AP EAPCET (AP EAMCET) 2024 Question Paper May 21 Shift 2 (Available): Download MPC Question Paper with Answer Key PDF

The range of the real valued function \( f(x) = \sin^{-1} \left( \frac{1 + x^2}{2x} \right) + \cos^{-1} \left( \frac{2x}{1 + x^2} \right) \) is:

• (1) \( \left\{ \frac{\pi}{2} \right\} \)
• (2) \( \mathbb{R} \)
• (3) \( \mathbb{Q} \)
• (4) \( \left\{ -\frac{\pi}{2}, \frac{\pi}{2} \right\} \)

The real valued function \( f: \mathbb{R} \to \left[ \frac{5}{2}, \infty \right) \) defined by \( f(x) = \left| 2x + 1 \right| + \left| x - 2 \right| \) is:

• (1) One-one function but not onto
• (2) Onto function but not one-one
• (3) Bijection
• (4) Neither one-one function nor onto

If \( 1 \cdot 3 \cdot 5 + 3 \cdot 5 \cdot 7 + 5 \cdot 7 \cdot 9 + \dots \) (n terms) = \( n(n + 1)f(n) - 3n \), then \( f(1) = \):

• (1) \( 9 \)
• (2) \( 8 \)
• (3) \( 7 \)
• (4) \( 6 \)

If
and \( AA^T = I \), then \( \frac{a}{b} + \frac{b}{a} = \):

• (1) \( -\frac{5}{2} \)
• (2) \( \frac{13}{6} \)
• (3) \( \frac{13}{6} \)
• (4) \( \frac{5}{2} \)

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

Assertion (A): If \( B \) is a \( 3 \times 3 \) matrix and \( |B| = 6 \), then \( | Adj(B) | = 36 \).

Reason (R): If \( B \) is a square matrix of order \( n \), then \( |Adj(B)| = |B|^n \).

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

Imaginary part of \( \frac{(1 - i)^3}{(2 - i)(3 - 2i)} \) is:

• (1) \( \frac{22}{65} \)
• (2) \( \frac{6}{65} \)
• (3) \( -\frac{6}{65} \)
• (4) \( -\frac{22}{65} \)

The square root of \( 7 + 24i \) is:

• (1) \( 4 - 3i \)
• (2) \( 3 + 4i \)
• (3) \( 3 - 4i \)
• (4) \( 4 + 3i \)

If \( n \) is an integer and \( Z = \cos \theta + i \sin \theta, \theta \neq (2n + 1)\frac{\pi}{2}, \) then: \[ \frac{1 + Z^{2n}}{1 - Z^{2n}} = ? \]

• (1) \( i \tan n\theta \)
• (2) \( i \cot n\theta \)
• (3) \( -i \tan n\theta \)
• (4) \( -i \cot n\theta \)

If \( x \) is real and \( \alpha, \beta \) are maximum and minimum values of \( \frac{x^2 - x + 1}{x^2 + x + 1} \) respectively, then \( \alpha + \beta = \):

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

If \( a \) is a common root of \( x^2 - 5x + \lambda = 0 \) and \( x^2 - 8x - 2\lambda = 0 \) (\( \lambda \neq 0 \)) and \( \beta, \gamma \) are the other roots of them, then \( a + \beta + \gamma + \lambda = \):

• (1) 0
• (2) -1
• (3) 1
• (4) 2

The equation \( x^4 - x^3 - 6x^2 + 4x + 8 = 0 \) has two equal roots. If \( \alpha, \beta \) are the other two roots of this equation, then \( \alpha^2 + \beta^2 = \):

• (1) 4
• (2) 5
• (3) 6
• (4) 7

The condition that the roots of \( x^3 - bx^2 + cx - d = 0 \) are in arithmetic progression is:

• (1) \( 9cb = 2b^3 + 27d \)
• (2) \( 9cb = 2d^3 + 27b \)
• (3) \( 9cd = 2b^3 + 27d \)
• (4) \( 9cd = 2d^3 + 27b \)

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arrangements is:

• (1) 270
• (2) 180
• (3) 540
• (4) 1080

If a five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4, and 5 without repetition, then the total number of ways this can be done is:

• (1) 120
• (2) 144
• (3) 192
• (4) 216

Four digit numbers with all digits distinct are formed using the digits 1, 2, 3, 4, 5, 6, 7 in all possible ways. If \( p \) is the total number of numbers thus formed and \( q \) is the number of numbers greater than 3400 among them, then \( p : q = \):

• (1) \( 3:2 \)
• (2) \( 4:3 \)
• (3) \( 6:5 \)
• (4) \( 7:4 \)

If the ratio of the terms equidistant from the middle term in the expansion of \( (1 + x)^{12} \) is \( \frac{1}{256} \), then the sum of all the terms of the expansion \( (1 + x)^{12} \) is:

• (1) \( 4^{12} \) or \( 6^{12} \)
• (2) \( 3^{12} \) or \( 5^{12} \)
• (3) \( 6^{12} \) or \( 7^{12} \)
• (4) \( 12^{12} \)

In the expansion of \( \frac{2x+1}{(1+x)(1-2x)} \), the sum of the coefficients of the first 5 odd powers of \( x \) is:

• (1) \( \frac{5}{3} + \frac{8}{9} (45 - 1) \)
• (2) \( \frac{5}{3} + \frac{8}{3} (45 - 1) \)
• (3) \( \frac{-5}{3} + \frac{8}{9} (45 - 1) \)
• (4) \( \frac{5}{3} + \frac{8}{12} (45 + 1) \)

If \[ \frac{x + 2}{(x^2 + 3)(x^4 + x^2)(x^2 + 2)} = \frac{Ax + B}{x^2 + 3} + \frac{Cx + D}{x^2 + 2} + \frac{Ex^3 + Fx^2 + Gx + H}{x^4 + x^2}, \]
then \[ (E + F)(C + D)(A) = \]

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

If \( A, B, C \) are the angles of a triangle, then \[ \sin 2A - \sin 2B + \sin 2C = \]

• (1) \( 4 \cos A \cos B \sin C \)
• (2) \( 4 \cos A \sin B \cos C \)
• (3) \( 4 \cos A \sin B \cos C - 1 \)
• (4) \( 4 \sin A \cos B \sin C \)

Assertion (A): If \( A = 10^\circ, B = 16^\circ, C = 19^\circ \), then: \[ \tan(2A) \tan(2B) + \tan(2B) \tan(2C) + \tan(2C) \tan(2A) = 1. \]
Reason (R): If \( A + B + C = 180^\circ \), then: \[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right) = \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right). \]

• (1) Both (A) and (R) are true and (R) is the correct explanation of (A).
• (2) Both (A) and (R) are true and (R) is NOT correct explanation of (A).
• (3) (A) is true, (R) is false.
• (4) (A) is false, (R) is true.

If \( a \) is in the 3rd quadrant, \( \beta \) is in the 2nd quadrant such that \( \tan \alpha = \frac{1}{7}, \sin \beta = \frac{1}{\sqrt{10}} \), then \[ \sin(2\alpha + \beta) = \]

• (1) \( \frac{3 \times \sqrt{10}}{25} \)
• (2) \( \frac{3}{\sqrt{10}} \)
• (3) \( \frac{3}{25\sqrt{10}} \)
• (4) \( \frac{\sqrt{10}}{3 \times 25} \)

Number of solutions of the trigonometric equation \[ 2 \tan 2\theta - \cot 2\theta + 1 = 0 \quad lying in the interval \quad [0, \pi] \]

• (1) 2
• (2) 3
• (3) 4
• (4) 5

The real values of \( x \) that satisfy the equation \[ \tan^{-1}x + \tan^{-1}2x = \frac{\pi}{4} \]
is:

• (1) \( \frac{-3 \pm \sqrt{17}}{4} \)
• (2) \( \frac{-1 \pm \sqrt{3}}{2} \)
• (3) \( \sqrt{3} - 1 \)
• (4) \( \frac{\sqrt{17} - 3}{4} \)

Evaluate the expression \[ 2 \cot h^{-1}(4) + \sec h^{-1}\left( \frac{3}{5} \right). \]

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

If 7 and 8 are the lengths of two sides of a triangle and \( a \) is the length of its smallest side. The angles of the triangle are in AP and \( a \) has two values \( a_1 \) and \( a_2 \) satisfying this condition. If \( a_1 < a_2 \), then \( 2a_1 + 3a_2 = \):

• (1) 15
• (2) 21
• (3) 24
• (4) 28

In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):

• (1) \( 2S \)
• (2) \( \Delta \)
• (3) \( S \)
• (4) \( 2A \)

In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):

• (1) \( a + b \)
• (2) \( c + a \)
• (3) \( 2 \sqrt{2} R \cos \left( \frac{C - A}{2} \right) \)
• (4) \( 2 \sqrt{2} R \cos \left( \frac{B - C}{2} \right) \)

If \( \vec{i} - 2\vec{j} + 3\vec{k}, 2\vec{i} + 3\vec{j} - \vec{k}, -3\vec{i} - \vec{j} - 2\vec{k} \) are the position vectors of three points A, B, C respectively, then A, B, C:

• (1) are collinear points
• (2) form an isosceles triangle which is not equilateral
• (3) form an equilateral triangle
• (4) form a scalene triangle

If \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) are position vectors of 4 points such that \( 2\vec{a} + 3\vec{b} + 5\vec{c} - 10\vec{d} = 0 \), then the ratio in which the line joining \( \vec{c} \) divides the line segment joining \( \vec{a} \) and \( \vec{b} \) is:

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

If \( \vec{a}, \vec{b}, \vec{c} \) are 3 vectors such that \( |\vec{a}| = 5, |\vec{b}| = 8, |\vec{c}| = 11 \) and \( \vec{a} + \vec{b} + \vec{c} = 0 \), then the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is:

• (1) \( \cos^{-1}\left( \frac{-2}{5} \right) \)
• (2) \( \cos^{-1}\left( \frac{10}{11} \right) \)
• (3) \( \cos^{-1}\left( \frac{41}{55} \right) \)
• (4) \( \frac{\pi}{3} \)

The angle between the planes \( \vec{r} \cdot (12\hat{i} + 4\hat{j} - 3\hat{k}) = 5 \) and \( \vec{r} \cdot (5\hat{i} + 3\hat{j} + 4\hat{k}) = 7 \) is:

• (1) \( \cos^{-1}\left( \frac{12}{13} \right) \)
• (2) \( \cos^{-1}\left( \frac{6\sqrt{2}}{13} \right) \)
• (3) \( \cos^{-1}\left( \frac{3\sqrt{2}}{13} \right) \)
• (4) \( \cos^{-1}\left( \frac{6}{13} \right) \)

The shortest distance between the skew lines \( \vec{r} = (2\hat{i} - \hat{j}) + t(\hat{i} + 2\hat{k}) \) and \( \vec{r} = (-2\hat{i} + \hat{k}) + s(\hat{i} - \hat{j} - \hat{k}) \) is:

• (1) \( \frac{3\sqrt{2}}{\sqrt{7}} \)
• (2) \( \frac{3}{\sqrt{7}} \)
• (3) \( \frac{3}{\sqrt{14}} \)
• (4) \( \frac{4}{\sqrt{14}} \)

The coefficient of variation for the frequency distribution is:

• (1) \( \frac{50}{\sqrt{3}} \)
• (2) \( \frac{125}{2\sqrt{3}} \)
• (3) \( \frac{100}{3\sqrt{2}} \)
• (4) \( \frac{100}{\sqrt{3}} \)

If all the letters of the word ‘SENSELESSNESS’ are arranged in all possible ways and an arrangement among them is chosen at random, then, the probability that all the E’s come together in that arrangement is:

• (1) \( \frac{1}{990} \)
• (2) \( \frac{2}{143} \)
• (3) \( \frac{1}{120} \)
• (4) \( \frac{1}{429} \)

If two numbers \(x\) and \(y\) are chosen one after the other at random with replacement from the set of numbers \( \{1, 2, 3, \ldots, 10\} \), then the probability that \( |x^2 - y^2| \) is divisible by 6 is:

• (1) \( \frac{8}{25} \)
• (2) \( \frac{6}{25} \)
• (3) \( \frac{3}{10} \)
• (4) \( \frac{13}{50} \)

Bag A contains 3 white and 4 red balls, bag B contains 4 white and 5 red balls, and bag C contains 5 white and 6 red balls. If one ball is drawn at random from each of these three bags, then the probability of getting one white and two red balls is:

• (1) \( \frac{268}{693} \)
• (2) \( \frac{310}{693} \)
• (3) \( \frac{38}{99} \)
• (4) \( \frac{26}{63} \)

Two persons A and B throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If A starts the game, then the probability that B wins the game is:

• (1) \( \frac{11}{23} \)
• (2) \( \frac{1}{2} \)
• (3) \( \frac{5}{11} \)
• (4) \( \frac{8}{17} \)

An urn contains 3 black and 5 red balls. If 3 balls are drawn at random from the urn, the mean of the probability distribution of the number of red balls drawn is:

• (1) \( \frac{45}{28} \)
• (2) \( \frac{15}{8} \)
• (3) \( \frac{2}{5} \)
• (4) \( \frac{3}{7} \)

If \( X \sim B(5, p) \) is a binomial variate such that \( p(X = 3) = p(X = 4) \), then \( P(|X - 3| < 2) = \dots \)

• (1) \( \frac{242}{243} \)
• (2) \( \frac{201}{243} \)
• (3) \( \frac{200}{243} \)
• (4) \( \frac{121}{243} \)

The perimeter of the locus of the point \( P \) which divides the line segment \( QA \) internally in the ratio 1:2, where \( A = (4, 4) \) and \( Q \) lies on the circle \( x^2 + y^2 = 9 \), is:

• (1) \( 8\pi \)
• (2) \( 4\pi \)
• (3) \( \pi \)
• (4) \( 9\pi \)

Suppose the axes are to be rotated through an angle \( \theta \) so as to remove the \( xy \) term from the equation \(3 x^2 + 2\sqrt{3}xy + y^2 = 0 \). Then in the new coordinate system, the equation \( x^2 + y^2 + 2xy = 2 \) is transformed to:

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

P is a point on \( x + y + 5 = 0 \), whose perpendicular distance from \( 2x + 3y + 3 = 0 \) is \( \sqrt{13} \), then the coordinates of P are:

• (1) \( (20, -25) \)
• (2) \( (1, -6) \)
• (3) \( (-6,1) \)
• (4) \( (\sqrt{13}, -5 - \sqrt{13}) \)

For \( \lambda, \mu \in \mathbb{R} \), the lines \[ (x - 2y - 1) + \lambda (3x + 2y - 11) = 0 \]
and \[ (3x + 4y - 11) + \mu (-x + 2y - 3) = 0 \]
represent two families of lines. If the equation of the line common to both families is given by \[ ax + by - 5 = 0, \]
then \( 2a + b = \) ?

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

If the pair of lines represented by \[ 3x^2 - 5xy + P y^2 = 0 \]
and \[ 6x^2 - xy - 5y^2 = 0 \]
have one line in common, then the sum of all possible values of \( P \) is:

• (1) \( \frac{33}{4} \)
• (2) \( \frac{17}{4} \)
• (3) \( -\frac{33}{4} \)
• (4) \( -\frac{17}{4} \)

The area of the region enclosed by the curves \[ 3x^2 - y^2 - 2xy + 4x + 1 = 0 \]
and \[ 3x^2 - y^2 - 2xy + 6x + 2y = 0 \]
is:

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

If the equation of the circle whose radius is 3 units and which touches internally the circle \[ x^2 + y^2 - 4x - 6y - 12 = 0 \]
at the point \( (-1, -1) \) is \[ x^2 + y^2 + px + qy + r = 0, \]
then \( p + q - r \) is:

• (1) \( 2 \)
• (2) \( \frac{5}{2} \)
• (3) \( \frac{26}{5} \)
• (4) \( 3 \)

The equation of the circle touching the circle \[ x^2 + y^2 - 6x + 6y + 17 = 0 \]
externally and to which the lines \[ x^2 - 3xy - 3x + 9y = 0 \]
are normal is:

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

The pole of the straight line \[ 9x + y - 28 = 0 \]
with respect to the circle \[ 2x^2 + 2y^2 - 3x + 5y - 7 = 0 \]
is:

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

The equation of a circle which touches the straight lines \[ x + y = 2, \quad x - y = 2 \]
and also touches the circle \[ x^2 + y^2 = 1 \]
is:

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

The radical axis of the circles \[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
and \[ 2x^2 + 2y^2 + 3x + 8y + 2c = 0 \]
touches the circle \[ x^2 + y^2 + 2x + 2y + 1 = 0. \]

Then:

• (1) \( g = \frac{3}{8} \) \textbf{or} \( f = 1 \)
• (2) \( g = \frac{2}{3} \) \textbf{or} \( f = 3 \)
• (3) \( g = \frac{1}{2} \) \textbf{or} \( f = 1 \)
• (4) \( g = \frac{3}{4} \) \textbf{or} \( f = 2 \)

If the ordinates of points \( P \) and \( Q \) on the parabola \[ y^2 = 12x \]
are in the ratio 1:2, then the locus of the point of intersection of the normals to the parabola at \( P \) and \( Q \) is:

• (1) \( y + 18 \left( \frac{x - 6}{21} \right)^{3/2} = 0 \)
• (2) \( y - 18 \left( \frac{x - 6}{12} \right)^{3/2} = 0 \)
• (3) \( y + 12 \left( \frac{x - 6}{14} \right)^{1/2} = 0 \)
• (4) \( y - 12 \left( \frac{x - 6}{18} \right)^{1/2} = 0 \)

The product of perpendiculars from the two foci of the ellipse \[ \frac{x^2}{9} + \frac{y^2}{25} = 1 \]
on the tangent at any point on the ellipse is:

• (1) \( 6 \)
• (2) \( 7 \)
• (3) \( 8 \)
• (4) \( 9 \)

The value of \( c \) such that the straight line joining the points \[ (0,3) \quad and \quad (5,-2) \]
is tangent to the curve \[ y = \frac{c}{x+1} \]
is:

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

The descending order of magnitude of the eccentricities of the following hyperbolas is:

A. A hyperbola whose distance between foci is three times the distance between its directrices.

B. Hyperbola in which the transverse axis is twice the conjugate axis.

C. Hyperbola with asymptotes \( x + y + 1 = 0, x - y + 3 = 0 \).

• (1) \( C, A, B \)
• (2) \( B, C, A \)
• (3) \( \) (No option provided)
• (4) \( A, C, B \)

If the plane \[ x - y + z + 4 = 0 \]
divides the line joining the points \[ P(2,3,-1) \quad and \quad Q(1,4,-2) \]
in the ratio \( l:m \), then \( l + m \) is:

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

If the line with direction ratios \[ (1, a, \beta) \]
is perpendicular to the line with direction ratios \[ (-1,2,1) \]
and parallel to the line with direction ratios \[ (\alpha,1,\beta), \]
then \( (\alpha, \beta) \) is:

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

Let \( P(x_1, y_1, z_1) \) be the foot of the perpendicular drawn from the point \[ Q(2, -2, 1) \]
to the plane \[ x - 2y + z = 1. \]
If \( d \) is the perpendicular distance from the point \( Q \) to the plane and \[ I = x_1 + y_1 + z_1, \]
then \( I + 3d^2 \) is:

• (1) \( 5 \)
• (2) \( 7 \)
• (3) \( 19 \)
• (4) \( 26 \)

Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}. \]

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

Evaluate the limit: \[ \lim_{x \to 1} \frac{x + x^2 + x^3 + \dots + x^n - n}{x - 1}. \]

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

If the function \[ f(x) = \frac{\sqrt{1+x} - 1}{x} \]
is continuous at \( x = 0 \), then \( f(0) \) is:

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

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \]
then \( f'(2) \) is:

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

If \[ \frac{d}{dx} \left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) = ax + b, \]
then \( (a,b) \) is:

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

If \[ y = \sin^{-1} x, \]
then \[ (1 - x^2)y_2 - xy_1 = 0. \]

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

If the percentage error in the radius of a circle is 3, then the percentage error in its area is:

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

The equation of the tangent to the curve \[ y = x^3 - 2x + 7 \]
at the point \( (1,6) \) is:

• (1) \( y = x + 5 \)
• (2) \( x + y = 7 \)
• (3) \( 2x + y = 8 \)
• (4) \( x + 2y = 13 \)

The distance \( s \) traveled by a particle in time \( t \) is given by: \[ s = 4t^2 + 2t + 3. \]
The velocity of the particle when \( t = 3 \) seconds is:

• (1) \( 26 \) unit/sec
• (2) \( 20 \) unit/sec
• (3) \( 24 \) unit/sec
• (4) \( 30 \) unit/sec

If \[ a^2 x^4 + b^2 y^4 = c^6, \]
then the maximum value of \( xy \) is:

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

Evaluate the integral \[ \int \frac{\sin^6 x}{\cos^8 x} \, dx. \]

• (1) \( \tan 7x + c \)
• (2) \( \frac{\tan^7 x}{7} + c \)
• (3) \( \frac{\tan 7x}{7} + c \)
• (4) \( \sec^7 x \)

Evaluate the integral \[ \int \frac{x^5}{x^2 + 1} dx. \]

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

Evaluate the integral \[ \int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx. \]

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

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

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

Evaluate the integral \[ \int e^x (x+1)^2 dx. \]

• (1) \( xe^x + c \)
• (2) \( e^x x^2 + c \)
• (3) \( e^x (x^2 + 1) + c \)
• (4) \( e^x (x + 1) + c \)

Evaluate the integral \[ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. \]

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

Evaluate the integral \[ I = \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx \]

• (1) \( \frac{4}{5} \)
• (2) \( \frac{8}{15} \)
• (3) \( \frac{14}{5} \)
• (4) \( \frac{16}{5} \)

Evaluate the integral \[ I = \int_{-1}^{1} \left( \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right) \, dx \]

• (1) \( 2 \)
• (2) \( 4 \)
• (3) \( 0 \)
• (4) \( 8 \)

Evaluate the integral \[ I = \int_1^5 \left( |x - 3| + |1 - x| \right) \, dx \]

• (1) \( 4 \)
• (2) \( 8 \)
• (3) \( 12 \)
• (4) \( 24 \)

The differential equation formed by eliminating arbitrary constants \( A \) and \( B \) from the equation \[ y = A \cos 3x + B \sin 3x \]
is:

• (1) \( \frac{d^2y}{dx^2} + y = 0 \)
• (2) \( \frac{d^2y}{dx^2} + 9y = 0 \)
• (3) \( \frac{d^2y}{dx^2} - 9y = 0 \)
• (4) \( \frac{d^2y}{dx^2} - y = 0 \)

If \[ \cos x \frac{dy}{dx} - y \sin x = 6x, \quad (0 < x < \frac{\pi}{2}) \quad and \quad y(\frac{\pi}{3}) = 0, \quad then \quad y(\frac{\pi}{6}) = \]

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

The solution of the differential equation \[ \frac{dy}{dx} = \frac{y + x \tan \left( \frac{y}{x} \right)}{x}. \] \[ \sin\frac{y}{x} = \]

• (1) \( cx^2 \)
• (2) \( cx \)
• (3) \( cx^3 \)
• (4) \( cx^4 \)

The length of the side of a cube is \( 1.2 \times 10^{-2} \) m. Its volume up to correct significant figures is:

• (1) \( 1.732 \times 10^{-6} \, m^3 \)
• (2) \( 1.73 \times 10^{-6} \, m^3 \)
• (3) \( 1.70 \times 10^{-6} \, m^3 \)
• (4) \( 1.7 \times 10^{-6} \, m^3 \)

The velocity of a particle is given by the equation \( v(x) = 3x^2 - 4x \), where \( x \) is the distance covered by the particle. The expression for its acceleration is:

• (1) \( (6x - 4) \)
• (2) \( 6(3x^2 - 4x) \)
• (3) \( (3x^2 - 4x) (6x - 4) \)
• (4) \( (6x - 4)^2 \)

The acceleration of a particle which moves along the positive \( x \)-axis varies with its position as shown in the figure. If the velocity of the particle is \( 0.8 \, m/s \) at \( x = 0 \), then its velocity at \( x = 1.4 \, m \) is:

• (1) \( 1.6 \, m/s \)
• (2) \( 1.2 \, m/s \)
• (3) \( 1.4 \, m/s \)
• (4) \( 0.8 \, m/s \)

The maximum height attained by the projectile is increased by 10% by keeping the angle of projection constant. What is the percentage increase in the time of flight?

• (1) 5%
• (2) 10%
• (3) 20%
• (4) 40%

A light body of momentum \( P_L \) and a heavy body of momentum \( P_H \), both have the same kinetic energy, then:

• (1) \( P_L > P_H \)
• (2) \( P_L < P_H \)
• (3) \( P_L = P_H \)
• (4) \( P_H = 2 P_L \)

A block of metal 4 kg is in rest on a frictionless surface. It was targeted by a jet releasing water of 2 kg/s at a speed of 10 ms\(^{-1}\). The acceleration of the block is:

• (1) 10 ms\(^{-2}\)
• (2) 15 ms\(^{-2}\)
• (3) 20 ms\(^{-2}\)
• (4) 5 ms\(^{-2}\)

A person climbs up a conveyor belt with a constant acceleration. The speed of the belt is \( \sqrt{\frac{g h}{6}} \) and the coefficient of friction is \( \frac{5}{3\sqrt{3}} \). The time taken by the person to reach from A to B with maximum possible acceleration is:

• (1) \( \frac{\sqrt{h g}}{6} \)
• (2) \( \sqrt{6gh} \)
• (3) \( \frac{2h}{\sqrt{g}} \)
• (4) \( \sqrt{\frac{6h}{g}}.\)

A machine with efficiency \( \frac{2}{3} \) used 12 J of energy in lifting a 2 kg block through a certain height and it is allowed to fall through the same. The velocity while it reaches the ground is:

• (1) \( \sqrt{2} \, ms^{-1} \)
• (2) \( 2 \, ms^{-1} \)
• (3) \( 2 \sqrt{2} \, ms^{-1} \)
• (4) \( 0.2 \, ms^{-1} \)

A solid cylinder rolls down on an inclined plane of height \( h \) and inclination \( \theta \). The speed of the cylinder at the bottom is:

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

Three particles of each mass \( m \) are kept at the three vertices of an equilateral triangle of side \( 1 \). The moment of inertia of the system of the particles about any side of the triangle is:

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

In a spring-block system as shown in the figure, if the spring constant \( K = 9 \, N/m \), then the time period of oscillation is:

• (1) 1 s
• (2) 3.14 s
• (3) 1.414 s
• (4) 0.5 s

A body is executing simple harmonic motion. At a displacement \( x \), its potential energy is \( E_1 \), and at a displacement \( y \), its potential energy is \( E_2 \). The potential energy \( E \) at a displacement \( (x + y) \) is:

• (1) \(\sqrt{E} = \sqrt{E_1} - \sqrt{E_2} \)
• (2) \(\sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \)
• (3) \(E = E_1 - E_2\)
• (4) \(E = E_1 + E_2\)

A particle is projected from the surface of the Earth with a velocity equal to twice the escape velocity. When the particle is far from the Earth, its speed will be:

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

A 4 kg stone is attached to a steel wire being whirled at a constant speed of \( 12 \) m/s in a horizontal circle. The wire is 4 m long with a diameter of 2 mm, and Young’s modulus is \( 2 \times 10^{11} \) Nm\(^2\). The strain in the wire is:

• (1) \( 2.3 \times 10^{-4} \)
• (2) \( 2.3 \times 10^{-5} \)
• (3) \( 4.6 \times 10^{-4} \)
• (4) \( 6.9 \times 10^{-4} \)

A spherical ball of radius \( 1 \times 10^{-4} \) m and density \( 10^4 \) kgm\(^{-3}\) falls freely under gravity before entering water. The distance \( h \) before velocity change in water is:

• (1) \( 20.4 \) cm
• (2) \( 20.4 \) mm
• (3) \( 20.4 \) m
• (4) \( 10.2 \) m

A metal block is made from a mixture of 2.4 kg of aluminium, 1.6 kg of brass, and 0.8 kg of copper. The metal block is initially at 20°C. If the heat supplied to the metal block is 44.4 calories, find the final temperature of the block if specific heats of aluminium, brass, and copper are 0.216, 0.0917, and 0.0931 cal.kg\(^{-1}\)°C\(^{-1}\) respectively.

• (1) \( 100^\circ C \)
• (2) \( 60^\circ C \)
• (3) \( 40^\circ C \)
• (4) \( 80^\circ C \)

An ideal gas is found to obey \( PV^{\frac{3}{2}} = constant \) during an adiabatic process. If such a gas initially at a temperature \( T \) is adiabatically compressed to \( \frac{1}{4} \)th of its volume, then its final temperature is:

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

The condition \( dw = dq \) holds good in the following process:

• (1) Adiabatic process
• (2) Isothermal process
• (3) Isochoric process
• (4) Isobaric process

The efficiency of a Carnot engine is found to increase from 25% to 40% on increasing the temperature (\(T_1\)) of the source alone through 100 K. The temperature (\(T_2\)) of the sink is given by:

• (1) \( 300 \, K \)
• (2) \( 250 \, K \)
• (3) \( 325 \, K \)
• (4) \( 125 \, K \)

Match the following (\( f \) is the number of degrees of freedom):

When a wave enters into a rarer medium from a denser medium, the property of the wave which remains constant is:

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

The focal length of the objective lens of a telescope is 30 cm and that of its eye lens is 3 cm. It is focused on a scale at a distance 2 m from it. The distance of the objective lens from the eye lens to see the clear image is:

• (1) \( 38.3 \) cm
• (2) \( 48.3 \) cm
• (3) \( 58.3 \) cm
• (4) \( 22.5 \) cm

In case of diffraction, if \( a \) is a slit width and \( \lambda \) is the wavelength of the incident light, then the required condition for diffraction to take place is:

• (1) \( \frac{a}{\lambda} = 1000 \)
• (2) \( \frac{a}{\lambda} \leq 1 \)
• (3) \( a \ll \lambda \)
• (4) \( a \gg \lambda \)

The electric field intensity (\(E\)) at a distance of 3 m from a uniform long straight wire of linear charge density 0.2 \(\mu C m^{-1}\) is:

• (1) \( 1.2 \times 10^3 \, Vm^{-1} \)
• (2) \( 0.6 \times 10^3 \, Vm^{-1} \)
• (3) \( 1.8 \times 10^3 \, Vm^{-1} \)
• (4) \( 2.4 \times 10^3 \, Vm^{-1} \)

When a parallel plate capacitor is charged up to 95 V, its capacitance is \( C \). If a dielectric slab of thickness 2 mm is inserted between plates and the plate separation is increased by 1.6 mm such that the potential difference remains constant, find the dielectric constant of the material:

• (1) \( 2.4 \)
• (2) \( 4.5 \)
• (3) \( 5.0 \)
• (4) \( 9.0 \)

The capacitance of an isolated sphere of radius \( r_1 \) is increased by 5 times when enclosed by an earthed concentric sphere of radius \( r_2 \). The ratio \( \frac{r_1}{r_2} \) is:

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

The charge \( q \) (in coulombs) passing through a \( 10 \Omega \) resistor as a function of time \( t \) (in seconds) is given by: \[ q = 3t^2 - 2t + 6 \]
The potential difference across the ends of the resistor at time \( t = 5 \) s is:

• (1) \( 120 \, V \)
• (2) \( 240 \, V \)
• (3) \( 140 \, V \)
• (4) \( 280 \, V \)

A cell of emf 1.2 V and internal resistance 2 \( \Omega \) is connected in parallel to another cell of emf 1.5 V and internal resistance 1 \( \Omega \). If the like poles of the cells are connected together, the emf of the combination of the two cells is:

• (1) \( 0.8 \) V
• (2) \( 3.9 \) V
• (3) \( 2.7 \) V
• (4) \( 1.4 \) V

A proton and an alpha particle moving with energies in the ratio \( 1:4 \) enter a uniform magnetic field of 37 T at right angles to the direction of the field. The ratio of the magnetic forces acting on the proton and the alpha particle is:

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

A charged particle moving along a straight-line path enters a uniform magnetic field of \( 4 \) mT at right angles to the direction of the magnetic field. If the specific charge of the charged particle is \( 8 \times 10^7 \) C/kg, the angular velocity of the particle in the magnetic field is:

• (1) \( 64 \times 10^4 \) rad \( s^{-1} \)
• (2) \( 32 \times 10^4 \) rad \( s^{-1} \)
• (3) \( 16 \times 10^4 \) rad \( s^{-1} \)
• (4) \( 48 \times 10^4 \) rad \( s^{-1} \)

At a place the horizontal component of earth’s magnetic field is \( 3 \times 10^{-5} \) T and the magnetic declination is \( 30^\circ \). A compass needle of magnetic moment \( 18 \) A\(m^2\) pointing towards geographic north at this place experiences a torque of:

• (1) \( 36 \times 10^{-5} \) Nm
• (2) \( 18 \times 10^{-5} \) Nm
• (3) \( 54 \times 10^{-5} \) Nm
• (4) \( 27 \times 10^{-5} \) Nm

The current passing through a coil of 120 turns and inductance \( 40 \) mH is \( 30 \) mA. The magnetic flux linked with the coil is:

• (1) \( 20 \times 10^{-6} \) Wb
• (2) \( 5 \times 10^{-6} \) Wb
• (3) \( 12 \times 10^{-6} \) Wb
• (4) \( 10 \times 10^{-6} \) Wb

A resistor of resistance \( R \), inductor of inductive reactance \( 2R \) and a capacitor of capacitive reactance \( X_C \) are connected in series to an A.C. source. If the series LCR circuit is in resonance, then the power factor of the circuit and the value \( X_C \) are respectively:

• (1) \( 0.5 \) and \( 4R \)
• (2) \( 1 \) and \( 2R \)
• (3) \( 0.5 \) and \( 2R \)
• (4) \( 1 \) and \( 4R \)

The RMS value of the electric field of an electromagnetic wave emitted by a source is \( 660 \) N/C. The average energy density of the electromagnetic wave is:

• (1) \( 1.75 \times 10^{-6} \) J/m\(^3\)
• (2) \( 2.75 \times 10^{-6} \) J/m\(^3\)
• (3) \( 4.85 \times 10^{-6} \) J/m\(^3\)
• (4) \( 3.85 \times 10^{-6} \) J/m\(^3\)

The maximum wavelength of light which causes photoelectric emission from a photosensitive metal surface is \( \lambda_0 \). Two light beams of wavelengths \( \frac{\lambda_0}{3} \) and \( \frac{\lambda_0}{9} \) incident on the metal surface. The ratio of the maximum velocities of the emitted photoelectrons is:

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

The electrostatic potential energy of the electron in an orbit of hydrogen is \( -6.8 \) eV. The speed of the electron in this orbit is (C is the speed of light in vacuum):

• (1) \( \frac{C}{137} \)
• (2) \( \frac{C}{274} \)
• (3) \( \frac{2C}{137} \)
• (4) \( \frac{3C}{137} \)

The surface areas of two nuclei are in the ratio \( 9:25 \). The mass numbers of the nuclei are in the ratio:

• (1) \( 27:125 \)
• (2) \( 9:25 \)
• (3) \( 3:5 \)
• (4) \( 1:1 \)

Pure silicon at 300K has equal electron and hole concentration of \( 1.5 \times 10^{16} \) m\(^{-3}\). If the hole concentration increases to \( 3 \times 10^{22} \) m\(^{-3}\), then the electron concentration in the silicon is:

• (1) \( 0.75 \times 10^9 \) m\(^{-3}\)
• (2) \( 750 \) m\(^{-3}\)
• (3) \( 75 \) m\(^{-3}\)
• (4) \( 7.5 \times 10^9 \) m\(^{-3}\)

In an \( npn \) transistor circuit, the collector current is \( 10 \) mA. If \( 95% \) of the electrons emitted reach the collector, then the base current is nearly:

• (1) \( 5.3 \) mA
• (2) \( 53 \) mA
• (3) \( 35 \) mA
• (4) \( 0.53 \) mA

A transmitter of power 10 kW emits radio waves of wavelength 500 m. The number of photons emitted per second by the transmitter is of the order of:

• (1) \( 10^{37} \)
• (2) \( 10^{31} \)
• (3) \( 10^{25} \)
• (4) \( 10^{43} \)

The difference in radii between fourth and third Bohr orbits of \( He^+ \) (in m) is:

• (1) \( 2.64 \times 10^{-10} \)
• (2) \( 1.85 \times 10^{-12} \)
• (3) \( 1.85 \times 10^{-10} \)
• (4) \( 1.85 \times 10^{-9} \)

If \( \lambda_0 \) and \( \lambda \) are respectively the threshold wavelength and wavelength of incident light, the velocity of photoelectrons ejected from the metal surface is:

• (1) \( \sqrt{\frac{2h}{m} (\lambda_0 - \lambda)} \)
• (2) \( \sqrt{\frac{2hc}{m} \left( \frac{\lambda_0 - \lambda}{\lambda_0} \right)} \)
• (3) \( \sqrt{\frac{2hc}{m} (\lambda_0 - \lambda)} \)
• (4) \( \sqrt{\frac{2h}{m} \left( \frac{1}{\lambda_0} - \frac{1}{\lambda} \right)} \)

The correct order of atomic radii of N, F, Al, Si is:

• (1) \( F > N > Si > Al \)
• (2) \( F > N > Al > Si \)
• (3) \( Al > Si > F > N \)
• (4) \( Al > Si > N > F \)

The correct order of covalent bond character of \( BCl_3, CCl_4, BeCl_2, LiCl \) is:

• (1) \( LiCl < BeCl_2 < BCl_3 < CCl_4 \)
• (2) \( CCl_4 < BeCl_2 < BCl_3 < LiCl \)
• (3) \( CCl_4 < BCl_3 < BeCl_2 < LiCl \)
• (4) \( LiCl < BCl_3 < BeCl_2 < CCl_4 \)

In which of the following pairs, both molecules possess dipole moment?

• (1) \( CO_2, BCl_3 \)
• (2) \( BCl_3, NF_3 \)
• (3) \( CO_2, SO_2 \)
• (4) \( SO_2, NF_3 \)

At \( T(K) \), the \( P, V \) and \( u_{rms} \) of 1 mole of an ideal gas were measured. The following graph is obtained. What is its slope (\( m \))? (x-axis = \( PV \); y-axis = \( u_{rms}^2 \); \( M \) = Molar mass)

• (1) \( \frac{3}{M} \)
• (2) \( \frac{M}{3} \)
• (3) \( \left(\frac{M}{3}\right)^{1/2} \)
• (4) \( \left(\frac{3}{M}\right)^{1/2} \)

Three layers of liquid are flowing over a fixed solid surface as shown below. The correct order of velocity of liquid in these layers is:

• (1) \( V_1 > V_2 > V_3 \)
• (2) \( V_1 = V_2 = V_3 \)
• (3) \( V_3 > V_2 > V_1 \)
• (4) \( V_3 > V_1 > V_2 \)

A flask contains 98 mg of H₂SO₄. If \( 3.01 \times 10^{20} \) molecules of H₂SO₄ are removed from the flask, the number of moles of H₂SO₄ remaining in the flask is (\( N = 6.02 \times 10^{23} \)):

• (1) \( 1 \times 10^{-4} \)
• (2) \( 5 \times 10^{-4} \)
• (3) \( 1.66 \times 10^{-3} \)
• (4) \( 9.95 \times 10^{-3} \)

Identify the correct equation relating \( \Delta H \), \( \Delta U \), and \( \Delta T \) for 1 mole of an ideal gas (R = gas constant):

• (1) \( (\Delta H)^2 = \Delta U + R \Delta T \)
• (2) \( \Delta H = (\Delta U)^2 + R \Delta T \)
• (3) \( \Delta U = \Delta H - R \Delta T \)
• (4) \( \Delta U = \Delta H + R \Delta T \)

The number of extensive properties in the following list is:
Enthalpy, density, volume, internal energy, temperature.

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

The equilibrium constant for the dissociation of HI at 773 K is:

• (1) \( 2 \times 10^{-2} \)
• (2) \( 50 \)
• (3) \( 2 \times 10^{-1} \)
• (4) \( 5.0 \)

The values of \( a \) and \( b \) in the solubility product equation for barium phosphate are:

• (1) \( 7, 5 \)
• (2) \( 5, 7 \)
• (3) \( 5, 5 \)
• (4) \( 7, 7 \)

Hydrated sodium aluminum silicate is called:

• (1) Calgon
• (2) Zeolite
• (3) Dead burnt plaster
• (4) Kaolinite

Which one of the following statements is NOT correct about the compounds of alkaline earth metals?

• (1) Basic nature increases from Mg(OH)₂ to Ba(OH)₂
• (2) Thermal stability decreases from BeCO₃ to BaCO₃
• (3) Solubility of sulphates in water decreases from BeSO₄ to BaSO₄
• (4) Nitrates of these on heating give oxides

Consider the following standard electrode potentials (\(E^\circ\) in volts) in aqueous solution:



Based on this data, which of the following statements is correct?

• (1) \( Tl^{3+} \) is more stable than \( Al^{3+} \)
• (2) \( Tl^{1+} \) is more stable than \( Al^{3+} \)
• (3) \( Al^{1+} \) is more stable than \( Al^{3+} \)
• (4) \( Tl^{1+} \) is more stable than \( Al^{1+} \)

Which of the allotropic forms of carbon is aromatic in nature?

• (1) Diamond
• (2) Graphite
• (3) Buckminster fullerene
• (4) Coke

The enamel present on teeth becomes much harder due to the conversion of hydroxyapatite into fluorapatite. What are \( X \) and \( Y \)?

• (1) \( X = Ca(OH)_2, Y = CaF_2 \)
• (2) \( X = Ca(OH)_2, Y = CaCl_2 \)
• (3) \( X = Ca(OH)_2, Y = NaCl \)
• (4) \( X = CaO, Y = CaCl_2 \)

Number of deactivating groups among: \( -Cl, -SO_3H, -OH, -NHC_2H_5, -COOCH_3, -CH_3 \)

• (1) 4
• (2) 3
• (3) 2
• (4) 1

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

• (1)
• (2)
• (3)
• (4)

Correct Answer: (4)
View Solution

Identify the incorrect set from the following:

• (1) SiO₂, covalent solid, insulator, high melting point
• (2) MgO, covalent solid, insulator, high melting point
• (3) H₂O (ice), molecular solid, insulator, low melting point
• (4) Ag(s), metallic solid, conductor, high melting point

What is the boiling point (in K) of the urea solution from the given graph?

• (1) \( T_1 \)
• (2) \( T_2 \)
• (3) \( T_3 \)
• (4) \( T_4 \)

Evaluate the correctness of the given statements.

• (1) Both Statement-I and Statement-II are correct
• (2) Both Statement-I and Statement-II are incorrect
• (3) Statement-I is correct but Statement-II is not correct
• (4) Statement-I is not correct but Statement-II is correct

The standard reduction potentials of \( 2H^+/H_2 \), \( Cu^{2+}/Cu \), \( Zn^{2+}/Zn \), and \( NO_3^-/HNO_3 \) are 0.0 V, 0.34 V, -0.76 V, and 0.97 V respectively. Identify the correct statements from the following:

I. \( H^+ \) does not oxidize \( Cu \) to \( Cu^{2+} \).

II. \( Zn \) reduces \( Cu^{2+} \) to \( Cu \).

III. \( NO_3^- \) oxidizes \( Cu \) to \( Cu^{2+} \).

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

A → P is a zero-order reaction. At 298 K, the rate constant of the reaction is \( 1 \times 10^{-3} \) mol L\(^{-1}\) s\(^{-1}\).
Initial concentration of 'A' is 0.1 mol L\(^{-1}\). What is the concentration of 'A' after 10 sec?

• (1) 0.09 mol L\(^{-1}\)
• (2) 0.099 mol L\(^{-1}\)
• (3) 0.087 mol L\(^{-1}\)
• (4) 0.011 mol L\(^{-1}\)

Match List - I with List - II:

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

Identify the method of preparation of a colloidal sol from the following:

• (1) Ultrafiltration
• (2) Peptisation
• (3) Dialysis
• (4) Electro-dialysis

The flux used in the preparation of wrought iron from cast iron in reverberatory furnace is:

• (1) SiO\(_2\)
• (2) CaCO\(_3\)
• (3) C
• (4) NaCN

X, Y are oxoacids of phosphorous. The number of P – OH bonds in X, Y respectively is:

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

Which of the following occurs with KMnO\(_4\) in neutral medium?

• (1) Oxidation of oxalate ion
• (2) Precipitation of sulfur from hydrogen sulfide
• (3) Oxidation of Fe\(^{2+}\) to Fe\(^{3+}\)
• (4) Oxidation of iodide to iodate

Cobalt (III) chloride forms a green-colored complex ‘X’ with NH\(_3\). Number of moles of AgCl formed when excess AgNO\(_3\) solution is added to 100 mL of 1M solution of ‘X’ is:

• (1) 0.3
• (2) 0.2
• (3) 0.1
• (4) 1

The correctly matched set of the following is:

• (1) Polystyrene – copolymer – thermoplastic
• (2) Bakelite – addition polymer – thermosetting
• (3) Nylon 6 – Homopolymer – fibre
• (4) Buna-N – Homopolymer – elastomer

Identify the correctly matched set from the following:

• (1) Vitamin A – Water soluble – Xerophthalmia
• (2) Vitamin B\(_6\) – Water soluble – Scurvy
• (3) Vitamin D – Fat soluble – Rickets
• (4) Vitamin C – Fat soluble – Convulsions

Given below are two statements:

I. Cytosine and guanine are formed in equal quantities in DNA hydrolysis.

II. Adenine and uracil are formed in equal quantities in RNA hydrolysis.

The correct answer is:

• (1) Statements I, II both are correct
• (2) Statements I, II both are incorrect
• (3) Statement I is correct but statement II is incorrect
• (4) Statement I is incorrect but statement II is correct

Identify the correctly matched pair from the following:

• (1)
• (2)
• (3) Salt of propanoic acid – Antioxidant
• (4) Veronal – Analgesic

Correct Answer:
View Solution

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

• (1)
• (2)
• (3)
• (4)

Correct Answer:
View Solution

Hydrolysis of an alkyl bromide X (C\(_4\)H\(_9\)Br) follows first-order kinetics. Reaction of X with Mg in dry ether followed by treatment of D\(_2\)O gave Y. What is Y?

• (1)
• (2)
• (3)
• (4)

An alcohol \( X \) (\( C_4H_{10}O \)) does not give turbidity with conc. HCl and ZnCl\(_2\) at room temperature. \( X \) on reaction with reagent \( Y \) gives \( Z \). What are \( X \), \( Y \), and \( Z \) respectively?

• (1)
• (2)
• (3)
• (4)

Correct Answer:
View Solution

Which of the following sets of reagents convert toluene to benzaldehyde?

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

What are X and Y respectively in the following reactions?

• (1) \( H^+, H^+ \)
• (2) \( H^+, Pyridine \)
• (3) \( Pyridine, H^+ \)
• (4) \( Pyridine, Pyridine \)

IUPAC names of the following compounds A and B are:

• (1) A = But-3-en-2-amine, B = 4-N, N-dimethylaminochlorobenzene
• (2) A = But-1-en-3-amine, B = 4-N, N-dimethylaminochlorobenzene
• (3) A = But-1-en-3-amine, B = 4-Chloro-N, N-dimethylbenzenamine
• (4) A = But-1-en-3-amine, B = 4-Chloro-N, N-dimethylbenzenamine