AP EAPCET (AP EAMCET) 2024 Question Paper May 20 Shift 2 (Available): Download MPC Question Paper with Answer Key PDF

AP EAPCET 2024 Question Paper with Solution
 

SECTION-A
Mathematics

Question 1:

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be defined by \( f(x + y) = f(x) + 12y \), for all \( x, y \in \mathbb{R} \). If \( f(1) = 6 \), then the value of \( \sum_{r=1}^n f(r) \) is:

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

The domain of the real valued function \( f(x) = \sqrt{2 + x} + \sqrt{3 - x} \) is:

• (A) \( (-2, 3) \)
• (B) \( [-2, 3] \)
• (C) \( (-2, 3] \)
• (D) \( [-2, 3] \)

If \(2.4^{2n+1} + 3^{3n+1}\) is divisible by \(k\) for all \(n \in \mathbb{N}\), then \(k\) is:

• (A) \(209\)
• (B) \(11\)
• (C) \(8\)
• (D) \(3\)

The determinant of the matrix \[ \begin{bmatrix} a & b & c
a^2 & b^2 & c^2
1 & 1 & 1 \end{bmatrix} \]
is not equal to:

• (A) \[ \begin{bmatrix} a + 1 & b + 1 & c + 1
  a^2 + 1 & b^2 + 1 & c^2 + 1
  1 & 1 & 1 \end{bmatrix} \]
• (B) \[ \begin{bmatrix} a - b & b - c & c
  a^2 - b^2 & b^2 - c^2 & c^2
  0 & 0 & 1 \end{bmatrix} \]
• (C) \[ \begin{bmatrix} a(a + 1) & b(b + 1) & c(c + 1)
  a + 1 & b + 1 & c + 1
  -1 & -1 & -1 \end{bmatrix} \]
• (D) \[ \begin{bmatrix} a + b & b + c & c + a
  a^2 + b^2 & b^2 + c^2 & c^2 + a^2
  2 & 2 & 2 \end{bmatrix} \]

Let \( A, B, C, D, \) and \( E \) be \( n \times n \) matrices, each with non-zero determinant. If \( ABCDE = I \), then \( C^{-1} \) is:

• (A) \( E^{-1}D^{-1}B^{-1}A^{-1} \)
• (B) \( DEAB \)
• (C) \( A^{-1}B^{-1}D^{-1}E^{-1} \)
• (D) \( ABDE \)

If \( A = [a_{ij}]\) where \( 1 \leq i, j \leq n \) with \( n \geq 2 \) and \( a_{ij} = i + j \) is a matrix, then the rank of \( A \) is:

• (A) 0
• (B) 1
• (C) 2
• (D) 4

If \( z_1 = 10 + 6i \), \( z_2 = 4 + 6i \) and \( z \) is any complex number such that the argument of \( \frac{z-z_1}{z-z_2} \) is \( \frac{\pi}{4} \), then the value of \( |z - 7 - 9i| \) is:

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

If \( \frac{3 - 2i \sin \theta}{1 + 2i \sin \theta}\) is a purely imaginary number, then \( \theta \) is:

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

If \( z = x+iy \), \( x^2+y^2 = 1 \) and \( z_1 = e^{i\theta} \), then the expression \( \frac{z_1^{2n-1} - 1}{z_1^{2^n-1} + 1} \) simplifies to:

• (A) \( -i \tan \left( n (\theta + \tan^{-1}(y/x)) \right) \)
• (B) \( i \cot \left( n (\theta + \tan^{-1}(x/y)) \right) \)
• (C) None of these
• (D) \( i \tan \left( n (\theta + \tan^{-1}(y/x)) \right) \)

Let \([r]\) denote the largest integer not exceeding \(r\) and the roots of the equation \[ 3z^2 + 6z + 5 + \alpha(x^2 + 2x + 2) = 0 \]
are complex numbers whenever \( \alpha > L \) and \( \alpha < M \). If \( (L-M) \) is minimum, then the greatest value of \([r]\) such that \( Ly^2 + My + r < 0 \) for all \( y \in \mathbb{R} \) is:

• (A) \( L \)
• (B) \( M \)
• (C) \( L + M \)
• (D) \( M - L \)

For any real value of \(x\), if \( \frac{11x^{2}+12x+6}{x^{2}+4x+2} \not\in (a, b) \), then the value for \(x\) for which \[ \frac{11x^{2}+12x+6}{x^{2}+4x+2} = b - a + 3 \]
is:

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

If the roots of \[ \sqrt{\frac{1-y}{y}} + \sqrt{\frac{y}{1-y}} = \frac{5}{2} \]
are \( \alpha \) and \( \beta \) (\( \beta > \alpha \)) and the equation \[ (\alpha + \beta)x^4 - 25\alpha \beta x^2 + (\gamma + \beta - \alpha) = 0 \]
has real roots, then a possible value of \( y \) is:

• (A) \( \frac{1}{2} \)
• (B) \( 4 \)
• (C) \( 2\pi \)
• (D) \( \sqrt{e + 13} \)

If the roots of the equation \(x^3 + ax^2 + bx + c = 0\) are in arithmetic progression, then:

• (A) \(a^3 - 3ab + c = 0\)
• (B) \(9ab = 2a^3 + 27c\)
• (C) \(a^2 - 2bc + c = 0\)
• (D) \(3ab - 3c - a^3 = 0\)

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct answers. If every student has attempted all the questions, then the maximum possible number of students who have written the test is:

• (A) \(80\)
• (B) \(63\)
• (C) \(15\)
• (D) \(11\)

The number of numbers lying between 1000 and 10000 such that every number contains the digits 3 and 7 only once without repetition is:

• (A) \( 1140 \)
• (B) \( 918 \)
• (C) \( 720 \)

The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is:

• (A) \( 1140 \)
• (B) \( 336 \)
• (C) \( 36 \)
• (D) \( 56 \)

If the coefficients of \(x^5\) and \(x^6\) are equal in the expansion of \( \left( a + \frac{x}{5} \right)^{65} \), then the coefficient of \(x^2\) in the expansion of \( \left( a + \frac{x}{5} \right)^4 \) is:

• (A) \( 1 \)
• (B) \( \frac{32}{25} \)
• (C) \( 2 \)

If \( |x| < \frac{2}{3} \), then the fourth term in the expansion of \( (3x - 2)^{2/3} \) is:

• (A) \( \frac{\sqrt[3]{4}}{6} x^3 \)
• (B) \( -\frac{\sqrt[3]{4}}{6} x^3 \)
• (C) \( \frac{\sqrt[3]{4}}{8} x^3 \)
• (D) \( -\frac{\sqrt[3]{4}}{8} x^3 \)

If \[ \frac{x^2+3}{x^4+2x^2+9} = \frac{Ax+B}{x^2+ax+b} + \frac{Cx+D}{x^2+cx+d} \]
then \( aA + bB + cC + dD = \)

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

If \(\sec \theta + \tan \theta = \frac{1}{3}\), then the quadrant in which \(2\theta\) lies is:

• (A) 1st quadrant
• (B) 2nd quadrant
• (C) 3rd quadrant
• (D) 4th quadrant

If \(540^\circ < A < 630^\circ\) and \(|\cos A| = \frac{5}{13}\), then \( \tan\frac{A}{2} \tan A =\)

• (A) \( \frac{18}{5} \)
• (B) \( -\frac{8}{5} \)
• (C) \( \frac{8}{5} \)
• (D) \( -\frac{18}{5} \)

If \((\alpha + \beta)\) is not a multiple of \(\frac{\pi}{2}\) and \(3 \sin(\alpha - \beta) = 5 \cos(\alpha + \beta)\), then \[ \tan\left(\frac{\pi}{4} + \alpha\right) + 4\tan\left(\frac{\pi}{4} + \beta\right) = \]

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

The general solution of the equation \( \sin^2 \theta + 3 \cos^2 \theta = 5 \sin \theta \) is:

• (A) \( n\pi + \frac{\pi}{3}, n \in \mathbb{Z} \)
• (B) \( n\pi + (-1)^n \frac{\pi}{6}, n \in \mathbb{Z} \)
• (C) \( n\pi \pm \frac{\pi}{6}, n \in \mathbb{Z} \)
• (D) \( n\pi + (-1)^n \frac{\pi}{3}, n \in \mathbb{Z} \)

If \( \cos^{-1}(2x) + \cos^{-1}(3x) = \frac{\pi}{3} \) and \(4x^2 = \frac{a}{b}\), then \(a + b =\)

• (A) \( 12 \)
• (B) \( 11 \)
• (C) \( 31 \)
• (D) \( 10 \)

If \( \theta = \sec^{-1}(\cosh u) \), then \( u = \)

• (A) \( \log_e \left( \cot \left( \frac{\theta}{2} - \frac{\pi}{4} \right) \right) \)
• (B) \( \log_e \left( \tan \left( \frac{\theta}{2} - \frac{\pi}{4} \right) \right) \)
• (C) \( \log_e \left( \tan \left( \frac{\pi}{4} - \frac{\theta}{2} \right) \right) \)
• (D) \( \log_e \left( \tan \left( \frac{\pi}{4} - \frac{\theta}{2} \right) \right) \)

In \(\triangle ABC\), if \(4r_1 = 5r_2 = 6r_3\), then \(\sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} =\)

• (A) \( \frac{19}{22} \)
• (B) \( \frac{25}{33} \)
• (C) \( \frac{74}{99} \)
• (D) \( \frac{28}{33} \)

In \(\triangle ABC\), \(r r_1 \cot^ \frac{A}{2} + r r_2 \cot^ \frac{B}{2} + r r_3 \cot^ \frac{C}{2} = \)

• (A) \( 3\Delta \)
• (B) \( 3S \)
• (C) \( \frac{S}{\Delta} \)
• (D) \( \Delta \)

In \(\triangle ABC\), \(bc - r_2 r_3 = \)

• (A) \( rr_1 \)
• (B) \( rr_2 \)
• (C) \( r_1 \)
• (D) \( a r_1 \)

The angle between the diagonals of the parallelogram whose adjacent sides are \(2i + 4j - 5k\) and \(i + 2j + 3k\) is

• (A) \( \cos^{-1} \left(\frac{7}{69}\right) \)
• (B) \( \cos^{-1} \left(\frac{1}{\sqrt{69}}\right) \)
• (C) \( \cos^{-1} \left(\frac{1}{7}\right) \)
• (D) \( \cos^{-1} \left(\frac{31}{7\sqrt{69}}\right) \)

If the points having the position vectors \( \mathbf{r}_1 = -i + 4j - 4k\), \( \mathbf{r}_2 = 3i + 2j - 5k\), \( \mathbf{r}_3 = -3i + 8j - 5k\) and \( -3i + 2j + \lambda k\) are coplanar, then \(\lambda = \)

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

If \( |\mathbf{\bar{f}}| = 10\), \( |\mathbf{\bar{g}}| = 14\) and \( |\mathbf{\bar{f}} - \mathbf{\bar{g}}| = 15\), then \( |\mathbf{\bar{f}} + \mathbf{\bar{g}}| =\)

• (1) \( 367 \)
• (2) \( \sqrt{367} \)
• (3) \( 400 \)
• (4) \( 20 \)

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three vectors such that \( |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = \sqrt{3} \) and \( ( \mathbf{a} + \mathbf{b} - \mathbf{c} )^2 + ( \mathbf{b} + \mathbf{c} - \mathbf{a} )^2 + ( \mathbf{c} + \mathbf{a} - \mathbf{b} )^2 = 36\), then \(|2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2 =\)

• (A) \( 15 \)
• (B) \( 25 \)
• (C) \( 147 \)
• (D) \( 75 \)

The angle between the line with the direction ratios \( (2, 5, 1) \) and the plane \( 8x + 2y - z = 4 \) is given by

• (A) \( \cos^{-1} \left(\frac{64}{\sqrt{9804}}\right) \)
• (B) \( \sin^{-1} \left(\frac{64}{\sqrt{9804}}\right) \)
• (C) \( \sin^{-1} \left(\frac{25}{\sqrt{2070}}\right) \)
• (D) \( \cos^{-1} \left(\frac{25}{\sqrt{2070}}\right) \)

If the mean deviation about the mean is \(m\) and the variance is \(\sigma^2\) for the following data, then \(m + \sigma^2 =\)

% Data table \[ \begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9
\hline f & 4 & 24 & 28 & 16 & 8
\hline \end{array} \]

• (A) \( 8 \)
• (B) \( 7.2 \)
• (C) \( \frac{28}{5} \)
• (D) \( 6 \)

If five-digit numbers are formed from the digits 0, 1, 2, 3, 4 using every digit exactly only once, then the probability that a randomly chosen number from those numbers is divisible by 4 is

• (A) \( \frac{5}{16} \)
• (B) \( \frac{3}{16} \)
• (C) \( \frac{3}{8} \)
• (D) \( \frac{7}{16} \)

Two natural numbers are chosen at random from 1 to 100 and are multiplied. If \(A\) is the event that the product is an even number and \(B\) is the event that the product is divisible by 4, then \(P(A \cap \bar B) = \)

• (A) \( \frac{25}{198} \)
• (B) \( \frac{49}{198} \)
• (C) \( \frac{25}{99} \)
• (D) \( \frac{50}{99} \)

A box \( P \) contains one white ball, three red balls and two black balls. Another box \( Q \) contains two white balls, three red balls and four black balls. If one ball is drawn at random from each one of the two boxes, then the probability that the balls drawn are of different color is

• (A) \( \frac{29}{54} \)
• (B) \( \frac{25}{42} \)
• (C) \( \frac{35}{54} \)
• (D) \( \frac{39}{52} \)

A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is

• (A) \( \frac{1}{37} \)
• (B) \( \frac{1}{5} \)
• (C) \( \frac{12}{37} \)
• (D) \( \frac{25}{37} \)

For a binomial variate \(X \sim B(n, p)\), the difference between the mean and variance is 1 and the difference between their squares is 11. If the probability of \(P(X = 2) = m\left(\frac{5}{6}\right)^n\) and \(n = 36\) then \(m:n\) is

• (A) \(6:5\)
• (B) \(7:10\)
• (C) \(36:1\)
• (D) \(42:25\)

The probability that a man failing to hit a target is \(\frac{1}{3}\). If he fires 4 times, then the probability that he hits the target at least thrice is

• (A) \(\frac{16}{27}\)
• (B) \(\frac{11}{27}\)
• (C) \(\frac{8}{81}\)
• (D) \(\frac{32}{81}\)

Let \( A(2, 3), B(1, -1) \) be two points. If \( P \) is a variable point such that \( \angle APB = 90^\circ \), then the locus of \( P \) is

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

If the origin is shifted to remove the first degree terms from the equation \(2x^2 - 3y^2 + 4xy + 4x + 4y - 14 = 0\), then with respect to this new co-ordinate system, the transformed equation of \(x^2 + y^2 - 3xy + 4y + 3 = 0\) is

• (A) \(x^2 + y^2 - 3xy - 2x + y + 6 = 0\)
• (B) \(x^2 + y^2 - 3xy - 2x + 7y + 3 = 0\)
• (C) \(x^2 + y^2 - 3xy - 2x + y + 4 = 0\)
• (D) \(x^2 + y^2 - 3xy - 2x + 7y + 4 = 0\)

The circumcentre of the triangle formed by the lines \(x + y + 2 = 0\), \(2x + y + 8 = 0\) and \(x - y - 2 = 0\) is

• (A) \((-5, 1)\)
• (B) \((-4, 0)\)
• (C) \((0, -2)\)
• (D) \(\left(-\frac{8}{3}, -\frac{2}{3}\right)\)

If the line \(2x - 3y + 5 = 0\) is the perpendicular bisector of the line segment joining \(1, -2\) and \((a, b)\), then \(a + b =\)

• (A) \(7\)
• (B) \(1\)
• (C) \(-1\)
• (D) \(-7\)

If the area of the triangle formed by the straight lines \(-15x^2 + 4xy + 4y^2 = 0\) and \(x = a\) is \(200 \, sq. units\), then \(|a| =\)

• (A) \(10\)
• (B) \(20\)
• (C) \(5\sqrt{2}\)
• (D) \(40\)

The equation of the straight line passing through the point of intersection of the lines represented by \(x^2 + 4xy + 3y^2 - 4x - 10y + 3 = 0\) and the point \((2,2)\) is:

• (A) \(2x + 3y - 10 = 0\)
• (B) \(3x + 2y - 10 = 0\)
• (C) \(2x + y - 6 = 0\)
• (D) \(x + 2y - 6 = 0\)

The largest among the distances from the point \(P(15,9)\) to the points on the circle \(x^2 + y^2 - 6x - 8y - 11 = 0\) is:

• (A) \(12\)
• (B) \(13\)
• (C) \(19\)
• (D) \(7\)

The circle \(x^2 + y^2 - 8x - 12y + \alpha = 0\) lies in the first quadrant without touching the coordinate axes. If \((6, 6)\) is an interior point to the circle, then the range of \(\alpha\) is:

• (A) \(4 < \alpha < 6\)
• (B) \(6 < \alpha < 16\)
• (C) \(16 < \alpha < 48\)
• (D) \(36 < \alpha < 48\)

The equation of the circle whose diameter is the common chord of the circles \(x^2 + y^2 - 6x - 7 = 0\) and \(x^2 + y^2 - 10x + 16 = 0\) is:

• (A) \(8x^2 + 8y^2 - 92x + 197 = 0\)
• (B) \(x^2 + y^2 - 23x + 197 = 0\)
• (C) \(x^2 + y^2 - \frac{23}{2}x + \frac{197}{4} = 0\)
• (D) \(4x^2 + 4y^2 - 46x + 197 = 0\)

If the locus of the mid points of the chords of the circle \(x^2 + y^2 = 25\) that subtend a right angle at the origin is given by \( \frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\), then \(|a| =\)

• (A) \( \frac{2}{5} \)
• (B) \( \frac{5}{\sqrt{2}} \)
• (C) \( \frac{2}{25} \)
• (D) \( 5\sqrt{2} \)

The radical center of the circles \(x^2 + y^2 + 2x + 3y + 1 = 0\), \(x^2 + y^2 - x + y + 3 = 0\), and \(x^2 + y^2 - 3x + 2y + 5 = 0\) is:

• (A) \( \left(-\frac{7}{38}, \frac{6}{19}\right) \)
• (B) \( \left(\frac{6}{19}, \frac{14}{19}\right) \)
• (C) \( \left(\frac{14}{19}, \frac{6}{19}\right) \)
• (D) \( \left(\frac{2}{19}, \frac{3}{19}\right) \)

Equation of a tangent line of the parabola \(y^2 = 8x\), which passes through the point \((1, 3)\) is:

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

If the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) having \((1,1)\) as its middle point is \( x + \alpha y = \beta \), then:

• (A) \( \alpha + \beta = 1 \)
• (B) \( \alpha + 1 = \beta \)
• (C) \( \alpha - 1 = \beta \)
• (D) \( 2\alpha - 1 = 3\beta \)

If a directrix of a hyperbola centered at the origin and passing through the point \( (4, -2\sqrt{3}) \) is \( \sqrt{5}x = 4 \) and \( e \) is its eccentricity, then \( e^2 = \)

• (A) \( \frac{\sqrt{7}}{2} \)
• (B) \( \frac{7}{2} \)
• (C) \( \frac{35}{4} \)
• (D) \( 2\sqrt{3} \)

If \( l_1 \) and \( l_2 \) are the lengths of the perpendiculars drawn from a point on the hyperbola \( 5x^2 - 4y^2 - 20 = 0 \) to its asymptotes, then \( \frac{l_1^2 l_2^2}{100} = \)

• (A) \( \frac{20}{9} \)
• (B) \( \frac{16}{81} \)
• (C) \( \frac{4}{81} \)
• (D) \( \frac{2}{9} \)

If \( O(0, 0, 0), A(3, 0, 0), B(0, 4, 0) \) form a triangle then the incenter of triangle OAB is:

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

The direction cosines of the line of intersection of the planes \( x + 2y + z - 4 = 0 \) and \( 2x - y + z - 3 = 0 \) are:

• (A) \( \left( \frac{3}{\sqrt{26}}, \frac{1}{\sqrt{26}}, \frac{-4}{\sqrt{26}} \right) \)
• (B) \( \left( \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}} \right) \)
• (C) \( \left( \frac{3}{\sqrt{35}}, \frac{1}{\sqrt{35}}, \frac{-5}{\sqrt{35}} \right) \)
• (D) \( \left( \frac{3}{\sqrt{22}}, \frac{-2}{\sqrt{22}}, \frac{3}{\sqrt{22}} \right) \)

If \( L_1 \) and \( L_2 \) are two lines which pass through origin and have direction ratios \( (3, 1, -5) \) and \( (2, 3, -1) \) respectively, then the equation of the plane containing \( L_1 \) and \( L_2 \) is:

• (A) \( 4x + 5y - 63 = 0 \)
• (B) \( 5x - y + 3z = 0 \)
• (C) \( 2x - y + z = 0 \)
• (D) \( x - 5y + 3z = 0 \)

Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{4\sqrt{2} - (\cos x + \sin x)^5}{1 - \sin 2x} = \]

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

Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x} - a - \log(1+x)}{\sin x} = 0, then find a. \]

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

Determine the values of \(a\) and \(b\) for which the function \(f(x)\) defined as: \[ f(x) = \begin{cases} 1 + |\sin x|^{(a/|\sin x|)} & if \frac{-\pi}{6} < x < 0,
b & if x = 0,
e^{\left(\frac{\tan 2x}{\tan 3x}\right)} & if 0 < x < \frac{\pi}{6} \end{cases} \]
is continuous at \(x = 0\).

• (A) \( a = 1, b = \frac{2}{3} \)
• (B) \( a = \frac{2}{3}, b = e^{\frac{2}{3}} \)
• (C) \( a = \frac{2}{3}, b = \frac{3}{2} \)
• (D) \( a = -1, b = e^{\frac{2}{3}} \)

If \(f(x) = \begin{cases} 2x+3, & x \leq 1
2ax + bx, & x > 1 \end{cases}\) is differentiable \(\forall x \in \mathbb{R}\), then \(f(2) =\)

• (A) \(5\)
• (B) \(4\)
• (C) \(-4\)
• (D) \(-10\)

If \(y = t^2 + t^3\) and \(x = t - t^4\), then \(\frac{d^2y}{dx^2}\) at \(t = 1\) is:

• (A) \(-\frac{2}{3}\)
• (B) \(-\frac{4}{3}\)
• (C) \(\frac{8}{3}\)
• (D) \(4\)

In the interval \([0, 3]\), the function \(f(x) = |x - 1| + |x - 2|\) is:

• (A) Discontinuous
• (B) Differentiable
• (C) Continuous but not differentiable at \(x = 2\) only
• (D) Continuous but not differentiable at \(x = 1\) and \(x = 2\)

If \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) respectively. If \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), then \( k_1 + k_2 = \)

• (A) \(7\)
• (B) \(6\)
• (C) \(5\)
• (D) \(4\)

The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:

• (A) \(1\)
• (B) \(\frac{1}{3}\)
• (C) \(\frac{1}{2}\)
• (D) \(0\)

The acute angle between the curves \(x^2 + y^2 = x + y\) and \(x^2 + y^2 = 2y\) is:

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

A value of \(c\) according to the Lagrange's mean value theorem for \(f(x) = (x - 1)(x - 2)(x - 3)\) in \([0,4]\) is:

• (A) \(2 + \frac{2}{\sqrt{3}}\)
• (B) \(2 - \frac{\sqrt{16}}{\sqrt{3}}\)
• (C) \(1 + \frac{\sqrt{5}}{4}\)
• (D) \(2 + \frac{\sqrt{8}}{3}\)

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

• (A) \(\log\left(\frac{x}{x^4 + 1}\right) + C\)
• (B) \(\frac{3}{4} \log(x^4 + 1) + C\)
• (C) \(\frac{1}{3} \log\left(\frac{x^3}{x^4 + 1}\right) + C\)
• (D) \(\frac{1}{4} \log\left(\frac{x^4}{x^4 + 1}\right) + C\)

Evaluate the integral \( \int \frac{dx}{\sqrt{\sin^3 x \cdot \cos (x - \alpha)}} \):

• (A) \( \frac{1}{\cos \alpha} \sqrt{\cot x + \tan \alpha} + C \)
• (B) \( \frac{1}{\cos \alpha} \sqrt{\cot x - \tan \alpha} + C \)
• (C) \( -\frac{1}{\sin \alpha} \sqrt{\cot x + \tan \alpha} + C \)
• (D) \( -\frac{2}{\cos \alpha} \sqrt{\cot x + \tan \alpha} + C \)

Evaluate the integral \( \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx \):

• (A) \(\frac{4}{7} (e^x + 1)^{1/4}(3e^x - 1) + C\)
• (B) \(\frac{2}{21} (e^x + 1)^{3/4}(3e^x - 7) + C\)
• (C) \(\frac{4}{21} (e^x + 1)^{3/4}(3e^x - 4) + C\)
• (D) \(\frac{8}{21} (e^x + 1)^{3/4}(3e^x - 1) + C\)

Evaluate the integral \( \int \frac{2 - \sin x}{2 \cos x + 3} \, dx \):

• (A) \(\frac{2}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right) - \log \sqrt{2 \cos x + 3} + C\)
• (B) \(\frac{4}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \sqrt{2 \cos x + 3} + C\)
• (C) \(\frac{3}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \sqrt{2} \cos x - 3 + C\)
• (D) \(\frac{1}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) - \log \sqrt{2 \cos x - 3} + C\)

Evaluate the integral \( \int \frac{\sin^{-1} \left(\frac{x}{\sqrt{a + x}}\right)}{\sqrt{a + x}} \, dx \):

• (A) \((a + x) \tan^{-1}\left(\frac{x}{\sqrt{a}}\right) - \sqrt{ax} + C\)
• (B) \(\frac{1}{a + x} \tan^{-1}\left(\frac{x}{\sqrt{a}}\right) - \sqrt{ax} + C\)
• (C) \((a + x) \tan^{-1}\left(\frac{a}{x}\right) + \sqrt{ax} + C\)
• (D) \(\sqrt{a} + x \tan^{-1}\left(\frac{x}{\sqrt{a}}\right) + ax + C\)

Evaluate the integral \( \int_{-\frac{1}{24}}^{\frac{1}{24}} \sec(x) \log\left(\frac{1-x}{1+x}\right) \, dx \):

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

If \([x]\) is the greatest integer function, then evaluate the integral \( \int_{0}^{5} [x] \, dx \):

• (A) 15
• (B) 2
• (C) 3
• (D) 10

Evaluate the integral \( \int_0^{\frac{\pi}{2}} \frac{1}{1 + \sqrt{\tan x}} \, dx \):

• (A) \(0\)
• (B) \(\frac{\pi}{2}\)
• (C) \(\frac{\pi}{3}\)
• (D) \(\frac{\pi}{4}\)

Evaluate the integral \( \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \):

• (A) \(0\)
• (B) \(\frac{\pi}{2}\)
• (C) \(\frac{\pi^2}{2}\)
• (D) \(\frac{\pi^2}{4}\)

Determine the order and degree of the differential equation \( \frac{d^3y}{dx^3} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} \):

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

Find the integrating factor of the differential equation \( \sin x \frac{dy}{dx} - y \cos x = 1 \):

• (A) \(\sin x\)
• (B) \(\cos x\)
• (C) \(\sec x\)
• (D) \(\csc x\)

The general solution of the differential equation \( (x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx \) is:

• (A) \(\cos \frac{x}{y} = \log_ex + c\)
• (B) \(\cos \frac{y}{x} = \log_ex + c\)
• (C) \(\cos \frac{x}{y} = \log_ey + c\)
• (D) \(\cos \frac{y}{x} = \log_ey + c\)

Find the dimension formula of \(\frac{a}{b}\) in the equation \( F = a\sqrt{x} + bt^2 \), where \(F\) is a force, \(x\) is distance and \(t\) is time.

• (A) \([M^0L^{-1/2}T^2]\)
• (B) \([M^0L^0T^{3/2}]\)
• (C) \([M^0L^1T^{-4}]\)
• (D) \([M^{0}L^{3/2}T^{4}]\)

The relation between time \( t \) and displacement \( x \) is \( t = \alpha x^2 + \beta x \), where \(\alpha\) and \(\beta\) are constants. If \( \nu \) is the velocity, the retardation is:

• (A) \( 2\alpha \nu^3 \beta^2 \)
• (B) \( 2\alpha \beta \nu^3 \)
• (C) \( -2\beta \nu^3 \)
• (D) \( 2\alpha \nu^3 \)

If two stones are projected at angle \( \theta \) and \( (90^\circ - \theta) \) respectively with horizontal with a speed of \( 20 \, m/s \). If the second stone rises \( 10 \, m \) higher than the first stone, then the angle of projection \( \theta \) is (acceleration due to gravity = \( 10 \, m/s^2 \)):

• (A) \( 45^\circ \)
• (B) \( 30^\circ \)
• (C) \( 60^\circ \)
• (D) \( 20^\circ \)

A particle revolving in a circular path travels the first half of the circumference in 4 s and the next half in 2 s. What is its average angular velocity?

• (A) \( \frac{4\pi}{9} \, rad/s \)
• (B) \( \frac{\pi}{6} \, rad/s \)
• (C) \( \frac{2\pi}{3} \, rad/s \)
• (D) \( \frac{\pi}{3} \, rad/s \)

A block of metal 2 kg is in rest on a smooth plane. It is struck by a jet releasing water of 1 kg s\(^{-1}\) at a speed of 5 m s\(^{-1}\), then the acceleration of the block is

• (A) \(2 \, ms^{-2}\)
• (B) \(2.5 \, ms^{-2}\)
• (C) \(0.25 \, ms^{-2}\)
• (D) \(50 \, ms^{-2}\)

An insect is crawling in a hemi-spherical bowl of radius \( R \). If the coefficient of friction between the insect and bowl is \( \mu \), then the maximum height to which the insect can crawl the bowl is

• (A) \( R \left[1 - \frac{1}{\sqrt{1 + \mu^2}}\right] \)
• (B) \( R \left[1 + \frac{1}{\sqrt{1 + \mu^2}}\right] \)
• (C) \( R \left[\frac{1}{\sqrt{1 + \mu^2}}\right] \)
• (D) \( R \left[\sqrt{1 - \mu^2}\right] \)

Two objects having masses in \(1:4\) ratio are at rest. When both of them are subjected to the same force separately, they achieved the same kinetic energy during times \(t_1\) and \(t_2\) respectively. The ratio of \(\frac{t_2}{t_1}\) is

• (A) \(4\)
• (B) \(2\)
• (C) \(2.5\)
• (D) \(1\)

An object of mass 'm' is projected with an initial velocity 'u' with an angle of '\(\theta\)' with the horizontal. The average power delivered by gravity in reaching the highest point is

• (A) \(\frac{mgu \sin^2 \theta}{2}\)
• (B) \(\frac{mu^2 \sin^2 \theta}{2g}\)
• (C) \(\frac{mg \sin \theta}{2u}\)
• (D) \(\frac{mgu \sin \theta}{2}\)

A small disc is on the top of a smooth hemisphere of radius 'R'. The smallest horizontal velocity 'V' that should be imparted to the disc so that the disc leaves the hemisphere surface without sliding down (there is no friction) is

• (A) \(V = \sqrt{g2R}\)
• (B) \(V = \sqrt{2gR}\)
• (C) \(V = \sqrt{gR}\)
• (D) \(V = \sqrt{\frac{g}{R}}\)

A block (\(P\)) is rotating in contact with the vertical wall of a rotor as shown in figures A, B, C. The relation between angular velocities \( \omega_A, \omega_B, \omega_C \) so that the block does not slide down. (Given: \( R_A < R_B < R_C \), where \( R \) denotes radius)

• (A) \( \omega_A < \omega_B < \omega_C \)
• (B) \( \omega_A = \omega_B = \omega_C \)
• (C) \( \omega_C < \omega_B < \omega_A \)
• (D) \( \omega_C = \omega_A + \omega_B \)

A horizontal board is performing simple harmonic oscillations horizontally with an amplitude of \(0.3 \, m\) and a period of \(4 \, s\). The minimum coefficient of friction between a heavy body placed on the board if the body is not to slip is:

• (A) \( \mu = 0.05 \)
• (B) \( \mu = 0.075 \)
• (C) \( \mu = 0.173 \)
• (D) \( \mu = 1.14 \)

A test tube of mass \(6 \, g\) and uniform area of cross section \(10 \, cm^2\) is floating in water vertically when \(10 \, g\) of mercury is in the bottom. The tube is depressed by a small amount and then released. The time period of oscillation is: (Acceleration due to gravity = \(10 \, m/s^2\))

• (A) \(0.75 \, s\)
• (B) \(0.5 \, s\)
• (C) \(0.25 \, s\)
• (D) \(0.85 \, s\)

What is the height from the surface of earth, where acceleration due to gravity will be \( \frac{1}{4} \) of that of the earth? (Re = 6400 km)

• (A) \(6400 \, km\)
• (B) \(3200 \, km\)
• (C) \(1600 \, km\)
• (D) \(640 \, km\)

Depth of a river is \(100 \, m\). Magnitude of compressibility of the water is \(0.5 \times 10^{-9} \, N^{-1}m^2\). The fractional compression in water at the bottom of the river is (Acceleration due to gravity = \(10 \, m/s^2\))

• (A) \(0.9 \times 10^3\)
• (B) \(0.5 \times 10^{-3}\)
• (C) \(2 \times 10^{-3}\)
• (D) \(1.3 \times 10^{-2}\)

Two mercury drops, each with same radius \( r \), merged to form a bigger drop. If \( T \) is the surface tension of mercury, then the surface energy of the bigger drop is given by

• (A) \( 2\pi r^2 T \)
• (B) \( \frac{5}{3} \pi r^2 T \)
• (C) \( 2\pi r^2 T^2 \)
• (D) \( \frac{8}{3} \pi r^2 T \)

The absorption coefficient value of a perfect black body is

• (A) zero
• (B) \( < 1 \)
• (C) \( > 1 \)
• (D) \( 1 \)

A certain volume of a gas at 300 K expands adiabatically until its volume is doubled. The resultant fall in temperature of the gas is nearly (The ratio of the specific heats of the gas is 1.5)

• (A) \( 88 \, K \)
• (B) \( 77 \, K \)
• (C) \( 67 \, K \)
• (D) \( 54 \, K \)

The efficiency of a Carnot's engine is 25%, when the temperature of sink is 300 K. The increase in the temperature of source required for the efficiency to become 50% is

• (A) \( 225 \, K \)
• (B) \( 400 \, K \)
• (C) \( 200 \, K \)
• (D) \( 100 \, K \)

When 100 J of heat is supplied to a gas, the increase in the internal energy of the gas is 60 J. Then the gas is/can

• (A) be triatomic or diatomic gas
• (B) Triatomic gas
• (C) Monoatomic gas
• (D) Diatomic gas

An ideal gas is kept in a cylinder of volume \(3 \, m^3\) at a pressure of \(3 \times 10^5 \, Pa\). The energy of the gas is

• (A) \(13.5 \times 10^6 \, J\)
• (B) \(1.35 \times 10^5 \, J\)
• (C) \(13.5 \times 10^5 \, J\)
• (D) \(135 \times 10^6 \, J\)

A pipe with 30 cm Length is open at both ends. Which harmonic mode of the pipe resonates a 1.65 kHz source? (Velocity of sound in air = 330 m/s)

• (A) \(2\)
• (B) \(3\)
• (C) \(3.5\)
• (D) \(2.5\)

An object is placed at a distance of 18 cm in front of a mirror. If the image is formed at a distance of 4 cm on the other side, then the focal length, nature of the mirror and nature of the image are respectively:

• (A) \(3.14 \, cm, concave mirror, and real image\)
• (B) \(3.14 \, cm, convex mirror, and real image\)
• (C) \(5.14 \, cm, convex mirror, and virtual image\)
• (D) \(5.14 \, cm, concave mirror, and virtual image\)

If a microscope is placed in air, the minimum separation of two objects seen as distinct is 6 µm. If the same is placed in a medium of refractive index 1.5, the minimum separation of the two objects to see as distinct is:

• (A) \(4 \, \mu m\)
• (B) \(6 \, \mu m\)
• (C) \(3 \, \mu m\)
• (D) \(9 \, \mu m\)

Three point charges \( +q, +2q, \) and \( +4q \) are placed along a straight line such that the charge \( +2q \) lies equidistant from the other two charges. The ratio of the net electrostatic force on charges \( +q \) and \( +4q \) is:

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

Three parallel plate capacitors of capacitances \(4 \mu F\), \(6 \mu F\), and \(12 \mu F\) are first connected in series and then in parallel. The ratio of the effective capacitances in the two cases is:

• (A) \(1 : 11\)
• (B) \(5 : 8\)
• (C) \(3 : 7\)
• (D) \(4 : 9\)

A particle of mass 2 g and charge 6 \(\mu\)C is accelerated from rest through a potential difference of 60 V. The speed acquired by the particle is:

• (A) \(0.6 \, ms^{-1}\)
• (B) \(1.2 \, ms^{-1}\)
• (C) \(1.8 \, ms^{-1}\)
• (D) \(0.3 \, ms^{-1}\)

A straight wire of resistance \(R\) is bent in the shape of a square. A cell of emf 12 V is connected between two adjacent corners of the square. The potential difference across any diagonal of the square is:

• (A) \(8 \, V\)
• (B) \(18 \, V\)
• (C) \(6 \, V\)
• (D) \(12 \, V\)

In the given circuit, if the potential at point B is 24 V, the potential at point A is:

• (A) \(-4.8 \, V\)
• (B) \(-2.4 \, V\)
• (C) \(-12 \, V\)
• (D) \(-14.4 \, V\)

Two long straight parallel conductors A and B carrying currents 4.5 A and 8 A respectively are separated by 25 cm in air. The resultant magnetic field at a point which is a distance of 15 cm from conductor A and 20 cm from conductor B is:

• (A) \(2 \times 10^{-5} \, N\)
• (B) \(2 \times 10^{-4} \, N\)
• (C) \(10^{-5} \, N\)
• (D) \(10^{-4} \, N\)

Two concentric thin circular rings of radii 50 cm and 40 cm each, carry a current of 3.5 A in opposite directions. If the two rings are coplanar, the net magnetic field due to the two rings at their centre is:

• (A) \(11 \times 10^{-7} \, T\)
• (B) \(22 \times 10^{-7} \, T\)
• (C) \(17 \times 10^{-7} \, T\)
• (D) \(8 \times 10^{-7} \, T\)

At a place where the magnitude of the earth's magnetic field is \(4 \times 10^{-5} \, T\), a short bar magnet is placed with its axis perpendicular to the earth's magnetic field direction. If the resultant magnetic field at a point at a distance of 40 cm from the center of the magnet on the normal bisector of the magnet is inclined at \(45^\circ\) with the earth's field, the magnetic moment of the magnet is:

• (A) \(38.4 \, Am^2\)
• (B) \(51.2 \, Am^2\)
• (C) \(12.8 \, Am^2\)
• (D) \(25.6 \, Am^2\)

The ratio of the number of turns per unit length of two solenoids A and B is \(1:3\) and the lengths of A and B are in the ratio \(1:2\). If the two solenoids have the same cross-sectional area, the ratio of the self-inductances of the solenoids A and B is:

• (A) \(1:12\)
• (B) \(1:6\)
• (C) \(1:18\)
• (D) \(1:9\)

An inductor and a resistor are connected in series to an AC source of voltage \( 144\sin(100\pi t + \frac{\pi}{2}) \) volts. If the current in the circuit is \( 6\sin(100\pi t + \frac{\pi}{2}) \) amperes, then the resistance of the resistor is:

• (A) \( 24 \, \Omega \)
• (B) \( 36 \, \Omega \)
• (C) \( 12 \, \Omega \)
• (D) \( 18 \, \Omega \)

Inner shell electrons in atoms moving from one energy level to another lower energy level produce:

• (A) Gamma rays
• (B) Microwaves
• (C) Radio waves
• (D) Ultraviolet rays

If the kinetic energy of a particle in motion is decreased by 36%, the increase in de Broglie wavelength of the particle is:

• (A) 18%
• (B) 25%
• (C) 20%
• (D) 32%

The speed of the electron in a hydrogen atom in the \( n = 3 \) level is:

• (A) \( 6.2 \times 10^5 \) ms\(^{-1}\)
• (B) \( 3.7 \times 10^5 \) ms\(^{-1}\)
• (C) \( 7.3 \times 10^5 \) ms\(^{-1}\)
• (D) \( 1.6 \times 10^5 \) ms\(^{-1}\)

One mole of radium has an activity of \( \frac{1}{6.3 \times 10^{37}} \) kilo curie. Its decay constant is:

• (A) \( \frac{1}{6} \times 10^{-10} \, s^{-1} \)
• (B) \( 10^{-10} \, s^{-1} \)
• (C) \( 10^{-11} \, s^{-1} \)
• (D) \( 10^{-8} \, s^{-1} \)

The voltage gain and current gain of a transistor amplifier in common emitter configuration are respectively 150 and 50. If the resistance in the base circuit is 850 \(\Omega\), then the resistance in the collector circuit is:

• (A) \( 1700 \, \Omega \)
• (B) \( 2250 \, \Omega \)
• (C) \( 2550 \, \Omega \)
• (D) \( 3000 \, \Omega \)

If the energy gap of a substance is 5.4 eV, then the substance is:

• (A) Insulator
• (B) Conductor
• (C) p-type semiconductor
• (D) n-type semiconductor

In amplitude modulation, the amplitude of the carrier wave is 10 V and the amplitude of one of the side bands is 2 V. The modulation index is:

• (A) 0.4
• (B) 0.6
• (C) 0.7
• (D) 0.5

If uncertainty in position and momentum of an electron are equal, then uncertainty in its velocity is:

• (A) \( \frac{1}{2m} \sqrt{\frac{\hbar}{\pi}} \)
• (B) \( \frac{1}{m} \sqrt{\frac{\hbar}{\pi}} \)
• (C) \( \sqrt{\frac{\hbar}{\pi}} \)
• (D) \( m \sqrt{\frac{\hbar}{\pi}} \)

The graph shown below represents the variation of probability density, \( \Psi(r) \), with distance \( r \) of the electron from the nucleus. This represents:

% Include graphics

• (A) 1s-orbital
• (B) 2s-orbital
• (C) 3s-orbital
• (D) 2p-orbital

Match the following elements with their correct classifications:


\begin{tabular{|c|l|l|
\hline
Element & List I & List II

\hline
A & Technetium & I.Non-metal

B & Fluorine & II.Transition metal

C & Tellurium & III.Lanthanoid

D & Dysprosium & IV.Metalloid

\hline
\end{tabular

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

Observe the following reactions. Identify the reaction in which the hybridisation of the underlined atom is changed:

• (A) & \mathrm{NH}_3 + \mathrm{H}^+ \rightarrow \mathrm{NH}_4^+
• (B) & \mathrm{PCl}_3 + 3\mathrm{H}_2\mathrm{O} \rightarrow \mathrm{H}_3\mathrm{PO}_3 + 3\mathrm{HCl}
• (C) & \mathrm{NaNO}_3 + \mathrm{H}_2\mathrm{SO}_4 \rightarrow \mathrm{NaHSO}_4 + \mathrm{HNO}_3
• (D) & \mathrm{XeF}_6 + \mathrm{H}_2\mathrm{O} \rightarrow \mathrm{XeOF}_4 + 2\mathrm{HF}

Among the following species, correct set of isomolecular pairs are:

\begin{align*
& \mathrm{XeO_3, \mathrm{CO_3^{2-, \mathrm{SO_3, \mathrm{H_3\mathrm{O^+, \mathrm{ClF_3
\end{align*


(1) (\mathrm{XeO_3, \mathrm{CO_3^{2-) & \quad (\mathrm{SO_3, \mathrm{H_3\mathrm{O^+)


(2) (\mathrm{XeO_3, \mathrm{SO_3) & \quad (\mathrm{CO_3^{2-, \mathrm{H_3\mathrm{O^+)


(3) (\mathrm{XeO_3, \mathrm{H_3\mathrm{O^+) & \quad (\mathrm{SO_3, \mathrm{CO_3^{2-)


(4) (\mathrm{SO_3, \mathrm{ClF_3) & \quad (\mathrm{XeO_3, \mathrm{CO_3^{2-)

What is the ratio of kinetic energies of \(3 \, g\) of hydrogen and \(4 \, g\) of oxygen at a certain temperature?


(1) \(3:4\)


(2) \(6:1\)


(3) \(12:1\)


(4) \(1:12\)

What is the kinetic energy (in \( J/mol \)) of one mole of an ideal gas (molar mass = \(0.01 \, kg/mol\)) if its rms velocity is \(4 \times 10^2 \, m/s\)?

• (A) \(2 \times 10^5\)
• (B) \(8 \times 10^4\)
• (C) \(8 \times 10^2\)
• (D) \(8 \times 10^3\)

At STP \(x\) g of a metal hydrogen carbonate (MHCO\(_3\)) (molar mass \(84 \, g/mol\)) on heating gives CO\(_2\), which can completely react with \(0.02 \, moles\) of MOH (molar mass \(40 \, g/mol\)) to give MHCO\(_3\). The value of \(x\) is:

• (A) \(67.2\)
• (B) \(33.6\)
• (C) \(11.2\)
• (D) \(22.4\)

The volume of an ideal gas contracts from 10.0 L to 2.0 L under an applied pressure of 2.0 atm. During contraction, the system also evolved 90 J of heat. The change in internal energy (in J) involved in the system is (1 L·atm = 101.3 J):

• (A) \(720.8\)
• (B) \(360.4\)
• (C) \(1620.8\)
• (D) \(810.4\)

The molar heats of fusion and vaporization of benzene are 10.9 and 31.0 kJ mol\(^{-1}\) respectively. The changes in entropy for the solid \(\rightarrow\) liquid and liquid \(\rightarrow\) vapor transitions for benzene are \(x\) and \(y\) J K\(^{-1}\) mol\(^{-1}\) respectively. The value of \(y(x)\) in J\(^2\) K\(^{-2}\) mol\(^{-2}\) is:

• (A) \(87.8\)
• (B) \(48.7\)
• (C) \(39.1\)
• (D) \(28.7\)

At \(T\) K, the equilibrium constant for the reaction \[ H_2(g) + Br_2(g) \rightleftharpoons 2 HBr(g) \]
is 1.6 \(\times\) 10\(^{1}\). If 10 bar of HBr is introduced into a sealed vessel at \(T\) K, the equilibrium pressure of HBr (in bar) is approximately:

• (1) \(10.20\)
• (2) \(10.95\)
• (3) \(9.95\)
• (4) \(11.95\)

Which of the following will make a basic buffer solution?

The hydrides of which group elements are examples of electron precise hydrides?

• (A) Group 14 elements
• (B) Group 13 elements
• (C) Group 15 elements
• (D) Group 16 elements

The correct order of density of Be, Mg, Ca, Sr is:

• (1) \( Sr > Be > Mg > Ca \)
• (2) \( Be > Mg > Ca > Sr \)
• (3) \( Mg > Ca > Sr > Be \)
• (4) \( Ca > Sr > Be > Mg \)

Which of the following orders is not correct against the given property?

• (A) \( Ga < In < Tl < Al < B \) - melting point
• (B) \( Al < Ga < In < Tl < B \) - Electronegativity
• (C) \( B < Al < Ga < In < Tl \) - Density
• (D) \( B < Al < Ga < In < Tl \) - Atomic Radius

Which of the following are correct?

i. Basic structural unit of silicates is –\( -RSiO- \)

ii. Silicones are biocompatible

iii. Producer gas contains CO and N_2

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

A metal catalyst (X) is used in the catalytic converter of automobiles. This prevents the release of gas Y into the atmosphere. What are X and Y respectively?

• (A) Pd, \(NO_2\)
• (B) Rh, \(CO_2\)
• (C) Pt, \(N_2\)
• (D) Ni, \(CH_4\)

A mixture of substances A, B, C, D is subjected to column chromatography. The degree of adsorption is in the order of \(D > B > C > A\). The column is eluted with a suitable solvent. Identify the correct statement with respect to the separation of the mixture.

• (A) D comes out first from the column
• (B) A comes out first from the column
• (C) C comes out after B from the column
• (D) B comes out after D from the column

What is X in the following reaction?


% Including image

• (A)
• (B)
• (C)
• (D)

Correct Answer: (D)
View Solution

The density of \(\beta\)-Fe is 7.6 g/cm\(^3\). It crystallizes in a cubic lattice with \( a = 290 \) pm.

What is the value of \( Z \)? (\( Fe = 56 \) g/mol, \( N_A = 6.022 \times 10^{23} \) mol\(^{-1}\))

• (A) \( 2 \)
• (B) \( 1 \)
• (C) \( 4 \)
• (D) \( 6 \)

The mass % of urea solution is 6. The total weight of the solution is 1000 g. What is its concentration in mol L\(^{-1}\)? (Density of water = 1.0 g mL\(^{-1}\))

(Given: C = 12u, N = 14u, O = 16u, H = 1u)

• (A) \( 1.5 \)
• (B) \( 1.064 \)
• (C) \( 1.12 \)
• (D) \( 0.80 \)

A non-volatile solute is dissolved in water. The \(\Delta T_f\) of the resultant solution is 0.052 K. What is the freezing point of the solution (in K)?

(Given: \( K_b \) of water = 0.52 K kg mol\(^{-1}\), \( K_f \) of water = 1.86 K kg mol\(^{-1}\), Freezing point of water = 273 K)

• (A) \( 272.628 \)
• (B) \( 273.186 \)
• (C) \( 273.000 \)
• (D) \( 272.814 \)

The standard reduction potentials of \(2H^+ / H_2\), \(Cu^{2+} / Cu\), \(Zn^{2+} / Zn\), and \(NO_3^- / HNO_2\) are 0.0, +0.34, -0.76, and +0.97 V respectively. Observe the following reactions:



I. \( Zn + HCl \rightarrow \)

II. \( Cu + HCl \rightarrow \)

III. \( Cu + HNO_3 \rightarrow \)


Which reactions do not liberate \(H_2\) gas?

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

At 298 K, the value of \( -\frac{d[Br^-]}{dt} \) for the reaction


\[ 5Br^- (aq) + BrO_3^- (aq) + 6H^+ (aq) \rightarrow 3Br_2 (aq) + 3H_2O (l) \]
is \( x \) mol \( L^{-1} \) min\(^{-1}\). What is the rate (in mol \( L^{-1} \) min\(^{-1}\)) of this reaction?

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

Which of the following general reactions is an example for heterogeneous catalysis?

• (A) \( A_2 (g) + B_2 (g) \xrightarrow{C (g)} 2AB (g) \)
• (B) \( A (s) + B (s) \xrightarrow{C (s)} D (s) \)
• (C) \( A (g) + B (g) \xrightarrow{C (s)} D (g) \)
• (D) \( A_2 (g) + B_2 (g) \xrightarrow{C (g)} D (s) \)

Match List I with List II and select the correct answer.


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

The type of iron obtained from the Blast furnace in the extraction of iron is:

• (A) Wrought iron
• (B) Pig iron
• (C) Cast iron
• (D) Steel

The ratio of Xe: F\textsubscript{2} required in the above reaction is:

\[ Xe(g) + 2F_2(g) \xrightarrow{873 K, 7 bar} XeF_4(s) \]

• (A) \( 1:2 \)
• (B) \( 1:5 \)
• (C) \( 1:20 \)
• (D) \( 1:12 \)

The transition metal with the highest melting point is:


% Image inclusion (if applicable)
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%
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• (A) \( Re \)
• (B) \( Cr \)
• (C) \( Mo \)
• (D) \( W \)

Arrange the following in the increasing order of number of unpaired electrons present in the central metal ion:


% Given complex ions
I. \([MnCl_6]^{4-}\)

II. \([FeF_6]^{3-}\)

III. \([Mn(CN)_6]^{3-}\)

IV. \([Fe(CN)_6]^{3-}\)

• (A) \( IV < I < III < II \)
• (B) \( I < III < II < IV \)
• (C) \( IV < III < I < II \)
• (D) \( I < II < III < IV \)

Which of the following polymerisation leads to the formation of neoprene?

• (A)
• (B)
• (C)
• (D)

Correct Answer: (C) (C)
View Solution

Which of the following represents the simplified version of nucleoside?

• (A) Base- sugar- phosphate
• (B) Sugar- base
• (C) Sugar- Phosphate
• (D) Base- Phosphate

Which of the following amino acids possess two chiral centres?

• (A) Leucine
• (B) Valine
• (C) Serine
• (D) Threonine

Which of the following sweeteners use is limited to soft drinks?

• (A) Aspartame
• (B) Saccharin
• (C) Sucralose
• (D) Alitame

Which of the following are general methods for the preparation of 1-iodopropane?



%

• (A) A,B
• (B) B,C
• (C) C,D
• (D) A,D

The product of which of the following reactions undergo hydrolysis by SN1 mechanism?

• (A) C,D ONLY
• (B) A,B,C ONLY
• (C) B,C ONLY
• (D) A,D ONLY

Styrene on reaction with reagent X gave Y, which on hydrolysis followed by oxidation gave Z. Z gives positive 2,4-DNP test but does not give iodoform test. What are X and Z respectively?

• (1) \( HBr : C_6H_5COCH_3 \)
• (2) \( HBr : C_6H_5CHO \)
• (3) \( HBr : (C_6H_5CO)_2O_2 : C_6H_5CH_2CHO \)
• (4) \( HBr : (C_6H_5CO)_2O_2 : C_6H_5COCH_3 \)

What are A and B in the following reaction sequence?

• (A)
• (B)
• (C)
• (D)

Correct Answer: (A)
View Solution

Which of the following sequence of reagents convert propene to 1-chloropropane?

• (A) (i)\( (BH_3)_2 (ii)H_2O_2 / OH^- \) ; \( HCl, ZnCl_2 \)
• (B) (i)\( (BH_3)_2 (ii)H_2O_2 / OH^- \) ; NaCl
• (C) (i)\( dil. H_2SO_4 \) ; \( HCl, ZnCl_2 \)
• (D) (i)\( dil. H_2SO_4 \) ; Conc. HCl

What are X and Y respectively in the following reactions?

• (A)
• (B)
• (C)
• (D) C

Correct Answer: (B)
View Solution