AP EAPCET (AP EAMCET) 2025 Question Paper May 22 Shift 2(Soon): Download Solutions with Answer Key

The set of all real values of \(x\) for which \(f(x) = \sqrt{\frac{|x|-2}{|x|-3}}\) is a well defined function is

• (1) \( (-3,-2] \cup (2,3] \)
• (2) \( \mathbb{R} - ([-3,-2) \cup (2,3]) \)
• (3) \( \mathbb{R} - [-3,3] \)
• (4) \( (-3,3) \)

\(f(x)\) is a quadratic polynomial satisfying the condition \( f(x) + f\left(\frac{1}{x}\right) = f(x)f\left(\frac{1}{x}\right) \). If \(f(-1)=0\), then the range of \(f\) is

• (1) \( [1, \infty) \)
• (2) \( [-1,1] \)
• (3) \( (-\infty, 1] \)
• (4) \( \mathbb{R} \)

\( \sum_{k=1}^{n} k(k+1)(k+2)...(k+r-1) = \)

(1) \( \frac{n(n+1)(n+2)...(n+r)}{r+1} \) 
(2) \( \frac{n(n+1)(n+2)...(n+r-1)}{r} \) 
(3) \( \frac{n(n+1)(n+2)...(n+r+1)}{r+1} \) 
(4) \( \frac{n(n+1)(n+2)...2n}{2n+1} \)

If \( A = \begin{pmatrix} 1 & 2 & 3
1 & 3 & 5
2 & 1 & 6 \end{pmatrix} \) and \( |adj(adj(A))|(adj A)^{-1} = kA \), then k =

(1) 1296 
(2) 216 
(3) 36 
(4) 432

If the values \( x = \alpha, y = \beta, z = \gamma \) satisfy all the 3 equations \(x+2y+3z=4\), \(3x+y+z=3\) and \(x+3y+3z=2\), then \(3\alpha + \gamma = \)

(1) \( \beta \) 
(2) \( 2\beta \) 
(3) \( 1-2\beta \) 
(4) \( 2\beta+1 \)

The number of solutions of the system of equations \(2x+y-z=7\), \(x-3y+2z=1\), \(x+4y-3z=5\) is

(1) 1 
(2) 0 
(3) Infinite 
(4) 2

The points in the Argand plane represented by the complex numbers \(4\hat{i}+3\hat{j}\), \(6\hat{i}-2\hat{j}-3\hat{k}\) and \(\hat{i}-\hat{j}-3\hat{k}\) form

• (1) a right - angled triangle
  (2) a right - angled isosceles triangle
  (3) an equilateral triangle
  (4) an isosceles triangle

If \( z = x+iy \) and \(x^2+y^2=1\), then \( \frac{1+x+iy}{1+x-iy} = \)

(1) \( \bar{z} \) 
(2) \( z \) 
(3) \( z+1 \) 
(4) \( z-1 \)

If \( x^6 = (\sqrt{3}-i)^5 \), then the product of all of its roots is

(1) \( 2^5(\sqrt{3}+i) \) 
(2) \( \frac{2^6}{\sqrt{3}+i} \) 
(3) \( 2^6(\sqrt{3}-i) \) 
(4) \( \frac{2^6}{\sqrt{3}-i} \)

Question 10:

If \( \alpha \neq 0 \) and zero are the roots of the equation \( x^2 - 5kx + (6k^2-2k) = 0 \), then \( \alpha = \)

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

The set of all real values of x satisfying the inequation \( \frac{8x^2-14x-9}{3x^2-7x-6} > 2 \) is
Options

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

When the roots of \( x^3 + \alpha x^2 + \beta x + 6 = 0 \) are increased by 1, if one of the resultant values is the least root of \( x^4 - 6x^3 + 11x^2 - 6x = 0 \), then
Options

• (1) \( \alpha - \beta + 5 = 0 \)
• (2) \( \alpha + \beta + 7 = 0 \)
• (3) \( 2\alpha + \beta + 7 = 0 \)
• (4) \( 2\alpha + 3\beta - 1 = 0 \) 

Let 'a' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation \( x^3 - ax^2 + ax - 1 = 0 \) is identical with this cubic equation, then 'a' =
Options

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

If \( {}^{27}P_{r+7} = 7722 \cdot {}^{25}P_{r+4} \), then r =
Options

• (1) \( 9 \)
• (2) \( 12 \)
• (3) \( 11 \)
• (4) \( 10 \) 

If the number of diagonals of a regular polygon is 35, then the number of sides of the polygon is
Options

• (1) \( 12 \)
• (2) \( 9 \)
• (3) \( 10 \)
• (4) \( 11 \) 

If four letters are chosen from the letters of the word ASSIGNMENT and are arranged in all possible ways to form 4 letter words (with or without meaning), then total number of such words that can be formed is
Options

• (1) \( 1680 \)
• (2) \( 2184 \)
• (3) \( 2196 \)
• (4) \( 2190 \) 

The terms containing \( x^r y^s \) (for certain r and s) are present in both the expansions of \( (x+y^2)^{13} \) and \( (x^2+y)^{14} \). If \( \alpha \) is the number of such terms, then the sum \( \sum_{r,s} \alpha (r+s) = \) (Note: The sum is over the common terms)
Options

• (1) \( 27 \)
• (2) \( 40 \)
• (3) \( 18 \)
• (4) \( 35 \) 

The coefficient of \( x^3 \) in the power series expansion of \( \frac{1+4x-3x^2}{(1+3x)^3} \) is
Options

• (1) \( -27 \)
• (2) \( 27 \)
• (3) \( 153 \)
• (4) \( -153 \) 

If \( \frac{ax+5}{(x^2+b)(x+3)} = \frac{x+21}{12(x^2+b)} + \frac{c}{12(x+3)} \), then \( b^2 = \)
Options

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

If \( \alpha, \beta \) are the acute angles such that \( \frac{\sin \alpha}{\sin \beta} = \frac{6}{5} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{9}{5\sqrt{5}} \) then \( \sin \alpha = \)
Options

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

If \(2\sin x - \cos 2x = 1\), then \( (3 - 2\sin^2x) = \)
Options

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

If \( \left(\frac{\sin 3\theta}{\sin \theta}\right)^2 - \left(\frac{\cos 3\theta}{\cos \theta}\right)^2 = a \cos b\theta \), then \( a : b = \)
Options

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

If \( x \ne (2n+1)\frac{\pi}{4} \), then the general solution of \( \cos x + \cos 3x = \sin x + \sin 3x \) is
Options

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

If \( \frac{1}{2} \sin^{-1} \left( \frac{3\sin 2\theta}{5+4\cos 2\theta} \right) = \tan^{-1} x \) then \( x = \)
Options

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

If \( sech^{-1}x = \log 2 \) and \( cosech^{-1}y = -\log 3 \), then \( (x+y) = \)
Options

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

If the sides a,b,c of the triangle ABC are in harmonic progression, then \( cosec^2 A/2, cosec^2 B/2, cosec^2 C/2 \) are in
Options

• (1) Arithmetic-geometric progression
• (2) Arithmetic progression
• (3) Geometric progression
• (4) Harmonic progression Correct Answer

In \( \triangle ABC \), if \( r = 3 \) and \( R = 5 \) then \( \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \)
Options

• (1) \( \frac{1}{30} \)
• (2) \( \frac{12}{15} \)
• (3) \( \frac{1}{15} \)
• (4) \( \frac{5}{36} \) 

An aeroplane is flying at a constant speed, parallel to the horizontal ground at a height of 5 kms. A person on the ground observed that the angle of elevation of the plane is changed from \(15^\circ\) to \(30^\circ\) in the duration of 50 seconds, then the speed of the plane (in kmph) is
Options

• (1) 100
• (2) 720
• (3) 360
• (4) 540 Correct Answer

If the vector \( \vec{v} = \vec{i} - 7\vec{j} + 2\vec{k} \) is along the internal bisector of the angle between the vectors \( \vec{a} \) and \( \vec{b} = -2\vec{i} - \vec{j} + 2\vec{k} \) and the unit vector along \( \vec{a} \) is \( \hat{a} = x\vec{i} + y\vec{j} + z\vec{k} \) then \( x = \)
Options

• (1) 0
• (2) \( \frac{7}{9} \)
• (3) \( \frac{1}{9} \)
• (4) \( \frac{5}{3} \) 

If \( \vec{a} = 2\vec{i} - \vec{j} + 6\vec{k} \); \( \vec{b} = \vec{i} - \vec{j} + \vec{k} \) and \( \vec{c} = 3\vec{j} - \vec{k} \), then \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \)
Options

• (1) \( 20\vec{i} + 3\vec{j} - 4\vec{k} \)
• (2) \( 20\vec{i} - 3\vec{j} + 4\vec{k} \)
• (3) \( 3\vec{i} + 20\vec{j} - 4\vec{k} \)
• (4) \( 4\vec{i} + 20\vec{j} - 3\vec{k} \) 

Let \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \) and \( \vec{b} = \vec{i} + \vec{j} \) be two vectors. If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}| \), \( |\vec{c} - \vec{a}| = 2\sqrt{2} \) and the angle between \( \vec{a} \times \vec{b} \) and \( \vec{c} \) is \( 30^\circ \), then \( |(\vec{a} \times \vec{b}) \times \vec{c}| = \)
Options

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

For a positive real number p, if the perpendicular distance from a point \( -\vec{i} + p\vec{j} - 3\vec{k} \) to the plane \( \vec{r} \cdot (2\vec{i} - 3\vec{j} + 6\vec{k}) = 7 \) is 6 units, then p =
Options

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

\( (\vec{a}+2\vec{b}-\vec{c}) \cdot ((\vec{a}-\vec{b}) \times (\vec{a}-\vec{b}-\vec{c})) = \)
Options

• (1) \( [\vec{a}\vec{b}\vec{c}] \)
• (2) \( 3[\vec{a}\vec{b}\vec{c}] \)
• (3) \( [\vec{a}\vec{b}\vec{c}]^2 \)
• (4) \( 2[\vec{a}\vec{b}\vec{c}] \) 

Variance of the following discrete frequency distribution is

Options

• (1) \( \frac{463}{15} \)
• (2) \( \frac{838}{15} \)
• (3) \( \frac{44}{5} \)
• (4) \( \frac{88}{15} \) 

An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is

(1) \( \frac{5}{256} \)
(2) \( \frac{5}{128} \) 
(3) \( \frac{5}{64} \) 
(4) \( \frac{5}{32} \)

A box contains twelve balls of which 4 are red, 5 are green, and 3 are white. If three balls are drawn at random, the probability that exactly 2 balls have the same color is
(1) \( \frac{27}{44} \)
(2) \( \frac{29}{44} \) 
(3) \( \frac{17}{22} \) 
(4) \( \frac{31}{44} \)

There are three families \( F_1, F_2, F_3 \). \( F_1 \) has 2 boys and 1 girl; \( F_2 \) has 1 boy and 2 girls; \( F_3 \) has 1 boy and 1 girl. A family is randomly chosen and a child is chosen from that family randomly. If it is known that the child is a girl, the probability that she is from \( F_3 \) is

(1) \( \frac{4}{9} \)
(2) \( \frac{2}{9} \) 
(3) \( \frac{3}{7} \) 
(4) \( \frac{5}{7} \)

An urn A contains 4 white and 1 black ball; urn B contains 3 white and 2 black balls; urn C contains 2 white and 3 black balls. One ball is transferred randomly from A to B; then one ball is transferred randomly from B to C. Finally, a ball is drawn randomly from C. Find the probability that it is black.

(1) \( \frac{7}{12} \)
(2) \( \frac{89}{180} \) 
(3) \( \frac{101}{180} \) 
(4) \( \frac{17}{36} \)

If the probability distribution of a discrete random variable X is given by \( P(X=k) = \frac{2^{-k}(3k+1)}{2^c} \), k = 0, 1, 2, ..., \( \infty \) then P(X \( \le \) c) = (The expression seems to be \( \frac{2^{-k}(3k+1)}{K} \) where K is a constant, or \(2^c\) is part of the constant. Assuming \(2^c\) is the normalization constant \(K\).)
Options

• (1) \( \frac{c}{5} \)
• (2) \( \frac{c}{4} \)
• (3) \( \frac{c+2}{5} \)
• (4) \( \frac{c-2}{7} \) 

In a binomial distribution, if \(n=4\) and \( P(X=0) = \frac{16}{81} \), then \( P(X=4) = \)
Options

• (1) \( \frac{1}{8} \)
• (2) \( \frac{1}{27} \)
• (3) \( \frac{1}{16} \)
• (4) \( \frac{1}{81} \)

If A(1,0), B(0,-2), C(2,-1) are three fixed points, then the equation of the locus of a point P such that area of \( \triangle PAB \) is equal to area of \( \triangle PAC \) is

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

The transformed equation of \( 3x^2 - 4xy = r^2 \) when the coordinate axes are rotated about the origin through an angle of \( \tan^{-1}(2) \) in positive direction is

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

A line \(L_1\) passing through the point of intersection of the lines \(x-2y+3=0\) and \(2x-y=0\) is parallel to the Line \(L_2\). If \(L_2\) passes through origin and also through the point of intersection of the lines \(3x-y+2=0\) and \(x-3y-2=0\), then the distance between the lines \(L_1\) and \(L_2\) is

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

If the lines \(x+y-2=0\), \(3x-4y+1=0\) and \(5x+ky-7=0\) are concurrent at \((\alpha, \beta)\), then equation of the line concurrent with the given lines and perpendicular to \(kx+y-k=0\) is

• (1) \( x-3y=-2 \)
• (2) \( x+4y=5 \)
• (3) \( x+6y=7 \)
• (4) \( x-2y=-1 \)

If two sides of a triangle are represented by \( 3x^2 - 5xy + 2y^2 = 0 \) and its orthocentre is (2,1), then the equation of the third side is

• (1) \( 2x+y-4=0 \)
• (2) \( 6x+3y-13=0 \)
• (3) \( 8x+4y-17=0 \)
• (4) \( 10x+5y-21=0 \)

If \( ax^2 + 2hxy - 2ay^2 + 3x + 15y - 9 = 0 \) represents a pair of lines intersecting at (1,1), then ah =

• (1) 14
• (2) -15
• (3) -7
• (4) 9

A circle passing through the point (1,0) makes an intercept of length 4 units on X-axis and an intercept of length \(2\sqrt{11}\) units on Y-axis. If the centre of the circle lies in the fourth quadrant, then the radius of the circle is

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

If \( \left(\frac{1}{10}, \frac{-1}{5}\right) \) is the inverse point of a point (-1, 2) with respect to the circle \( x^2+y^2-2x+4y+c=0 \) then c =

• (1) 4
• (2) -4
• (3) 2
• (4) -2

If the equation of the circle lying in the first quadrant, touching both the coordinate axes and the line \( \frac{x}{3} + \frac{y}{4} = 1 \) is \( (x-c)^2+(y-c)^2=c^2 \), then c =

• (1) 1 or 4
• (2) 2 or 3
• (3) 1 or 6
• (4) 2 or 5

If the point of contact of the circles \( x^2+y^2-6x-4y+9=0 \) and \( x^2+y^2+2x+2y-7=0 \) is \( (\alpha, \beta) \), then \( 7\beta = \)

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

If the circles \( x^2+y^2-2\lambda x - 2y - 7 = 0 \) and \( 3(x^2+y^2) - 8x + 29y = 0 \) are orthogonal, then \( \lambda = \)

• (1) 4
• (2) 3
• (3) 2
• (4) 1

If the perpendicular distance from the focus of a parabola \(y^2=4ax\) to its directrix is \( \frac{3}{2} \), then the equation of the normal drawn at \( (4a, -4a) \) is

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

Let \( A_1 \) be the area of the given ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Let \( A_2 \) be the area of the region bounded by the curve which is the locus of mid point of the line segment joining the focus of the ellipse and a point P on the given ellipse, then \( A_1 : A_2 = \)

• (1) 3:2
• (2) a:b
• (3) 4:1
• (4) 2a:3b

If the equation of the tangent of the hyperbola \( 5x^2 - 9y^2 - 20x - 18y - 34 = 0 \) which makes an angle \( 45^\circ \) with the positive X-axis in positive direction is \( x+by+c=0 \) then \( b^2+c^2 = \)

• (1) 2 or 13
• (2) 5 or 26
• (3) 2 or 26
• (4) 26 or 28

If the distance between the foci of a hyperbola H is 26 and distance between its directrices is \( \frac{50}{13} \), then the eccentricity of the conjugate hyperbola of the hyperbola H is

• (1) \( \frac{13}{12} \)
• (2) \( \frac{25}{17} \)
• (3) \( \frac{13}{7} \)
• (4) \( \frac{25}{13} \)

If Q \( (\alpha, \beta, \gamma) \) is the harmonic conjugate of the point P(0,-7,1) with respect to the line segment joining the points (2,-5,3) and (-1,-8,0), then \( \alpha - \beta + \gamma = \)

• (1) 4
• (2) 3
• (3) 2
• (4) 1

On a line with direction cosines l, m, n, \( A(x_1, y_1, z_1) \) is a fixed point. If \( B=(x_1+4kl, y_1+4km, z_1+4kn) \) and \( C=(x_1+kl, y_1+km, z_1+kn) \) (\(k>0\)) then the ratio in which the point B divides the line segment joining A and C is

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

If the line of intersection of the planes \(2x+3y+z=1\) and \(x+3y+2z=2\) makes an angle \( \alpha \) with the positive x-axis, then \( \cos \alpha = \)

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

\([x]\) denotes the greatest integer less than or equal to x. If \(\{x\}=x-[x]\) and \( \lim_{x\to 0} \frac{\sin^{-1}(x+[x])}{2-\{x\}} = \theta \), then \( \sin\theta + \cos\theta = \)

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

\( \lim_{n\to\infty} \frac{1}{n^3} \sum_{k=1}^{n} k^2 x = \)

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

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} a - \frac{\sin[x-1]}{x-1} & , if x > 1
1 & , if x = 1
b - \frac{\sin([x-1] - [x-1]^3)}{([x-1]^2)} & , if x < 1 \end{cases} \]
where \([t]\) denotes the greatest integer less than or equal to t. If f is continuous at \(x=1\), then \(a+b=\)

• (1) 0
• (2) 1
• (3) 2
• (4) 3

If g is the inverse of the function f(x) and \( g(x) = x + \tan x \) then, \( f'(x) = \)

• (1) \( 1+\sec^2x \)
• (2) \( \frac{1}{1+\sec^2f(x)} \)
• (3) \( \frac{1}{1+\sec^2g(x)} \)
• (4) \( 1+\sec^2f(x) \)

If \( \sqrt{x-xy} + \sqrt{y-xy} = 1 \), then \( \frac{dy}{dx} = \)

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

If \( y = \tan^{-1}\left(\frac{x}{1+2x^2}\right) + \tan^{-1}\left(\frac{x}{1+6x^2}\right) \), then \( \frac{dy}{dx} = \)

• (1) \( \frac{4}{16x^2+1} - \frac{3}{9x^2+1} \)
• (2) \( \frac{3}{9x^2+1} - \frac{1}{x^2+1} \)
• (3) \( \frac{3}{9x^2+1} - \frac{2}{4x^2+1} \)
• (4) \( \frac{1}{9x^2+1} - \frac{1}{x^2+1} \)

If the tangent drawn at the point \( (x_1,y_1) \), \(x_1,y_1 \in N \) on the curve \( y = x^4 - 2x^3 + x^2 + 5x \) passes through origin, then \( x_1+y_1 = \)

• (1) 5
• (2) 4
• (3) 7
• (4) 6

Which one of the following functions is monotonically increasing in its domain?

• (1) \( f(x) = \log(1+x) - x + \frac{x^2}{2} \)
• (2) \( g(x) = 2 \tan^{-1}x - x - 1 \)
• (3) \( h(x) = 4\cos x + x \)
• (4) \( u(x) = \log(1+x) - \frac{x}{x+1} \)

If \( \beta \) is an angle between the normals drawn to the curve \( x^2+3y^2=9 \) at the points \( (3\cos\theta, \sqrt{3}\sin\theta) \) and \( (-3\sin\theta, \sqrt{3}\cos\theta) \), \( \theta \in \left(0, \frac{\pi}{2}\right) \), then

• (1) \( \tan\beta = \frac{1}{\sqrt{3}} \sec 2\theta \)
• (2) \( \cot\beta = \sqrt{3} \operatorname{cosec} 2\theta \)
• (3) \( \sqrt{3}\cot\beta = \sin 2\theta \)
• (4) \( \cot\beta = \frac{1}{\sqrt{2}} \sec 2\theta \)

If the area of a right angled triangle with hypotenuse 5 is maximum, then its perimeter is

• (1) 12
• (2) \( 2\sqrt{3}+\sqrt{13}+5 \)
• (3) \( 7+\sqrt{21} \)
• (4) \( 5(\sqrt{2}+1) \)

\( \int \left( \sum_{r=0}^{\infty} \frac{x^r 2^r}{r!} \right) dx = \)

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

\( \int \frac{dx}{12\cos x + 5\sin x} = \)

• (1) \( \frac{1}{13}\log\left|\tan\left(\frac{\pi}{4} + \frac{x}{2} - \frac{1}{2}\tan^{-1}\frac{5}{12}\right)\right|+c \)
• (2) \( \frac{5}{12}\log\left|\tan\left(\frac{\pi}{4} + \frac{x}{2} - \frac{1}{2}\tan^{-1}\frac{5}{12}\right)\right|+c \)
• (3) \( \frac{1}{13}\log\left|\tan\left(\frac{\pi}{4} + \frac{x}{2} + \frac{1}{2}\tan^{-1}\frac{5}{12}\right)\right|+c \)
• (4) \( \frac{5}{12}\log\left|\tan\left(\frac{\pi}{4} + \frac{x}{2} + \frac{1}{2}\tan^{-1}\frac{5}{12}\right)\right|+c \)

If \( \int \frac{\cos^3 x}{\sin^2 x + \sin^4 x} dx = c - \operatorname{cosec} x - f(x) \), then \( f\left(\frac{\pi}{2}\right) = \)

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

\( \int \frac{13\cos 2x - 9\sin 2x}{3\cos 2x - 4\sin 2x} dx = \)

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

\( \int \sqrt{x^2+x+1} \ dx \)

• (1) \( \frac{(2x+1)}{4}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• (2) \( \frac{x+1}{4}\sqrt{x^2+x+1} + \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• (3) \( \frac{x+1}{4}\sqrt{x^2+x+1} - \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)
• (4) \( \frac{(2x+1)}{4}\sqrt{x^2+x+1} - \frac{3}{8}\sinh^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c \)

If \( k \in N \) then \( \lim_{n\to\infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \dots + \frac{1}{kn} \right] = \) (Note: The last term should be \( \frac{1}{n+ (k-1)n} = \frac{1}{kn} \) or sum up to \(n+(k-1)n\). The given form \(1/kn\) as the endpoint of the sum means sum from \(r=1\) to \((k-1)n\). The sum is usually \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \). If the last term is \( \frac{1}{kn} \), it means \( n+r = kn \implies r = (k-1)n \). So it's \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \).)
Let's assume the sum goes up to \( \frac{1}{n+(k-1)n} = \frac{1}{kn} \).
So the sum is \( \sum_{r=1}^{(k-1)n} \frac{1}{n+r} \). No, this seems to be \( \frac{1}{n+1} + \dots + \frac{1}{n+(kn-n)} \).
The sum should be written as \( \sum_{i=1}^{(k-1)n} \frac{1}{n+i} \). The dots imply the denominator goes up.
The last term is \( \frac{1}{kn} \). This means the sum is actually \( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{n+(k-1)n} \).
The number of terms is \( (k-1)n \).

• (1) \( \log(k+1) \)
• (2) \( \log k \)
• (3) \( \log(k+5) \)
• (4) \( \log(k+1) - \log 6 \)

\( \int_{-1}^{4} \sqrt{\frac{4-x}{x+1}} \ dx = \)

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

\( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4\sin^2 x} dx = \)

• (1) \( \frac{\pi}{2} - \frac{1}{3}\tan^{-1}2 \)
• (2) \( \frac{\pi}{4} - \frac{4}{3}\tan^{-1}2 \)
• (3) \( \frac{\pi}{6} + \frac{2}{3}\tan^{-1}2 \)
• (4) \( \frac{\pi}{12} + \frac{2}{3}\tan^{-1}2 \) 

\( \int_{5\pi}^{25\pi} |\sin 2x + \cos 2x| \ dx = \)

• (1) \( 20\sqrt{2} \)
• (2) \( 10\sqrt{2} \)
• (3) \( 40\sqrt{2} \)
• (4) \( 80\sqrt{2} \)

The differential equation of the family of circles passing through the origin and having centre on X-axis is

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

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

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

The general solution of the differential equation \( \frac{dy}{dx} + \frac{\sec x}{\cos x + \sin x}y = \frac{\cos x}{1+\tan x} \) is

• (1) \( (\cos x + \sin x)y = \sin x + c \)
• (2) \( (\cos x + \sin x)y = \cos x + c \)
• (3) \( (1+\tan x)y = \cos x + c \)
• (4) \( \sec x(\cos x + \sin x)y = \sin x + c \)

The number of significant figures in the simplification of \( \frac{0.501}{0.05}(0.312-0.03) \) is

• (1) 1
• (2) 3
• (3) 2
• (4) 5

If the displacement 'x' of a body in motion in terms of time 't' is given by \(x = A\sin(\omega t + \theta)\), then the minimum time at which the displacement becomes maximum is

• (1) \( \frac{1}{\omega}\left[\frac{\pi}{2} - \theta\right] \)
• (2) \( \frac{1}{\omega}\left[\frac{2\omega}{\pi} - \theta\right] \)
• (3) \( \frac{1}{\omega}\left[\frac{\pi}{2} - 1\right] \)
  (4) \( \frac{1}{\omega}\left[\theta - \frac{\omega}{\pi^2}\right] \)

If the magnitude of a vector \( \vec{p} \) is 25 units and its y-component is 7 units, then its x-component is

• (1) 24 units
• (2) 18 units
• (3) 32 units
• (4) 16 units

The height of ceiling in an auditorium is 30 m. A ball is thrown with a speed of \( 30 \, m s^{-1} \) from the entrance such that it just moves very near to the ceiling without touching it and then it reaches the ground at the end of the auditorium. Then the length of auditorium is [Acceleration due to gravity \( = 10 \, m s^{-2} \)]

• (1) \( 60\sqrt{2} \, m \)
• (2) \( 30\sqrt{2} \, m \)
• (3) \( 70\sqrt{2} \, m \)
• (4) \( 100\sqrt{2} \, m \)

A balloon with mass 'm' is descending vertically with an acceleration 'a' (where a \( < \) g). The mass to be removed from the balloon, so that it starts moving vertically up with an acceleration 'a' is

• (1) \( \frac{ma}{g+a} \)
• (2) \( \frac{ma}{g-a} \)
• (3) \( \frac{2ma}{g+a} \)
• (4) \( \frac{2ma}{g-a} \)

A conveyor belt is moving horizontally with a velocity of \( 2 \, m s^{-1} \). If a body of mass 10 kg is kept on it, then the distance travelled by the body before coming to rest is (The coefficient of kinetic friction between the belt and the body is 0.2 and acceleration due to gravity is \( 10 \, m s^{-2} \))

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

Two bodies A and B of masses 20 kg and 5 kg respectively are at rest. Due to the action of a force of 40 N separately, if the two bodies acquire equal kinetic energies in times \( t_A \) and \( t_B \) respectively, then \( t_A : t_B = \)

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

A crane of efficiency 80% is used to lift 8000 kg of coal from a mine of depth 108 m. If the time taken by the crane to lift the coal is one hour, then the power of the crane (in kW) is (Acceleration due to gravity \( = 10 \, m s^{-2} \))

• (1) 5
• (2) 4
• (3) 6
• (4) 3

Three blocks A, B and C are arranged as shown in the figure such that the distance between two successive blocks is 10 m. Block A is displaced towards block B by 2 m and block C is displaced towards block B by 3 m. The distance through which the block B should be moved so that the centre of mass of the system does not change is

• (1) 1.4 m, towards block C
• (2) 1.5 m, towards block A
• (3) 2 m, towards block A
• (4) 1 m, towards block C

A solid sphere of mass 4 kg and radius 28 cm is on an inclined plane. If the acceleration of the sphere when it rolls down without sliding is \( 3.5 \, m s^{-2} \), then the acceleration of the sphere when it slides down without rolling is

• (1) \( 2.5 \, m s^{-2} \)
• (2) \( 3.5 \, m s^{-2} \)
• (3) \( 1.7 \, m s^{-2} \)
• (4) \( 4.9 \, m s^{-2} \)

If the maximum velocity and maximum acceleration of a particle executing simple harmonic motion are respectively \( 5 \, m s^{-1} \) and \( 10 \, m s^{-2} \), then the time period of oscillation of the particle is

• (1) \( \pi \, s \)
• (2) \( 2\pi \, s \)
• (3) \( 2 \, s \)
  (4) \( 1 \, s \)

A body of mass 1 kg is suspended from a spring of force constant \( 600 \, N m^{-1} \). Another body of mass 0.5 kg moving vertically upwards hits the suspended body with a velocity of \( 3 \, m s^{-1} \) and embedded in it. The amplitude of motion is

• (1) 5 cm
• (2) 15 cm
• (3) 10 cm
• (4) 8 cm

Two satellites A and B are revolving around the earth in orbits of heights \(1.25R_E\) and \(19.25R_E\) from the surface of earth respectively, where \(R_E\) is the radius of the earth. The ratio of the orbital speeds of the satellites A and B is

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

When a wire made of material with Young's modulus Y is subjected to a stress S, the elastic potential energy per unit volume stored in the wire is

• (1) \( \frac{YS}{2} \)
• (2) \( \frac{S^2Y}{2} \)
• (3) \( \frac{S^2}{2Y} \)
• (4) \( \frac{S}{2Y} \)

An aeroplane of mass \( 4.5 \times 10^4 \) kg and total wing area of \( 600 \, m^2 \) is travelling at a constant height. The pressure difference between the upper and lower surfaces of its wings is (Acceleration due to gravity \( = 10 \, m s^{-2} \))

• (1) \( 500 \, N m^{-2} \)
• (2) \( 825 \, N m^{-2} \)
• (3) \( 600 \, N m^{-2} \)
• (4) \( 750 \, N m^{-2} \)

If the wavelengths of maximum intensity of radiation emitted by two black bodies A and B are \( 0.5 \, \mu m \) and \( 0.1 \, mm \) respectively, then ratio of the temperatures of the bodies A and B is

• (1) 5
• (2) 25
• (3) 100
• (4) 200

Water of mass 5 kg in a closed vessel is at a temperature of \(20 \, ^\circ C\). If the temperature of the water when heated for a time of 10 minutes becomes \(30 \, ^\circ C\), then the increase in the internal energy of the water is (Specific heat capacity of water \( = 4200 \, J kg^{-1} K^{-1} \))

• (1) 100 kJ
• (2) 420 kJ
• (3) 510 kJ
• (4) 210 kJ

A Carnot engine A working between temperatures 600 K and T (T \( < \) 600 K) and another Carnot engine B working between temperatures T (T \( > \) 400 K) and 400 K are connected in series. If the work done by both the engines is same, then T =

• (1) 550 K
• (2) 500 K
• (3) 575 K
• (4) 525 K

When an ideal diatomic gas is heated at constant pressure, the fraction of the heat utilised to increase the internal energy of the gas is

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

If the degrees of freedom of a gas molecule is 6, then the total internal energy of the gas molecule at a temperature of \( 47 \, ^\circ C \) (in eV) is (Boltzmann constant \( = 1.38 \times 10^{-23} \, J K^{-1} \))

• (1) \( 414 \times 10^{-4} \)
• (2) \( 828 \times 10^{-4} \)
• (3) \( 927 \times 10^{-4} \)
• (4) \( 572 \times 10^{-4} \)

When a stretched wire of fundamental frequency f is divided into three segments, the fundamental frequencies of these three segments are \(f_1\), \(f_2\) and \(f_3\) respectively. Then the relation among \(f, f_1, f_2, f_3\) and f is (Assume tension is constant) % "and f is" seems redundant

• (1) \( \sqrt{f} = \sqrt{f_1} + \sqrt{f_2} + \sqrt{f_3} \)
• (2) \( f = f_1 + f_2 + f_3 \)
• (3) \( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \)
• (4) \( \frac{1}{\sqrt{f}} = \frac{1}{\sqrt{f_1}} + \frac{1}{\sqrt{f_2}} + \frac{1}{\sqrt{f_3}} \)

Images of same size are formed by a convex lens when an object is placed either at 20 cm or 10 cm distance from the lens. The focal length of the lens is

• (1) 12 cm
• (2) 40 cm
• (3) 18 cm
• (4) 15 cm

In Young's double slit experiment, the wavelength of monochromatic light is increased by 20% and the distance between the two slits is decreased by 25%. If the initial fringe width is 0.3 mm, then the final fringe width is

• (1) 0.72 mm
• (2) 0.60 mm
• (3) 0.16 mm
• (4) 0.48 mm

Two charged conducting spheres of radii 5 cm and 10 cm have equal surface charge densities. If the electric field on the surface of the smaller sphere is E, then the electric field on the surface of the larger sphere is

• (1) 2E
• (2) 4E
• (3) 0.5E
• (4) E

As shown in the figure, if the values of the electric potential at three points A, B and C in a uniform electric field (\( \vec{E} \)) are \(V_A\), \(V_B\), and \(V_C\) respectively, then

• (1) \( V_A > V_B > V_C \)
• (2) \( V_A > V_C > V_B \)
• (3) \( V_C > V_B > V_A \)
• (4) \( V_C > V_A > V_B \)

As shown in the figure, the work done to move the charge 'Q' from point C to point D along the semi-circle CRD is

• (1) \( \frac{qQ}{4\pi\epsilon_0 d} \)
• (2) \( \frac{qQ}{2\pi\epsilon_0 d} \)
• (3) \( \frac{-qQ}{6\pi\epsilon_0 d} \)
• (4) \( \frac{-qQ}{4\pi\epsilon_0 d} \)

The length and area of cross-section of a copper wire are respectively 30 m and \( 6 \times 10^{-7} \, m^2 \). If the resistivity of copper is \( 1.7 \times 10^{-8} \, \Omega \, m \), then the resistance of the wire is

• (1) \( 0.51 \, \Omega \)
• (2) \( 0.68 \, \Omega \)
• (3) \( 0.85 \, \Omega \)
• (4) \( 0.75 \, \Omega \)

If current of 80 A is passing through a straight conductor of length 10 m, then the total momentum of electrons in the conductor is (mass of electron \( = 9.1 \times 10^{-31} \) kg and charge of electron \( = 1.6 \times 10^{-19} \) C)

• (1) \( 910 \times 10^{-9} \, Ns \)
• (2) \( 910 \times 10^{-11} \, Ns \)
• (3) \( 455 \times 10^{-9} \, Ns \)
• (4) \( 455 \times 10^{-11} \, Ns \)

In a wire of radius 1 mm, a steady current of 2 A uniformly distributed across the cross-section of the wire is flowing. Then the magnetic field at a point 0.25 mm from the centre of the wire is

• (1) \( 100 \, \mu T \)
• (2) \( 200 \, \mu T \)
• (3) \( 300 \, \mu T \)
• (4) \( 400 \, \mu T \)

The magnetic field at the centre of a current carrying circular coil of radius R is \( B_C \) and the magnetic field at a point on its axis at a distance R from its centre is \( B_A \). The value of \( \frac{B_C}{B_A} \) is

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

A short bar magnet of magnetic moment \( 10^4 \, J T^{-1} \) is free to rotate in a horizontal plane. The work done in rotating the magnet slowly from the direction parallel to a horizontal magnetic field of \( 4 \times 10^{-5} \, T \) to a direction \( 60^\circ \) to the direction of the field is

• (1) 0.2 J
• (2) 2.6 J
• (3) 0.4 J
• (4) 6.2 J

A metallic disc of radius 0.3 m is rotating with a constant angular speed of \( 60 \, rad s^{-1} \) in a plane perpendicular to a uniform magnetic field of \( 5 \times 10^{-2} \, T \). The emf induced between a point on the rim and the centre of the disc is

• (1) 0.06 V
• (2) 0.612 V
• (3) 1.35 V
• (4) 0.135 V

A resistor of \(450 \, \Omega\) and an inductor are connected in series to an ac source of frequency \( \frac{75}{\pi} \, Hz \). If the power factor of the circuit is 0.6, then the inductance connected in the circuit is

• (1) 6 mH
• (2) 4 H
• (3) 4 mH
• (4) 6 H

If the rms value of the electric field of electromagnetic waves at a distance of 3 m from a point source is \( 3 \, N C^{-1} \), then the power of the source is

• (1) 10.8 W
• (2) 8.1 W
• (3) 5.4 W
• (4) 2.7 W

If the threshold wavelength of light for photoelectric emission to take place from a metal surface is \( 6250 \, AA \), then the work function of the metal is (Planck's constant \( = 6.6 \times 10^{-34} \, Js \))

• (1) 3.98 eV
• (2) 1.98 eV
• (3) 2.98 eV
• (4) 4.98 eV

The ratio of the wavelengths of the first Lyman line and the second Balmer line of hydrogen atom is

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

Each nuclear fission of \( {}^{235}U \) releases 200 MeV of energy. If a reactor generates 1 MW power, then the rate of fission in the reactor is

• (1) \( 3.125 \times 10^6 \)
• (2) \( 3.125 \times 10^8 \)
• (3) \( 3.125 \times 10^{10} \)
• (4) \( 3.125 \times 10^{16} \)

When three NAND logic gates are connected as shown in the figure, then the logic gate equivalent to the circuit is

• (1) NOT
• (2) AND
• (3) OR
• (4) NOR

The device used for voltage regulation is

• (1) Zener diode
• (2) photo diode
• (3) light emitting diode
• (4) solar cell

For transmitting a signal of frequency 1000 kHz, the minimum length of the antenna is

• (1) 30 m
• (2) 50 m
• (3) 75 m
• (4) 1500 m

The difference between the radii of \(3^{rd}\) and \(2^{nd}\) orbit of H-atom is x pm. The difference between the radii of \(4^{th}\) and \(3^{rd}\) orbit of \( Li^{2+} \) ion is y pm. \(y:x\) is equal to

• (1) 15:7
• (2) 7:15
• (3) 3:1
• (4) 1:3

The de Broglie wavelength of an electron in the third Bohr orbit of H-atom is

• (1) \( 3\pi \times 5.29 \, pm \)
• (2) \( 4\pi \times 52.9 \, pm \)
• (3) \( 6\pi \times 52.9 \, pm \)
• (4) \( 2\pi \times 5.29 \, pm \)

The correct order of the non-metallic character among the elements B, C, N, F and Si is

• (1) B > C > Si > N > F
• (2) Si > C > B > N > F
• (3) F > N > C > B > Si
• (4) F > N > C > Si > B

How many of the following molecules have two lone pairs of electrons on central atom? SF\(_6\), BF\(_3\), ClF\(_3\), PCl\(_5\), BrF\(_5\), XeF\(_4\), H\(_2\)O, SF\(_4\)

• (1) 5
• (2) 4
• (3) 3
• (4) 2

The pair of molecules / ions with the same bond order value is

• (1) B\(_2\), C\(_2\)
• (2) O\(_2\), C\(_2\)
• (3) O\(_2^+\), O\(_2^-\)
• (4) H\(_2^+\), Li\(_2\)

At what temperature (in K) the rms velocity of SO\(_2\) molecules is equal to rms velocity of O\(_2\) molecules at \(27 \, ^\circ C\)?

• (1) 300
• (2) 1200
• (3) 600
• (4) 900

For one mole of an ideal gas an isochore is obtained. The slope of the isochore is \(0.082 \, atm K^{-1}\). What will be its pressure (in atm) when the temperature is 12.2 K? (R = \(0.082 \, L atm mol^{-1} K^{-1}\)).

• (1) 10.0
• (2) 0.1
• (3) 1.0
• (4) 0.5

Consider the following
A) 0.0025  B) 500.0  C) 2.0034
Number of significant figures in A, B and C respectively, are

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

Consider the following reaction
A(g) + 3B(g) \( \longrightarrow \) 2C(g); \( \Delta H^\ominus = -24 \) kJ.
At \(25 \, ^\circ C\) if \( \Delta G^\ominus \) of the reaction is -9 kJ, the standard entropy change (in JK\(^{-1}\)) of the same reaction at same temperature is

• (1) -5.33
• (2) -50.33
• (3) -500.33
• (4) -0.533

One mole of \( C_2H_5OH(l) \) was completely burnt in oxygen to form \( CO_2(g) \) and \( H_2O(l) \). The standard enthalpy of formation (\( \Delta_f H^\ominus \)) of \( C_2H_5OH(l) \), \( CO_2(g) \) and \( H_2O(l) \) is x, y, z kJ mol\(^{-1}\) respectively. What is \( \Delta_r H^\ominus \) (in kJ mol\(^{-1}\)) for this reaction?

• (1) \( (2y + 3z + x) \)
• (2) \( (2y - 3z + x) \)
• (3) \( (x - 2y - 3z) \)
• (4) \( (2y + 3z - x) \)

At \(25 ^\circ C\), \(K_a\) of formic acid is \(1.8 \times 10^{-4}\). What is the \(K_b\) of \( HCOO^- \)?

• (1) \( 1.8 \times 10^{-10} \)
• (2) \( 5.55 \times 10^{-4} \)
• (3) \( 5.55 \times 10^{-11} \)
• (4) \( 5.55 \times 10^{-12} \)

At T(K), the following gaseous equilibrium is established.
W + X \( \rightleftharpoons \) Y + Z
The initial concentration of W is two times to the initial concentration of X. The system is heated to T(K) to establish the equilibrium. At equilibrium the concentration of Y is four times to the concentration of X. What is the value of \(K_c\)?

• (1) 0.375
• (2) 1.333
• (3) 2.666
• (4) 5.333

4 mL of 'X volume' \( H_2O_2 \) on heating gives 80 mL of oxygen at STP. The value of X is

• (1) 10
• (2) 20
• (3) 15
• (4) 40

Compound 'X' is prepared commercially by the electrolysis of brine solution. Which of the following is not the use of 'X'?

• (1) Manufacture of paper
• (2) Petroleum refining
• (3) Antichlor
• (4) Mercirising cotton fabrics

Consider the following
Statement-I : \( Al_2O_3 \) is amphoteric in nature.
Statement-II : \( Tl_2O_3 \) is more basic than \( Ga_2O_3 \).
The correct answer is

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

Identify the incorrect order against the stated property.

• (1) Ge > Sn > Pb - Ionization enthalpy
• (2) Ge > Pb > Sn - Melting point
• (3) Pb > Sn > Ge - Density
• (4) Ge > Pb > Sn - Electrical resistivity

Question 137:

Among the following compounds, which one is not responsible for the depletion of ozone layer?

• (1) CH\(_4\)
• (2) CFC\(l_3\)
• (3) NO
• (4) Cl\(_2\)

Which method is used to purify liquids having very high boiling points and liquids which decompose at or below their boiling point?

• (1) Distillation
• (2) Fractional distillation
• (3) Distillation under reduced pressure
• (4) Steam distillation

What are X, Y, Z in the following reaction sequence?
But-2-ene \( \xrightarrow{X} \) Ethanoic acid \( \xrightarrow{Y} \) Ethanoyl chloride \( \xrightarrow{Benzene, Anhy. AlCl_3} \) Z

• (1) KMnO\(_4\) / H\(^+\); SOCl\(_2\); Acetophenone
• (2) KMnO\(_4\) / H\(^+\); Cl\(_2\); Propiophenone
• (3) Cold KMnO\(_4\); SOCl\(_2\); Propiophenone
• (4) Cold KMnO\(_4\); Cl\(_2\); Acetophenone

An element (atomic weight = 250 u) crystallises in a simple cubic lattice. If the density of the unit cell is \( 7.2 \, g cm^{-3} \), what is the radius (in \( AA \)) of the atom of the element? (\(N = 6.02 \times 10^{23} \, mol^{-1}\))

• (1) 4.04
• (2) 2.93
• (3) 1.93
• (4) 3.04

1.95 g of non-volatile and non-electrolyte solute dissolved in 100 g of benzene lowered the freezing point of it by 0.64 K. The molar mass of the solute (in g mol\(^{-1}\)) (\(K_f(C_6H_6) = 5.12 \, K kg mol^{-1}\)) is

• (1) 240
• (2) 156
• (3) 165
• (4) 265

At 298 K, 0.714 moles of liquid A is dissolved in 5.555 moles of liquid B. The vapour pressure of the resultant solution is 475 torr. The vapour pressure of pure liquid A at the same temperature is 280.7 torr. What is the vapour pressure of pure liquid B in torr?

• (1) 486
• (2) 550
• (3) 514
• (4) 500

The resistance of a conductivity cell filled with 0.1 M KCl solution is \(100 \, \Omega\). If the resistance of the same cell when filled with 0.02 M KCl solution is \(520 \, \Omega\), the molar conductivity of 0.02 M solution (in S cm\(^2\) mol\(^{-1}\)) is (Given: conductivity of 0.1 M KCl solution = \(1.29 \, S m^{-1}\))

• (1) 124
• (2) 186
• (3) 248
• (4) 104

In a first order reaction, the concentration of the reactant is reduced to 1/8 of the initial concentration in 75 minutes. The \(t_{1/2}\) of the reaction (in minutes) is (\(\log 2 = 0.30, \log 3 = 0.47, \log 4 = 0.60\))

• (1) 60.2
• (2) 50.2
• (3) 25.1
• (4) 75.1

In a colloidal solution, both the dispersed phase and dispersion medium are in liquid phase. What is the type of colloid?

• (1) gel
• (2) emulsion
• (3) foam
• (4) aerosol

The equation which represents Freundlich adsorption isotherm is (x = amount of gas, m = mass of solid)

• (1) \( \log\frac{x}{m} = \log p + \frac{1}{n}\log k \)
• (2) \( \log\frac{x}{m} = \log k + \frac{1}{n}\log p \)
• (3) \( \frac{x}{m} = k + \frac{1}{n}\log p \)
• (4) \( \frac{x}{m} = \log p + \frac{1}{n}\log k \)

Which of the following is used as froth stabilizer in froth floatation process?

• (1) xanthate
• (2) aniline
• (3) pine oil
• (4) NaCN

White phosphorus on heating with concentrated NaOH solution in an inert atmosphere of CO\(_2\) gives a salt 'X' and gas 'Y'. The oxidation state of central atom in X and Y is respectively

• (1) -3, +1
• (2) +1, -3
• (3) 0, -3
• (4) +1, +2

For which of the following the \( E^\ominus (M^{3+}/M^{2+}) \) is negative?

• (1) Mn
• (2) Co
• (3) Fe
• (4) Cr

In \( Fe_x[Fe_y(CN)_6]_3 \), x, y respectively, are

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

The correct statement regarding X and Y in the following set of reactions is \[ Y \xrightarrow{(C_2H_5)_3Al / TiCl_4, \, 333-343 \, K, \, 6-7 \, atm} nCH_2=CH_2 \xrightarrow{(C_6H_5COO)_2, \, 350-570 \, K, \, 1000-2000 \, atm} X \]

• (1) X is HDP and Y is LDP
• (2) X is LDP and Y is HDP
• (3) X is used in the preparation of flexible pipes and Y is used in manufacturing squeeze bottles
• (4) X is used in insulation of electricity carrying wires, Y is used in manufacturing of bottles

Consider the following
Statement-I : Lactose is composed of \( \alpha \)-D-glucose and \( \beta \)-D-glucose.
Statement-II : Lactose is a reducing sugar.
The correct answer is

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

Match the following

The correct answer is

• (1) A-II, B-III, C-IV, D-I
• (2) A-IV, B-I, C-II, D-III
• (3) A-IV, B-III, C-II, D-I
• (4) A-IV, B-I, C-III, D-II % Based on image, option 4 has same initial parts as 2

The synthetic detergents of the following are

Correct answer is (only = options)

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

In the given reaction sequence conversion of Y to Z is


 

• (1) Wurtz reaction
• (2) Wurtz-Fittig reaction
• (3) Fittig reaction
• (4) Swarts reaction

The preferred reagent for the preparation of pure alkyl chloride from alcohol is

• (1) HCl + ZnCl\(_2\)
• (2) PCl\(_5\)
• (3) SOCl\(_2\)
• (4) PCl\(_3\)

What are X and Y respectively in the following set of reactions?

• (1) m-Bromobenzyl alcohol for X, m-Bromobenzyl alcohol for Y
• (2) m-Bromobenzaldehyde for X, m-Bromobenzyl alcohol for Y
• (3) m-Bromobenzaldehyde for X, m-Bromotoluene for Y
• (4) m-Bromobenzyl alcohol for X, m-Bromotoluene for Y

Match the following

The correct answer is

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

Consider the reaction sequence
Dimethyl ketone \( \xrightarrow{(i)CH_3MgCl (ii)H_2O} \) X \( \xrightarrow{(i)Na (ii)CH_3Br} \) Y
How many sp\(^3\) carbons are present in Y?

• (1) 5
• (2) 4
• (3) 3
• (4) 6

What are X and Y respectively in the following reaction sequence? (\(anhy = anhydrous\)) \[ C_6H_5N_2^+X^- \xrightarrow{C_2H_5OH} X \xrightarrow{CO, HCl, anhy. AlCl_3} Y \]

• (1) Benzene for P1 (X in Q), Benzaldehyde for Y
• (2) Benzene for P1, Benzoic acid for Y
• (3) Phenol for P1, Salicylic acid (o-Hydroxybenzoic acid) for Y
• (4) Phenol for P1, Salicylaldehyde (o-Hydroxybenzaldehyde) for Y