TS EAMCET 2024 Question Paper May 11 Shift 1: Download MPC Question Paper with Solutions PDF

If \(f(x)\) is a quadratic function such that \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\), then \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =\)

• (1) \(\frac{25}{12}\)
• (2) \(\frac{10}{3}\)
• (3) \(\frac{13}{6}\)
• (4) \(\frac{41}{20}\)

If \( f(x) = ax^2 + bx + c \) is an even function and \( g(x) = px^3 + qx^2 + rx \) is an odd function, and if \( h(x) = f(x) + g(x) \) and \( h(-2) = 0 \), then \( 8p + 4q + 2r = \)?

• (1) \( 4a + 3b + 2c \)
• (2) \( a + b + c \)
• (3) \( 4a + 2b + c \)
• (4) \( 8a + 4b + 2c \)

If \(1\cdot3\cdot5 + 3\cdot5\cdot7 + 5\cdot7\cdot9 + \ldots\) to \(n\) terms is \(n(n+1)f(n)\), then \(f(2) =\)

• (1) 12
• (2) 42
• (3) 18
• (4) 20

Given matrices \( A = \begin{bmatrix} 1 & 2
2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} x & y
1 & 2 \end{bmatrix} \), where \((A + B)(A - B) = A^2 - B^2\). If \( C = \begin{bmatrix} x & 2
1 & y \end{bmatrix} \), then the trace of \( C \) is:

• (1) 3
• (2) 5
• (3) 7
• (4) 9

If \( x = k \) satisfies the equation \( \begin{vmatrix} x-2 & 3x-3 & 5x-5
x-4 & 3x-9 & 5x-25
x-8 & 3x-27 & 5x-125 \end{vmatrix} = 0 \), then \( x = k \) also satisfies the equation:

• (1) \( x^2 + x - 2 = 0 \)
• (2) \( x^2 - x - 6 = 0 \)
• (3) \( x^2 - 2x - 8 = 0 \)
• (4) \( x^2 + 2x - 3 = 0 \)

If \(A\) is a non singular matrix, then \(Adj(A^{-1}) =\)

• (1) \((Adj A)^{-1}\)
• (2) \(\frac{1}{|A|} A^{-1}\)
• (3) \(|A| A^{-1}\)
• (4) \(|A| A\)

If the homogeneous system of linear equations \(x - 2y + 3z = 0\), \(2x + 4y - 5z = 0\), \(3x + \lambda y + \mu z = 0\) has a non-trivial solution, then \(8\mu + 11\lambda =\)

• (1) 2
• (2) 6
• (3) -6
• (4) -2

If \( z = \frac{(2-i)(1+i)^3}{(1-i)^2} \), then \(Arg(z) =\)

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

If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane. If the amplitude of \( \frac{2z-i}{z+2i} \) is \( \frac{\pi}{4} \), then the equation of the locus of \( P \) is:

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

If \( \alpha, \beta \) are the roots of the equation \( x^2 + 2x + 4 = 0 \). If the point representing \( \alpha \) in the Argand diagram lies in the 2nd quadrant and \( \alpha^{2024} - \beta^{2024} = ik \), where \( i = \sqrt{-1} \), then \( k = \)

• (1) \(-2^{2025} \sqrt{3}\)
• (2) \(2^{2025} \sqrt{3}\)
• (3) \(-2^{2024} \sqrt{3}\)
• (4) \(2^{2024} \sqrt{3}\)

If \( \alpha \) is a root of the equation \( x^2 - x + 1 = 0 \), then \(\left(\alpha + \frac{1}{\alpha}\right)^3 + \left(\alpha^2 + \frac{1}{\alpha^2}\right)^3 + \left(\alpha^3 + \frac{1}{\alpha^3}\right)^3 + \left(\alpha^4 + \frac{1}{\alpha^4}\right)^3 =\)

• (1) 0
• (2) 1
• (3) -3
• (4) -9

If \( \alpha, \beta \) are the real roots of the equation \( x^2 + ax + b = 0 \). If \( \alpha + \beta = \frac{1}{2} \) and \( \alpha^3 + \beta^3 = \frac{37}{8} \), then \( {a}-{\frac{1}{b}} =\)

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

The solution set of the inequality \( \sqrt{x^2 + x - 2} > (1 - x) \) is:

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

If \( \alpha, \beta, \gamma \) are the roots of the equation \( 4x^3 - 3x^2 + 2x - 1 = 0 \), then \( \alpha^3 + \beta^3 + \gamma^3 = \)

• (1) \(\frac{2}{27}\)
• (2) \(\frac{1}{8}\)
• (3) \(\frac{3}{64}\)
• (4) \(\frac{27}{128}\)

The equation \( 16x^4 + 16x^3 - 4x - 1 = 0 \) has a multiple root. If \( \alpha, \beta, \gamma, \delta \) are the roots of this equation, then \( \frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} = \)

• (1) \(\frac{1}{64}\)
• (2) \(\frac{1}{32}\)
• (3) \(\frac{1}{32}\)
• (4) \(\frac{1}{64}\)

The sum of all the 4-digit numbers formed by taking all the digits from 0, 3, 6, 9 without repetition is:

• (1) 119592
• (2) 115992
• (3) 211599
• (4) 119952

The number of ways in which 6 distinct things can be distributed into 2 boxes so that no box is empty is:

• (1) 36
• (2) 64
• (3) 62
• (4) 34

The number of ways in which the number 831600 can be split into two factors which are relatively prime is:

• (1) 8
• (2) 64
• (3) 32
• (4) 16

The coefficient of \(xy^2z^3\) in the expansion of \((x - 2y + 3z)^6\) is:

• (1) 6480
• (2) 3240
• (3) 1620
• (4) 810

The set of all real values of \(x\) for which the expansion of \( \left( 125x^2 - \frac{27}{x} \right)^{\frac{-2}{3}} \) is valid, is:

• (1) \(\left(-\frac{3}{5}, \frac{3}{5}\right)\)
• (2) \(\left(-\infty, -\frac{3}{5}\right) \cup \left(\frac{3}{5}, \infty\right)\)
• (3) \(\left(-\frac{5}{3}, \frac{5}{3}\right)\)
• (4) \(\left(-1, -\frac{1}{3}\right) \cup \left(\frac{1}{3}, 1\right)\)

If \(\frac{x^2}{2x^2 + 7x + 6} = \frac{Ax + B}{x^2 + a} + \frac{Cx + D}{ax^2 + 3}\), then \(A + B + C - 2D =\)

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

Given \((\sin \theta - \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 5\) and \(\theta\) lies in the third quadrant, find \((\sin \theta + \cos \theta)^3\).

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

Given \( \cos B - \sin A = \frac{\sqrt{3}+1}{4\sqrt{2}} \) and \( 2 \cos A \cos B = \frac{1+\sqrt{2}+\sqrt{3}}{2\sqrt{2}} \), calculate \( \cos^2 \frac{4B}{3} - \sin^2 \frac{4A}{5} \):

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

If \( \theta \) is an acute angle and \( 2\sin^2 \theta = \cos^4 \frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \cos^4 \frac{5\pi}{8} + \sin^4 \frac{7\pi}{8} \), then \( \theta \) is:

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

If \(2 \tan^2 \theta - 4 \sec \theta + 3 = 0\), then \(2 \sec \theta\) is:

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

If \( \sin^{-1} x - \cos^{-1} 2x = \sin^{-1} \left(\frac{\sqrt{3}}{2}\right) - \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) \), then \( \tan^{-1} x + \tan^{-1} \left(\frac{x}{x+1}\right) = \)

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

If \( sech^{-1}\left(\frac{3}{5}\right) - tanh^{-1}\left(\frac{3}{5}\right) = \)

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

In a triangle ABC, if \(a = 5\), \(b = 3\), and \(c = 7\), then the ratio \(\sqrt{\frac{\sin(A-B)}{\sin(A+B)}}\) is:

• (1) \(\frac{4}{7}\)
• (2) 16
• (3) 36
• (4) \(\frac{4}{5}\)

In a triangle ABC, if \( r_1 = 6 \), \( r_2 = 9 \), \( r_3 = 18 \), then \( \cos A \) is:

• (1) \(\frac{5}{13}\)
• (2) \(\frac{4}{5}\)
• (3) \(\frac{5}{7}\)
• (4) \(\frac{7}{25}\)

Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):

• (1) \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} - 3\mathbf{k})\)
• (2) \(\frac{1}{\sqrt{266}}(4\mathbf{i} - 13\mathbf{j} + 9\mathbf{k})\)
• (3) \(\frac{1}{\sqrt{42}}(8\mathbf{i} - 5\mathbf{j} + 17\mathbf{k})\)
• (4) \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k})\)

Given the position vectors of points \(A\) and \(B\) as \( \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \), and \(C\) divides \(AB\) in the ratio 3:2. If \( \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) is the position vector of point \(D\), find the unit vector in the direction of \( \mathbf{CD} \):

• (1) \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} - 3\mathbf{k})\)
• (2) \(\frac{1}{\sqrt{266}}(4\mathbf{i} - 13\mathbf{j} + 9\mathbf{k})\)
• (3) \(\frac{1}{\sqrt{42}}(8\mathbf{i} - 5\mathbf{j} + 17\mathbf{k})\)
• (4) \(\frac{1}{\sqrt{7}}(8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k})\)

A unit vector \( \mathbf{e} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) is coplanar with the vectors \( \mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \) and \( 3\mathbf{i} + \mathbf{j} - 5\mathbf{k} \) and is perpendicular to the vector \( \mathbf{i} + \mathbf{j} + \mathbf{k} \). Calculate \(2a^2 + 3b^2 + 4c^2\):

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

Given vectors \( \mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k} \), \( \mathbf{b} = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} \). If \( \mathbf{d} \) is a normal to the plane of \( \mathbf{a} \) and \( \mathbf{b} \) and satisfies \( \mathbf{d} \cdot \mathbf{c} = 2 \), find the magnitude of \( \mathbf{d} \):

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

Given two planes with equations \( \mathbf{r} \cdot (\mathbf{i} - \mathbf{j} + \mathbf{k}) = 5 \) and \( \mathbf{r} \cdot (2\mathbf{i} + \mathbf{j} - \mathbf{k}) = 3 \). A plane \( \pi \) passing through the line of intersection of these two planes also passes through the point (0,1,2). If the equation of \( \pi \) is \( \mathbf{r} \cdot (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) = m \), determine the value of \( \frac{bc}{a^2} \):

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

Calculate the variance of the data set: 1, 2, 3, 5, 8, 13, 17.

• (1) 31.14
• (2) 29.57
• (3) 30.62
• (4) 32.71

The numbers 2, 3, 5, 7, 11, 13 are written on six distinct paper chits. If 3 of them are chosen at random, calculate the probability that the sum of the numbers on the obtained chits is divisible by 3.

• (1) \(\frac{7}{20}\)
• (2) \(\frac{6}{20}\)
• (3) \(\frac{5}{20}\)
• (4) \(\frac{1}{5}\)

If 4 letters are selected at random from the letters of the word PROBABILITY, compute the probability that at least one letter is repeated in the selection.

• (1) \(\frac{43}{170}\)
• (2) \(\frac{19}{61}\)
• (3) \(\frac{57}{184}\)
• (4) \(\frac{29}{155}\)

If two dice are rolled, determine the probability that the sum of the numbers on the top faces is a multiple of 3, given that their sum is an odd number.

• (1) \(\frac{1}{6}\)
• (2) \(\frac{11}{36}\)
• (3) \(\frac{1}{3}\)
• (4) \(\frac{7}{18}\)

If a random variable \( X \) has the following probability distribution, compute its variance:

• (1) \(\frac{9}{4}\)
• (2) \(\frac{25}{8}\)
• (3) \(\frac{27}{16}\)
• (4) \(\frac{15}{16}\)

The mean and variance of a binomial variate \(X\) are \(\frac{16}{5}\) and \(\frac{48}{25}\) respectively. Given that \(P(X \geq 1) = 1 - K \left(\frac{3}{5}\right)^7\), find \(5K\):

• (1) 19
• (2) 3
• (3) 2
• (4) 11

P and Q are the points of trisection of the line segment joining the points \( (3, -7) \) and \( (-5,3) \). If line PQ subtends a right angle at a variable point R, then the locus of R is:

• (1) A circle with radius \( \frac{\sqrt{41}}{3} \)
• (2) A circle with radius \( \sqrt{409} \)
• (3) A pair of straight lines passing through \((-1, -2)\)
• (4) A pair of straight lines passing through \( (1, 2) \)

(a,b) is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation \(2x^2 - 3xy + 4y^2 + 5y - 6 = 0\). If the angle by which the axes are to be rotated in positive direction about the origin to remove the \(xy\)-term from the equation \(ax^2 + 23abxy + by^2 = 0\) is \( \theta \), then \(2\theta =\)

• (1) \( \frac{\pi}{4} \)
• (2) \(60^\circ\)
• (3) \( \frac{\pi}{3} \)
• (4) \(15^\circ\)

If \( A(1, -2), B(-2, 3), C(-1, -3) \) are the vertices of a triangle ABC. \( L_1 \) is the perpendicular drawn from \( A \) to \( BC \) and \( L_2 \) is the perpendicular bisector of \( AB \). If \( (l, m) \) is the point of intersection of \( L_1 \) and \( L_2 \), then \( 26m - 3 = \)

• (1) 26L
• (2) 89L
• (3) 13L
• (4) 43L

The area of the parallelogram formed by the lines \( L_1: \lambda x + 4y + 2 = 0 \), \( L_2: 3x + 4y - 3 = 0 \), \( L_3: 2x + \mu y + 6 = 0 \), and \( L_4: 2x + y + 3 = 0 \), where \( L_1 \) is parallel to \( L_2 \) and \( L_3 \) is parallel to \( L_4 \), is

• (1) 9
• (2) 7
• (3) 5
• (4) 3

If \( A(1,2) \), \( B(2,1) \) are two vertices of an acute angled triangle and \( S(0,0) \) is its circumcenter, then the angle subtended by \( AB \) at the third vertex is

• (1) \(\tan^{-1}\left(\frac{1}{3}\right)\)
• (2) \(\tan^{-1}\left(\frac{1}{2}\right)\)
• (3) \(\frac{\pi}{4}\)
• (4) \(\frac{\pi}{6}\)

If the angle between the pair of lines given by the equation \(ax^2 + 4xy + 2y^2 = 0\) is \(45^\circ\), then the possible values of 'a' are

• (1) \(-3\) or \(21\)
• (2) \(-6 + 4\sqrt{3}\)
• (3) \(-6 + 24\sqrt{2}\)
• (4) do not exist

A circle passing through the points \( (1, 1) \) and \( (2, 0) \) touches the line \( 3x - y - 1 = 0 \). If the equation of this circle is \( x^2 + y^2 + 2gx + 2fy + c = 0 \), then a possible value of \( g \) is

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

A circle passes through the points \( (2, 0) \) and \( (1, 2) \). If the power of the point \( (0, 2) \) with respect to this circle is 4, then the radius of the circle is

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

If \( x - 2y - 6 = 0 \) is a normal to the circle \( x^2 + y^2 + 2gx + 2fy - 8 = 0 \) and the line \( y = 2 \) touches this circle, then the radius of the circle can be:

• (1) \( \sqrt{32} \)
• (2) 6
• (3) 4
• (4) \( \sqrt{18} \)

The line \( x + y + 1 = 0 \) intersects the circle \( x^2 + y^2 - 4x + 2y - 4 = 0 \) at the points A and B. If \( M(a, b) \) is the midpoint of AB, then \( a - b = \)?

• (1) 0
• (2) 1
• (3) 2
• (4) 3

A circle \( S \) passes through the points of intersection of the circles \( x^2 + y^2 - 2x - 3 = 0 \) and \( x^2 + y^2 - 2y = 0 \). If \( x + y + 1 = 0 \) is a tangent to the circle \( S \), then the equation of \( S \) is:

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

If the common chord of the circles \( x^2 + y^2 - 2x + 2y + 1 = 0 \) and \( x^2 + y^2 - 2x - 2y - 2 = 0 \) is the diameter of a circle \( S \), then the centre of the circle \( S \) is:

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

If \( (1,1) \) is the vertex and \( x + y + 1 = 0 \) is the directrix of a parabola. If \( (a, b) \) is its focus and \( (c, d) \) is the point of intersection of the directrix and the axis of the parabola, then \( a + b + c + d \) is:

• (1) 6
• (2) 5
• (3) 4
• (4) 3

The axis of a parabola is parallel to Y-axis. If this parabola passes through the points \( (1,0), (0,2), (-1,-1) \) and its equation is \( ax^2 + bx + cy + d = 0 \), then \( \frac{ad}{bc} \) is:

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

If the focus of an ellipse is \((-1,-1)\), equation of its directrix corresponding to this focus is \(x + y + 1 = 0\) and its eccentricity is \(\frac{1}{\sqrt{2}}\), then the length of its major axis is:

• (1) 2
• (2) 1
• (3) 4
• (4) 3

If the normal drawn at the point \((2, -1)\) to the ellipse \(x^2 + 4y^2 = 8\) meets the ellipse again at \((a, b)\), then \(17a\) is:

• (1) 23
• (2) 14
• (3) 37
• (4) 9

P(\( \theta \)) is a point on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{9} = 1 \), S is its focus lying on the positive X-axis and Q = (0,1). If SQ = \( \sqrt{26} \) and SP = 6, then \( \theta \) is:

• (1) \( \frac{\pi}{6} \)
• (2) \( \frac{\pi}{4} \)
• (3) \( \frac{\pi}{3} \)
• (4) \( \cos^{-1}\left(\frac{2}{3}\right) \)

If \(A(-2,4,a)\), \(B(1,3,b)\), \(C(0,4,c)\), and \(D(-5,6,1)\) are collinear points, then \(a+b+c=\)

• (1) 4
• (2) 8
• (3) 12
• (4) -4

A(1, -2, 1) and B(2, -1, 2) are the end points of a line segment. If \(D(\alpha, \beta, \gamma)\) is the foot of the perpendicular drawn from \(C(1, 2, 3)\) to AB, then \(\alpha^2 + \beta^2 + \gamma^2 =\)

• (1) \(\sqrt{18}\)
• (2) 14
• (3) 9
• (4) 27

The foot of the perpendicular drawn from the point \((-2, -1, 3)\) to a plane is \((1, 0, -2)\). If \(a, b, c\) are the intercepts made by the plane \(\pi\) on \(X, Y, Z\)-axes respectively, then \(3a+b+5c =\)

• (1) 39
• (2) 26
• (3) 13
• (4) 0

Calculate the limit as \(x\) approaches \(-\frac{3}{2}\): \(\lim_{x \to -\frac{3}{2}} \frac{4x^2 - 6x)(4x^2 + 6x + 9)}{\sqrt{2x - \sqrt{3}}}\)

• (1) \(\frac{3\sqrt{17}}{2}\)
• (2) \(\frac{3\sqrt{16}}{2}\)
• (3) \(\frac{3\sqrt{15}}{2}\)
• (4) \(\frac{3\sqrt{14}}{2}\)

For the real-valued function \( f(x) \) defined as \[ f(x) = \begin{cases} \frac{(4^x-1)^4 \cot(x \log 4)}{\sin(x \log 4) \log(1 + x^2 \log 4)}, & if x \neq 0
{k,} & if x = 0 \end{cases} \]
if \( f(x) \) is continuous at \( x = 0 \), find \( e^k \).

• (1) 1
• (2) 4
• (3) \( e \)
• (4) 2

A function \( f : \mathbb{R} \rightarrow \mathbb{R} \) is such that \( yf(x+y) + \cos mxy = 1 + yf(x) \). If \( m = 2 \), find \( f'(x) \).

• (1) \(-2 \sin 2xy\)
• (2) \(4x\)
• (3) \(\frac{2 \sin 2xy}{y}\)
• (4) \(2x^2\)

If \( y = \sqrt{\sin(\log(2x)) + \sqrt{\sin(\log(2x))} + \sqrt{\sin(\log(2x))} + \ldots \infty} \), then find \( \frac{dy}{dx} \).

• (1) \(\frac{\cos(\log(2x))}{2x(2y-1)}\)
• (2) \(\frac{\cos(\log(2x))}{(2y-1)}\)
• (3) \(\frac{\cos(\log(2x))}{x(2y-1)}\)
• (4) \(\frac{\sin(\log(2x))}{(2y-1)}\)

If \( y = \tan^{-1}\left(\frac{\sin^3(2x) - 3x^2 \sin(2x)}{3x \sin(2x) - x^3}\right), \) then find \( \frac{dy}{dx} \).

• (1) \(\frac{6x \cos(2x) - 3 \sin(2x)}{x^2 - \sin^2(2x)}\)
• (2) \(\frac{6x \sin(2x) - 3 \cos(2x)}{x^2 + \sin^2(2x)}\)
• (3) \(\frac{2x \cos(2x) - \sin(2x)}{x^2 + \sin^2(2x)}\)
• (4) \(\frac{6x \cos(2x) - 3 \sin(2x)}{x^2 + \sin^2(2x)}\)

Derivative of \((\sin x)^x\) with respect to \(x\) is:

• (1) \(\frac{(\sin x)^{x-1}[(\sin x)\log(\sin x) + x\cos x]}{x^{\sin x-1}[x\cos x(\log x) + \sin x]}\)
• (2) \(\frac{(\sin x)^x[(\sin x)(\log(\sin x)) + x\cos x]}{x^{\sin x}[x\cos(\log x) + \sin x]}\)
• (3) \(\frac{(\sin x)^{x-1}[x\cos x(\log x) + \sin x]}{(\sin x)^{x-1}[(\sin x)\log(\sin x) + x\cos x]}\)
• (4) \(\frac{(\sin x)^{x\sin x}[x\cos x(\log x) + \sin x]}{(\sin x)^{x}[(\sin x)\log(\sin x) + x\cos x]}\)

For a given function \( y = f(x) \), \( \delta y \) denotes the actual error in \( y \) corresponding to actual error \( \delta x \) in \( x \) and \( dy \) denotes the approximate value of \( \delta y \). If \( y = f(x) = 2x^2 - 3x + 4 \) and \( \delta x = 0.02 \), then the value of \( \delta y - dy \) when \( x = 5 \) is:

• (1) \( 0.0008 \)
• (2) \( 0.008 \)
• (3) \( 0.0004 \)
• (4) \( 0.004 \)

The length of the normal drawn at \( t = \frac{\pi}{4} \) on the curve \( x = 2(\cos 2t + t \sin 2t) \), \( y = 4(\sin 2t + t \cos 2t) \) is:

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

If water is poured into a cylindrical tank of radius 3.5 ft at the rate of 1 cubic ft/min, then the rate at which the level of the water in the tank increases (in ft/min) is:

• (1) \( \frac{1}{154} \)
• (2) (Missing option)
• (3) \( \frac{2}{77} \)
• (4) \( \frac{1}{11} \)

The function \( y = 2x^3 - 8x^2 + 10x - 4 \) is defined on \([1,2]\). If the tangent drawn at a point \( (a,b) \) on the graph of this function is parallel to the X-axis and \( a \in (1,2) \), then \( a = \) ?

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

If \( m \) and \( M \) are respectively the absolute minimum and absolute maximum values of a function \( f(x) = 2x^3 + 9x^2 + 12x + 1 \) defined on \([-3,0]\), then \( m + M \) is:

• (1) \( -7 \)
• (2) \( 0 \)
• (3) \( 1 \)
• (4) \( 5 \)

Evaluate the integral: \[ \int \frac{\sec x}{3(\sec x + \tan x) + 2} \,dx \]

• (1) \( \frac{1}{2} \log \left| \frac{\tan \frac{x}{2} + 1}{\tan \frac{x}{2} + 5} \right| + C \)
• (2) \( \frac{2}{\sqrt{11}} \tan^{-1} \left( \frac{3\tan \frac{x}{2} + 4}{\sqrt{11}} \right) + C \)
• (3) \( \log |3 \sec x + 2 \tan x| + C \)
• (4) \( \log |3 \tan x + 2 \sec x| + C \)

Evaluate the integral: \[ \int \frac{dx}{4 + 3\cot x} \]

• (1) \( -\frac{3}{25} \log |4 + 3\cot x| + \frac{4}{25} x + C \)
• (2) \( -\frac{3}{25} \log |4\sin x + 3\cos x| + \frac{4}{25} x + C \)
• (3) \( \frac{4}{25} \log |4\sin x + 3\cos x| - \frac{3}{25} x + C \)
• (4) \( \frac{4}{25} \log |4 + 3\cot x| - \frac{3}{25} x + C \)

Evaluate the integral: \[ \int \frac{dx}{(x+1)\sqrt{x^2 + 4}} \]

• (1) \( \frac{1}{2} \frac{x+1}{\sqrt{x+2}} + C \)
• (2) \( \log \left| \frac{x+2}{x+1} \right| + C \)
• (3) \( \frac{1}{\sqrt{5}} \sinh^{-1} \left( \frac{4-x}{2(x+1)} \right) + C \)
• (4) \( \frac{1}{\sqrt{5}} \cosh^{-1} \left( \frac{4+x}{2(x-1)} \right) + C \)

If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \]
then \( f(1) \) is:

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

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \]

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

Evaluate the integral: \[ I = \int_{-\frac{\pi}{15}}^{\frac{\pi}{15}} \frac{\cos 5x}{1 + e^{5x}} \, dx \]

• (1) \( \frac{1}{5} \)
• (2) \( \frac{\sqrt{3}}{10} \)
• (3) \( \frac{1}{15} \)
• (4) \( \frac{1}{10} \)

The area of the region (in square units) enclosed by the curves \( y = 8x^3 - 1 \), \( y = 0 \), \( x = -1 \), and \( x = 1 \) is:

• (1) \( \frac{15}{4} \)
• (2) \( \frac{15}{8} \)
• (3) \( \frac{19}{4} \)
• (4) \( \frac{19}{8} \)

If the equation of the curve which passes through the point \( (1,1) \) satisfies the differential equation: \[ \frac{dy}{dx} = \frac{2x - 5y + 3}{5x + 2y - 3}, \]
then the equation of the curve is:

• (1) \( x^2 + 5xy - y^2 + 3x - 3y - 5 = 0 \)
• (2) \( x^2 + 5xy - y^2 + 3x + 3y - 11 = 0 \)
• (3) \( x^2 - 5xy - y^2 - 3x - 3y + 11 = 0 \)
• (4) \( x^2 - 5xy - y^2 + 3x + 3y - 1 = 0 \)

The general solution of the differential equation: \[ (6x^2 - 2xy - 18x + 3y) dx - (x^2 - 3x) dy = 0 \]

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

The range of gravitational forces is:

• (1) \( 10^{-15} \) m
• (2) \( 10^{-39} \) m
• (3) infinity
• (4) \( 10^{-2} \) m

In a simple pendulum experiment for the determination of acceleration due to gravity, the error in the measurement of the length of the pendulum is 1% and the error in the measurement of the time period is 2%. The error in the estimation of acceleration due to gravity is:

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

The position \( x \) (in meters) of a particle moving along a straight line is given by: \[ x = t^3 - 12t + 3 \]
where \( t \) is time (in seconds). The acceleration of the particle when its velocity becomes \( 15 \, ms^{-1} \) is:

• (1) \( 15 \, ms^{-2} \)
• (2) \( 24 \, ms^{-2} \)
• (3) \( 18 \, ms^{-2} \)
• (4) \( 12 \, ms^{-2} \)

The maximum horizontal range of a ball projected from the ground is 32 m. If the ball is thrown with the same speed horizontally from the top of a tower of height 25 m, the maximum horizontal distance covered by the ball is:

(Acceleration due to gravity \( g = 10 \, m/s^2 \))

• (1) \( 40 \) m
• (2) \( 57 \) m
• (3) \( 60 \) m
• (4) \( 75 \) m

A block of mass 5 kg is kept on a smooth horizontal surface. A horizontal stream of water coming out of a pipe of area of cross-section \( 5 \, cm^2 \) hits the block with a velocity of \( 5 \, ms^{-1} \) and rebounds back with the same velocity. The initial acceleration of the block is:

(Density of water is \( 1 \, g/cc \))

• (1) \( 10 \, ms^{-2} \)
• (2) \( 2.5 \, ms^{-2} \)
• (3) \( 12.5 \, ms^{-2} \)
• (4) \( 5 \, ms^{-2} \)

A constant force of \[ \mathbf{F} = (8\hat{i} - 2\hat{j} + 6\hat{k}) N \]
acts on a body of mass 2 kg, displacing it from \[ \mathbf{r_1} = (2\hat{i} + 3\hat{j} - 4\hat{k}) m to \mathbf{r_2} = (4\hat{i} - 3\hat{j} + 6\hat{k}) m. \]
The work done in the process is:

• (1) \( 72 \) J
• (2) \( 88 \) J
• (3) \( 44 \) J
• (4) \( 36 \) J

A ball 'A' of mass 1.2 kg moving with a velocity of 8.4 m/s makes a one-dimensional elastic collision with a ball 'B' of mass 3.6 kg at rest. The percentage of kinetic energy transferred by ball 'A' to ball 'B' is:

• (1) \( 25% \)
• (2) \( 50% \)
• (3) \( 75% \)
• (4) \( 60% \)

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 9 g, are kept one above the other at the 10 cm mark, the scale is found to be balanced at 35 cm. The mass of the metre scale is:

• (1) \( 15 \) g
• (2) \( 30 \) g
• (3) \( 45 \) g
• (4) \( 60 \) g

A body of mass \( m \) and radius \( r \) rolling horizontally with velocity \( V \), rolls up an inclined plane to a vertical height \( \frac{V^2}{g} \). The body is:

• (1) a sphere
• (2) a circular disc
• (3) a circular ring
• (4) a solid cylinder

A massless spring of length \( l \) and spring constant \( k \) oscillates with a time period \( T \) when loaded with a mass \( m \). The spring is now cut into three equal parts and connected in parallel. The frequency of oscillation of the combination when it is loaded with a mass \( 4m \) is:

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

An object of mass \( m \) at a distance of \( 20R \) from the center of a planet of mass \( M \) and radius \( R \) has an initial velocity \( u \). The velocity with which the object hits the surface of the planet is:

(G - Universal gravitational constant)

• (1) \( \left[ u^2 + \frac{19GM}{10R} \right]^{\frac{1}{2}} \)
• (2) \( \left[ u^2 + \frac{19Gm}{10R} \right]^{\frac{1}{2}} \)
• (3) \( \left[ u^2 - \frac{19GM}{10R} \right]^{\frac{1}{2}} \)
• (4) \( \left[ u^2 - \frac{19Gm}{10R} \right]^{\frac{1}{2}} \)

A simple pendulum is made of a metal wire of length \( L \), area of cross-section \( A \), material of Young's modulus \( Y \), and a bob of mass \( m \). This pendulum is hung in a bus moving with a uniform speed \( V \) on a horizontal circular road of radius \( R \). The elongation in the wire is:

• (1) \( \frac{mL}{RAY} \sqrt{g^2 R^2 + V^4} \)
• (2) \( \frac{mgL}{AY} \)
• (3) \( \frac{mL V^2}{RAY} \)
• (4) \( \frac{L}{AY} \sqrt{mg + \frac{mV^2}{R}} \)

If the excess pressures inside two soap bubbles are in the ratio \( 2:3 \), then the ratio of the volumes of the soap bubbles is:

• (1) \( 3:2 \)
• (2) \( 9:4 \)
• (3) \( 27:8 \)
• (4) \( 81:16 \)

The velocities of air above and below the surfaces of a flying aeroplane wing are 50 m/s and 40 m/s respectively. If the area of the wing is 10 m² and the mass of the aeroplane is 500 kg, then as time passes by (density of air = 1.3 kg/m³), the aeroplane will:

• (1) the aeroplane will gain altitude
• (2) the aeroplane will experience weightlessness
• (3) the aeroplane will fly horizontally
• (4) the aeroplane will lose altitude

A pendulum clock loses 10.8 seconds a day when the temperature is \( 38^\circ C \) and gains 10.8 seconds a day when the temperature is \( 18^\circ C \). The coefficient of linear expansion of the metal of the pendulum clock is:

• (1) \( 7 \times 10^{-5} \, ^\circ C^{-1} \)
• (2) \( 1.25 \times 10^{-5} \, ^\circ C^{-1} \)
• (3) \( 5 \times 10^{-5} \, ^\circ C^{-1} \)
• (4) \( 2.5 \times 10^{-5} \, ^\circ C^{-1} \)

A liquid cools from a temperature of 368 K to 358 K in 22 minutes. In the same room, the same liquid takes 12.5 minutes to cool from 358 K to 353 K. The room temperature is:

• (1) \( 27.5^\circ C \)
• (2) \( 27.5 K \)
• (3) \( 30.5^\circ C \)
• (4) \( 30.5 K \)

For a gas in a thermodynamic process, the relation between internal energy (U), the pressure (P), and the volume (V) is given by:
\[ U = 3 + 1.5PV \]

The ratio of the specific heat capacities of the gas at constant volume and constant pressure is:

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

At a pressure \( P \) and temperature \( 127^\circ C \), a vessel contains 21 g of a gas. A small hole is made into the vessel so that the gas leaks out. At a pressure of \( \frac{2P}{3} \) and a temperature of \( t^\circ C \), the mass of the gas leaked out is 5 g. Then \( t \) is:

• (1) \( 273^\circ C \)
• (2) \( 77^\circ C \)
• (3) \( 350^\circ C \)
• (4) \( 87^\circ C \)

The tension applied to a metal wire of one metre length produces an elastic strain of 1%.
The density of the metal is \( 8000 \, kg/m^3 \) and Young's modulus of the metal is \( 2 \times 10^{11} \, Nm^{-2} \).
The fundamental frequency of the transverse waves in the metal wire is:

• (1) \( 500 \) Hz
• (2) \( 375 \) Hz
• (3) \( 250 \) Hz
• (4) \( 125 \) Hz

Two closed pipes when sounded simultaneously in their fundamental modes produce 6 beats per second.
If the length of the shorter pipe is 150 cm, then the length of the longer pipe is:
\[ (Speed of sound in air = 336 ms^{-1}) \]

• (1) 168 cm
• (2) 184 cm
• (3) 176 cm
• (4) 192 cm

An object placed at a distance of 24 cm from a concave mirror forms an image at a distance of 12 cm from the mirror.
If the object is moved with a speed of 12 ms\(^{-1}\), then the speed of the image is:

• (1) 24 ms\(^{-1}\)
• (2) 3 ms\(^{-1}\)
• (3) 6 ms\(^{-1}\)
• (4) 12 ms\(^{-1}\)

When the object and the screen are 90 cm apart, it is observed that a clear image is formed on the screen when a convex lens is placed at two positions separated by 30 cm between the object and the screen.
The focal length of the lens is:

• (1) 21.4 cm
• (2) 20 cm
• (3) 30 cm
• (4) 30.8 cm

When a monochromatic light is incident on a surface separating two media, both the reflected and refracted lights have the same:

• (1) Frequency
• (2) Wavelength
• (3) Velocity
• (4) Amplitude

The electric flux due to an electric field \[ \vec{E} = (8\hat{i} + 13\hat{j}) NC^{-1} \]
through an area 3 m\(^2\) lying in the XZ plane is:

• (1) 39 Wb
• (2) 24 Wb
• (3) 63 Wb
• (4) 15 Wb

A capacitor of capacitance ‘C’ is charged to a potential ‘V’ and disconnected from the battery. Now if the space between the plates is completely filled with a substance of dielectric constant ‘K’, the final charge and the final potential on the capacitor are respectively:

• (1) \( KCV \) and \( \frac{V}{K} \)
• (2) \( CV \) and \( \frac{V}{K} \)
• (3) \( \frac{CV}{K} \) and \( KV \)
• (4) \( \frac{CV}{K} \) and \( \frac{V}{K} \)

A voltmeter of resistance \(400 \, \Omega\) is used to measure the emf of a cell with an internal resistance of \(4 \, \Omega\). The error in the measurement of emf of the cell is:

• (1) 1.01%
• (2) 2.01%
• (3) 1.99%
• (4) 0.99%

When two wires are connected in the two gaps of a meter bridge, the balancing length is 50 cm. When the wire in the right gap is stretched to double its length and again connected in the same gap, then the new balancing length from the left end of the bridge wire is:

• (1) 80 cm
• (2) 20 cm
• (3) 33.3 cm
• (4) 66.6 cm

A magnetic field is applied in y-direction on an \( \alpha \)-particle traveling along x-direction. The motion of the \( \alpha \)-particle will be:

• (1) along x-axis
• (2) a circle in xz plane
• (3) a circle in yz plane
• (4) a circle in xy plane

A straight wire carrying a current of \( 2\sqrt{2} \, A \) is making an angle of \( 45^\circ \) with the direction of a uniform magnetic field of \( 3 \, T \). The force per unit length on the wire due to the magnetic field is:

• (1) \( 4 \, Nm^{-1} \)
• (2) \( 8 \, Nm^{-1} \)
• (3) \( 6 \, Nm^{-1} \)
• (4) \( 3 \, Nm^{-1} \)

The magnetizing field which produces a magnetic flux of \(22 \times 10^{-6} \, Wb\) in a metal bar of area of cross-section \(2 \times 10^{-5} \, m^2\) is (susceptibility of the metal = 699):

• (1) 2500 A/m
• (2) 1250 A/m
• (3) 3750 A/m
• (4) 5000 A/m

The energy stored in a coil of inductance 80 mH carrying a current of 2.5 A is:

• (1) 1.25 J
• (2) 0.75 J
• (3) 0.25 J
• (4) 0.50 J

A capacitor and a resistor are connected in series to an ac source. If the ratio of the capacitive reactance of the capacitor and the resistance of the resistor is 4:3, then the power factor of the circuit is:

• (1) 0.3
• (2) 0.8
• (3) 0.6
• (4) 0.5

For the displacement current through the plates of a parallel plate capacitor of capacitance 30 µF to be 150 µA, the potential difference across the plates of the capacitor has to vary at the rate of:

• (1) 10 V/s
• (2) 5 V/s
• (3) 15 V/s
• (4) 20 V/s

The work functions of two photosensitive metal surfaces A and B are in the ratio 2:3. If x and y are the slopes of the graphs drawn between the stopping potential and frequency of incident light for the surfaces A and B respectively, then \(x : y\) is:

• (1) 1:1
• (2) 2:3
• (3) 4:9
• (4) 2:5

In a hydrogen atom, the frequency of the photon emitted when an electron jumps from the second orbit to the first orbit is 'f'. The frequency of the photon emitted when an electron jumps from the third excited state to the first excited state is:

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

If the ratio of the radii of nuclei \( \mathbf{^{52}X_A} \) and \( \mathbf{^{13}Al^{27}} \) is 5:3, then the number of neutrons in the nucleus \( X \) is:

Choose the correct answer from the options given below:

Match the devices given in List-I with their uses given in List-II:


\renewcommand{\arraystretch{1.5
\begin{tabular{|c|c|c|c|
\hline
\multicolumn{2{|c|{List I & \multicolumn{2{c|{List II

\hline
a & Transistor & e & Filter circuit

b & Diode & f & Voltage regulator

c & Zener diode & g & Rectifier

d & Capacitor & h & Amplifier

\hline
\end{tabular

To get output 1 for the following logic circuit, the correct choice of the inputs is:

The maximum distance between the transmitting and receiving antennas is D. If the heights of both transmitting and receiving antennas are doubled, then the maximum distance between the two antennas is:

Choose the correct answer from the options given below:

If \(n\) and \(l\) represent the principal and azimuthal quantum numbers respectively, the formula used to know the number of radial nodes possible for a given orbital is:

• (1) \(n - l\)
• (2) \(n - l + 1\)
• (3) \(n - l - 1\)
• (4) \(n - 2\)

If the radius of the first orbit of the hydrogen-like ion is \(1.763 \times 10^{-2}\) nm, the energy associated with that orbit (in J) is:

• (1) \(+1.962 \times 10^{-17}\)
• (2) \(-1.962 \times 10^{-17}\)
• (3) \(-0.872 \times 10^{-17}\)
• (4) \(-2.18 \times 10^{-18}\)

If first ionization enthalpy (\(\Delta H\)) values of Na, Mg and Si are respectively 496, 737, and 786 kJ/mol, the first ionization enthalpy value of Al (in kJ/mol) will be:

• (1) 575
• (2) 760
• (3) 400
• (4) 790

Among the oxides SiO\(_2\), SO\(_2\), Al\(_2\)O\(_3\), and P\(_2\)O\(_5\), the correct order of acidic strength is:

• (1) \(SiO_2 < SO_2 < Al_2O_3 < P_2O_5\)
• (2) \(SO_2 < P_2O_5 < Al_2O_3 < SiO_2\)
• (3) \(Al_2O_3 < SiO_2 < P_2O_5 < SO_2\)
• (4) \(Al_2O_3 < P_2O_5 < SiO_2 < SO_2\)

According to molecular orbital theory, which of the following statement is not correct?

• (1) \(C_2\) molecule is diamagnetic in nature
• (2) Bond order of \(C_2\) molecule is 2
• (3) \(C_2^-\) ion is paramagnetic in nature
• (4) \(C_2\) consists of 1 sigma and 1 pi bond

The melting point of o-hydroxybenzaldehyde (A) is lower than that of p-hydroxybenzaldehyde (B). This is because:

• (1) Both (A) and (B) have intermolecular H-bonding
• (2) (A) has intermolecular H-bonding and (B) has intramolecular H-bonding
• (3) Both (A) and (B) have intramolecular H-bonding
• (4) (A) has intramolecular H-bonding and (B) has intermolecular H-bonding

At what temperature will the RMS velocity of sulfur dioxide molecules at 400 K be the same as the most probable velocity of oxygen molecules?

• (1) 600 K
• (2) 200 K
• (3) 400 K
• (4) 300 K

At 300 K, 3.0 moles of an ideal gas at 3.0 atm pressure is compressed isothermally to one half of its volume by an external pressure of 6.0 atm. The work done (in kJ) is:

• (1) 7.476
• (2) 11.214
• (3) 3.738
• (4) 14.952

The equilibrium constants for the following two reactions are given below at T(K): \[ 2A(g) \leftrightarrow B(g) + C(g), \quad K = 16 \quad and \quad 2B(g) \leftrightarrow D(g), \quad K = 25. \]
What is the value of the equilibrium constant (K) for the reaction given below at T(K)? \[ \frac{1}{2} A(g) \leftrightarrow B(g) \]

• (1) 100
• (2) 50
• (3) 20
• (4) 75

At T(K), the equilibrium constants for the following two reactions are given below: \[ 2A(g) \leftrightarrow B(g) + C(g), \quad K = 16 \quad and \quad 2B(g) \leftrightarrow D(g), \quad K = 25. \]
What is the value of the equilibrium constant (K) for the reaction given below at T(K)? \[ \frac{1}{2} A(g) \leftrightarrow \frac{1}{2} B(g) \]

• (1) 100
• (2) 50
• (3) 20
• (4) 75

Identify the pair of hydrides which have polymeric structure:

• (1) LiH, NaH
• (2) BeH\(_2\), MgH\(_2\)
• (3) NH\(_3\), CH\(_4\)
• (4) B\(_2\)H\(_6\), H\(_2\)O

Match the following:

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

The hydroxide of which of the following metal reacts with both acid and alkali?

• (1) Mg
• (2) Na
• (3) Be
• (4) Ca

The correct formula of borax is Na\(_2\)[B\(_4\)O\(_5\)(OH)\(_x\)]·yH\(_2\)O. The sum of \(x\) and \(y\) is:

• (1) 14
• (2) 09
• (3) 12
• (4) 10

Formic acid on heating with concentrated H\(_2\)SO\(_4\) at 373 K gives X, a colourless substance and Y, a good reducing agent. The number of \(\sigma\) and \(\pi\) bonds in X, Y are respectively:

• (1) X = 2, 0; Y = 1, 2
• (2) X = 1, 2; Y = 2, 2
• (3) X = 2, 1; Y = 1, 1
• (4) X = 1, 2; Y = 3, 3

Eutrophication can lead to:

• (1) Decrease in nutrients
• (2) Increase in dissolved salts
• (3) Decrease in dissolved oxygen
• (4) Decrease in water pollution

In which of the following options, the IUPAC name is not correctly matched with the structure of the compound?

• (1) 3,4-Dimethylphenol
• (2) 4-Chloro-1,3-dinitrobenzene
• (3) 2-Chloro-1-methyl-4-nitrobenzene
• (4) 4-Ethyl-2-methylaniline

Arrange the above carbocations in the order of decreasing stability:

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

Consider the following reaction sequence. What are A and B?

• (1) \(CH_3CH=O, CH_3CH=O\)
• (2) \((CH_3)_2C=O, CH_2=O\)
• (3) \((CH_3)_2C=O, CH_3CH=O\)
• (4) \(CH_3CH=O, CH_2=O\)

Identify the end product (Z) in the sequence of the following reactions:

• (1)
• (2)
• (3)
• (4)

Correct Answer: (2)
View Solution

In bcc lattice containing X and Y type of atoms, X type of atoms are present at the corners and Y type of atoms are present at the centers. In its unit cell, if three atoms are missing in the corners, the formula of the compound is:

• (1) \( X_5Y_8 \)
• (2) \( X_8Y_5 \)
• (3) \( X_3Y_5 \)
• (4) \( X_5Y_3 \)

At 300 K, the vapour pressure of toluene and benzene are 3.63 kPa and 9.7 kPa, respectively. What is the composition of vapour in equilibrium with the solution containing 0.4 mole fraction of toluene?

• (1) 0.40
• (2) 0.60
• (3) 0.80
• (4) 0.20

0.592 g of copper is deposited in 60 minutes by passing 0.5 amperes current through a solution of copper (II) sulphate. The electro chemical equivalent of copper (II) in g/C is:

• (1) \(3.3 \times 10^{-3}\)
• (2) \(3.3 \times 10^{-4}\)
• (3) \(6.6 \times 10^{-3}\)
• (4) \(6.6 \times 10^{-4}\)

For the gaseous reaction \( N_2O_5 \rightarrow 2NO_2 + \frac{1}{2}O_2 \), the rate can be expressed as follows:



The correct relation between \( K_1, K_2 \) and \( K_3 \) is

• (1) \( k_1 = 2k_2 = 4k_3 \)
• (2) \( 2k_1 = k_2 = 4k_3 \)
• (3) \( 2k_1 = 3k_2 = 4k_3 \)
• (4) \( 4k_1 = 2k_2 = k_3 \)

Match the following industrial processes with the catalysts used:

• (1) Ostwald's process – \( CuCl_2 \)
• (2) Haber's process – Zeolites
• (3) Deacon's process – Pt gauge
• (4) Cracking of hydrocarbons – Fe

Copper matte is a mixture of:

• (1) Oxides of \(Cu\) and \(Fe\)
• (2) Carbonates of \(Cu\) and \(Fe\)
• (3) Sulphides of \(Cu\) and \(Fe\)
• (4) Silicates of \(Cu\) and \(Fe\)

In the following reaction: \[ C + Conc. \, H_2SO_4 \xrightarrow{\Delta} X + Y + H_2O \]
X and Y in the above reaction are:

• (1) \(CO, \, SO_3\)
• (2) \(CO_2, \, SO_2\)
• (3) \(CO, \, SO_2\)
• (4) \(C_3O_2, \, SO_2\)

Which among the following oxoacids of phosphorus will have P-O-P bonds?

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

The bond angles H-O-N and O-N-O in the planar structure of nitric acid molecule are respectively:

• (1) 130°, 102°
• (2) 102°, 130°
• (3) 134°, 100°
• (4) 100°, 134°

Observe the following f-block elements:

Eu (Z = 63); Pu (Z = 94); Cf (Z = 98); Sm (Z = 62); Gd (Z = 64); Cm (Z = 96)

How many of the above have half-filled f-orbitals in their ground state?

• (1) 3
• (2) 4
• (3) 2
• (4) 5

Which one of the following complex ions has geometrical isomers?

• (1) \(\left[Co(Cl)_2(en)_2\right]^+\)
• (2) \(\left[Cr(NH_3)_4(en)\right]^{3+}\)
• (3) \(\left[Co(en)_3\right]^{3+}\)
• (4) \(\left[Ni(NH_3)_5\right]Br\)

Which one of the following is not an example of a condensation polymer?

• (1) Terylene
• (2) Nylon 6,6
• (3) Bakelite
• (4) Polystyrene

What is the IUPAC name of the product Y in the given reaction sequence?

• (1) 2,3,4,5,6,7 - hexahydroxyheptanoic acid
• (2) 2,3,4,5,6,7 - pentahydroxyhexanoic acid
• (3) 3,4,5,6 - trihydroxyheptanoic acid
• (4) 3,4,5 - trihydroxyhexanoic acid

What is the value of 'n' in 'Z' of the following sequence?

• (1) 10
• (2) 12
• (3) 16
• (4) 14

The organic halide, which does not undergo hydrolysis by SN1 mechanism is:

• (1) \(C_6H_5CH_2Cl\)
• (2) \(CH_2CH - CH_2Cl\)
• (3) \((CH_3)_3 C - Cl\)
• (4) \(CH_3 - CH = CH - Cl\)

What is 'Z' in the given sequence of reactions?

• (1)
• (2)
• (3)
• (4) C

What is the percentage of carbon in the product 'Z' formed in the reaction?

• (1) 40
• (2) 50
• (3) 70
• (4) 60

Match the following reactions with their corresponding products:

What are Y and Z respectively in the given reaction sequence?

• (1)
• (2)
• (3)
• (4) C

What is 'C' in the given sequence of reactions?

• (1)
• (2)
• (3)
• (4) C

Correct Answer: (2)
View Solution