AP EAPCET (AP EAMCET) 2024 Question Paper May 19 Shift 2 (Available): Download MPC Question Paper with Answer Key PDF

If a real valued function \( f: [a, \infty) \to [b, \infty) \) is defined by \( f(x) = 2x^2 - 3x + 5 \) and is a bijection, then find the value of \( 3a + 2b \):

• (1) 20
• (2) 10
• (3) 12
• (4) 6

The domain of the real valued function \( f(x) = \frac{1}{\sqrt{log_{0.5}(2x-3)}} + \sqrt{4 - 9x^2} \) is:

• (1) \( \left[ \frac{2}{3}, \frac{3}{2} \right] \)
• (2) Null Set
• (3) \( \left[ \frac{2}{3}, 2 \right) \)
• (4) \( \left( -\frac{2}{3}, \frac{3}{2} \right) \)

Find the sum of the first 10 terms of the sequence \( 2.5 + 5.9 + 8.13 + 11.17 + \cdots \ to \ 10 \ terms =\):

• (1) 3355
• (2) 4555
• (3) 1375
• (4) 1380

Evaluate the following determinant: \( \begin{vmatrix} 1 & 1 & 1
a^2 & {b^2} & {c^2}
{a^3} & {b^3} & {c^3}
\end{vmatrix} \)

• (1) \( (a - b)(b - c)(c - a)(a + b + c) \)
• (2) \( (a - b)(b - c)(c - a)(ab + bc + ca) \)
• (3) \( (a - b)(b - c)(c - a)(a + b + c) \)
• (4) \( (a - b)(b - c)(c - a)(ab + bc + ca) \)

If \( A = \begin{pmatrix} 1 & 2
-2 & -5 \end{pmatrix} \) and \( \alpha^2 + \beta A = 21 \) for some \( \alpha, \beta \in \mathbb{R} \), then find \( \alpha + \beta \):

• (1) 7
• (2) 10
• (3) 12
• (4) 5

The system of equations \( x + 2y + 3z = 6 \), \( x + 3y + 5z = 9 \), \( 2x + 5y + az = 12 \) has no solution when \( a = \):

• (1) 5
• (2) 6
• (3) 7
• (4) 8

If \( m, n \) are respectively the least positive and greatest negative integer values of such that \( (\frac{1-i}{1+i})^k = -i\), then \( m - n = \):

• (1) 4
• (2) 0
• (3) 6
• (4) 2

If a complex number \( z \) is such that \( \frac{z-2i}{z-2} \) and the locus of \( z \) is a closed curve, then the area of the region bounded by that closed curve and lying in the first quadrant is:

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

The real part of \( \frac{\left( \cos a + i \sin a \right)^6}{\left( \sin b + i \cos b \right)^8} \) is:

• (1) \( \sin (6a - 8b) \)
  \
• (2) \( \cos (6a - 8b) \)
• (3) \( \sin (6a + 8b) \)
• (4) \( \cos (6a + 8b) \)

Simplify the expression: \( 4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \cdots}}} \)

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

If \(x^2 + 5ax + 6 = 0\) and \(x^2 + 3ax + 2 = 0\) have a common root, then that common root is:

• (A) \(3 \quad or \quad -3\)
• (B) \(2 \quad or \quad -2\)
• (C) \(-2 \quad or \quad 3\)
• (D) \(-3 \quad or \quad 2\)

If \( \alpha, \beta, \gamma \) are roots of the equation \( x^3 + ax^2 + bx + c = 0 \), then \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \) is:

• (A) \( \frac{a}{c} \)
• (B) \( \frac{-b}{c} \)
• (C) \( \frac{c}{a} \)
• (D) \( \frac{b}{a} \)

If the roots of the equation \( x^3 - 13x^2 + Kx - 27 = 0 \) are in geometric progression, then \( K = \):

• (A) \(-30\)
• (B) \(30\)
• (C) \(39\)
• (D) \(-39\)

If all the letters of the word MASTER are permuted in all possible ways and words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word MASTER is:

• (A) 357
• (B) 527
• (C) 257
• (D) 752

If set \( A \) contains 8 elements, then the number of subsets of \( A \) which contain at least 6 elements is:

• (A) 28
• (B) 73
• (C) 37
• (D) 82

The number of different permutations that can be formed by taking 4 letters at a time from the letters of the word "REPETITION" is:

• (A) 1380
• (B) 1218
• (C) 1398
• (D) 1286

Numerically greatest term in the expansion of \( (5 + 3x)^6 \), when \( x = 1 \), is:

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

The sum of the series \( 1 - \frac{2}{3} + \frac{2.4}{3.6} - \frac{2.4.6}{3.6.9} + \cdots \infty \) is:

• (A) \( \frac{3}{5} \)
• (B) \( \left( \frac{2}{5} \right)^{2/3} \)
• (C) \( \frac{2}{5} \)
• (D) \( \left( \frac{3}{5} \right)^{2/3} \)

If \( \frac{1}{x^4 + 1} = \frac{Ax + B}{x^2 + \sqrt{2}x + 1} + \frac{Cx + D}{x^2 - \sqrt{2}x + 1} \), then \( BD - AC = \):

• (A) \( \frac{3}{8} \)
• (B) \( \frac{1}{8} \)
• (C) \( 1 \)
• (D) \( 0 \)

The smallest positive value (in degrees) of \( \theta \) for which \( \tan(\theta + 100^\circ) = \tan(\theta + 50^\circ) \tan(\theta - 50^\circ) \) is valid, is:

• (A) \( 60^\circ \)
• (B) \( 45^\circ \)
• (C) \( 30^\circ \)
• (D) \( 15^\circ \)

The value of \( 5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) + 3 \) lies between:

• (A) -2 and 5
• (B) -1 and 8
• (C) -3 and 6
• (D) -4 and 10

Statement (S1): \( \sin 55^\circ + \sin 53^\circ - \sin 19^\circ - \sin 17^\circ = \cos 2^\circ \)

Statement (S2): The range of \( \frac{1}{3 - \cos 2x} \) is \( \left[ \frac{1}{4}, \frac{1}{2} \right] \)

Which one of the following is correct?

• (A) Both (S1) and (S2) are true
• (B) Both (S1) and (S2) are false
• (C) (S1) is true, (S2) is false
• (D) (S1) is false, (S2) is true

The general solution of \( 4 \cos 2x - 4 \sqrt{3} \sin 2x + \cos 3x - \sqrt{3} \sin 3x + \cos x - \sqrt{3} \sin x = 0 \) is:

• (A) \( \frac{n\pi}{2}-\frac{\pi}{3} \)
• (B) \( \frac{n\pi}{2} + \frac{\pi}{6} \)
• (C) \( \frac{n\pi}{2} + \frac{\pi}{12} \)
• (D) \( \frac{n\pi}{2} - \frac{\pi}{12} \)

The general solution of \( 2 \cos^2 x - 2 \tan x + 1 = 0 \) is:

• (A) \( n\pi + \frac{\pi}{4}, \, n \in \mathbb{Z} \)
• (B) \( 2n\pi + \frac{\pi}{4}, \, n \in \mathbb{Z} \)
• (C) \( 2n\pi \pm \frac{\pi}{3}, \, n \in \mathbb{Z} \)
• (D) \( n\pi \pm \frac{\pi}{3}, \, n \in \mathbb{Z} \)

The value of \( \cosh \left( \sin^{-1} \left( \sqrt{8} \right) + \cosh^{-1} 5 \right) \) is:

• (A) \( \sqrt{6} + 4\sqrt{2} \)
• (B) \( 15 + 8\sqrt{3} \)
• (C) \( 6\sqrt{6} + 10\sqrt{2} \)
• (D) \( 8 - 15\sqrt{3} \)

In a triangle ABC, if \( r_1 = 2r_2 = 3r_3 \), then \(\sin A\): \(\sin B\): \(\sin C\) =

Options:

In \(\Delta ABC\) if \(B = 90^\circ\) then \(2(r + R) = \)

• (1) \(a + b\)
• (2) \(b + c\)
• (3) \(a + c\)
• (4) \(0\)

In a triangle ABC, if \( (a-b)(s-c) = (b-c)(s-a) \), then \( r_1 + r_3 = \):

• (A) \( r_2 - r_3 \)
• (B) \( 3r_2 \)
• (C) \( 2r_2 \)
• (D) \( 3(r_1 + r_2) \)

If \( L, M, N \) are the midpoints of the sides \overline{PQ, QR, and RP of triangle \( \Delta PQR \), then \( \overline{QM} + \overline{LN} + \overline{ML} + \overline{RN} - \overline{MN} - \overline{QL} = \):

• (A) \( \overline{PQ} + \overline{QR} + \overline{LM} + \overline{MN} \)
• (B) \( \overline{LP} + \overline{PM} + \overline{MQ} \)
• (C) \( \overline{PQ} + \overline{QR} - \overline{PR} \)
• (D) \( \overline{LM} + \overline{MN} + \overline{NR} \)

Let \( \vec{a} \times \vec{b} = 7\hat{i} - 5\hat{j} - 4\hat{k} \) and \( \vec{a} = \hat{i} + 3\hat{j} - 2\hat{k} \), if the length of projection of \( \vec{b} \) on \( \vec{a} \) is \( \frac{8}{\sqrt{14}} \), then \( |\vec{b}| \) is:

• (A) \( 121 \)
• (B) \( \sqrt{12} \)
• (C) \( \sqrt{11} \)
• (D) \( 144 \)

Let ABC be an equilateral triangle of side \(a\). M and N are two points on the sides AB and AC respectively such that \(AN = K \cdot AC\) and \(AB = 3 \cdot AM\). If the vectors \(BN\) and \(CM\) are perpendicular, then \(K = \) ?

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

Let \( \mathbf{a} \) and \( \mathbf{b} \) be two non-collinear vectors of unit modulus. If \( \mathbf{u} = \mathbf{a} - (\mathbf{a} \cdot \mathbf{b})\mathbf{b} \) and \( \mathbf{v} = \mathbf{a} \times \mathbf{b} \), then \( \lVert \mathbf{v} \rVert = \) ?

• (A) \( \lVert \mathbf{u} \rVert + \lVert \mathbf{u} \cdot \mathbf{v} \rVert \)
• (B) \( \frac{\lVert \mathbf{u} \rVert}{2} \)
• (C) \( \lVert \mathbf{u} \rVert + \frac{\lVert \mathbf{u} \cdot \mathbf{b} \rVert}{2} \)
• (D) \( \frac{\lVert \mathbf{u} \rVert}{5} \)

Find the shortest distance between the skew lines \(\vec{r} = (-\hat{i} - 2\hat{j} - 3\hat{k}) + t(3\hat{i} - 2\hat{j} - 2\hat{k})\) and \(\vec{r} = (7\hat{i} + 4\hat{k}) + s(\hat{i} - 2\hat{j} + 2\hat{k})\).

• (A) \( 15 \)
• (B) \( 0 \)
• (C) \( 9 \)
• (D) \( 16 \)

If \(m\) and \(M\) denote the mean deviations about mean and about median respectively of the data 20, 5, 15, 2, 7, 3, 11, then the mean deviation about the mean of \(m\) and \(M\) is:

• (A) \(\frac{1}{7}\)
• (B) \(\frac{38}{7}\)
• (C) \(\frac{36}{7}\)
• (D) \(\frac{37}{7}\)

If 7 different balls are distributed among 4 different boxes, then the probability that the first box contains 3 balls is:

• (A) \(\frac{35}{128}{(\frac{3}{4})}^{3}\)
• (B) \(\frac{35}{64}{(\frac{3}{4})}^{4}\)
• (C) \(\frac{7}{8}(\frac{3}{4})^{7}\)
• (D) \(\frac{5}{16}(\frac{3}{4})^{5}\)

Out of the first 5 consecutive natural numbers, if two different numbers \(x\) and \(y\) are chosen at random, then the probability that \(x^4 - y^4\) is divisible by 5 is:

• (A) \(\frac{2}{5}\)
• (B) \(\frac{4}{5}\)
• (C) \(\frac{3}{5}\)
• (D) \(\frac{1}{5}\)

A bag contains 2 white, 3 green, and 5 red balls. If three balls are drawn one after the other without replacement, then the probability that the last ball drawn was red is:

• (A) \(\frac{2}{3}\)
• (B) \(\frac{3}{4}\)
• (C) \(\frac{5}{9}\)
• (D) \(\frac{1}{2}\)

There are 2 bags each containing 3 white and 5 black balls and 4 bags each containing 6 white and 4 black balls. If a ball drawn randomly from a bag is found to be black, then the probability that this ball is from the first set of bags is:

• (A) \(\frac{25}{57}\)
• (B) \(\frac{25}{41}\)
• (C) \(\frac{2}{5}\)
• (D) \(\frac{3}{5}\)

If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability distribution of number of kings is:

• (A) \(\frac{215}{221}\)
• (B) \(\frac{2}{13}\)
• (C) \(\frac{188}{221}\)
• (D) \(\frac{13}{2}\)

In a consignment of 15 articles, it is found that 3 are defective. If a sample of 5 articles is chosen at random from it, then the probability of having 2 defective articles is:

• (A) \(\frac{256}{625}\)
• (B) \(\frac{64}{625}\)
• (C) \(\frac{128}{625}\)
• (D) \(\frac{512}{625}\)

If a variable straight line passing through the point of intersection of the lines \(x - 2y + 3 = 0\) and \(2x - y - 1 = 0\) intersects the X and Y axes at A and B respectively, then the equation of the locus of a point which divides the segment AB in the ratio -2 : 3 is:

• (A) \(14x^2 + 3xy - 15y^2 = 0\)
• (B) \(xy = 14x + 15y\)
• (C) \(x^2 + xy - y^2 = 0\)
• (D) \(14x + 3xy - 15y = 0\)

Point (-1, 2) is changed to (a, b) when the origin is shifted to the point (2, -1) by translation of axes. Point (a, b) is changed to (c, d) when the axes are rotated through an angle of 45\(^{\circ}\) about the new origin. (c, d) is changed to (e, f) when (c, d) is reflected through y = x. Then (e, f) = ?

• (A) (-3, 3)
• (B) \((0, 3\sqrt{2})\)
• (C) \((3\sqrt{2}, 0)\)
• (D) (1, 2)

The point (a, b) is the foot of the perpendicular drawn from the point (3, 1) to the line x + 3y + 4 = 0. If (p, q) is the image of (a, b) with respect to the line 3x - 4y + 11 = 0, then \(\frac{p}{a} + \frac{q}{b} = \)

• (A) \(-3\)
• (B) \(-5\)
• (C) \(3\)
• (D) \(7\)

A ray of light passing through the point (2, 3) reflects on the Y-axis at a point P. If the reflected ray passes through the point (3, 2) and P = (a, b), then 5b = ?

• (A) \(a - 5\)
• (B) \(a - 13\)
• (C) \(a + 13\)
• (D) \(a + 5\)

The area (in square units) of the triangle formed by the lines \(6x^2 + 13xy + 6y^2 = 0\) and \(x + 2y + 3 = 0\) is:

• (A) \(\frac{9}{2}\)
• (B) \(\frac{45}{4}\)
• (C) \(\frac{9}{8}\)
• (D) \(\frac{45}{8}\)

The angle subtended by the chord \(x + y - 1 = 0\) of the circle \(x^2 + y^2 - 2x + 4y + 4 = 0\) at the origin is:

• (A) \(\cos^{-1}\left(\frac{6}{\sqrt{34}}\right)\)
• (B) \(\frac{\pi}{2}\)
• (C) \(\cos^{-1}\left(\frac{2}{\sqrt{13}}\right)\)
• (D) \(\frac{\pi}{3}\)

Let P be any point on the circle \(x^2 + y^2 = 25\). Let L be the chord of contact of P with respect to the circle \(x^2 + y^2 = 9\). The locus of the poles of the lines L with respect to the circle \(x^2 + y^2 = 36\) is:

• (A) \(y^2 = 20x\)
• (B) \(\frac{x^2}{9} + \frac{y^2}{36} = 1\)
• (C) \(x^2 + y^2 = 400\)
• (D) \(\frac{x^2}{25} - \frac{y^2}{16} = 1\)

If the circles \(S = x^2 + y^2 - 14x + 6y + 33 = 0\) and \(S' = x^2 + y^2 - a^2 = 0\) (\(a \in \mathbb{N}\)) have 4 common tangents, then the possible number of values of \(a\) is:

• (A) \(13\)
• (B) \(5\)
• (C) \(14\)
• (D) \(2\)

If the area of the circum-circle of the triangle formed by the line \(2x + 5y + a = 0\) and the positive coordinate axes is \(\frac{29\pi}{4}\) sq. units, then \(|a| = \)

• (A) \(25\)
• (B) \(10\)
• (C) \(20\)
• (D) \(400\)

The circle \(S \equiv x^2 + y^2 - 2x - 4y + 1 = 0\) cuts the y-axis at A, B (OA > OB). If the radical axis of \(S \equiv 0\) and \(S' \equiv x^2 + y^2 - 4x - 2y + 4 = 0\) cuts the y-axis at C, then the ratio in which C divides AB is:

• (A) \(7 + 2\sqrt{3} : -7 + 2\sqrt{3}\)
• (B) \(\sqrt{3} + 2 : \sqrt{3} - 2\)
• (C) \(6 - 2\sqrt{3} : 2\sqrt{3} - 6\)
• (D) \(-3 : \sqrt{3}\)

If the circle \(S = 0\) cuts the circles \(x^2 + y^2 - 2x + 6y = 0\), \(x^2 + y^2 - 4x - 2y + 6 = 0\), and \(x^2 + y^2 - 12x + 2y + 3 = 0\) orthogonally, then the equation of the tangent at (0, 3) on \(S = 0\) is:

• (A) \([)x + y - 3 = 0\)
• (B) \(y = 3\)
• (C) \(x = 0\)
• (D) \(x - y + 3 = 0\)

The normal drawn at a point \( (2, -4) \) on the parabola \( y^2 = 8x \) cuts again the same parabola at \( (\alpha, \beta) \). Then \( \alpha + \beta \) is:

• (A) \( 8 \)
• (B) \( 16 \)
• (C) \( 24 \)
• (D) \( 30 \)

If a tangent of slope 2 to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) touches the circle \(x^2 + y^2 = 4\), then the maximum value of \(ab\) is:

• (A) \(4\)
• (B) \(12\)
• (C) \(5\)
• (D) \(7\)

The locus of the midpoints of the chords of the hyperbola \( x^2 - y^2 = a^2 \) which touch the parabola \( y^2 = 4ax \) is:

• (A) \( x(y^2 - x^2) = ay^2 \)
• (B) \( x(x^2 + y^2) = y^2 + x \)
• (C) \( ax^3 + y^3 = 3x \)
• (D) (Not given)

If the product of the eccentricities of the ellipse \( \frac{x^2}{16} + \frac{y^2}{b^2} = 1 \) and the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) is 1, then the value of \( b^2 \) is:

• (A) \( \frac{12}{25} \)
• (B) \( 144 \)
• (C) \( 25 \)
• (D) \( \frac{144}{25} \)

If \( A(1,2,0), B(2,0,1), C(-3,0,2) \) are the vertices of \( \triangle ABC \), then the length of the internal bisector of \( \angle BAC \) is:

• (A) \( 3\sqrt{6} \)
• (B) \( \frac{2\sqrt{14}}{3} \)
• (C) \( 6\sqrt{14} \)
• (D) \( \frac{2\sqrt{6}}{3} \)

The perpendicular distance from the point \( (-1,1,0) \) to the line joining the points \( (0,2,4) \) and \( (3,0,1) \) is:

• (A) \( 10 \)
• (B) \( \frac{2\sqrt{5}}{5} \)
• (C) \( \frac{5}{\sqrt{2}} \)
• (D) \( 8 \)

A line \( L \) passes through \( (1,2,-3) \) and \( (3,3,-1) \), and a plane \( \pi \) passes through \( (2,1,-2), (-2,-3,6), (0,2,-1) \). If \( \theta \) is the angle between \( L \) and \( \pi \), then \( 27 \cos^2 \theta = \) ?

• (A) \( 25 \)
• (B) \( 9 \)
• (C) \( 5 \)
• (D) \( 2 \)

\( \lim_{x \to 3} \frac{x^3 - 27}{x^2 - 9}. \)

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

If \( f(x) \) is given as:
\( f(x) = \begin{cases} 3ax - 2b, & x > 1
ax + b + 1, & x \(<\) 1
\end{cases
\)

and \( \lim_{x \to 1} f(x) \) exists, then the relation between \( a \) and \( b \) is:

• (A) \( 3a - 2b = 1 \)
• (B) \( 2a - 3b = 1 \)
• (C) \( 2a + 3b = 1 \)
• (D) \( 2a + 3b = -1 \)

The function \( f(x) \) is given by:
\[ f(x) = \begin{cases} \frac{2}{5 - x}, & x \(<\) 3
5 - x, & x \geq 3 \end{cases} \]

Which of the following is true?

• (A) left discontinuous at \( x = 3 \)
• (B) left continuous at \( x = 3 \)
• (C) right discontinuous at \( x = 5 \)
• (D) discontinuous at \( x = 5 \)

If \( y = f(x) \) is a thrice differentiable function and a bijection, then \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = ? \]

• (A) \( y \)
• (B) \( -y \)
• (C) \( x \)
• (D) \( 0 \)

If \[ f(x) = \begin{cases} x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0
0, & x = 0 \end{cases} \]

Which of the following is true?

• (A) \( f(x) \) is continuous and differentiable if \( 0 \leq \alpha \(<\) 1 \)
• (B) \( f(x) \) is discontinuous and not differentiable if \( 0 \leq \alpha \(<\) 1 \)
• (C) \( f(x) \) is continuous and differentiable for \( \alpha > 1 \)
• (D) \( f(x) \) is discontinuous and differentiable for \( \alpha > 1 \)

If \[ f(x) = \min \{ x, x^2 \} \]

Which of the following is true?

• (A) \( f(x) \) is continuous for all \( x \)
• (B) \( f(x) \) is differentiable for all \( x \)
• (C) \( f'(x) = 2 \) for all \( x > 1 \)
• (D) \( f(x) \) is not differentiable at three values of \( x \)

If \[ y = (1 + a + a^2 + \dots)e^{nx} \]
then the relative error in \( y \) is:

• (A) Error in \( x \)
• (B) Percentage error in \( x \)
• (C) \( n \times \) (error in \( x \))
• (D) \( n \times \) (relative error in \( x \))

If the equation of the tangent at (2, 3) on \(y^2 = ax^3 + b\) is \(y = 4x - 5\), then the value of \(a^2 + b^2\) is:

• (A) \(51\)
• (B) \(53\)
• (C) \(58\)
• (D) \(25\)

If Rolle's theorem is applicable for the function \(f(x) = x(x+3)e^{-x/2}\) on \([-3, 0]\), then the value of \(c\) is:

• (A) \(3\)
• (B) \(3 and -2\)
• (C) \(-2\)
• (D) \(-1\)

For all \(x \in [0, 2024]\) assume that \(f(x)\) is differentiable. \(f(0) = -2\) and \(f'(x) \ge 5\). Then the least possible value of \(f(2024)\) is:

• (A) \(10,120\)
• (B) \(10,118\)
• (C) \(10,122\)
• (D) \(2024\)

\(\int\frac{2x^{2}\cos(x^{2})-\sin(x^{2})}{x^{2}}dx=\)

• (A) \(\frac{\sin(x^{2})}{x^{2}}+c\)
• (B) \(\frac{\cos(x^{2})}{x^{2}}+c\)
• (C) \(\sin(x^{2})+c\)
• (D) \(\frac{\sin(x^{2})}{x}+c\)

If \(\int \frac{\log(1+x^4)}{x^3} dx = f(x) \log(\frac{1}{g(x})) + \tan^{-1}(h(x)) + c\), then \(h(x) [f(x) + f(\frac{1}{x})] = \)

• (A) \(h(x)g(-x)\)
• (B) \(\frac{g(x)}{2}\)
• (C) \(g(x) + g(-x)\)
• (D) \(g(x)h(x)\)

Let \(f(x) = \int \frac{x}{(x^2+1)(x^2+3)} dx\). If \(f(3) = \frac{1}{4} \log(\frac{5}{6})\), then \(f(0) = \)

• (A) \(\frac{1}{4}\log\left(\frac{1}{3}\right)\)
• (B) \(0\)
• (C) \(\frac{1}{2}\log\left(\frac{1}{3}\right)\)
• (D) \(\log\left(\frac{1}{3}\right)\)

\(\int\frac{2\cos 2x}{(1+\sin 2x)(1+\cos 2x)}dx=\)

• (A) \(2\tan x + \log(1+\tan x) + c\)
• (B) \(\tan x - 2\log(1+\tan x) + c\)
• (C) \(2\log(1+\tan x) + \tan x + c\)
• (D) \(2\log(1+\tan x) - \tan x + c\)

\(\int\left(\frac{x}{x\cos x - \sin x}\right)^2 dx = \)

• (A) \(\frac{x \csc x}{x\cos x - \sin x} + \cot x + c\)
• (B) \(\frac{x \csc x}{x\cos x - \sin x} - \cot x + c\)
• (C) \(\frac{x \csc x}{x\cos x + \sin x} + \cot x + c\)
• (D) \(\frac{x}{x\cos x - \sin x} - \cot x + c\)

If \(\lim_{n\rightarrow\infty}[(1+\frac{1}{n^{2}})(1+\frac{4}{n^{2}})(1+\frac{9}{n^{2}})\cdots(1+\frac{n^{2}}{n^{2}})]^{\frac{1}{n}}=ae^{b}\), then \(a+b=\)

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

\(\int_{0}^{\pi}x \sin^4 x \cos^6 x dx = \)

• (A) \(\frac{3\pi^2}{512}\)
• (B) \(\frac{3\pi^2}{256}\)
• (C) \(\frac{\pi^2}{256}\)
• (D) \(\frac{\pi^2}{512}\)

If \(I_{n}=\int_{0}^{\pi/4}\tan^{n}x~dx,\) then \(I_{13}+I_{11}=\)

• (A) \(\frac{1}{13}\)
• (B) \(\frac{1}{12}\)
• (C) \(\frac{1}{10}\)
• (D) \(\frac{1}{11}\)

The area (in square units) of the smaller region lying above the X-axis and bounded between the circle \[ x^2 + y^2 = 2ax \]
and the parabola \[ y^2 = ax \]

• (A) \( 2a^2 \left(\frac{\pi}{4} - \frac{2}{3} \right) \)
• (B) \( a^2 \left(\frac{\pi}{4} - \frac{2}{3} \right) \)
• (C) \( a^2 \left(\frac{\pi}{4} + \frac{2}{3} \right) \)
• (D) \( a^2 \left(\frac{\pi^2}{4} - \frac{1}{3} \right) \)

The difference of the order and degree of the differential equation \[ \left(\frac{d^2y}{dx^2} \right)^{-7/2} - \left(\frac{d^3y}{dx^3} \right)^2 - \left(\frac{d^2y}{dx^2} \right)^{-5/2} - \left(\frac{d^4y}{dx^4} \right) = 0 \]

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

If the differential equation \[ x dy + (y + y^2 x) dx = 0 \]
with condition \( y = 1 \) at \( x = 1 \), then the solution is:

• (A) \( y = \frac{x}{1 + \log x} \)
• (B) \( y = \frac{1 + \log x}{x} \)
• (C) \( y = x(1 + \log x) \)
• (D) \( y = \frac{1}{x(1 + \log x)} \)

The solution of the differential equation \[ x dy - y dx = \sqrt{x^2 + y^2} dx \]
when \( y(\sqrt{3}) = 1 \) is:

• (A) \( y^2 + \sqrt{x^2 + y^2} = x^2 \)
• (B) \( 5y - \sqrt{x^2 + y^2} = x^2 \)
• (C) \( y + \sqrt{x^2 + y^2} = x^2 \)
• (D) \( 5y^2 - \sqrt{x^2 + y^2} = x \)

The percentage error in the measurement of mass and velocity are 3% and 4% respectively. The percentage error in the measurement of kinetic energy is:

• (A) \( 11% \)
• (B) \( 12% \)
• (C) \( 14% \)
• (D) \( 8% \)

A car travelling at 80 kmph can be stopped at a distance of 60 m by applying brakes. If the same car travels at 160 kmph and the same braking force is applied, the stopping distance is:

• (A) \( 240 \) m
• (B) \( 170 \) m
• (C) \( 360 \) m
• (D) \( 480 \) m

A 2 kg ball is thrown vertically upward and another 3 kg ball is projected with a certain angle (\( \theta \neq 90^\circ \)). Both will have the same time of flight. The ratio of their maximum heights is:

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

In a sport event a disc is thrown such that it reaches its maximum range of 80 m, the distance travelled in first 3 s is (g = 10ms\(^2\))

• (1) 80 m
• (2) 60 m
• (3) 72 m
• (4) 74 m

A block of mass 18.5 kg kept on a smooth horizontal surface is pulled by a rope of 3 m length by a horizontal force of 40 N applied to the other end of the rope. If the linear density of the rope is 0.5 kgm\(^-1\) and initially the block is at rest, the time in which the block moves a distance of 9 m is

• (1) 3 s
• (2) 5 s
• (3) 7 s
• (4) 9 s

A block of mass 1.5 kg kept on a rough horizontal surface is given a horizontal velocity of 10 ms\(^{-1}\). If the block comes to rest after travelling a distance of 12.5 m, the coefficient of kinetic friction between the surface and the block is (Acceleration due to gravity = 10 ms\(^{-2}\))

• (1) 0.2
• (2) 0.4
• (3) 0.8
• (4) 0.6

A force of \( (6x^2 - 4x + 3) \, N \) acts on a body of mass 0.75 kg and displaces it from \( x = 5 \, m \) to \( x = 2 \, m \). The work done by the force is

• (1) 201 J
• (2) 215 J
• (3) 229 J
• (4) 307 J

A ball falls freely from rest on to a hard horizontal floor and repeatedly bounces. If the velocity of the ball just before the first bounce is 7 m/s and the coefficient of restitution is 0.75, the total distance travelled by the ball before it comes to rest (acceleration due to gravity = 10 ms\(^{-2}\)) is

• (1) 10.75 m
• (2) 9.75 m
• (3) 8.75 m
• (4) 11.75 m

A solid cylinder rolls down an inclined plane without slipping. If the translational kinetic energy of the cylinder is 140 J, the total kinetic energy of the cylinder is

• (1) 105 J
• (2) 70 J
• (3) 210 J
• (4) 280 J

Two blocks of masses \( m \) and \( 2m \) are connected by a massless string which passes over a fixed frictionless pulley. If the system of blocks is released from rest, the speed of the centre of mass of the system of two blocks after a time of 5.4 s is (Acceleration due to gravity = 10 ms\(^{-2}\))

• (1) 6 ms\(^{-1}\)
• (2) 8 ms\(^{-1}\)
• (3) 4 ms\(^{-1}\)
• (4) 12 ms\(^{-1}\)

The displacement of a particle executing simple harmonic motion is \( y = A \sin(2\pi t + \phi) \, m \), where \( t \) is time in seconds and \( \phi \) is the phase angle. At time \( t = 0 \), the displacement and velocity of the particle are 2 m and 4 ms\(^{-1}\). The phase angle, \( \phi \) =

• (1) 60\(^\circ\)
• (2) 30\(^\circ\)
• (3) 45\(^\circ\)
• (4) 90\(^\circ\)

The displacement of a damped oscillator is \( x(t) = \exp(-0.2t) \cos(3.2t + \phi) \), where \( t \) is time in seconds. The time required for the amplitude of the oscillator to become \( \frac{1}{e^{1.2}} \) times its initial amplitude is

• (1) 3 s
• (2) 6 s
• (3) 2 s
• (4) 8 s

Maximum height reached by a rocket fired with a speed equal to 50% of the escape speed from the surface of the earth is (R – Radius of the earth)

• (1) \( \frac{R}{2} \)
• (2) \( \frac{16R}{9} \)
• (3) \( \frac{R}{3} \)
• (4) \( \frac{R}{8} \)

If the work done in stretching a wire by 1 mm is 2 J, the work necessary for stretching another wire of same material but with double radius of cross section and half the length by 1 mm is

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

If \( S_1 \), \( S_2 \), and \( S_3 \) are the tensions at liquid-air, solid-air and solid-liquid interfaces respectively, and \( \theta \) is the angle of contact at the solid-liquid interface, then

• (1) \( S_1 \cos \theta + S_2 \sin \theta = S_3 \)
• (2) \( S_1 \cos \theta + S_3 = S_2 \)
• (3) \( S_2 \cos \theta + S_3 = S_1 \)
• (4) \( S_3 \cos \theta + S_1 = S_2 \)

If ambient temperature is 300 K, the rate of cooling at 600 K is H. In the same surroundings, the rate of cooling at 900 K is

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

An ideal heat engine operates in Carnot cycle between 127\(^\circ\)C and 27\(^\circ\)C. It absorbs \( 5 \times 10^4 \) cal of heat at higher temperature. Amount of heat converted to work is

• (1) \( 4.8 \times 10^4 \) cal
• (2) \( 2.4 \times 10^4 \) cal
• (3) \( 1.25 \times 10^4 \) cal
• (4) \( 6 \times 10^4 \) cal

One mole of a gas having \( \gamma = \frac{7}{5} \) is mixed with one mole of a gas having \( \gamma = \frac{4}{3} \). The value of \( \gamma \) for the mixture is ( \( \gamma \) is the ratio of the specific heats of the gas)

• (1) \( \frac{5}{11} \)
• (2) \( \frac{11}{15} \)
• (3) \( \frac{15}{11} \)
• (4) \( \frac{5}{13} \)

A Carnot heat engine has an efficiency of 10%. If the same engine is worked backward to obtain a refrigerator, then the coefficient of performance of the refrigerator is

• (1) 8
• (2) 9
• (3) 5
• (4) 6

The rms velocity of a gas molecule of mass \( m \) at a given temperature is proportional to

• (1) \( m^0 \)
• (2) \( m \)
• (3) \( \sqrt{m} \)
• (4) \( \frac{1}{\sqrt{m}} \)

The speed of a wave on a string is 150 ms\(^{-1}\) when the tension is 120 N. The percentage increase in the tension in order to raise the wave speed by 20% is

• (1) 44
• (2) 40
• (3) 22
• (4) 20

The minimum deviation produced by a hollow prism filled with a certain liquid is found to be 30\(^\circ\). The light ray is also found to be refracted at an angle of 30\(^\circ\). Then the refractive index of the liquid is

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

In Young's double slit experiment, the intensity at a point where the path difference is \( \frac{\lambda}{6} \) ( \( \lambda \) being the wavelength of the light used) is \( I \). If \( I_0 \) denotes the maximum intensity, \( \frac{I}{I_0} \) is equal to

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

Two particles of equal mass \( m \) and equal charge \( q \) are separated by a distance of 16 cm. They do not experience any force. The value of \( \frac{q}{m} \) is ______ (if \( G \) is the universal gravitational constant and \( g \) is the acceleration due to gravity).

• (1) \( \sqrt{4 \pi \epsilon_0 G} \)
• (2) \( \sqrt{\frac{G}{4 \pi \epsilon_0}} \)
• (3) \( \sqrt{\frac{\pi \epsilon_0}{G}} \)
• (4) \( \sqrt{4 \pi \epsilon_0 g} \)

In the following diagram, the work done in moving a point charge from point P to point A, B and C are \( W_A, W_B, W_C \) respectively. Then (A, B, C are points on semicircle and point charge \( q \) is at the centre of semicircle)

• (1) \( W_A = W_B = W_C \neq 0 \)
• (2) \( W_A = W_B = W_C = 0 \)
• (3) \( W_A > W_B > W_C \)
• (4) \( W_A \(<\) W_B \(<\) W_C \)

Four condensers each of capacitance 8 \(\mu\)F are joined as shown in the figure. The equivalent capacitance between the points A and B will be

• (1) 32 \(\mu\)F
• (2) 2 \(\mu\)F
• (3) 8 \(\mu\)F
• (4) 16 \(\mu\)F

The resistance between points A and C in the given network is

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

A steady current is flowing in a metallic conductor of non-uniform cross section. The physical quantity which remains constant is

• (1) Electricity current density
• (2) Drift velocity
• (3) Electricity current density and drift velocity
• (4) Electric current

A wire shaped in a regular hexagon of side 2 cm carries a current of 4 A. The magnetic field at the centre of the hexagon is.

• (1) \( 4\sqrt{3} \times 10^{-5} \, T \)
• (2) \( 8\sqrt{3} \times 10^{-5} \, T \)
• (3) \( \sqrt{3} \times 10^{-5} \, T \)
• (4) \( 6\sqrt{3} \times 10^{-5} \, T \)

A tightly wound coil of 200 turns and of radius 20 cm carrying current 5 A. Magnetic field at the centre of the coil is.

• (1) \( 3.14 \times 10^{-3} \, T \)
• (2) \( 3.14 \times 10^{-2} \, T \)
• (3) \( 6.28 \times 10^{-4} \, T \)
• (4) \( 6.28 \times 10^{-3} \, T \)

The domain in ferromagnetic material is in the form of a cube of side 2 \(\mu\)m. Number of atoms in that domain is \(9 \times 10^{10}\) and each atom has a dipole movement of \(9 \times 10^{-24} \, Am^2\). The magnetisation of the domain is (approximately).

• (1) \( 10 \times 10^4 \, Am^{-1} \)
• (2) \( 8 \times 10^4 \, Am^{-1} \)
• (3) \( 12 \times 10^4 \, Am^{-1} \)
• (4) \( 9 \times 10^4 \, Am^{-1} \)

Magnetic field at a distance \(r\) from z axis is \( B = B_0 r \, kt \) present in the region. \( B_0 \) is constant and \(t\) is time. The magnitude of induced electric field at a distance \(r\) from z-axis is.

• (1) \( \frac{B_0 r^3}{3} \)
• (2) \( \frac{2 \pi B_0 r}{3} \)
• (3) \( \frac{B_0 r^2}{2 \pi} \)
• (4) \( \frac{B_0 r^2}{3} \)

A series LCR circuit is shown in the figure. Where the inductance of 10 H, capacitance 40 \(\mu\)F and resistance 60 Ω are connected to a variable frequency 240 V source. The current at resonating frequency is.

• (1) 4 A
• (2) 2 A
• (3) 5.4 A
• (4) 5.8 A

An electromagnetic wave travels in a medium with a speed of \( 2 \times 10^8 \, ms^{-1} \). The relative permeability of the medium is 1. Then the relative permittivity is.

• (1) 1.75
• (2) 2
• (3) 2.25
• (4) 2.75

The longest wavelength of light that can initiate photo electric effect in the metal of work function 9 eV is

• (1) \( 1.37 \times 10^{-7} \, m \)
• (2) \( 1.5 \times 10^{-7} \, m \)
• (3) \( 3.7 \times 10^{-7} \, m \)
• (4) \( 4 \times 10^{-7} \, m \)

A hydrogen atom falls from \(n^{th}\) higher energy orbit to first energy orbit (\(n = 1\)). The energy released is equal to 12.75 eV. The \(n^{th}\) orbit is

• (1) \( n = 4 \)
• (2) \( n = 3 \)
• (3) \( n = 6 \)
• (4) \( n = 5 \)

The decrease in each day in the Uranium mass of the material in a Uranium reactor operating at a power of 12 MW is (Energy released in one \(^{92}U\) fission is about 200 MeV)

• (1) \( 12.64 \times 10^{-2} \) kg
• (2) \( 11.50 \times 10^{-2} \) g
• (3) \( 12.64 \) kg
• (4) \( 12.64 \) g

When a signal is applied to the input of a transistor it was found that output signal is phase-shifted by 180\(^\circ\). The transistor configuration is

• (1) CB - configuration
• (2) CE - configuration
• (3) CC - configuration
• (4) Both CB and CC - configuration

The voltage \( V_o \) in the network shown is

• (1) \( V_o = 11.3 \) V
• (2) \( V_o = 9.8 \) V
• (3) \( V_o = 12.0 \) V
• (4) \( V_o = 0.7 \) V

A message signal of 3 kHz is used to modulate a carrier signal frequency 1 MHz, using amplitude modulation. The upper side band frequency and band width respectively are

• (1) 1.003 MHz and 6KHz
• (2) 0.997 MHz and 6KHz
• (3) 1.003 MHz and 3KHz
• (4) 1.003 MHz and 2MHz

In the ground state of hydrogen atom, electron absorbs 1.5 times energy than the minimum energy \( (2.18 \times 10^{-18} J)\) to escape from the atom. The wavelength of the emitted electron (in m) is \((m_e = 9 \times 10^{-31} kg)\)

• (1) \( \frac{h \times 10^{24}}{\sqrt{1.962}} \)
• (2) \( \frac{h}{\sqrt{1.962}} \times 10^{23} \)
• (3) \( \frac{h \times 10^{25}}{\sqrt{1.962}} \)
• (4) \( \frac{h}{\sqrt{1.962}} \times 10^{22} \)

A golf ball of mass ‘m’ has a speed of 50 m/s. If the speed can be measured within accuracy of 2%, the uncertainty in the position is

• (1) \( \frac{h}{4\pi \ m} \)
• (2) \( \frac{h}{16\pi \ m} \)
• (3) \( \frac{h}{4 \pi \ m} \times 10^3 \)
• (4) \( \frac{h}{16 \pi \ m} \times 10^3 \)

If the first ionisation enthalpy of Li, Be and C respectively are 520, 899, 1086 kJ/mol, the first ionisation enthalpy (in kJ/mol) of B will be

• (1) 487
• (2) 950
• (3) 801
• (4) 1402

In which of the following sets of molecules, the central atoms of molecules have same hybridisation?

• (1) NH3,\( CIF_3\)
• (2) H2O, \( SO_3\)
• (3) SF4, CH4
• (4) XeF6, IF7

The correct increasing order of number of lone pair of electrons on the central atom of \(SnCl_2\), \(XeF_2\),\( CIF_3\) and \( SO_3\) is

• (1) \( SO_3\) \(<\)\( CIF_3\) \(<\) \(SnCl_2\) \(<\) \(XeF_2\)
• (2) \( SO_3\) \(<\) \(SnCl_2\) \(<\)\( CIF_3\) \(<\) \(XeF_2\)
• (3) \(XeF_2\) \(<\) \(SnCl_2\) \(<\)\( CIF_3\) \(<\) \( SO_3\)
• (4) \(XeF_2\) \(<\)\( CIF_3\) \(<\) \(SnCl_2\) \(<\) \( SO_3\)

Identify the correct statements from the following:

For an ideal gas, the compressibility factor is 1.0.
The kinetic energy of NO (g) (molar mass = 30 g mol\(^{-1}\)) at T(K) is \( x \) J mol\(^{-1}\). The kinetic energy of N\(_2\)O\(_4\) (g) (molar mass = 92 g mol\(^{-1}\)) at T(K) is \( 2x \) J mol\(^{-1}\).
The rate of diffusion of a gas is inversely proportional to the square root of its density.

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

The following graph is obtained for a gas at different temperatures (T1, T2, T3). What is the correct order of temperature? (x-axis = velocity; y-axis = number of molecules)

• (1) \( T_2 > T_1 > T_3 \)
• (2) \( T_2 > T_3 > T_1 \)
• (3) \( T_3 > T_1 > T_2 \)
• (4) \( T_3 > T_2 > T_1 \)

Observe the following stoichiometric equation

P_4 + 3 \text{OH^- + 3 \text{H_2\text{O \rightarrow \text{PH_3 + 3 \text{OH^-.

What is the conjugate acid of \text{OH^- ?

• (1) Phosphorous acid
• (2) Hypophosphorous acid
• (3) Phosphoric acid
• (4) Pyrophosphoric acid

Given below are two statements

Statement - I: For isothermal irreversible change of an ideal gas, \[ q = -w = P_{ext}(V_{final} - V_{initial}) \]
Statement - II: For adiabatic change, \[ \Delta U = W_{adiabatic} \]
The correct answer is:

• (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

A thermodynamic process (B \(\rightarrow\) E) was completed as shown below. The work done is equal to area under the limits.

• (1)
• (2)
• (3)
• (4)

In a one litre flask, 2 moles of \( A_2 \) was heated to \( T(K) \) and the above equilibrium is reached. The concentrations at equilibrium of \( A_2 \) and \( B_2 \) are \( C_1(A_2) \) and \( C_2(B_2) \) respectively. Now, one mole of \( A_2 \) was added to flask and heated to \( T(K) \) to establish the equilibrium again. The concentrations of \( A_2 \) and \( B_2 \) are \( C_3(A_2) \) and \( C_4(B_2) \) respectively. What is the value of \( C_3(A_2) \) in mol L\(^{-1}\)?

• (1) \( 1.98 \)
• (2) \( 0.01 \)
• (3) \( 0.03 \)
• (4) \( 2.97 \)

What is the conjugate base of chloric acid?

• (A) \( ClO_4^- \)
• (B) \( ClO^- \)
• (C) \( ClO_2^- \)
• (D) \( ClO_3^- \)

The correct statements among the following are:

i. Saline hydrides produce \( H_2 \) gas when reacted with water.

ii. Presently ~77% of the industrial dihydrogen is produced from coal.

iii. Commercially marketed \( H_2 O_2 \) contains 3% \( H_2 O_2 \).

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

The correct order of decomposition temperature of \(MgCO_3\) (X), \(BaCO_3\) (Y), \(CaCO_3\) (Z) is:

• (A) \( Y > Z > X \)
• (B) \( X > Y > Z \)
• (C) \( Y > X > Z \)
• (D) \( X > Z > Y \)

Identify the correct statements from the following:

• (i) Oxidation of NaBH\(_4\) with \(I_2\) gives \(B_2H_6\)
  (ii) \(B_2H_6\) burns in oxygen and releases an enormous amount of energy
  (iii) \(B_2H_6\) on hydrolysis gives a tribasic acid
• (A) i, ii, iii
• (B) i, iii only
• (C) ii, iii only
• (D) i, ii only

Which one of the following is used as piezoelectric material?

• (A) Tridymite
• (B) Quartz
• (C) Zeolite
• (D) Mica

Two statements are given below:
I. In dry cleaning, the solvent \(Cl_2C\) = \(CCl_2\) was earlier used and now it is replaced by liquefied \(CO_2\).

II. In bleaching of paper, \(H_2O_2\) was used earlier and now it is replaced by chlorine gas.

• (A) Statements I, II both are correct
• (B) Statements I, II both are incorrect
• (C) Statement I is correct but statement II is incorrect
• (D) Statement I is incorrect but statement II is correct

Tropolone is an example for which of the following class of compounds?

• (A) Benzenoid aromatic compound
• (B) Non-Benzenoid aromatic compound
• (C) Alicyclic compound
• (D) Heterocyclic aromatic compound

What are X and Y respectively in the following reaction sequence?

• (A)
• (B)
• (C)
• (D)

Some substances are given below
Ag: CO\(_2\) (s); SiO\(_2\) (s); ZnS (s)
SO\(_2\) (s); A/N: HCl (s); H\(_2\)O (s)
The number of molecular solids and network solids in the above list is respectively.

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

The \(\Delta T_b\) value for 0.01 m KCl solution is 0.01 K. What is the Van’t Hoff factor?
(Kb for water = 0.52 K kg mol\(^{-1}\))

• (1) 1.92
• (2) 1.72
• (3) 0.96
• (4) 0.86

200 g of 20% w/w urea solution is mixed with 400 g of 40% w/w urea solution. What is the weight percentage (w/w %) of resultant solution?

• (1) 30.33
• (2) 33.33
• (3) 36.33
• (4) 28.33

2.644 g of metal (M) was deposited when 8040 coulombs of electricity was passed through molten MF\(_2\) salt. What is the atomic mass of M? (F = 96500 C mol\(^{-1}\))

• (1) 63.47 u
• (2) 65.54 u
• (3) 31.74 u
• (4) 61.48 u

The first order reaction \( A(g) \rightarrow B(g) + 2C(g) \) occurs at 25\(^\circ\)C. After 24 minutes the ratio of the concentration of products to the concentration of the reactant is 1:3. What is the half-life of the reaction (in min)? (log 1.11 = 0.046)

• (1) 150.5
• (2) 142.2
• (3) 157.8
• (4) 15.78

Which of the following has maximum coagulating power in the coagulation of positively charged sol?

• (1) \( Cl^{-} \)
• (2) \( SO_4^{2-} \)
• (3) \( PO_4^{3-} \)
• (4) \( [Fe(CN)_6]^{4-} \)

Identify the autocatalytic reaction from the following:

• (1) \( N_2 + 3H_2 \xrightarrow{Fe, Mo} 2NH_3 \)
• (2) \( 2KClO_3 \xrightarrow{MnO_2} 2KCl + 3O_2 \)
• (3) \( CH_3COOC_2H_5 + H_2O \rightarrow CH_3COOH + C_2H_5OH \)
• (4) \( AgNO_3 + KCl \rightarrow AgCl + KNO_3 \)

The anode and cathode used in electrolytic refining of copper respectively are:

• (1) Pure copper, impure copper
• (2) Impure copper, pure copper
• (3) Pure copper, pure zinc
• (4) Impure copper, pure zinc

The disproportionation products of ortho phosphorous acid are:

• (1) \( H_3PO_4, PH_3 \)
• (2) \( H_3PO_2, H_3PO_3 \)
• (3) \( H_3PO_4, HPO_3 \)
• (4) \( H_3PO_2, P_2H_4 \)

In neutral medium potassium permanganate oxidizes \( I^- \) to \( X \). Identify \( X \).

• (1) Iodine
• (2) Iodate
• (3) Per iodate
• (4) Hypo iodite

The spin-only magnetic moments of the complexes \([Mn(CN)_6]^{3-}\) and \([Co(C_2O_4)_3]^{3-}\) are respectively:

• (1) \( 2.84 \) BM, \( 0 \) BM
• (2) \( 0 \) BM, \( 2.84 \) BM
• (3) \( 0 \) BM, \( 3.87 \) BM
• (4) \( 5.92 \) BM, \( 2.84 \) BM

PHBV is a biodegradable polymer of two monomers X and Y. X and Y respectively are:

• (1) \( X = C_2H_5-CH(OH)-CH_2CO_2H, Y = C_2H_5-CH(OH)-CO_2H \)
• (2) \( X = CH_3-CH(OH)-CH_2CO_2H, Y = C_2H_5-CH(OH)-CH_2CO_2H \)
• (3) \( X = CH_3-CH(OH)-CH_2OH, Y = C_2H_5-CH(OH)-CH_2CO_2H \)
• (4) \( X = H_2N-(CH_2)_5-CO_2H, Y = CH_3-CH(OH)-CH_2CO_2H \)

The carbohydrate which does not react with ammoniacal \( AgNO_3 \) solution is:

• (1) Sucrose
• (2) Maltose
• (3) Lactose
• (4) Fructose

Identify the amino acid which has:

• (1) Alanine
• (2) Arginine
• (3) Asparagine
• (4) Aspartic acid

The structure given below represents:

• (1) Salvarsan
• (2) Penicillin
• (3) Prontosil
• (4) Sulphapyridine

The major product (X) formed in the given reaction is an example of:

• (1) Secondary alkyl halide
• (2) Primary alkyl halide
• (3) Tertiary alkyl halide
• (4) Benzylic halide

Identify the Swarts reaction from the following:

• (1) \( R-CH_2-Br + NaI \rightarrow R-CH_2-I + NaBr \)
• (2) \( 2R-CH_2-Br + 2Na \rightarrow R-(CH_2)_2-R + 2NaBr \)
• (3) \( 2C_6H_5Cl + 2Na \rightarrow C_6H_5-C_6H_5 + 2NaCl \)
• (4) \( 2R-CH_2-Br + CoF_2 \rightarrow 2R-CH_2-F + CoBr_2 \)

An alcohol X (\( C_4H_{10}O \)) reacts with conc. HCl at room temperature to give Y (\( C_4H_9Cl \)). Reaction of X with Cu at 573 K gave Z. What is Z?

• (1)
• (2)
• (3)
• (4)

What is Y in the following reaction sequence?

• (1)
• (2)
• (3)
• (4)

A carbonyl compound X (\( C_3H_6O \)) on oxidation gave a carboxylic acid Y (\( C_3H_6O_2 \)). Oxime of X is:

• (1)
• (2)
• (3)
• (4)

The correct sequence of reactions involved in the following conversion is:

• (1) Bromination, Reduction, Carbylamine Reaction
• (2) Reduction, Bromination, Carbylamine Reaction
• (3) Bromination, Reduction, Oxidation
• (4) Reduction, Bromination, Oxidation