AP EAPCET (AP EAMCET) 2025 Question Paper May 26 Shift 1 (Available): Download Solutions with Answer Key

The domain of the real valued function \( f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x) \) is

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

A real valued function \( f: A \to B \) defined by \( f(x) = \frac{4 - x^2}{4 + x^2} \ \forall x \in A \) is a bijection.
If \( -4 \in A \), then \( A \cap B = \)

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

If \( S_n = 1^3 + 2^3 + \ldots + n^3 \) and \( T_n = 1 + 2 + \ldots + n \), then

• (1) \( S_n = T_n^3 \)
• (2) \( S_n = T_n^3 \)
• (3) \( S_n = T_n^2 \)
• (4) \( S_n = T_n^2 \)

If \( A = \begin{bmatrix} -1 & x & -3
2 & 4 & z
y & 5 & -6 \end{bmatrix} \) is symmetric and \( B = \begin{bmatrix} 0 & 2 & q
p & 0 & 4
-3 & r & s \end{bmatrix} \) is skew-symmetric, then find \( |A| + |B| - |AB| \)

• (1) \( xyz + pqr \)
• (2) \( xyz + q + r \)
• (3) \( \frac{xyz}{pq} \)
• (4) \( xyz + pq + rs \)

If the inverse of \[ \begin{bmatrix} -x & 14x & 7x
0 & 1 & 0
x & -4x & -2x \end{bmatrix} \]
is \[ \begin{bmatrix} 2 & 0 & 7
0 & 1 & 0
1 & -2 & 1 \end{bmatrix} \]
then the value of \[ \begin{vmatrix} x & x+1 & x+2
x+1 & x+2 & x+3
x+2 & x+3 & x+4 \end{vmatrix} \]

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

If the system of equations \( 2x + 3y - 3z = 3,\ x + 2y + \alpha z = 1,\ 2x - y + z = \beta \) has infinitely many solutions, then \( \frac{\alpha}{\beta} = \frac{\beta}{\alpha} \)

• (1) \( \frac{53}{14} \)
• (2) \( \frac{45}{14} \)
• (3) \( -\frac{53}{14} \)
• (4) \( -\frac{45}{14} \)

If a complex number \( z = x + iy \) represents a point \( P \) on the Argand plane and \[ Arg \left( \frac{z - 3 + 2i}{z + 2 - 3i} \right) = \frac{\pi}{4} \]
then the locus of \( P \) is

• (1) Circle with the line \( x + y = 12 \) as its diameter
• (2) Circle with radius \( \sqrt{11} \)
• (3) Circle with the line \( x - y = 6 \) as its diameter
• (4) Circle with radius 5

By taking \( \sqrt{a \pm ib} = x + iy, x > 0 \), if we get \[ \frac{\sqrt{21} + 12\sqrt{2}i}{\sqrt{21} - 12\sqrt{2}i} = a + ib, \]
then \( \frac{b}{a} = \) ?

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

Two values of \( (-8 - 8\sqrt{3}i)^{1/4} \) are

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

Let \( f(x) = x^2 + 2bx + 2c^2 \) and \( g(x) = -x^2 - 2cx + b^2 \), \( x \in \mathbb{R} \).
If \( b \) and \( c \) are non-zero real numbers such that \( \min f(x) > \max g(x) \), then \[ \left| \frac{c}{b} \right| \]
lies in the interval

• (1) \( \left( \frac{1}{2}, \frac{1}{\sqrt{2}} \right) \)
• (2) \( \left( \frac{1}{\sqrt{2}}, \sqrt{2} \right) \)
• (3) \( \left( \sqrt{2}, \infty \right) \)
• (4) \( (0, 1) \)

If \( x^2 - 4x + 5 + a > 0 \) for all \( x \in \mathbb{R} \) whenever \( a \in (\alpha, \beta) \), then \( 4\beta + \alpha = \)

• (1) 0
• (2) 4
• (3) 5
• (4) 8

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 - 12x^2 + kx - 18 = 0 \) and one of them is thrice the sum of the other two, then \[ \alpha^2 + \beta^2 + \gamma^2 - k = ? \]

• (1) 115
• (2) 41
• (3) 56
• (4) 57

The polynomial equation of degree 5 whose roots are the roots of the equation \[ x^5 - 3x^4 + 11x^2 - 12x + 4 = 0 \]
each increased by 2 is

• (1) \( x^5 - 13x^4 + 63x^3 - 135x^2 - 108x = 0 \)
• (2) \( x^5 - 13x^4 + 63x^3 + 135x^2 + 108x = 0 \)
• (3) \( x^5 - 13x^4 + 63x^3 - 135x^2 + 108x = 0 \)
• (4) \( x^5 - 13x^4 - 63x^3 - 135x^2 - 108x = 0 \)

The number of positive integers less than 10000 which contain the digit 5 at least once is

• (1) 3168
• (2) 3420
• (3) 3439
• (4) 5832

5 men and 4 women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is \( \alpha \), and the number of arrangements in which they are not together is \( \beta \), then \( \frac{\alpha}{\beta} = \)

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

If a team of 4 persons is to be selected out of 4 married couples to play mixed doubles tennis game, then the number of ways of forming a team in which no married couple appears is

• (1) 12
• (2) 8
• (3) 6
• (4) 24

In the binomial expansion of \( (p - q)^{14} \), if the sum of 7^th and 8^th terms is zero, then \[ \frac{p + q}{p - q} = ? \]

• (1) 14
• (2) 15
• (3) 16
• (4) 13

The numerically greatest term in the expansion of \( (x + 3y)^{13} \), when \( x = \frac{1}{2},\ y = \frac{1}{3} \), is

• (1) \( \binom{13}{9} \left( \frac{1}{3} \right)^4 \)
• (2) \( \binom{13}{4} \left( \frac{1}{2} \right)^9 \)
• (3) \( \binom{13}{9} \left( \frac{1}{2} \right)^4 \)
• (4) \( \binom{13}{10} \left( \frac{1}{2^4} \right) \)

If \( \frac{x^4}{(x-1)(x-2)} = \frac{A}{x - 1} + \frac{B}{x - 2} \), then \[ f(-2) + A + B = ? \]

• (1) 32
• (2) 28
• (3) 22
• (4) 20

Evaluate: \[ \sin \frac{\pi}{12} \cdot \sin \frac{2\pi}{12} \cdot \sin \frac{3\pi}{12} \cdot \sin \frac{4\pi}{12} \cdot \sin \frac{5\pi}{12} \cdot \sin \frac{6\pi}{12} \]

• (1) \( \frac{\sqrt{3}}{16\sqrt{2}} \)
• (2) \( \frac{\sqrt{3}}{8\sqrt{2}} \)
• (3) \( \frac{1}{32} \)
• (4) \( \frac{1}{16} \)

If \( \tan\left( \frac{\pi}{4} + \alpha \right) = \tan^3\left( \frac{\pi}{4} + \beta \right) \), then compute: \[ \tan(\alpha + \beta) \cot(\alpha - \beta) = ? \]

• (1) \( \sec^2 2\beta + \tan^2 2\beta \)
• (2) \( \csc^2 2\beta + \cot^2 2\beta \)
• (3) \( 2(\sec^2 2\beta + \tan^2 2\beta) \)
• (4) \( 4(\sec^2 2\beta + \tan^2 2\beta) \)

If \( A + B + C + D = 2\pi \), then \[ \sin A + \sin B + \sin C + \sin D = ? \]

• (1) \( 4\sin\left( \frac{A + B}{4} \right)\sin\left( \frac{A + C}{4} \right)\sin\left( \frac{A + D}{4} \right) \)
• (2) \( 4\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A + C}{4} \right)\cos\left( \frac{A + D}{4} \right) \)
• (3) \( 4\sin\left( \frac{A + B}{2} \right)\sin\left( \frac{A + C}{2} \right)\sin\left( \frac{A + D}{2} \right) \)
• (4) \( 4\sin\left( \frac{A + B}{2} \right)\sin\left( \frac{A + C}{4} \right)\sin\left( \frac{A + D}{4} \right) \)

If \( 0 \le x \le 3,\ 0 \le y \le 3 \), then the number of solutions \((x, y)\) for the equation: \[ \left( \sqrt{\sin^2 x - \sin x + \frac{1}{2}} \right)^{\sec^2 y} = 1 \]

• (1) 5
• (2) 2
• (3) 6
• (4) 1

If \( \theta = \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{13} \right) + \tan^{-1} \left( \frac{1}{21} \right) + \tan^{-1} \left( \frac{1}{31} \right) \), then \( \tan \theta = ? \)

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

If \( \tanh^{-1} x = \coth^{-1} y = \log \sqrt{5} \), then find \( \tan^{-1}(xy) = ? \)

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

In triangle \( ABC \), if \( C = 120^\circ \), \( c = \sqrt{19} \), and \( b = 3 \), then \( a = ? \)

• (1) 4
• (2) 5
• (3) 2

In triangle \( ABC \), \( 2A + C = 300^\circ \). If the circumradius is 8 times the inradius, then \( \sin\frac{C}{2} = ? \)

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

In triangle \( ABC \), if \( a = 5,\ b = 4,\ \cos(A - B) = \frac{31}{32} \), then \( c = ? \)

• (1) 8
• (2) \( \sqrt{41} \)
• (3) 6
• (4) \( \sqrt{24} \)
   

If the line joining points \( \vec{r}_1 = \hat{i} + 2\hat{j} \) and \( \vec{r}_2 = \hat{j} - 2\hat{k} \) intersects the plane through the points \( \vec{A} = 2\hat{i} - \hat{j},\ \vec{B} = -2\hat{j} + 3\hat{k},\ \vec{C} = \hat{k} - 2\hat{i} \) at \( T \), then find \( \vec{r}_T \cdot (\hat{i} + \hat{j} + \hat{k}) \)

• (1) 15
• (2) 5
• (3) 3

Let vectors: \( \vec{A} = \hat{i} - 2\hat{j} + \hat{k},\ \vec{B} = \hat{i} + \hat{j} - 2\hat{k},\ \vec{C} = 2\hat{i} - \hat{j},\ \vec{D} = \hat{i} + \hat{j} + \hat{k} \)

If \( P \) divides \( AB \) in ratio 2:1 internally, and \( Q \) divides \( CD \) in ratio 1:2 externally, find the ratio in which the point \( 5\hat{i} - 6\hat{j} - 5\hat{k} \) divides line \( PQ \)

• (1) 2:1
• (2) -2:1
• (3) 2:3

The vector equation of a plane passing through the line of intersection of the planes \[ \vec{r} \cdot (\hat{i} - 2\hat{k}) = 3,\quad \vec{r} \cdot (\hat{j} + \hat{k}) = 5 \]
and also passing through the point \( \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} \) is:

• (1) \( \vec{r} \cdot (\hat{i} + 4\hat{j}) = 13 \)
• (2) \( \vec{r} \cdot (\hat{i} + 6\hat{j} + \hat{k}) = 18 \)
• (3) \( \vec{r} \cdot (\hat{i} + 2\hat{j} - \hat{k}) = 8 \)
• (4) \( \vec{r} \cdot (\hat{i} + 8\hat{j} + 2\hat{k}) = 23 \)

If \( \vec{a} = \hat{i} + \hat{j}, \vec{b} = 2\hat{j} - \hat{k} \) are two vectors such that \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a}, \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \), then the unit vector in the direction of \( \vec{r} \) is:

• (1) \( \frac{1}{\sqrt{11}}(\hat{i} + 3\hat{j} - \hat{k}) \)
• (2) \( \frac{1}{\sqrt{11}}(\hat{i} - 3\hat{j} + \hat{k}) \)
• (3) \( \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \)

If \( \vec{a}, \vec{b}, \vec{c} \) are three unit vectors such that \( \vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2} \vec{b} + \frac{1}{2} \vec{c} \), and \( \alpha, \beta \) are the angles between \( \vec{a}, \vec{c} \) and \( \vec{a}, \vec{b} \) respectively, then \( \alpha + \beta = ? \)

• (1) \( \frac{\pi}{2} \)
• (2) \( \frac{7\pi}{6} \)
• (3) \( \frac{\pi}{6} \)

Find the variance of the following frequency distribution:

• (1) 16
• (2) \( \frac{44}{3} \)
• (3) 23
• (4) \( \frac{22}{3} \)

From the word "CURVE", how many 3-letter words can be formed out of all 2-letter or more combinations (with all distinct letters)? Find probability of getting a 3-letter word.

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

Three numbers are chosen from 1 to 30. Find the probability that they are NOT 3 consecutive numbers.

• (1) \( \frac{1}{145} \)
• (2) \( \frac{142}{145} \)
• (3) \( \frac{143}{145} \)

If \( P(\bar{A}) = 0.3,\ P(B) = 0.4,\ P(A \cap \bar{B}) = 0.5 \), then find \( P(B / (A \cup \bar{B})) \)

• (1) 0.25
• (2) 0.6
• (3) 0.45

Two candidates A and B attended an interview for two jobs. The probability that A gets the job is 0.8, and for B it is 0.7. What is the probability that at least one of them gets a job?

• (1) 0.96
• (2) 0.94
• (3) 0.92
• (4) 0.9

X denotes the number of heads in \( n \) tosses of a fair coin. If \( P(X = 4),\ P(X = 5),\ P(X = 6) \) are in arithmetic progression, find the largest possible value of \( n \).

• (1) 7
• (2) 14
• (3) 21
• (4) 28

Find the mean (expected value) of \( X \).

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

Let \( A(4, 3), B(2, 5) \) be two points. If \( P \) is a variable point on the same side of the origin as that of line \( AB \) and at most 5 units from the midpoint of \( AB \), then the locus of \( P \) is:

• (1) \( x^2 + y^2 - 6x - 8y = 0 \)
• (2) \( x^2 + y^2 - 6x - 8y \le 0, \quad x + y - 7 < 0 \)
• (3) \( x^2 + y^2 + 6x + 8y - 25 = 0, \quad x + y - 7 \le 0 \)
• (4) \( x^2 + y^2 - 6x + 8y \ge 0, \quad x + y - 7 < 0 \)

By shifting the origin to the point (2, 3) through translation of axes, if the equation of the curve \[ x^2 + 3xy - 2y^2 + 4x - y - 20 = 0 \]
is transformed to the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \]
then find \( D + E + F \).

• (1) -1
• (2) 1
• (3) -15
• (4) 15

The points \( (2,3) \) and \( \left(-4, \frac{4}{3}\right) \) lie on opposite sides of the line \[ L = 5x - 6y + k = 0, \]
and \( k \) is an integer. If the points \( (1, 2) \) and \( (4, 5) \) lie on the same side of the line, then the perpendicular distance from the origin to the line \( L = 0 \) is?

• (1) \( \frac{7}{\sqrt{61}} \)
• (2) \( \frac{9}{\sqrt{61}} \)
• (3) \( \frac{10}{\sqrt{61}} \)
• (4) \( \frac{11}{\sqrt{61}} \)

If the incentre of the triangle formed by lines \[ x - 2 = 0, \quad x + y - 1 = 0, \quad x - y + 3 = 0 \]
is \( (\alpha, \beta) \), then find \( \beta \).

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

If the equation of the pair of straight lines intersecting at \( (a, b) \) and perpendicular to the pair \[ 3x^2 - 4xy + 5y^2 = 0 \]
is \[ lx^2 + 2hxy + my^2 = 0, \]
then find \[ \frac{a + b + c}{l + h + m}. \]

• (1) \( \frac{38}{5} \)
• (2) \( \frac{17}{2} \)
• (3) \( \frac{15}{6} \)
• (4) -

PQR is a right angled isosceles triangle with right angle at \( P(2, 1) \). If the equation of the line \( QR \) is \[ 2x + y = 3, \]
then the equation representing the pair of lines \( PQ \) and \( PR \) is:

• (1) \( 3x^2 - 3y^2 - 8xy - 10x - 15y - 20 = 0 \)
• (2) \( 3x^2 - 3y^2 + 8xy + 20x + 10y + 25 = 0 \)
• (3) \( 3x^2 - 3y^2 + 8xy - 20x - 10y + 25 = 0 \)
• (4) \( 3x^2 - 3y^2 + 8xy + 10x + 15y + 20 = 0 \)

The circles \[ x^2 + y^2 - 2x - 4y - 4 = 0 \]
and \[ x^2 + y^2 + 2x + 4y - 11 = 0 \]
...

• (1) Cut each other orthogonally
  (2) Do not meet
  (3) Intersect at points lying on the line \( 4x + 8y - 7 = 0 \)
  (4) Touch each other at the point lying on the line \( 4x + 8y - 7 = 0 \)

If the line \[ 4x - 3y + 7 = 0 \]
touches the circle \[ x^2 + y^2 - 6x + 4y - 12 = 0 \]
at \( (\alpha, \beta) \), then find \( \alpha + 2\beta \).

• (1) 3
  (2) -1
  (3) 1
  (4) -3

The slope of the common tangent drawn to the circles \[ x^2 + y^2 - 4x + 12y - 216 = 0 \]
and \[ x^2 + y^2 + 6x - 12y + 36 = 0 \]
is:

• (1) 1
  (2) -1
  (3) \( \frac{5}{12} \)
  (4) \( \frac{12}{7} \)

If \( r_1 \) and \( r_2 \) are radii of two circles touching all the four circles \[ (x \pm r)^2 + (y \pm r)^2 = r^2, \]
then find the value of \[ \frac{r_1 + r_2}{r}. \]

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

If the equation of the circle having the common chord to the circles \[ x^2 + y^2 + x - 3y - 10 = 0 \]
and \[ x^2 + y^2 + 2x - y - 20 = 0 \]
as its diameter is \[ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, \]
then find \( \alpha + 2\beta + \gamma \).

• (1) 0
• (2) 1
• (3) -1
• (4) 2

If \( x - y - 3 = 0 \) is a normal drawn through the point \( (5, 2) \) to the parabola \( y^2 = 4x \), then the slope of the other normal that can be drawn through the same point to the parabola is?

• (1) 0
• (2) -1
• (3) 2
• (4) -2

If the normal drawn at the point \[ P \left(\frac{\pi}{4}\right) \]
on the ellipse \[ x^2 + 4y^2 - 4 = 0 \]
meets the ellipse again at \( Q(\alpha, \beta) \), then find \( \alpha \).

• (1) \( \sqrt{2} \)
• (2) \( \frac{-23}{17\sqrt{2}} \)
• (3) \( \frac{7\sqrt{2}}{17} \)
• (4) \( \frac{1}{\sqrt{2}} \)

If \( \theta \) is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity \[ \frac{2}{\sqrt{7} - \sqrt{3}}, \]
then find \( \sin \theta \).

• (1) \( \frac{1}{2} \tan \frac{\theta}{2} \)
• (2) \( 2 \cos \frac{\theta}{2} \)
• (3) \( \frac{1}{\sin \frac{\theta}{2} + \cos \frac{\theta}{2}} \)
• (4) \( 1 - \cos \frac{\theta}{2} \)

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola \[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
meets the x-axis and y-axis at \( A \) and \( B \) respectively. If \( O \) is the origin, find \[ (OA)^2 - (OB)^2. \]

• (1) \( \frac{20}{9} \)
• (2) \( \frac{16}{9} \)
• (3) \( -\frac{4}{9} \)
• (4) \( \frac{4}{3} \)

The points \( A(-1, 2, 3), B(2, -3, 1), C(3, 1, -2) \)
are:

• (1) are collinear
• (2) form an isosceles triangle
• (3) form a right angled triangle
• (4) form a scalene triangle

The direction cosines of the line making angles \[ \frac{\pi}{4}, \frac{\pi}{3} \]
and \( \theta \) (where \( 0 < \theta < \frac{\pi}{2} \)) with X, Y, and Z axes respectively are:

• (1) \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2} \)
• (2) \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{\sqrt{3}}{2} \)
• (3) \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}} \)
• (4) None of these

If the equation of the plane passing through point \( (3, 2, 5) \) and perpendicular to the planes \[ 2x - 3y + 5z = 7, \quad 5x + 2y - 3z = 11 \]
is \[ x + by + cz + d = 0, \]
then find \( 2b + 3c + d \).

• (1) 0
• (2) 35
• (3) 1
• (4) 20

Evaluate: \[ \lim_{x \to \infty} \left[ x - \log(\cosh x) \right] \]

• (1) 2
• (2) 0
• (3) Not exist
• (4) \(\log 2\)

Evaluate: \[ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + 4x^2} - \sqrt{x^2 - 3x} \right) \]

• (1) \(\frac{17}{6}\)
• (2) \(\frac{25}{6}\)
• (3) \(-\frac{1}{6}\)
• (4) \(\frac{37}{6}\)

If a real valued function \[ f(x) = \begin{cases} \frac{\sin a(x - [x])}{e^{x - [x]}}, & x < 1
b + 1, & x = 1
\frac{|x^2 + x - 2|}{x - 1}, & x > 1 \end{cases} \]
is continuous at \( x=1 \), then find \( b \).
Here, \( [x] \) denotes the greatest integer function.

• (1) 6
• (2) 4
• (3) \( \log_e 9 \)
• (4) \( \log_e 2 \)

If \[ \sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0, \]
then find \[ \frac{dy}{dx}. \]

• (1) \( \tan x \)
• (2) 1
• (3) -1
• (4) \( -\cot x \)

If \[ f(x) = 2 + |\sin^{-1} x|, \]
and \[ A = \{ x \in \mathbb{R} \mid f'(x) exists \}, \]
then find \( A \).

• (1) \( \{0\} \)
• (2) \( [-1,1] \)
• (3) \( (-\infty, -1) \cup (1, \infty) \)
• (4) \( (-1, 0) \cup (0,1) \)

If \[ y = (\log_x \sin x)^x, \]
then find \[ \frac{dy}{dx}. \]

• (1) \( y \left[ \frac{x \sin x}{\log x \cos x} + \frac{1}{\log x} - \log(\log x) \right] \)
• (2) \( y \left[ \frac{x \cos x}{\log \sin x} - \log(\log \sin x) + \frac{1}{\log x} \right] \)
• (3) \( y \left[ \frac{x \cot x}{\log \sin x} + \log(\log \sin x) - \frac{1}{\log x} \right] \)
• (4) \( y \left[ \frac{x \cot x}{\log \sin x} - \log(\log \sin x) + \frac{1}{\log x} \right] \)

If the area of a square is 575 square units, then the approximate value of its side is:

• (1) 23.9792
• (2) 23.7992
• (3) 23.8687
• (4) 23.7868

If the tangent of the curve \[ 4y^3 = 3ax^2 + x^3 \]
drawn at the point \( (a, a) \) forms a triangle of area \(\frac{25}{24}\) sq. units with the coordinate axes, then find \( a \).

• (1) \( \pm 10 \)
• (2) \( \pm 5 \)
• (3) 6
• (4) 3

If the function \[ f(x) = \sin x - \cos^2 x \]
is defined on the interval \( [-\pi, \pi] \), then \( f \) is strictly increasing in the interval:

• (1) \( \left(-\frac{5\pi}{6}, -\frac{\pi}{6}\right) \cup \left(-\frac{\pi}{6}, \frac{\pi}{2}\right) \)
• (2) \( \left(-\frac{\pi}{2}, -\frac{\pi}{6}\right) \)
• (3) \( \left(-\frac{5\pi}{6}, \frac{\pi}{2}\right) \)
• (4) \( \left(-\frac{5\pi}{6}, -\frac{\pi}{2}\right) \cup \left(-\frac{\pi}{6}, \frac{\pi}{2}\right) \)

If Lagrange's mean value theorem is applied to the function \[ f(x) = e^x \]
defined on the interval \( [1, 2] \) and the value of \( c \in (1, 2) \) is \( k \), then find \( e^{k-1} \).

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

If \[ \int \frac{x^4 + 1}{x^2 + 1} dx = Ax^3 + Bx^2 + Cx + D \tan^{-1} x + E, \]
then find \( A + B + C + D \).

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

If \[ \int \frac{x^2 - x + 2}{x^2 + x + 2} dx = x - \log(f(x)) + \frac{2}{\sqrt{7}} \tan^{-1}(g(x)) + c, \]
then find \[ f(-1) + \sqrt{7} g(-1). \]

• (1) 1
• (2) 0
• (3) -1
• (4) 2

Evaluate the integral: \[ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx \]

• (1) \[ \log \left| \frac{\sec \left(x - \frac{\pi}{3}\right)}{\sec \left(x + \frac{\pi}{6}\right)} \right| + c \]
• (2) \[ \log \left| \frac{\cos \left(x - \frac{\pi}{3}\right)}{\cos \left(x + \frac{\pi}{6}\right)} \right| + c \]
• (3) \[ \log \left| \frac{\csc \left(x - \frac{\pi}{3}\right)}{\csc \left(x + \frac{\pi}{6}\right)} \right| + c \]
• (4) \[ \log \left| \frac{\sin \left(x - \frac{\pi}{3}\right)}{\sin \left(x + \frac{\pi}{6}\right)} \right| + c \]

If \[ \int \frac{a \cos x + 3 \sin x}{5 \cos x + 2 \sin x} dx = \frac{26}{29} x - \frac{k}{29} \log |5 \cos x + 2 \sin x| + c, \]
then find \( |a + k| \).

• (1) 3
• (2) 11
• (3) 12
• (4) 2

If \[ \int \frac{dx}{1 - \sin^4 x} = A \tan x + B \tan^{-1}(\sqrt{2} \tan x) + C, \]
then find \( A^2 - B^2 \).

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

Evaluate: \[ \int_0^1 x \sin^{-1} x \, dx \]

• (1) \( \frac{\pi}{8} \)
• (2) \( \frac{\pi}{4} \)
• (3) \( \frac{\pi}{12} \)
• (4) \( \frac{\pi}{3} \)

Evaluate: \[ \int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx \]
where \([x]\) is the greatest integer function.

• (1) 0
• (2) \(3(1 - \cos 1) + \sin 2 - \sin 1\)
• (3) \(3(1 - \cos 1) + \cos 2 - \sin 1\)
• (4) \( \cos 2 - \sin 2 \)

Evaluate the integral: \[ \int_0^2 x^2 (2 - x)^5 \, dx \]

• (1) \(\frac{128}{21}\)
• (2) \(\frac{64}{7}\)
• (3) \(\frac{32}{21}\)
• (4) \(\frac{16}{7}\)

If \( f(x) = \max \{ x^3 - 4, x^4 - 4 \} \) and \( g(x) = \min \{ x^2, x^3 \} \), evaluate: \[ \int_{-1}^1 (f(x) - g(x)) \, dx \]

• (1) \(-\frac{151}{20}\)
• (2) \(\frac{9}{20}\)
• (3) \(\frac{131}{22}\)
• (4) \(-\frac{67}{9}\)

If \[ y = A t^2 + \frac{B}{t} \quad (A, B constants) \]
is a general solution of the differential equation \[ f(t) y'' + g(t) y' + h(t) y = 0, \]
then find the relation between \( g(t), f(t), h(t) \).

• (1) \( g(t) - h(t) \)
• (2) \( g(t) + f(t) \)
• (3) \( g(t) f(t) \)
• (4) \( (f(t))^{g(t)} \)

Find the general solution of: \[ (2x - y)^2 dy - 2(2x - y)^2 dx - 2 dx = 0 \]

• (1) \(\log (2x - y) = 2x + c\)
• (2) \((2x - y)^3 + 4 y = c\)
• (3) \((2x - y)^3 + 6 x = c\)
• (4) \(\log (2x - y) = 2 y + c\)

Find the general solution of the differential equation: \[ x \log x \, dy = (x \log x - y) dx \]

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

The number of significant figures in 0.03240 is

• (1) 5
• (2) 4
• (3) 6
• (4) 3

A ball projected vertically upwards with velocity 'v' passes through a point P in its upward journey in a time of 'x' seconds. Then, the time in which the ball again passes through the same point P is

• (1) \(\frac{v}{2g}\)
• (2) \(\frac{2v}{g} - x\)
• (3) \(\frac{v}{2g} - x\)
• (4) \(2\left(\frac{v}{g} - x\right)\)

Three vectors each of magnitude \(3\sqrt{1.5}\) units are acting at a point. If the angle between any two vectors is \(\frac{\pi}{3}\), then the magnitude of the resultant vector of the three vectors is

• (1) \(9\sqrt{3}\) units
• (2) 9 units
• (3) \(\sqrt{6}\) units
• (4) 3 units

A vector perpendicular to the vector \(\left(4\hat{i} - 3\hat{j}\right)\) is

• (1) \(4\hat{i} + 3\hat{j}\)
• (2) \(6\hat{i}\)
• (3) \(3\hat{i} - 4\hat{j}\)
• (4) \(7\hat{k}\)

If the breaking strength of a rope is \(\frac{4}{3}\) times the weight of a person, then the maximum acceleration with which the person can safely climb up the rope is (g - acceleration due to gravity)

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

A block of mass 2 kg is placed on a rough horizontal surface. If a horizontal force of 20 N acting on the block produces an acceleration of 7 m/s\(^2\) in it, then the coefficient of kinetic friction between the block and the surface is (Acceleration due to gravity = 10 m/s\(^2\))

• (1) 0.2
• (2) 0.3
• (3) 0.4
• (4) 0.5

If a position dependent force \((3x^2 - 2x + 7)N\) acting on a body of mass 2 kg displaces it from \(x = 0\, m\) to \(x = 5\, m\), then the work done by the force is

• (1) 165 J
• (2) 115 J
• (3) 150 J
• (4) 135 J

Two smooth inclined planes A and B each of height 20 m have angles of inclination \(30^\circ\) and \(60^\circ\) respectively. If \(t_1\) and \(t_2\) are the times taken by two blocks to reach the bottom of the planes A and B from the top, then find the value of \(t_1 - t_2\). (Acceleration due to gravity \(g = 10\, m/s^2\))

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

The moment of inertia of a solid cylinder of mass 2.5 kg and radius 10 cm about its axis is

• (1) 0.0725 kg m\(^2\)
• (2) 12500 kg m\(^2\)
• (3) 0.0125 kg m\(^2\)
• (4) 72500 kg m\(^2\)

A body of mass 2 kg is moving towards north with a velocity of 20 m/s and another body of mass 3 kg is moving towards east with a velocity of 10 m/s. The magnitude of the velocity of the centre of mass of the system of the two bodies is

• (1) 20 m/s
• (2) 10 m/s
• (3) 15 m/s
• (4) \(2 \sqrt{5}\) m/s

If the function \(\sin^2 \omega t\) (where \(t\) is time in seconds) represents a periodic motion, then the period of the motion is

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

On a smooth inclined plane, a block of mass \(M\) is fixed to two rigid supports using two springs, each having spring constant \(k\), as shown in the figure. If the masses of the springs are neglected, then the period of oscillation of the block is

• (1) \(2\pi \sqrt{\frac{M}{2k}}\)
• (2) \(2\pi \sqrt{\frac{2M}{k}}\)
• (3) \(2\pi \sqrt{\frac{Mg \sin \theta}{2k}}\)
• (4) \(2\pi \sqrt{\frac{2Mg}{k}}\)

The acceleration due to gravity at a height of \((\sqrt{2} - 1)R\) from the surface of the earth is (where \(g = 10\, m/s^2\) and \(R\) is the radius of the earth)

• (1) \(2.5\, m/s^2\)
• (2) \(7.5\, m/s^2\)
• (3) \(5\, m/s^2\)
• (4) \(10\, m/s^2\)

If the given graph shows the load (W) attached to and the elongation (\(\Delta l\)) produced in a wire of length 1 meter and cross-sectional area 1 mm\(^2\), then the Young's modulus of the material of the wire is

• (1) \(20 \times 10^{10}\, N/m^2\)
• (2) \(2 \times 10^{10}\, N/m^2\)
• (3) \(10 \times 10^{10}\, N/m^2\)
• (4) \(4 \times 10^{10}\, N/m^2\)

A wire of length 20 cm is placed horizontally on the surface of water and is gently pulled up with a force of \(1.456 \times 10^{-2}\, N\) to keep the wire in equilibrium. The surface tension of water is

• (1) \(0.00364\, N/m\)
• (2) \(0.0364\, N/m\)
• (3) \(0.00464\, N/m\)
• (4) \(0.0864\, N/m\)

If some heat is given to a metal of mass 100 g, its temperature rises by 20 \(^\circ\)C. If the same heat is given to 20 g of water, the change in its temperature (in \(^\circ\)C) is (The ratio of specific heat capacities of metal and water is 1:10)

• (1) 5
• (2) 10
• (3) 12
• (4) 15

The ratio of the efficiencies of two Carnot engines A and B is 1.25 and the temperature difference between the source and the sink is the same in both engines. The ratio of the absolute temperatures of the sources of the engines A and B is

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

The heat supplied to a gas at a constant pressure of \(5 \times 10^5\, Pa\) is 1000 kJ. If the volume of gas changes from 1 m\(^3\) to 2.5 m\(^3\), then the change in internal energy of the gas is

• (1) 250 kJ
• (2) 225 kJ
• (3) 200 kJ
• (4) 175 kJ

When an ideal diatomic gas undergoes adiabatic expansion, if the increase in its volume is 0.5%, then the change in the pressure of the gas is

• (1) +0.5%
• (2) -0.5%
• (3) -0.7%
• (4) +0.7%

To increase the RMS speed of gas molecules by 25%, the percentage increase in absolute temperature of the gas is to be

• (1) 42.75
• (2) 56.25
• (3) 36.75
• (4) 18.25

When both the source of sound and observer approach each other with a speed equal to 10% of the speed of sound, then the percentage change in frequency heard by the observer is nearly

• (1) 33.3%
• (2) 12.2%
• (3) 22.2%
• (4) 11.1%

According to Rayleigh, when sunlight travels through atmosphere, the amount of scattering is proportional to \(n^{th}\) power of wavelength of light. Then the value of \(n\) is

• (1) 4
• (2) -4
• (3) 3
• (4) -3

In Young's double slit experiment, if the distance between the slits is 2 mm and the distance of the screen from the slits is 100 cm, the fringe width is 0.36 mm. If the distance between the slits is decreased by 0.5 mm and the distance of the screen from the slits is increased by 50 cm, the fringe width becomes

• (1) 0.84 mm
• (2) 0.96 mm
• (3) 0.48 mm
• (4) 0.72 mm

An electric dipole with dipole moment \(2 \times 10^{-10}\, C \cdot m\) is aligned at an angle \(30^\circ\) with the direction of a uniform electric field of \(10^4\, N/C\). The magnitude of the torque acting on the dipole is

• (1) \(10^{-6}\, N \cdot m\)
• (2) \(10^{-5}\, N \cdot m\)
• (3) \(10^{-4}\, N \cdot m\)
• (4) \(10^{-3}\, N \cdot m\)

If a dielectric slab of dielectric constant 3 is introduced between the plates of a capacitor having electric field \(1.5\, N/C\), then the electric displacement is

• (1) \(125 \times 10^{-12}\, C/m^2\)
• (2) \(125 \times 10^{-9}\, C/m^2\)
• (3) \(250 \times 10^{-12}\, C/m^2\)
• (4) \(250 \times 10^{-9}\, C/m^2\)

An electric charge \(10^{-3}\, \mu C\) is placed at the origin of the x-y plane. The potential difference between points A and B located at \((\sqrt{2}\, m, \sqrt{2}\, m)\) and \((2m, 0m)\) respectively is

• (1) 4.5 V
• (2) 9 V
• (3) 0 V
• (4) 2 V

If each resistance in the given figure is 9 \(\Omega\), then the reading of the ammeter (A) is

• (1) 8 A
• (2) 5 A
• (3) 2 A
• (4) 9 A

The area of cross-section of a copper wire is \(4 \times 10^{-7} m^2\) and the number of electrons per cubic meter in copper is \(8 \times 10^{28}\). If the wire carries a current of 6.4 A, then the drift velocity of the electrons (in \(10^{-3} m/s\)) is

• (1) 0.25
• (2) 2.5
• (3) 0.125
• (4) 1.25

In a solenoid, if the current of 15 A passes through the solenoid of length 25 cm, radius 2 cm, and number of turns 500, then the magnetic moment of the solenoid is

• (1) 6 J T\(^{-1}\)
• (2) 3 J T\(^{-1}\)
• (3) 3\(\pi\) J T\(^{-1}\)
• (4) 15 J T\(^{-1}\)

The maximum magnetic field produced by a current of 12 A passing through a copper wire of diameter 1.2 mm is

• (1) 2 mT
• (2) 4 mT
• (3) 1.5 mT
• (4) 8 mT

Two moving coil galvanometers A and B having identical springs are placed in magnetic fields of 0.25 T and 0.5 T respectively. If the number of turns in A and B are 36 and 48, the areas of the coils A and B are 2.4 \(\times 10^{-3}\) m\(^2\) and 4.8 \(\times 10^{-3}\) m\(^2\) respectively, then the ratio of the current sensitivities of the galvanometers A and B is

• (1) 3 : 16
• (2) 16 : 3
• (3) 4 : 3
• (4) 3 : 4

The self-inductance of an air-cored solenoid of length 40 cm, diameter 7 cm having 200 turns is

• (1) 484 \(\mu\)H
• (2) 242 \(\mu\)H
• (3) 121 \(\mu\)H
• (4) 96 \(\mu\)H

A coil of inductive reactance \(\frac{1}{\sqrt{3}} \Omega\) and a resistance 1 \(\Omega\) are connected in series to a 200 V, 50 Hz ac source. The time lag between voltage and current is

• (1) 1200 s
• (2) 1 s
• (3) \(\frac{\pi}{600}\) s
• (4) \(\frac{\pi}{1800}\) s

If the magnetic field in a plane progressive wave is represented by the equation \(B = 2 \times 10^{-8} \sin (0.5 \times 10^3 t + 1.5 \times 10^4 x)\) T, then the frequency of the wave is

• (1) 75 \(\times 10^6\) Hz
• (2) 150 \(\times 10^6\) Hz
• (3) 75 \(\times 10^4\) Hz
• (4) 75 \(\times 10^3\) Hz

When photons of energy 8 \(\times 10^{-19}\) J incident on a photosensitive material, the work function of the photosensitive material is nearly 10 eV, then the maximum kinetic energy of the photoelectrons emitted is



(1) 3.5 eV

(2) 2.5 eV

(3) 2.0 eV

(4) 1.0 eV

The minimum wavelength of X-rays produced by 20 keV electrons is nearly

• (1) 0.62 AA
• (2) 1.8 AA
• (3) 3.2 AA
• (4) 6.5 AA

If the half-life of a radioactive material is 10 years, then the percentage of the material decayed in 30 years is

• (1) 87.5
• (2) 78.5
• (3) 58.7
• (4) 48

At absolute zero temperature, an intrinsic semiconductor behaves as

• (1) conductor
• (2) superconductor
• (3) insulator
• (4) semiconductor

The logic gate equivalent to the combination of logic gates shown in the figure is

• (1) AND
• (2) NOR
• (3) OR
• (4) NAND

The heights of the transmitting and receiving antennas are respectively \(\frac{1}{2000}\) and \(\frac{1}{5000}\) times the radius of the Earth. The maximum distance between these two antennas for satisfactory communication in line of sight is

(Radius of the Earth = 6.4 \(\times 10^6\) m)

• (1) 48 km
• (2) 96 km
• (3) 72 km
• (4) 192 km

The work function of Cu is 7.66 \(\times 10^{-19}\) J. If photons of wavelength 221 nm are made to strike the surface (h = 6.63 \(\times 10^{-34}\) Js), the kinetic energy (in J) of the ejected electrons will be

• (1) 2.64 \(\times 10^{-18}\)
• (2) 1.32 \(\times 10^{-19}\)
• (3) 2.64 \(\times 10^{-19}\)
• (4) 5.28 \(\times 10^{-19}\)

In an element with atomic number (Z) 25, the number of electrons with (n + l) value equal to 3 and 4 are x and y respectively. The value of (x + y) is

• (1) 21
• (2) 12
• (3) 14
• (4) 16

Among the ions Mg\(^{2+}\), O\(^{2-}\), Al\(^{3+}\), F\(^{-}\), Na\(^{+}\), and N\(^{3-}\), the ion with the largest size and the ion with the smallest size are respectively

• (1) N\(^{3-}\), Mg\(^{2+}\)
• (2) O\(^{2-}\), F\(^{-}\)
• (3) Al\(^{3+}\), N\(^{3-}\)
• (4) O\(^{2-}\), Al\(^{3+}\)

The correct order of increasing bond lengths of C--H, O--H, C--C, and H--H is

• (1) O--H \(<\) H--H \(<\) C--C \(<\) C--H
• (2) C--C \(<\) C--H \(<\) H--H \(<\) O--H
• (3) C--C \(<\) O--H \(<\) H--H \(<\) C--H
• (4) H--H \(<\) O--H \(<\) C--H \(<\) C--C

The sum of the bond orders of O\(_2^+\), O\(_2^-\), O\(_2\), O\(_2^{2-}\), and the sum of the unpaired electrons in them respectively are

• (1) 10, 4
• (2) 10, 6
• (3) 8, 4
• (4) 8, 6

2.0 g of H\(_2\) diffuses through a porous container in 10 minutes. How many grams of O\(_2\) will diffuse from the same container in the same time under identical conditions?

• (1) 2.0
• (2) 4.0
• (3) 16.0
• (4) 8.0

At T(K), the \(v_{rms}\) of CO\(_2\) is 412 m/s\(^{-1}\). What is its kinetic energy (in kJ mol\(^{-1}\)) at the same temperature? (CO\(_2\) = 44 u)

• (1) 3.7343
• (2) 7.4687
• (3) 14.9374
• (4) 2.7343

100 mL of aqueous solution of 0.05 M Cu\(^{2+}\) is added to 1 L of 0.1 M KI solution. The resultant solution was titrated with 0.1 M Na\(_2\)S\(_2\)O\(_3\) solution using starch indicator until blue color disappeared. What is the volume (in mL) of Na\(_2\)S\(_2\)O\(_3\) used?

• (1) 2000
• (2) 1000
• (3) 500
• (4) 100

Consider the following:

Statement-I: Both internal energy (U) and work (w) are state functions.

Statement-II: During the free expansion of an ideal gas into vacuum, the work done is zero.

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

The signs of \(\Delta H^\circ\) and \(\Delta S^\circ\) for a reaction to be spontaneous at all temperatures respectively are

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

The conjugate base of phosphorus acid is \( x \). The conjugate base of oleum is \( y \).
What are \( x \) and \( y \), respectively?

• (1) \(\mathrm{H_2PO_4^-},\ \mathrm{HS_2O_7^-}\)
• (2) \(\mathrm{H_2PO_4^-},\ \mathrm{HSO_5^-}\)
• (3) \(\mathrm{H_2PO_3^-},\ \mathrm{HS_2O_7^-}\)
• (4) \(\mathrm{H_2PO_3^-},\ \mathrm{HSO_4^-}\)

At temperature \( T(K) \), the equilibrium constant \( K_c \) for the reaction: \[ \mathrm{AO_2(g) + BO_2(g) \rightleftharpoons AO_3(g) + BO(g)} \]
is 16. In a 1 L flask, one mole each of \(\mathrm{AO_2}\), \(\mathrm{BO_2}\), \(\mathrm{AO_3}\), and \(\mathrm{BO}\) are taken and heated to \( T(K) \).
Identify the correct statements:

Total number of moles at equilibrium is 4
At equilibrium, the ratio of moles of \(\mathrm{AO_2}\) and \(\mathrm{AO_3}\) is 1:4
Total number of moles of \(\mathrm{AO_2}\) and \(\mathrm{BO_2}\) at equilibrium is 0.8

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

Identify the hydride which is \textbf{not} correctly matched with the example given in brackets.

• (1) Saline hydride -- (NaH)
• (2) Electron rich hydride -- (H₂O)
• (3) Electron deficient hydride -- (B₂H6)
• (4) Electron precise hydride -- (HF)

In Solvay process, \(\mathrm{NH_3}\) is recovered when the solution containing \(\mathrm{NH_4Cl}\) is treated with compound ‘X’. What is ‘X’?

• (1) \(\mathrm{Ca(OH)_2}\)
• (2) \(\mathrm{CaCl_2}\)
• (3) \(\mathrm{NaOH}\)
• (4) \(\mathrm{NaCl}\)

Which of the following reactions give \(\mathrm{H_2}\) as one of the products?
(Reactions are not balanced.)

\(\mathrm{NaBH_4 + I_2}\)
\(\mathrm{B_2H_6 + N(CH_3)_3}\)
\(\mathrm{Al + NaOH + H_2O}\)
\(\mathrm{BF_3 + NaH}\)

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

Consider the following:

Statement I: CCl₄ does not undergo hydrolysis. But SiCl₄ undergoes hydrolysis.

Statement II: Thermal and chemical stability of GeX₄ is more than GeX2.

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

Which one of the following gases is the major contributor to global warming?

• (1) CO
• (2) CO₂
• (3) CH4
• (4) N₂O

Match the Following

• (1) A–III, B–II, C–I
• (2) A–II, B–III, C–I
• (3) A–II, B–I, C–III
• (4) A–I, B–II, C–III

What is 'Z' in the following reaction sequence?
(Alcohol = colorless)
\[ \mathrm{C_3H_6} \xrightarrow[CCl_4]{\mathrm{Br_2}} \mathrm{X} \xrightarrow[\Delta]{i) KOH/alcohol
ii) NaNH_2} \mathrm{Y} \xrightarrow[\mathrm{Hg}^{2+}, \mathrm{H}^+]{333\,K} \mathrm{Z} \]

• (1) Acetone
• (2) Propanal
• (3) Propanol-2
• (4) Methoxy ethane

A metal crystallizes in simple cubic lattice. The radius of the metal atom is \(x\, pm\). What is the volume of the unit cell in \(pm^3\)?

• (1) \(x^3\)
• (2) \(4x^3\)
• (3) \(8x^3\)
• (4) \(16x^3\)

At \(T(K)\), the vapor pressure of water is \(x\) kPa. What is the vapor pressure (in kPa) of 1 molal solution containing non-volatile solute?

• (1) \(1.018x\)
• (2) \(0.8x\)
• (3) \(0.972x\)
• (4) \(0.982x\)

Elements X and Y form two non-volatile compounds (XY and XY\(_2\)). When 10 g of XY is dissolved in 50 g of ethanol, the depression in freezing point (\(\Delta T_f\)) was 5.333 K. When 10 g of XY\(_2\) is dissolved in 50 g of ethanol, the \(\Delta T_f\) was 2.2857 K. What are the atomic weights of X and Y respectively?

(\(K_f = 2 \, kg mol^{-1}\))

• (1) 50 u, 50 u
• (2) 25 u, 25 u
• (3) 75 u, 100 u
• (4) 25 u, 50 u

In a cell, a copper electrode was used as a cathode. What is the electrode potential (in V) of the copper electrode dipped in 0.1 M Cu\(^{2+}\) solution at 298 K?
\[ E^{\circ} = 0.34\, V, \quad \frac{2.303RT}{F} = 0.059\, V \]

• (1) 0.34
• (2) 0.31
• (3) 0.37
• (4) 0.40

\(R \to P\) is a first-order reaction. The concentration of R changed from 0.04 to 0.03 mol L\(^{-1}\) in 40 minutes. What is the average velocity of the reaction in mol L\(^{-1}\) s\(^{-1}\)?

• (1) \(2.5 \times 10^{-4}\)
• (2) \(4.167 \times 10^{-6}\)
• (3) \(4.167 \times 10^{-5}\)
• (4) \(2.5 \times 10^{-5}\)

Choose the incorrect statement from the following:

• (1) Brownian movement and Tyndall effect are shown by colloidal systems.
• (2) Hardy-Schulze rule is related with coagulation.
• (3) Gold number is a measure of the protection power of a lyophilic colloid.
• (4) Aerosol is a colloidal system in which gas is dispersed in liquid.

Which of the following statements regarding adsorption theory of heterogeneous catalysis is not correct?

• (1) The reactant molecules get adsorbed on the surface of the catalyst
• (2) The chemical reaction occurs at the surface of the catalyst
• (3) The product molecules remain permanently bound to the catalyst surface
• (4) The catalyst remains unchanged in mass and chemical composition at the end of the reaction

Which of the following are carbonate ores?


I. Siderite

II. Kaolinite

III. Calamine

IV. Sphalerite

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

Orthophosphorus acid on disproportionation gives PH\(_3\), and another oxoacid of phosphorus X. The basicity of X is

• (1) 2
• (2) 1
• (3) 3
• (4) 4

Identify the incorrect statement regarding the interstitial compounds.

• (1) They have high melting points
• (2) They lose electrical conductivity during the formation from metal
• (3) They are chemically inert
• (4) They are very hard

Which of the following exhibit ionization isomerism?


I) [Cr(NH\(_3\))\(_4\)Cl\(_2\)]Cl

II) [Ti(H\(_2\)O)\(_5\)Cl]Cl(NO\(_3\))\(_2\)

III) [Pt(en)(NH\(_3\))\(_4\)]Cl(NO\(_3\))

IV) [Co(NH\(_3\))\(_4\)(NO\(_3\))\(_2\)]NO\(_3\)

• (1) II & III only
• (2) I & II only
• (3) II & IV only
• (4) III & IV only

A polymer sample contains 10 molecules each with molecular mass 5,000 and 5 molecules each with molecular mass 50,000. The number average molecular mass of the polymer sample is

• (1) \(2 \times 10^4\)
• (2) \(3 \times 10^4\)
• (3) \(2 \times 10^5\)
• (4) \(3 \times 10^5\)

Which of the following do not reduce Tollens' reagent?


a) Fructose

b) Sucrose

c) Lactose

d) Cellulose

• (1) Fructose, Sucrose
• (2) Sucrose, Cellulose
• (3) Fructose, Lactose
• (4) Lactose, Cellulose

Consider the following statements:

Statement-I: Lysine, arginine are essential and basic amino acids.

Statement-II: Leucine, phenyl alanine are non-essential and neutral amino acids.

Which of the following is correct?

• (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

Consider the following statements:

Statement-I: Shaving soaps contain glycerol to prevent rapid drying.

Statement-II: Laundry soaps contain sodium carbonate as filler.

Which of the following is correct?

• (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

Which of the following sets of reagents convert aniline to chlorobenzene?

• (1) NaNO\(_2\) / HCl, 273 – 278 K; Cu\(_2\)Cl\(_2\) / HCl
• (2) NaNO\(_2\) / HCl, 293 – 298 K; Cu\(_2\)Cl\(_2\) / HCl
• (3) NaNO\(_2\) / HCl, 273 – 278 K; SOCl\(_2\)
• (4) NaNO\(_2\) / HCl, 273 – 278 K; Cl\(_2\)

Identify the compound which is least reactive towards nucleophilic substitution reactions.

An alcohol X \(\mathrm{C_5H_{12}O}\) on dehydration gives Y (major product). Reaction of Y with HBr gave Z (\(\mathrm{C_5H_{11}Br}\), major product). Z undergoes nucleophilic substitution in two steps. What are X and Y?

What are X and Y respectively in the following reaction sequence?
(G = ethanol, dil. = dilute HCl)

The carboxylic acid with highest \(pK_a\) and lowest \(pK_a\) values of the following respectively are:

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

The percentage of carbon in 'Z' is (At.wt. C = 12 u, H = 1 u, N = 14 u, O = 16 u)

• (1) 71.3%
• (2) 51.3%
• (3) 61.3%
• (4) 48.3%