3H}{4} \) and \( \frac{8H}{9} \) from the ground is:

• (1) \( 4:9 \)
• (2) \( 27:32 \)
• (3) \( 3:2 \)
• (4) \( 3:8 \)

The angle made by the resultant vector of two vectors \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( 2\hat{i} - 7\hat{j} - 4\hat{k} \) with the x-axis is:

• (1) \( 60^\circ \)
• (2) \( 45^\circ \)
• (3) \( 90^\circ \)
• (4) \( 120^\circ \)

The equation of projectile motion is given by \( y = 3x - 0.8x^2 \). The time of flight of the projectile is (Acceleration due to gravity \( g = 10 \) m/s²):

• (1) \( 1.5 \) s
• (2) \( 3 \) s
• (3) \( 2 \) s
• (4) \( 2.5 \) s

A 100 kg gun fires a ball of 1 kg horizontally from a cliff of height 500 m. It falls on the ground at a distance of 400 m from the bottom of the cliff. The recoil velocity of the gun is (Acceleration due to gravity \( g = 10 \) ms\(^{-2}\)):

• (1) \( 0.6 \) ms\(^{-1}\)
• (2) \( 0.8 \) ms\(^{-1}\)
• (3) \( 0.2 \) ms\(^{-1}\)
• (4) \( 0.4 \) ms\(^{-1}\)

A block of mass 5 kg is placed on a rough horizontal surface with a coefficient of friction 0.5. If a horizontal force of 60 N is acting on it, then the acceleration of the block is (Acceleration due to gravity \( g = 10 \) ms\(^{-2}\)):

• (1) \( 7 \) ms\(^{-2}\)
• (2) \( 5 \) ms\(^{-2}\)
• (3) \( 10 \) ms\(^{-2}\)
• (4) \( 15 \) ms\(^{-2}\)

The average power generated by a 90 kg mountain climber who climbs a summit of height 600 m in 90 minutes is (Acceleration due to gravity = 10 m/s\(^2\)):

• (A) \(100 W \)
• (B) \(25 W \)
• (C) \(200 W \)
• (D) \(50 W \)

A boy weighing 50 kg finished a long jump at a distance of 8 m. Considering that he moved along a parabolic path and his angle of jump is \( 45^\circ \), his initial kinetic energy is:

• (A) \(960 J \)
• (B) \(1560 J \)
• (C) \(2460 J \)
• (D) \(1960 J \)

The moment of inertia of a rod about an axis passing through its centre and perpendicular to its length is \( \frac{1}{12} ML^2 \), where \( M \) is the mass and \( L \) is the length of the rod. The rod is bent in the middle so that the two halves make an angle of \( 60^\circ \). The moment of inertia of the bent rod about the same axis would be:

• (A) \( \frac{1}{48} ML^2 \)
• (B) \( \frac{1}{12} ML^2 \)
• (C) \( \frac{1}{24} ML^2 \)
• (D) \( \frac{1}{8\sqrt{3}} ML^2 \)

A uniform rod of length \( 2L \) is placed with one end in contact with the earth and is then inclined at an angle \( \alpha \) to the horizontal and allowed to fall without slipping at the contact point. When it becomes horizontal, its angular velocity will be:

• (A) \( \sqrt{\frac{3g \sin \alpha}{2L}} \)
• (B) \( \sqrt{\frac{2L}{3g \sin \alpha}} \)
• (C) \( \sqrt{\frac{6g \sin \alpha}{L}} \)
• (D) \( \sqrt{\frac{L}{g \sin \alpha}} \)

Two simple harmonic motions are represented by \(y_1 = 5 \left[ \sin 2\pi t + \sqrt{3} \cos 2\pi t \right] \) and \(y_2 = 5 \sin \left[ 2\pi t + \frac{\pi}{4} \right] \). The ratio of their amplitudes is:

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

When a mass \( m \) is connected individually to the springs \( s_1 \) and \( s_2 \), the oscillation frequencies are \( v_1 \) and \( v_2 \). If the same mass is attached to the two springs as shown in the figure, the oscillation frequency would be:

• (A) \( v_1 + v_2 \)
• (B) \( \sqrt{v_1^2 + v_2^2} \)
• (C) \( \left( \frac{1}{v_1} + \frac{1}{v_2} \right)^{-1} \)
• (D) \( \sqrt{v_1^2 - v_2^2} \)

A satellite moving around the Earth in a circular orbit has kinetic energy \( E \). Then, the minimum amount of energy to be added so that it escapes from the Earth is:

• (A) \( \frac{E}{4} \)
• (B) \( E \)
• (C) \( \frac{E}{2} \)
• (D) \( 2E \)

The elongation of a copper wire of cross-sectional area \( 3.5 \) mm\(^2\), in the figure shown, is \[ (Y_{copper} = 10 \times 10^{10} Nm^{-2} and g = 10 m/s^2) \]

• (A) \( 10^{-4} m \)
• (B) \( 10^{-3} m \)
• (C) \( 10^{-6} m \)
• (D) \( 10^{-2} m \)

Water is flowing in a streamline manner in a horizontal pipe. If the pressure at a point where cross-sectional area is \( 10 \) cm\(^2\) and velocity \( 1 \) m/s is \( 2000 \) Pa, then the pressure of water at another point where the cross-sectional area is \( 5 \) cm\(^2\) is:

• (A) \( 2500 \) Pa
• (B) \( 2000 \) Pa
• (C) \( 1000 \) Pa
• (D) \( 500 \) Pa

A metal ball of mass 100 g at \( 20^\circ C \) is dropped in 200 ml of water at \( 80^\circ C \). If the resultant temperature is \( 70^\circ C \), then the ratio of specific heat of the metal to that of water is:

• (A) \( \frac{5}{2} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{2}{5} \)
• (D) \( \frac{2}{1} \)

The efficiency of a heat engine that works between the temperatures where Celsius-Fahrenheit scales coincide and Kelvin-Fahrenheit scales coincide is (approximately):

• (A) \( 45% \)
• (B) \( 35% \)
• (C) \( 60% \)
• (D) \( 50% \)

Initially the pressure of 1 mole of an ideal gas is \( 10^5 \) Nm\(^2\) and its volume is 16 liters. When it is adiabatically compressed, its final volume is 2 liters. Work done on the gas is (molar specific heat at constant volume \( C_V = \frac{3R}{2} \)):

• (A) \( 72 \) kJ
• (B) \( 7.2 \) kJ
• (C) \( 720 \) kJ
• (D) \( 360 \) kJ

An ideal gas is taken around ABCA as shown in the P-V diagram. The work done during a cycle is:

• (A) \( 2PV \)
• (B) \( PV \)
• (C) \( \frac{1}{2} PV \)
• (D) Zero

The ratio of kinetic energy of a diatomic gas molecule at a high temperature to that of NTP is:

• (A) \( \frac{3}{2} \)
• (B) \( \frac{5}{3} \)
• (C) \( \frac{5}{7} \)
• (D) \( \frac{7}{5} \)

The vibrations of four air columns are shown below. The ratio of frequencies is:

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

A person can see objects clearly when they lie between 40 cm and 400 cm from his eye. In order to increase the maximum distance of distant vision to infinity, the type of lens and power of correction lens required respectively are:

• (A) Convex, \( 0.25 \) Diopter
• (B) Concave, \( -0.25 \) Diopter
• (C) Concave, \( -0.5 \) Diopter
• (D) Convex, \( 0.5 \) Diopter

If a slit of width \( x \) was illuminated by red light having wavelength \( 6500 \) Å, the first minima was obtained at \( \theta = 30^\circ \). Then the value of \( x \) is:

• (A) \( 1.4 \times 10^{-4} \) µm
• (B) \( 1.2 \times 10^{-5} \) µm
• (C) \( 1.3 \) µm
• (D) \( 1.2 \) µm

A neutral ammonia (NH\(_3\)) molecule in its vapour state has an electric dipole moment of magnitude \( 5 \times 10^{-30} \) C·m. How far apart are the molecule's centers of positive and negative charge?

• (A) \( 4.125 \times 10^{-12} \) m
• (B) \( 3.125 \times 10^{-12} \) m
• (C) \( 3.125 \times 10^{-6} \) m
• (D) \( 4.125 \times 10^{-6} \) m

If four charges \( q_1 = +1 \times 10^{-8} C \), \( q_2 = -2 \times 10^{-8} C \), \( q_3 = +3 \times 10^{-8} C \), and \( q_4 = +2 \times 10^{-8} C \) are kept at the four corners of a square of side 1 m, then the electric potential at the centre of the square is:

• (A) \( 300 \) V
• (B) \( 200 \) V
• (C) \( 510 \) V
• (D) \( 410 \) V

Eight capacitors each of capacity \( 2 \) µF are arranged as shown in the figure. The effective capacitance between A and B is:

• (A) \( 10 \) µF
• (B) \( 12 \) µF
• (C) \( 16 \) µF
• (D) \( 4 \) µF

If \( E_1 = 4V \) and \( E_2 = 12V \), the current in the circuit and potential difference between the points P and Q respectively are:

• (A) \( 1A, 8V \)
• (B) \( 1A, 6V \)
• (C) \( 0.8A, 6.4V \)
• (D) \( 0.8A, 8V \)

Two identical cells gave the same current through an external resistance of \( 2 \)\(\omega\) regardless of whether the cells are grouped in series or parallel. The internal resistance of the cells is:

• (A) \( 1 \) \(\omega\)
• (B) \( 0.5 \) \(\omega\)
• (C) \( 1.5 \) \(\omega\)
• (D) \( 2.0 \) \(\omega\)

Two toroids with number of turns 400 and 200 have average radii respectively 30 cm and 60 cm.
If they carry the same current, the ratio of magnetic fields in these two toroids is:

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

Three rings, each with equal radius \( r \), are placed mutually perpendicular to each other and each having centre at the origin of the coordinate system. If \( I \) is the current passing through each ring, the magnetic field value at the common centre is:

• (A) \( 0 \)
• (B) \( (\sqrt{3} - 1) \frac{\mu_0 I}{2\pi r} \)
• (C) \( \frac{\sqrt{3} \mu_0 I}{2r} \)
• (D) \( \frac{\sqrt{2} \mu_0 I}{2r} \)

One bar magnet is in simple harmonic motion with time period \( T \) in an earth’s magnetic field. If its mass is increased by 9 times, the time period becomes:

• (A) \( 3T \)
• (B) \( 9T \)
• (C) \( 4T \)
• (D) \( \sqrt{3} T \)

A coil of inductance \( L \) is divided into 6 equal parts. All these parts are connected in parallel. The resultant inductance of this combination is:

• (A) \( \frac{L}{6} \)
• (B) \( \frac{L}{36} \)
• (C) \( \frac{L}{24} \)
• (D) \( 6L \)

A 50 Hz AC circuit has a 10 mH inductor and a 2 \(\omega\) resistor in series. The value of capacitance to be placed in series in the circuit to make the circuit power factor unity is:

• (A) \( 1.014 \times 10^{-6} F \)
• (B) \( 1.014 \times 10^{-3} F \)
• (C) \( 2.6 \times 10^{-3} F \)
• (D) \( 4.125 \times 10^{-3} F \)

The structure of solids is investigated by using:

• (A) Cosmic rays
• (B) \( \beta \)-rays
• (C) X-rays
• (D) \( \gamma \)-rays

The surface of a metal is first illuminated with a light of wavelength 300 nm and later illuminated by another light of wavelength 500 nm. It is observed that the ratio of maximum velocities of photoelectrons in two cases is 3. The work function of the metal value is close to:

• (A) \( 6.48 eV \)
• (B) \( 1.23 eV \)
• (C) \( 4.17 eV \)
• (D) \( 2.28 eV \)

The ratio of the minimum wavelength of the Balmer series to the maximum wavelength in the Brackett series in the hydrogen spectrum is:

• (A) \( 25:16 \)
• (B) \( 4:36 \)
• (C) \( 9:100 \)
• (D) \( 100:9 \)

The half-life period of a radioactive element A is 62 years. It decays into another stable element B. An archaeologist found a sample in which A and B are in 1 : 15 ratio. The age of the sample is:

• (1) 248 years
• (2) 186 years
• (3) 124 years
• (4) 310 years

The current gain of a transistor in a common emitter configuration is 80. The resistances in collector and base sides of the circuit are 5 k\(\omega\) and 1 k\(\omega\) respectively. If the input voltage is 2 mV, the output voltage is:

• (A) \( 4V \)
• (B) \( 0.4V \)
• (C) \( 0.8V \)
• (D) \( 8V \)

Four logic gates are connected as shown in the figure. If the inputs are \( A = 0, B = 1, C = 1 \), then the values of \( y_1 \) and \( y_2 \) respectively are:

• (A) \( 1,0 \)
• (B) \( 1,1 \)
• (C) \( 0,1 \)
• (D) \( 0,0 \)

The maximum distance between the transmitting and receiving antennas for satisfactory
communication in line of sight mode is 57.6 km. If the height of the receiving antenna is 80 m, the
height of the transmitting antenna is (Radius of Earth = \( 6.4 \times 10^6 \) m):

• (A) \( 28.8 \) m
• (B) \( 51.2 \) m
• (C) \( 25.6 \) m
• (D) \( 14.4 \) m

If the longest wavelength of the spectral line of the Paschen series of \( Li^{2+} \) ion spectrum is \( x \) Å, then the longest wavelength (in Å) of the Lyman series of the hydrogen spectrum is:

• (A) \( \frac{12}{7} x \)
• (B) \( \frac{7x}{12} \)
• (C) \( \frac{20x}{27} \)
• (D) \( \frac{27x}{20} \)

If \( v_0 \) is the threshold frequency of a metal X, the correct relation between de Broglie wavelength \( \lambda \) associated with photoelectron and frequency \( v \) of the incident radiation is:

• (A) \( \lambda \propto \frac{1}{\sqrt{v - v_0}} \)
• (B) \( \lambda \propto \frac{1}{(v - v_0)^{1/4}} \)
• (C) \( \lambda \propto \frac{1}{(v - v_0)^{3/4}} \)
• (D) \( \lambda \propto \sqrt{v - v_0} \)

In which of the following sets, elements are not correctly arranged with the property shown in brackets?

• (A) \( S \(>\) Se \(>\) O \) (Electron gain enthalpy)
• (B) \( F \(>\) O \(>\) Cl \) (Electronegativity)
• (C) \( Na \(>\) Li \(>\) Al \) (Metallic radius)
• (D) \( Na \(>\) K \(>\) Ba \) (Metallic nature)

In which of the following cases, there is no change in hybridization of the central atom?

• (A) \( NH_3 + H^+ \rightarrow NH_4^+ \)
• (B) \( BF_3 + F^- \rightarrow BF_4^- \)
• (C) \( PCl_5 + Cl^- \rightarrow PCl_6^- \)
• (D) \( ClF_3 + F^- \rightarrow ClF_4^- \)

In which of the following sets, the sum of bond orders of three species is maximum?

• (A) \( B_2, CN^-, O_2^{2-} \)
• (B) \( O_2, F_2, O_2^+ \)
• (C) \( O_2, N_2, O_2^+ \)
• (D) \( C_2, O_2, He_2^{2+} \)

At 240.55 K, for one mole of an ideal gas, a graph of \( P \) (on y-axis) and \( V^{-1} \) (on x-axis) gave a straight line passing through the origin. Its slope (m) is 2000 J mol\(^{-1}\). What is the kinetic energy (in J mol\(^{-1}\)) of the ideal gas?

• (A) \( 2000 \)
• (B) \( 3000 \)
• (C) \( 6000 \)
• (D) \( 1500 \)

At STP, a closed vessel contains 1 mole each of He and CH\(_4\). Through a small hole, 2L of He and 1L of CH\(_4\) escaped from the vessel in \( t \) minutes. What are the mole fractions of He and CH\(_4\) respectively remaining in the vessel? (Assume He and CH\(_4\) as ideal gases. At STP one mole of an ideal gas occupies 22.4 L of volume).

• (A) \( 0.512, 0.488 \)
• (B) \( 0.5, 0.5 \)
• (C) \( 0.329, 0.671 \)
• (D) \( 0.488, 0.512 \)

10 g of a metal (M) reacts with oxygen to form 11.6 g of oxide. What is the equivalent weight of \( M \)?

• (A) \( 50 \) g
• (B) \( 0.02 \)
• (C) \( 0.02 \) g
• (D) \( 50 \)

What is the enthalpy change (in J) for converting 9 g of H\(_2\)O (l) at \( +10^\circ C \) to H\(_2\)O (l) at \( +20^\circ C \)? Given \( C_p \) of water = 75 J/mol K and density of water = 1 g/mL.

• (A) \( 750 \)
• (B) \( 75 \)
• (C) \( 37.5 \)
• (D) \( 375 \)

A, B, C and D are some compounds. The enthalpy of formation of A(g), B(g), C(g) and D(g) is 9.7, -110, 81 and -393 kJ mol\(^{-1}\) respectively. What is \( \Delta H \) (in kJ mol\(^{-1}\)) for the given reaction?
\[ A(g) + 3B(g) \rightarrow C(g) + 3D(g) \]

• (A) \(-777.7\)
• (B) \(+777.7\)
• (C) \(-1418.3\)
• (D) \(+1418.3\)

At equilibrium for the reaction \[ A_2 (g) + B_2 (g) \rightleftharpoons 2AB (g) \]
The concentrations of \( A_2 \), \( B_2 \), and \( AB \) respectively are \( 1.5 \times 10^{-3}M \), \( 2.1 \times 10^{-3}M \), and \( 1.4 \times 10^{-3}M \). What will be \( K_p \) for the decomposition of AB at the same temperature?

• (A) \(0.62\)
• (B) \(1.6\)
• (C) \(0.44\)
• (D) \(2.27\)

Which of the following when added to 20 mL of a 0.01 M solution of HCl would decrease its pH?

• (A) \( 20 mL of 0.02 M HCl \)
• (B) \( 20 mL of 0.005 M HCl \)
• (C) \( 20 mL of 0.01 M HCl \)
• (D) \( 40 mL of 0.005 M HCl \)

Identify the incorrect statement:

• (A) Saline hydrides on electrolysis liberate dihydrogen gas at anode
• (B) \(CH_4\) is an electron precise hydride
• (C) Chromium hydride conducts heat and electricity
• (D) Hydrides of group 15 elements behave as Lewis acids

Which one of the following alkaline earth metals does not form hydride when it is heated with hydrogen directly?

• (1) Be
• (2) Mg
• (3) Ca
• (4) Sr

In the given structure of Diborane, \( \theta_1, \theta_2 \) are respectively:

\begin{figure
\centering

\end{figure

• (A) \( 101^\circ, 118^\circ \)
• (B) \( 118^\circ, 101^\circ \)
• (C) \( 97^\circ, 120^\circ \)
• (D) \( 120^\circ, 97^\circ \)

In which of the following sets, allotropes of carbon are correctly matched with their uses?

(i) Graphite - Crucibles

(ii) Activated Charcoal - Water filters

(iii) Carbon Black - Fuel

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

Which of the following is/are estimated by titrating polluted water with potassium dichromate solution in acidic medium?

• (A) I only
• (B) II only
• (C) II \& III only
• (D) I, II, III

The number of isomers possible for a dibromo derivative (Molecular weight = 186 u) of an alkene is (Br = 80 u):

• (1) 2
• (2) 3
• (3) 4
• (4) 6

In Kolbe's electrolysis of sodium propanoate, products formed at anode and cathode are respectively:

• (1) \( C_2H_6, H_2 \)
• (2) \( C_3H_8, H_2 \)
• (3) \( C_4H_{10}, H_2 \)
• (4) \( H_2, C_4H_{10} \)

Zinc oxide (white) is heated to high temperature for some time. Observe the following statement regarding above process:

I. Zinc oxide colour changes to pale yellow

II. The type of defect formed is ‘metal deficiency’

III. Some \( Zn^{2+} \) and \( e^- \) are present in interstitial places

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

Benzoic acid undergoes dimerization in benzene. \( x \) g of benzoic acid (molar mass 122 g mol\(^{-1}\)) is dissolved in 49 g of benzene. The depression in freezing point is 1.12 K. If degree of association of acid is 88%, what is the value of \( x \)? (K\(_f\) for benzene = 4.9 K kg mol\(^{-1}\))

• (A) 2.44
• (B) 1.22
• (C) 3.66
• (D) 4.88

At \( T(K) \), two liquids A and B form an ideal solution. The vapour pressures of pure liquids A and B at that temperature are 400 and 600 mm Hg respectively. If the mole fraction of liquid B is 0.3 in the mixture, the mole fractions of A and B in vapour phase respectively are:

• (A) 0.391, 0.609
• (B) 0.509, 0.491
• (C) 0.609, 0.391
• (D) 0.491, 0.509

In which of the following Galvanic cells, emf is maximum?
(Given: \( E^\circ_{Mg} = -2.36 \, V \) and \( E^\circ_{Cl_2/Cl^-} = +1.36 \, V \))

• (A) Mg | Mg\(^{2+} \) (1 M) || 2Cl\(^{1-}\) (1 M) | Cl\(_2\) (1 atm), Pt
• (B) Mg | Mg\(^{2+} \) (0.01 M) || 2Cl\(^{1-}\) (1 M) | Cl\(_2\) (1 atm), Pt
• (C) Mg | Mg\(^{2+} \) (1 M) || 2Cl\(^{1-}\) (0.01 M) | Cl\(_2\) (1 atm), Pt
• (D) Mg | Mg\(^{2+} \) (0.01 M) || 2Cl\(^{1-}\) (0.01 M) | Cl\(_2\) (1 atm), Pt

Isomerisation of gaseous cyclobutane to butadiene is a first-order reaction. At \( T(K) \), the rate constant of the reaction is \( 3.3 \times 10^{-4} \, s^{-1} \). What is the time required (in min) to complete 90% of the reaction at the same temperature? (log 2 = 0.3)

• (A) 116.67
• (B) 233.34
• (C) 58.34
• (D) 350.0

Match List-I with List-II:

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

The following data is obtained for coagulating a positively charged sol in 2 hours:


\begin{figure
\centering

\end{figure

What is the coagulating value of electrolyte for this sol?

• (1) \( 7 \times 10^{-5} \)
• (2) \( 7 \times 10^{-2} \)
• (3) \( 5 \times 10^{-2} \)
• (4) \( 9 \times 10^{-2} \)

In which of the following metals extraction, impurities are removed as slag?

(i) Al \hspace{1cm (ii) Fe \hspace{1cm (iii) Cu \hspace{1cm (iv) Zn

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

Two of the products formed by the reaction of \(X\) with HCl are gases. What is \(X\)?
HCl with \(X\) forms gaseous products, two of which are identified.

• (1) NaNO\(_2\)
• (2) Na\(_2\)S
• (3) Ca\(_3\)P\(_2\)
• (4) Na\(_2\)SO\(_3\)

The correct order of oxidizing power of the given ions is
Given: \( E^{\circ}_{Mg^{2+}/Mg} = -2.36 \, V \) and \( E^{\circ}_{Cl_2/Cl^-} = +1.36 \, V\)

• (1) VO\(_2^+\) \(<\) Cr\(_2\)O\(_7^{2-}\) \(<\) MnO\(_4^-\)
• (2) VO\(_2^+\) \(<\) MnO\(_4^-\) \(<\) Cr\(_2\)O\(_7^{2-}\)
• (3) MnO\(_4^-\) \(<\) Cr\(_2\)O\(_7^{2-}\) \(<\) VO\(_2^+\)
• (4) Cr\(_2\)O\(_7^{2-}\) \(<\) VO\(_2^+\) \(<\) MnO\(_4^-\)

Match the complexes in list-I with their hybridization in list-II.

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

Match the following:

• (A) A - III, B - I, C - IV, D - II
• (B) A - III, B - IV, C - I, D - II
• (C) A - I, B - II, C - III, D - IV
• (D) A - IV, B - III, C - II, D - I

Which of the following is not an essential amino acid?

• (A) Lysine
• (B) Histidine
• (C) Glutamine
• (D) Methionine

Which one of the following is NOT a disaccharide?

• (A) Sucrose
• (B) Fructose
• (C) Maltose
• (D) Lactose

Which of the following molecules contain sulfur atom in their structures?

I. Morphine

II. Heroin

III. Penicillin

IV. Terpinenol

V. Cimetidine

• (A) I, IV
• (B) II, III
• (C) III, V
• (D) IV, V

In Wurtz-Fitting reaction, a compound X reacts with alkyl halide. What is X?

• (A)
• (B)
• (C)
• (D)

Correct Answer: (2)
View Solution

The product (C) in the following reaction sequence is:

• (A)
• (B)
• (C)
• (D)

Correct Answer: (4)
View Solution

An organic compound (X) has an empirical formula C\(_4\)H\(_8\)O. This gives a pale yellow precipitate with iodine in NaOH solution. What is X?

• (1) \( CH_3 CH_2 CHO \)
• (2) \( CH_2 = CH CH(OH) CH_3 \)
• (3) \( CH_3 CH_2 COOH \)
• (4) \( CH_3 CH_2 O CH_2 \)

Arrange the following in the correct order of their acidic strength:
I. \( C_6 H_4 (OH) \) (I)
II. \( C_6 H_5 (OH) \) (II)
III. \( C_6 H_4 (NO_2)(OH) \) (III)
IV. \( C_6 H_4 (NO_2)(OH) \) (IV)

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

What is ‘Y’ in the given reaction sequence?

• (1)
• (2)
• (3)
• (4)

Correct Answer: (4)
View Solution

Identify B in the given reaction sequence:

• (1)
• (2)
• (3)
• (4)