AP EAPCET (AP EAMCET) 2025 Question Paper May 21 Shift 2: Download Solutions with Answer Key

The set of all real values of \(x\) such that \[ f(x) = \frac{[x] - 1}{\sqrt{[x]^2 - [x] - 6}} \]
is a real valued function is

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

If a function \(f : \mathbb{Z} \to \mathbb{Z}\) is defined by \(f(x) = x - (-1)^x\), then \(f(x)\) is

• (1) one-one, but not onto
• (2) onto, but not one-one
• (3) both one-one and onto
• (4) neither one-one nor onto

If \(2.5 + 5.9 + 8.13 + 11.17 + \ldots\) to \(n\) terms = \(an^3 + bn^2 + cn + d\), then find \(a - b - c - d\)

• (1) 7
• (2) 5
• (3) -3
• (4) -1

If \[ A = \begin{bmatrix} 1 & 2 & -2
2 & -1 & 2
-1 & 1 & -2 \end{bmatrix}, \]
then find \(A + 2A^{-1}\).

• (1) \(\begin{bmatrix} 1 & 4 & 0
  4 & -5 & -4
  0 & -2 & -7 \end{bmatrix}\)
• (2) \(\begin{bmatrix} 0 & 2 & 2
  2 & -4 & -6
  2 & -3 & -5 \end{bmatrix}\)
• (3) \(\begin{bmatrix} 0 & 2 & 1
  2 & -4 & -3
  2 & -6 & -5 \end{bmatrix}\)
• (4) \(\begin{bmatrix} 1 & 4 & -1
  4 & -5 & -1
  1 & -5 & -7 \end{bmatrix}\)

If \[ A = \begin{bmatrix} a & b & c
d & e & f
l & m & n \end{bmatrix} \]
is a matrix such that \(|A| > 0\) and \[ Adj(A) = \begin{bmatrix} 0 & 4 & -6
10 & 8 & 0
2 & 4 & -4 \end{bmatrix}, \]
then find the value of \[ \frac{cd}{fb} + \frac{ln}{em}. \]

• (1) \(2a\)
• (2) \(a + m\)
• (3) \(a + b\)
• (4) \(a\)

In solving a system of linear equations \(AX = B\) by Cramer's rule, in the usual notation, if \[ \Delta_1 = \begin{vmatrix} -11 & 1 & -7
-4 & 1 & -2
5 & 1 & 1 \end{vmatrix} \quad and \quad \Delta_3 = \begin{vmatrix} 4 & 1 & -11
3 & 1 & -4
4 & 1 & 5 \end{vmatrix}, \quad then X = ? \]

• (1) \(\begin{bmatrix} -1
  1
  2 \end{bmatrix}\)
• (2) \(\begin{bmatrix} 2
  1
  -1 \end{bmatrix}\)
• (3) \(\begin{bmatrix} 1
  -1
  2 \end{bmatrix}\)
• (4) \(\begin{bmatrix} 1
  2
  -1 \end{bmatrix}\)

If \(a = \ln \left( \frac{1}{z^2} \right)\) and \(z\) is any non-zero complex number such that \(|z| = 1\), then which of the following is the correct expression for \(a\)?

• (1) \(Re(z) Im(z)\)
• (2) \(Re(z)\)
• (3) \(-Re(z)\)
• (4) \(Re(z) + Im(z)\)

If \((3 + 4i)^{2025} = 5^{2023}(x + iy)\), then find \(\sqrt{x^2 + y^2}\).

• (1) 5
• (2) 25
• (3) 125
• (4) 625

If \[ \left(\frac{\cos \theta + i \sin \theta}{\sin \theta + i \cos \theta}\right)^{2024} + \left(\frac{1 + \cos \theta + i \sin \theta}{1 - \cos \theta + i \sin \theta}\right)^{2025} = x + iy, \]
and \(x + y\) at \(\theta = \frac{\pi}{2}\) is

• (1) 1
• (2) -1
• (3) 2
• (4) 2024

The roots \(\alpha, \beta\) of the equation \[ x^2 - 6(k-1)x + 4(k-2) = 0 \]
are equal in magnitude but opposite in sign. If \(\alpha > \beta\), then the product of the roots of the equation \[ 2x^2 - \alpha x + 6\beta (\alpha + 1) = 0 \]
is

• (1) 12
• (2) -12
• (3) 16
• (4) -18

If \( ax^2 + bx + e > 0 \) for all \( x \in \mathbb{R} \) and the expressions \( cx^2 + ax + b \) and \( ax^2 + bx + c \) have their extreme values at the same point \( x \), then for the expression \( cx^2 + ax + b \), find the correct statement regarding its extreme value.

• (1) Minimum value = \(\frac{4b}{3}\)
• (2) Minimum value = \(\frac{4b}{3}\)
• (3) Maximum value = \(\frac{4a}{3}\), Minimum value = \(\frac{3a}{4}\)
• (4) Maximum value = \(\frac{3b}{4}\)

If \( a \pm bi \) and \( b \pm ai \) are roots of \( x^4 - 10x^3 + 50x^2 - 130x + 169 = 0 \), then find the value of \( \frac{a}{b} + \frac{b}{a} \).

• (1) \(\frac{25}{12}\)
• (2) \(\frac{5}{2}\)
• (3) \(\frac{13}{6}\)
• (4) \(\frac{34}{15}\)

If \( x^2 - 5x + 6 \) is a factor of \( f(x) = x^4 - 17x^3 + kx^2 - 247x + 210 \), find the other quadratic factor of \( f(x) \).

• (1) \( x^2 + 12x + 35 \)
• (2) \( x^2 - 12x + 35 \)
• (3) \( x^2 - 6x + 35 \)
• (4) \( x^2 + 6x + 35 \)

If all letters of the word COMBINATION are arranged to form 11-letter words with \( C \) and \( N \) at the ends and no vowel in the middle position, find the number of such words.

• (1) \(\frac{5}{2} (8!)\)
• (2) \(4 (8!)\)
• (3) \(2 (8!)\)
• (4) \(36 (7!)\)

The number of ways of distributing 3 dozen fruits (no two fruits are identical) to 9 persons such that each gets the same number of fruits is

• (1) \(\frac{36!}{(9!)^4}\)
• (2) \(\frac{36!}{(4!)^9}\)
• (3) \(^{36}P_9 \times 4!\)
• (4) \(\frac{36!}{4!(9!)^4}\)

If \[ \binom{p}{q} = \binom{p}{q} \quad and \quad \sum_{i=0}^m \binom{10}{i} \binom{20}{m-i} is maximum, then find m. \]

• (1) 10
• (2) 12
• (3) 15
• (4) 20

Coefficient of \(x^2\) in the expansion of \((x^2 + x - 2)^5\) is

• (1) 800
• (2) 756
• (3) 0
• (4) 512

If \(P_n\) denotes the product of the binomial coefficients in the expansion of \((1 + x)^n\), then find \[ \frac{P_{n+1}}{P_n}. \]

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

The coefficient of \(x^3\) in the expansion of \(\frac{x^4 + 1}{(x^2 + 1)(x - 1)}\) when it is expressed in terms of positive integral powers of \(x\), is

• (1) 0
• (2) 1
• (3) 16
• (4) 24

If the polynomial \( f(x) = x^4 + ax^3 + bx^2 + cx + d \) is divided by \( x - 1 \) and \( x + 1 \), the remainders are 5 and 3 respectively. If \( f(x) \) is divided by \( x^2 - 1 \), then the remainder is

• (1) 2x + 3
• (2) 2x - 3
• (3) x + 2
• (4) x - 2

Evaluate the expression:



\[ \cos^3 \left( \frac{3\pi}{8} \right) \cos \left( \frac{3\pi}{8} \right) + \sin^3 \left( \frac{3\pi}{8} \right) \sin \left( \frac{3\pi}{8} \right) \]

 

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

If \(A + B + C = \dfrac{\pi}{4}\), then \(\sin 4A + \sin 4B + \sin 4C =\)

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

If \( A + B + C = \frac{\pi}{4} \), then evaluate the expression:



\[ \sin 4A + \sin 4B + \sin 4C \]

 

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

If \(x\) is a real number, then the number of solutions of \(\tan^{-1}\left(\sqrt{x(x+1)}\right) + \sin^{-1}\left(\sqrt{x^2 + x + 1}\right) = \dfrac{\pi}{2}\) is

• (1) 1
• (2) 2
• (3) 3
• (4) 4

Domain of the real-valued function \(f(x) = \log(x^2 - 1) + x \, \coth^{-1}x\) is

• (1) \(\mathbb{R}\)
• (2) \((-1,1)\)
• (3) \(\mathbb{R} - [-1,1]\)
• (4) \([0,1)\)

In a triangle ABC, if \(\sin\frac{A}{2} = \dfrac{1}{4}\sqrt{\dfrac{5}{\sqrt{5}}}, a = 2, c = 5\), and \(b\) is an integer, then the area (in sq. units) of triangle ABC is

• (1) \(\dfrac{\sqrt{297}}{4}\)
• (2) \(\dfrac{\sqrt{231}}{4}\)
• (3) \(\dfrac{\sqrt{385}}{4}\)
• (4) \(\dfrac{\sqrt{185}}{4}\)

In \(\triangle ABC\), if \(a + c = 5b\), then \(\cot\dfrac{A}{2} \cdot \cot\dfrac{C}{2} =\)

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

In a triangle ABC, if \(r_1 = 3, r_2 = 4, r_3 = 6\), then \(b =\)

• (1) \(2\sqrt{6}\)
• (2) \(\dfrac{5\sqrt{6}}{3}\)
• (3) \(\dfrac{7\sqrt{6}}{3}\)
• (4) \(3\sqrt{6}\)

Let the position vectors of the vertices of triangle ABC be \(\vec{a}, \vec{b}, \vec{c}\). If a point \(P\) on the plane of triangle has a position vector \(\vec{r}\) such that \(\vec{r} - \vec{b} = \vec{a} - \vec{c}\) and \(\vec{r} - \vec{c} = \vec{a} - \vec{b}\), then \(P\) is the

• (1) Centroid
• (2) Circumcentre
• (3) Incentre
• (4) Orthocentre

The point of intersection of the lines represented by \(\vec{r} = (\hat{i} - 6\hat{j} + 2\hat{k}) + t(\hat{i} + 2\hat{j} + \hat{k})\) and \(\vec{r} = (4\hat{j} + \hat{k}) + s(2\hat{i} + \hat{j} + 2\hat{k})\) is

• (1) \(8\hat{i} + 9\hat{j} + 10\hat{k}\)
• (2) \(8\hat{i} + 8\hat{j} + 7\hat{k}\)
• (3) \(8\hat{i} + 9\hat{j} + 8\hat{k}\)
• (4) \(8\hat{i} + 8\hat{j} + 9\hat{k}\)

If \(|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5, |\vec{a} + \vec{b} + \vec{c}| = \sqrt{69}\) and angle between \((\vec{a}, \vec{b}) = \dfrac{\pi}{3}\), then angle between \((\vec{c}, \vec{a}) =\)

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

If the points A, B, C, D with position vectors \(\vec{i} + \vec{j} - \vec{k}, -\vec{i} + 2\vec{k}, \vec{i} - 2\vec{j} + \vec{k}, 2\vec{i} + \vec{j} + \vec{k}\) form a tetrahedron, then angle between faces ABC and ABD is

• (1) \(\cos^{-1}\left(\dfrac{-4}{\sqrt{29}}\right)\)
• (2) \(\cos^{-1}\left(\dfrac{-4}{5}\right)\)
• (3) \(\cos^{-1}\left(\dfrac{3}{5}\right)\)
• (4) \(\cos^{-1}\left(\dfrac{\sqrt{29}}{\sqrt{33}}\right)\)

If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and \(\vec{a} \perp \vec{b}\), and \((\vec{a} - \vec{c}) \cdot (\vec{b} + \vec{c}) = 0\), and \(\vec{c} = l\vec{a} + m\vec{b} + n(\vec{a} \times \vec{b})\), then \(n^2 =\)

• (1) \(l^2 + m^2\)
• (2) \(-\dfrac{2}{m}\)
• (3) \(2l - 2m\)
• (4) \(\dfrac{l}{m} + l + m\)

If the variance of the first \(n\) natural numbers is 10 and the variance of the first \(m\) even natural numbers is 16, then \(n : m =\)

• (1) \(9 : 5\)
• (2) \(7 : 3\)
• (3) \(11 : 7\)
• (4) \(5 : 8\)

Given \(f(x) = x^2 - 5x + 4\). Out of first 20 natural numbers, if a number \(x\) is chosen at random, then the probability that the chosen \(x\) satisfies the inequality \(f(x) > 10\) is

• (1) \(\dfrac{1}{2}\)
• (2) \(\dfrac{3}{4}\)
• (3) \(\dfrac{7}{20}\)
• (4) \(\dfrac{13}{20}\)

A problem in Algebra is given to two students A and B whose chances of solving it are \(\dfrac{2}{5}\) and \(\dfrac{3}{5}\) respectively. The probability that the problem is solved if both try independently is

• (1) \(\dfrac{17}{20}\)
• (2) \(\dfrac{3}{20}\)
• (3) \(\dfrac{1}{2}\)
• (4) \(\dfrac{13}{20}\)

Three dice are thrown simultaneously and the sum of the numbers is noted. If A = getting sum greater than 14 and B = getting sum divisible by 3, then \(P(A \cap B) + P(A \cup B) =\)

• (1) \(\dfrac{35}{108}\)
• (2) \(\dfrac{17}{54}\)
• (3) \(\dfrac{45}{108}\)
• (4) \(\dfrac{5}{12}\)

A manufacturing company has 3 units A, B, and C which produce 25%, 35%, 40% of bulbs respectively. 5%, 4%, and 2% of their production is defective. If a bulb is found defective, the probability it came from B is

• (1) \(\dfrac{28}{69}\)
• (2) \(\dfrac{28}{71}\)
• (3) \(\dfrac{29}{67}\)

Given the PMF: \(P(X=x) = \alpha\) for \(x = 1,2\), \(= \beta\) for \(x = 4,5\), and \(= 0.3\) for \(x = 3\), with mean \(\mu = 4.2\). Find \(\sigma^2 + \mu^2\)

• (1) 20.4
• (2) 10.8
• (3) 16.4
• (4) 21.4

A student has probability \(\dfrac{2}{3}\) of getting distinction in a test. Out of 5 tests, the probability that he gets distinction in at least 3 tests is

• (1) \(\dfrac{112}{243}\)
• (2) \(\dfrac{17}{81}\)
• (3) \(\dfrac{131}{243}\)
• (4) \(\dfrac{64}{81}\)

If \(P\) is a variable point which is at a distance of 2 units from the line \(2x - 3y + 1 = 0\) and \(\sqrt{13}\) units from the point (5, 6), then the equation of the locus of \(P\) is

• (1) \(4x^2 + 12xy - 5y^2 - 44x - 42y + 245 = 0\)
• (2) \(12xy - 5y^2 - 44x - 42y + 243 = 0\)
• (3) \(8x^2 + 12xy - 5y^2 - 44x - 42y + 243 = 0\)
• (4) \(12xy - 13y^2 - 44x - 42y + 245 = 0\)

If the equation \(3x^2 + 4y^2 - xy + k = 0\) is the transformed equation of \(3x^2 + 4y^2 - xy - 5x - 7y + 2 = 0\) after shifting the origin to \((\alpha, \beta)\), then \(\alpha + \beta = k =\)

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

If the intercept of a line \(L\) made between the straight lines \(5x - y - 4 = 0\) and \(3x + 4y - 4 = 0\) is bisected at the point (1, 5), then the equation of \(L\) is

• (1) \(35x - 83y + 92 = 0\)
• (2) \(83x + 35y - 72 = 0\)
• (3) \(63x - 35y + 82 = 0\)
• (4) \(83x - 35y + 92 = 0\)

A line \(L\) passes through point \(P(1, 2)\) and makes an angle of \(60^\circ\) with OX in positive direction. A and B are points on line \(L\), 4 units from P. If O is origin, then area of \(\triangle OAB\) is

• (1) \(4 - 2\sqrt{3}\)
• (2) \(8 - 4\sqrt{3}\)
• (3) \(4 + 2\sqrt{3}\)
• (4) \(8 + 4\sqrt{3}\)

The equation \((2p - 3)x^2 + 2pxy - y^2 = 0\) represents a pair of distinct lines

• (1) Only when \(p = 0\)
• (2) \(p \in \mathbb{R} - \{-3,1\}\)
• (3) For all values of \(p \in \mathbb{R} - [-3,1]\)
• (4) For all values of \(p \in \mathbb{R}\)

The equation of chord AB of ellipse \(2x^2 + y^2 = 1\) is \(x - y + 1 = 0\). If O is the origin, then \(\angle AOB =\)

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

If a circle S passes through the origin and makes intercept 4 units on line \(x = 2\), then the equation of curve on which center of S lies is

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

A circle touches the line \(2x + y - 10 = 0\) at (3, 4) and passes through the point (1, -2). Then a point that lies on the circle is

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

If (a, b) is the common point of the circles \(x^2 + y^2 - 4x + 4y - 1 = 0\) and \(x^2 + y^2 + 2x - 4y + 1 = 0\), then \(a^2 + b^2 =\)

• (1) \(\dfrac{1}{5}\)
• (2) \(5\)
• (3) \(25\)
• (4) \(\dfrac{1}{25}\)

The angle between the tangents drawn from the point (2, 2) to the circle \(x^2 + y^2 + 4x + 4y + c = 0\) is \(\cos^{-1} \left( \frac{7}{16} \right)\). If two such circles exist, then the sum of values of \(c\) is

• (1) 16
• (2) 20
• (3) -20
• (4) -16

If the circle \(S_1 = x^2 + y^2 + 2gx + 4y + 1 = 0\) bisects the circumference of circle \(x^2 + y^2 - 2x - 3 = 0\), then the radius of circle \(S_1\) is

• (1) \(5\)
• (2) \(\sqrt{12}\)
• (3) \(25\)
• (4) \(12\)

The angle between the tangents drawn from point (1, 4) to parabola \(y^2 = 4x\) is

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

The square of the slope of a common tangent to the circle \(4x^2 + 4y^2 = 25\) and ellipse \(4x^2 + 9y^2 = 36\) is

• (1) \(1\)
• (2) \(\dfrac{9}{11}\)
• (3) \(\dfrac{2}{3}\)
• (4) \(2\)

The tangents drawn to the hyperbola \(5x^2 - 9y^2 = 90\) through a variable point \(P\) make angles \(\alpha\) and \(\beta\) with its transverse axis. If \(\alpha\) and \(\beta\) are complementary angles, then the locus of \(P\) is

• (1) \(x^2 + y^2 = 8\)
• (2) \(x^2 - y^2 = 8\)
• (3) \(x^2 - y^2 = 28\)
• (4) \(x^2 + y^2 = 28\)

If \(\theta\) is the acute angle between the asymptotes of a hyperbola \(7x^2 - 9y^2 = 63\), then \(\cos \theta =\)

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

If \(O(0,0,0), A(1,2,1), B(2,1,3)\), and \(C(-1,1,2)\) are the vertices of a tetrahedron, then the acute angle between its face \(OAB\) and edge \(BC\) is

• (1) \(\cos^{-1} \left( \dfrac{6\sqrt{2}}{5\sqrt{7}} \right)\)
• (2) \(\sin^{-1} \left( \dfrac{6\sqrt{2}}{5\sqrt{7}} \right)\)
• (3) \(\tan^{-1} \left( \dfrac{6\sqrt{2}}{5\sqrt{7}} \right)\)
• (4) \(\dfrac{\pi}{2}\)

If the angles between the sides of triangle ABC formed by A(2,3,5), B(-1,2,3), and C(3,5,-2) are \(\alpha, \beta, \gamma\), then \(\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma =\)

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

If the four points (6,2,4), (1,3,5), (1,-2,3), and (6,k,2) are coplanar, then \(k =\)

• (1) -5
• (2) 4
• (3) -3
• (4) 1

Evaluate \(\lim\limits_{x \to \infty} \dfrac{5x^3 - x^2 \sin 5x}{x^3 \cos 4x + 7|x|^3 - 4|x| + 3}\)

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

If \(\lim\limits_{x \to a^-} f(x) = p\), \(\lim\limits_{x \to a^+} f(x) = m\), and \(f(a) = k\), then which one of the following is true?

• (1) \(p - k = 0\) and \(m - k = 0\)
• (2) \(p - k = 0\) and \(m - k \neq 0\)
• (3) \(p - k \neq 0\) and \(m - k = 0\)
• (4) \(p - m = 0\) and \(p - k \neq 0\)

If a function defined by \[ f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x < 0
a, & x = 0
\dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x} - 4}}, & x > 0 \end{cases} \]
is continuous at \(x = 0\), then \(a =\)

• (1) 8
• (2) 4
• (3) 3
• (4) 2

If \( y = \tanh^{-1} \left( \dfrac{1 - x}{1 + x} \right) \), then \( \dfrac{dy}{dx} = \)

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

If \(x^2 + y^2 = \dfrac{1}{t} and x^4 + y^4 = t^2 + \dfrac{1}{t^2},\) then \(\dfrac{dy}{dx} =\)

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

If \(y = (ax + b)\cos x\), then \(y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = \)

• (1) \(y_2 \cos^2 x\)
• (2) \(y_2 \sin^2 x\)
• (3) \(y_1 \sin^2 x\)
• (4) \(y \sin^2 x\)

If the normal drawn at the point P on the curve \(y = x \log x\) is parallel to the line \(2x - 2y = 3\), then P =

• (1) \((e, e)\)
• (2) \(\left(\dfrac{1}{e}, -1\right)\)
• (3) \(\left(\dfrac{1}{e}, -\dfrac{2}{e^2}\right)\)
• (4) \((e^3, 3e^3)\)

If the curves \(y^2 = 16x and 9x^2 + \alpha y^2 = 25\) intersect at right angles, then \(\alpha =\)

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

If the function \(y = \sin(x)(1 - \cos x)\) is defined in the interval \([-\pi, \pi]\), then y is strictly increasing in the interval

• (1) \(\left(-\pi, -\dfrac{\pi}{3} \right) \cup \left(\dfrac{\pi}{3}, \pi \right)\)
• (2) \(\left( \dfrac{\pi}{6}, \dfrac{\pi}{2} \right)\)
• (3) \(\left(-\dfrac{\pi}{3}, 0 \right) \cup \left(0, \dfrac{\pi}{3} \right)\)
• (4) \(\left(-\dfrac{\pi}{6}, 0 \right) \cup \left(0, \dfrac{\pi}{6} \right)\)

If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is

• (1) constant
• (2) inversely proportional to its velocity
• (3) proportional to its velocity
• (4) proportional to its displacement

If \( \int e^{\sin x}(1 + \sec x \tan x)\, dx = e^{\sin x}f(x) + c \), then in \( 0 \leq x \leq 2\pi \), the number of solutions of \( f(x) = 1 \) is

• (1) 4
• (2) 0
• (3) 2
• (4) 3

If \( \int \frac{dx}{(x-1)^2(x-3)^2} = \sqrt{f(x)} + c \), then \( f(-1) - f(0) = \)

• (1) -3
• (2) -4
• (3) -2
• (4) -1

\( \int \frac{x}{(1-x^2)\sqrt{2 - x^2}} dx = \)

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

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

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

If \( \int x^2 \cos^3 x\, dx = \frac{1}{6}f(x) + g(x) \sin 2x + h(x) \cos 2x + c \), then \( f(1) + g(2) + h\left(\frac{1}{2}\right) = \)

• (1) 0
• (2) 2
• (3) 1
• (4) -1

Evaluate the integral \(\displaystyle \int_0^{\frac{\pi}{2}} \log(\tan x + \cot x)\, dx\)

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

Evaluate the integral \[ \int_{0}^{\pi} x \cdot \sin x \cdot \int_{x}^{5} \frac{\cos x}{x} \cdot dx = \]

• (1) \( \dfrac{16\pi}{693} \)
• (2) \( \dfrac{8\pi}{693} \)
• (3) \( \dfrac{4\pi}{693} \)
• (4) \( \dfrac{2\pi}{693} \)

Evaluate the integral \[ \int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \frac{1}{\left(x + \sqrt{1 - x^2}\right) \cdot \left(1 - x^2\right)} \, dx = \]

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

The area of the region (in sq. units) enclosed between the curves \( y = |x| \), \( y = [x] \) and the ordinates \( x = -1, x = 0, x = 1 \) is

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

The general solution of the differential equation \(\frac{dy}{dx} + xy = 4x - 2y + 8\) is

• (1) \(y = 4 - ce^{-\frac{(x+2)^2}{2}}\)
• (2) \(y = 8 + ce^{-\frac{x^2}{2} - 2x}\)
• (3) \(y = ce^{-(x+2)^2} + x\)
• (4) \(y + 2x = ce^{-\frac{x}{2} - 2x}\)

The general solution of the differential equation \((x+2y)^3\frac{dy}{dx} = y = 0, y > 0\) is

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

The general solution of the differential equation \(\frac{dy}{dx} = \frac{x + y + 1}{x - 3y + 5}\) is

• (1) \(3(y - 1)^2 - 2(x + 2)(y - 1) - (x + 2)^2 = c\)
• (2) \(x^2 - 3y^2 - 4xy - 2x - 10y = c\)
• (3) \(3(y + 1)^2 + 2(x - 2)(y + 1) - (x - 2)^2 = c\)
• (4) \(x^2 + 3y^2 + 4xy + 2x + 10y = c\)

If the maximum and minimum temperatures at a place on a day are measured as \(44^\circ C \pm 0.5^\circ C\) and \(22^\circ C \pm 0.5^\circ C\) respectively, then the temperature difference is

• (1) \(22^\circ C \pm 1^\circ C\)
• (2) \(22^\circ C \pm 0.5^\circ C\)
• (3) \(22^\circ C \pm 0.25^\circ C\)
• (4) \(22^\circ C \pm 1.5^\circ C\)

If a ball projected vertically upwards with certain initial velocity from the ground crosses a point at a height of 25 m twice in a time interval of 4 s, then the initial velocity of the ball is

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

• (1) \(20~m/s\)
• (2) \(30~m/s\)
• (3) \(40~m/s\)
• (4) \(25~m/s\)

If a particle of mass 'm' covers half of the horizontal circle with constant speed 'v', then the change in its kinetic energy is

• (1) \(mv^2\)
• (2) Zero (శూన్యం)
• (3) \(2mv^2\)
• (4) \(\dfrac{1}{2}mv^2\)

A car is moving with a velocity of \(4~m/s\) towards east. After a time of \(4~s\), it is heading north-east with a velocity of \(4\sqrt{2}~m/s\). Then the average velocity of the car is

• (1) \(2\sqrt{5}~m/s\)
• (2) \(3\sqrt{5}~m/s\)
• (3) \(4\sqrt{3}~m/s\)
• (4) \(5\sqrt{3}~m/s\)

A body of mass \(5~kg\) starts from the origin with an initial velocity \(\vec{v}_0 = (30\hat{i} + 40\hat{j})~m/s\).

If a constant force \(\vec{F} = -(i + 5j)~N\) acts on the body, then the time in which the y-component of its velocity becomes zero is

• (1) \(5~s\)
• (2) \(20~s\)
• (3) \(40~s\)
• (4) \(80~s\)

A block of mass \(10~kg\) moving with a speed of \(5~m/s\) on a frictionless horizontal surface suddenly explodes into two pieces.

If one piece with mass \(4~kg\) moves with a speed of \(10~m/s\), then the velocity of the second piece is

• (1) \(7.67~m/s\)
• (2) \(1.67~m/s\)
• (3) \(6.67~m/s\)
• (4) \(2.67~m/s\)

The bob of a simple pendulum of length \(200~cm\) is released from horizontal position.

If \(10%\) of its initial energy is lost to air resistance, then the speed of bob at the mean position is

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

• (1) \(6~m/s\)
• (2) \(3~m/s\)
• (3) \(12~m/s\)
• (4) \(2~m/s\)

A steel sphere of radius \(1.2~cm\) collides with another steel sphere at rest.

If the collision is elastic and the first sphere moves with \(\dfrac{7}{9}\) of its initial velocity after collision,

then the radius of the second sphere is

• (1) \(1.8~cm\)
• (2) \(2.4~cm\)
• (3) \(1.2~cm\)
• (4) \(0.6~cm\)

Ratio of angular velocity of hour hand of a watch and the angular velocity of rotation of earth is

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

Two bodies of masses \(2~kg\) and \(3~kg\) move at right angles with velocities \(20~m/s\) and \(10~m/s\) respectively.

Then the velocity of the centre of mass of the system is

• (1) \(5~m/s\)
• (2) \(30~m/s\)
• (3) \(10~m/s\)
• (4) \(14~m/s\)

The kinetic energy of a particle executing simple harmonic motion at a displacement of \(3~cm\) from the mean position is \(4~mJ\). If the amplitude of the particle is \(5~cm\), then the maximum force acting on the particle is

• (1) \(0.25~N\)
• (2) \(0.50~N\)
• (3) \(0.75~N\)
• (4) \(1.25~N\)

A body of mass \(1~kg\) is attached to the lower end of a vertically suspended spring of force constant \(600~N/m\). If another body of mass \(0.5~kg\) moving vertically upward hits the suspended body with a velocity \(3~m/s\) and is embedded in it, then the frequency of the oscillation is

• (1) \(\dfrac{5}{\pi}~Hz\)
• (2) \(\dfrac{10}{\pi}~Hz\)
• (3) \(\dfrac{\pi}{5}~Hz\)
• (4) \(\pi~Hz\)

If the angular velocity of a planet about its axis is halved, the distance of the stationary satellite of this planet from the centre of the planet becomes \(2^n\) times the initial distance. Then the value of \(n\) is

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

When a wire of length \(L\) clamped at one end is pulled by a force \(F\) from the other end, its length increases by \(L'\). If the radius and the applied force are halved, then the increase in its length is

• (1) \(3L\)
• (2) \(4L\)
• (3) \(1.5L\)
• (4) \(2L\)

A liquid drop of diameter \(D\) splits into \(3375\) small identical drops. If \(S\) is the surface tension of the liquid, then the change in the surface energy in the process is

• (1) \(44\pi D^2 S\)
• (2) \(44\pi D^3 S\)
• (3) \(56 D^3 S\)
• (4) \(56\pi D^2 S\)

When a sphere is taken to the bottom of a sea of depth \(1~km\), it contracts in volume by \(0.01%\). Then the bulk modulus of the material of the sphere is (Acceleration due to gravity = \(10~m/s^2\))

• (1) \(10 \times 10^6~N/m^2\)
• (2) \(1.2 \times 10^{10}~N/m^2\)
• (3) \(10 \times 10^{10}~N/m^2\)
• (4) \(10 \times 10^{11}~N/m^2\)

If a gas of volume \(400~cc\) at an initial pressure \(P\) is suddenly compressed to \(100~cc\), then its final pressure is (The ratio of specific heats \(\gamma = 1.5\))

• (1) \(\dfrac{P}{32}\)
• (2) \(8P\)
• (3) \(32P\)
• (4) \(16P\)

A Carnot engine having efficiency \(60%\) receives heat from a source at temperature \(600~K\). For the same sink temperature, to increase its efficiency to \(80%\), the temperature of the source is

• (1) \(300~K\)
• (2) \(900~K\)
• (3) \(1200~K\)
• (4) \(720~K\)

A gaseous mixture consists of \(2\) moles of oxygen and \(4\) moles of argon at temperature \(T\). Neglecting all vibrational modes, the total internal energy of the mixture is

• (1) \(4RT\)
• (2) \(15RT\)
• (3) \(9RT\)
• (4) \(11RT\)

The average translational kinetic energy of oxygen molecules at a temperature of \(127^\circ C\) is (Boltzmann constant \(= 1.38 \times 10^{-23}~J/K\))

• (1) \(4.07 \times 10^{-21}~J\)
• (2) \(2.07 \times 10^{-21}~J\)
• (3) \(8.28 \times 10^{-21}~J\)
• (4) \(8.00 \times 10^{-21}~J\)

The speed of a stationary wave represented by the equation \(y = 0.75 \sin\left(\dfrac{7\pi}{4}x\right)\cos(350\pi t)\) is

• (1) \(100~m/s\)
• (2) \(150~m/s\)
• (3) \(160~m/s\)
• (4) \(200~m/s\)

Two thin convex lenses are kept in contact coaxially. If the focal length of the combination is \(4~cm\) and the sum of the focal lengths of the two lenses is \(18~cm\), then the focal length of the lens of low power is

• (1) \(8~cm\)
• (2) \(10~cm\)
• (3) \(6~cm\)
• (4) \(12~cm\)

For an observer on the earth, if a spectral line of wavelength \(6600~\mathring{A}\) emitted by a star is found to be redshifted by \(22~\mathring{A}\), then the star is

• (1) Receding away with \(9 \times 10^5~m/s\)
• (2) Receding away with \(10^6~m/s\)
• (3) Moving towards earth with \(9 \times 10^5~m/s\)
• (4) Moving towards earth with \(10^6~m/s\)

Three particles of each charge \(q\) are placed at the vertices of an equilateral triangle of side \(L\). The work to be done to decrease the side of the triangle to \(\dfrac{L}{2}\) is

• (1) \(\dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{q^2}{L}\)
• (2) \(\dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{2q^2}{L}\)
• (3) \(\dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{3q^2}{L}\)
• (4) \(\dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{3q^2}{2L}\)

The radii of inner and outer spheres of a spherical capacitor are \(8~cm\) and \(9~cm\) respectively. The outer sphere is earthed and the inner sphere is charged. If the space is filled with a dielectric constant \(K = 5\), the capacitance is

• (1) \(400~pF\)
• (2) \(40~pF\)
• (3) \(400~\mu F\)
• (4) \(40~\mu F\)

If 27 charged water droplets, each of radius \(10^{-6}~m\) and charge \(10^{-12}~C\) coalesce to form a single spherical drop, then the potential of the big drop is

• (1) \(9~V\)
• (2) \(27~V\)
• (3) \(39~V\)
• (4) \(81~V\)

A straight wire of resistance \(18~\Omega\) is bent in the form of an equilateral triangle. The effective resistance between any two vertices of the triangle is

• (1) \(6~\Omega\)
• (2) \(3~\Omega\)
• (3) \(1~\Omega\)
• (4) \(4~\Omega\)

The power dissipated by a uniform wire of resistance \(100~\Omega\) when a potential difference of \(120~V\) is applied across its ends is

• (1) \(122~W\)
• (2) \(144~W\)
• (3) \(160~W\)
• (4) \(200~W\)

If a straight current-carrying wire of linear density \(0.12~kg/m\) is suspended in mid-air by a uniform horizontal magnetic field of \(0.5~T\) normal to the length of the wire, then the current through the wire is (Neglect earth’s magnetic field)

• (1) \(2.4~A\)
• (2) \(1.2~A\)
• (3) \(0.6~A\)
• (4) \(4.8~A\)

Two concentric loops \(A\) and \(B\) of same radius \(2\pi~cm\) are placed at right angles to each other. If the currents flowing through \(A\) and \(B\) are \(3~A\) and \(4~A\) respectively, then the net magnetic field at their common center is

• (1) \(0.75 \times 10^{-5}~T\)
• (2) \(25 \times 10^{-5}~T\)
• (3) \(5 \times 10^{-5}~T\)
• (4) \(2.5 \times 10^{-5}~T\)

A short bar magnet is placed in a uniform magnetic field of \(2~T\) such that the axis of the magnet makes an angle of \(45^\circ\) with the field. If the torque acting is \(0.36~Nm\), the magnetic moment of the magnet is

• (1) \(0.54~J/T\)
• (2) \(0.18~J/T\)
• (3) \(0.72~J/T\)
• (4) \(0.36~J/T\)

A horizontal telegraph wire of length \(30~m\) fell from a height of \(20~m\). If resistance is \(40~\Omega\) and horizontal component of Earth’s magnetic field is \(2 \times 10^{-5}~T\), the induced current is

• (1) \(0.3~mA\)
• (2) \(3~mA\)
• (3) \(3~A\)
• (4) \(0.03~A\)

In an LCR series circuit, if the potential differences across inductor, capacitor, and resistor are \(60~V\), \(30~V\), and \(40~V\) respectively, then the net voltage applied to the circuit is

• (1) \(50~V\)
• (2) \(70~V\)
• (3) \(130~V\)
• (4) \(60~V\)

A plane EM wave of frequency \(25~MHz\) propagates in vacuum. If electric field is \(6.3~V/m\), then magnitude of magnetic field is

• (1) \(2.1 \times 10^{-8}~T\)
• (2) \(4.2 \times 10^{-8}~T\)
• (3) \(6.3 \times 10^{-8}~T\)
• (4) \(8.4 \times 10^{-8}~T\)

A particle of mass \(8~\mu g\) collides with another stationary particle of mass \(4~\mu g\). If the collision is perfectly elastic and one dimensional, the ratio of de Broglie wavelengths after collision is

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

The difference between the frequencies of the first and second Lyman lines of hydrogen atom is (R - Rydberg constant and c - speed of light in vacuum)

• (1) \(\dfrac{9Rc}{28}\)
• (2) \(\dfrac{7Rc}{12}\)
• (3) \(\dfrac{3Rc}{8}\)
• (4) \(\dfrac{5Rc}{36}\)

If the half-life of a radioactive element is \(12.5\) hours, then the time taken to disintegrate \(256~g\) of the substance into \(1~g\) is (in hours)

• (1) \(12.5\)
• (2) \(25\)
• (3) \(37.5\)
• (4) \(100\)

A transistor works as an amplifier when

• (1) Emitter-base junction is forward biased and base-collector junction is reverse biased
• (2) Both emitter-base and base-collector junctions are forward biased
• (3) Both emitter-base and base-collector junctions are reverse biased
• (4) Emitter-base junction is reverse biased and base-collector junction is forward biased

If five logic gates are connected as shown in the figure, then the values of \(y_1\), \(y_2\), and \(y_3\) are respectively

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

In amplitude modulation of waves, the maximum amplitude is \(30~mV\) and minimum amplitude is \(5~mV\), then the modulation index is

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

The uncertainty in the position of electron (\(\Delta x\)) is approximately 100 pm. The uncertainty in momentum (in kg m s\(^{-1}\)) of an electron is [h = 6.626 \(\times 10^{-34}\) Js]

• (1) \(1.104 \times 10^{-22}\)
• (2) \(0.527 \times 10^{-27}\)
• (3) \(0.527 \times 10^{-24}\)
• (4) \(1.055 \times 10^{-24}\)

Which of the following statements are correct? (only correct)

I) The energy of hydrogen atom in its ground state is -13.6 eV.
II) On the basis of Bohr's model, the radius of the 3\(^{rd}\) orbit of hydrogen atom is 158.7 pm.
III) The order of radius of the first orbit of H, He\(^+\), Li\(^{2+}\) and Be\(^{3+}\) is H \(>\) He\(^+\) \(>\) Li\(^{2+}\) \(>\) Be\(^{3+}\)

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

Which of the following orders is not correct about the property shown against it?

(1) \(N > O > P > S\) - First ionisation enthalpy

(2) \(N > O > P > S\) - Negative electron gain enthalpy

(3) \(F > Cl > O > S\) - Negative electron gain enthalpy

(4) \(Fe^{3+} < Fe^{2+} < Fe\) - Size

Consider the following changes I and II

Change I: {O₂ -> O₂^2-}
Change II: {O₂ -> O₂^-}

The correct statements about these changes (I) and (II) in accordance with MO theory are (Note:  only =  energy)

• (A) In (I) bond order increases by 0.5 from the existing value
• (B) In (I) bond order decreases by 0.5 from the existing value
• (C) In (II) bond order decreases by 0.5 from the existing value
• (D) In both (I) and (II) magnetic property is not changed
  (1) A, B & C only
  (2) A & C only
  (3) A & D only
  (4) B & C only

The increasing order of number of lone pair electrons on the central atom of the following molecules is which of the following?

I) {ClF₃}   II) {XeF₂}    III) {SF₄}    IV) {SiH₄}

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

Which of the following is the correct statement for an ideal gas (constant = energy)?

At 256 K, the rms speed of {SO₂} molecules is 3.16\(\times 10^2\) m/s. What is the most probable velocity (in m/s) of the same gas molecules at the same temperature?

• (1) 2.911\(\times 10^2\)
• (2) 2.58\(\times 10^2\)
• (3) 3.16\(\times 10^2\)
• (4) 1.29\(\times 10^2\)

209 g of an element reacts with chlorine to form 315.5 g of its chloride. What is the weight of oxygen that reacts with 418 g of the same element? (Cl = 35.5, O = 16)

• (1) 24
• (2) 48
• (3) 96
• (4) 192

Consider the following

Statement-I: During isothermal expansion of an ideal gas its enthalpy decreases.

Statement-II: When 2.0 L of an ideal gas expands isothermally into vacuum, \(\Delta U = 0\).

• (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 energy required to increase the temperature of 180 g of liquid water from 10\(^\circ\)C to 15\(^\circ\)C is 3765 J. What is \(C_p\) of water in J mol\(^{-1}\) K\(^{-1}\)? (\(H_2O\) = 18 u)

• (1) 75.3
• (2) 376.5
• (3) 753
• (4) 37.65

At 25\(^\circ\)C, the percentage of ionization of x M acetic acid is 4.242. What is the pH of the acetic acid solution?

Given: \(\log 4.242 = 0.6275\), \(\log 0.04242 = -1.372\), \(K_a = 1.8 \times 10^{-5}\)

• (1) 3.37
• (2) 1.70
• (3) 1.37
• (4) 2.37

At 298 K, the value of \(K_c\) for the reaction A\(_2\)O\(_4\)(g) \(\rightleftharpoons\) 2AO\(_2\)(g) is \(x\) mol L\(^{-1}\). What is the approximate \(K_p\) value for this reaction?

Given: \(R = 0.082\) L atm mol\(^{-1}\) K\(^{-1}\)

• (1) 24.4x
• (2) 12.2x
• (3) \(\dfrac{x}{24.4}\)
• (4) \(\dfrac{24.4}{x}\)

\(H_2O_2\) with \(KMnO_4\) in acidic medium gives a manganese compound 'X' and in basic medium gives another manganese compound 'Y'. The oxidation states of manganese in X and Y respectively are:

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

Which of the following orders are correct against the stated property?

I) NaO\(_2 <\) KO\(_2 <\) RbO\(_2 <\) CsO\(_2\) — stability
II) Mg(OH)\(_2 <\) Ca(OH)\(_2 <\) Sr(OH)\(_2\) — basic strength
III) MgCO\(_3 <\) CaCO\(_3 <\) SrCO\(_3\) — thermal stability

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

In the structure of diborane, the number of 2-centre-2-electron bonds is X and 3-centre-2-electron bonds is Y. The value of (X + Y) is:

• (1) 5
• (2) 6
• (3) 4
• (4) 8

Match the following:
 

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

Identify the air pollutant which in high concentration leads to stiffness of flower buds?

• (1) CO\(_2\)
• (2) SO\(_2\)
• (3) CO
• (4) CH\(_4\)

The number of primary (1\(^\circ\)), secondary (2\(^\circ\)), and tertiary (3\(^\circ\)) alcohols possible for the formula C\(_5\)H\(_{12}\)O respectively are:

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

The catalyst used for the isomerisation of n-alkanes to branched chain alkanes is:

• (1) Anhy. AlCl\(_3\)/HCl
• (2) Mo\(_2\)O\(_3\)
• (3) FeCl\(_3\)
• (4) TiCl\(_4\) + R\(_3\)Al

An element crystallizes in bcc lattice. The atomic radius of the element is 2.598 AA. What is the volume (in cm\(^3\)) of one unit cell?

• (1) \(6.4 \times 10^{-22}\)
• (2) \(2.16 \times 10^{-20}\)
• (3) \(2.16 \times 10^{-22}\)
• (4) \(2.16 \times 10^{-24}\)

A centi molar solution of acetic acid is 50% dissociated at 27\(^\circ\)C. The osmotic pressure of the solution (in atm) is (R = 0.083 L atm K\(^{-1}\) mol\(^{-1}\))

• (1) 0.37
• (2) 3.7
• (3) 0.037
• (4) 0.73

At 300 K, vapour pressure of pure liquid A is 70 mm Hg. It forms an ideal solution with liquid B. Mole fraction of B = 0.2 and total vapour pressure of solution = 84 mm Hg. What is vapour pressure (in mm) of pure B?

• (1) 140
• (2) 70
• (3) 280
• (4) 560

The specific conductance of 0.05 M NaOH solution is 0.0115 S cm\(^{-1}\). What is its molar conductance (\(\Lambda_m\)) in S cm\(^2\) mol\(^{-1}\)?

• (1) 23
• (2) \(5.75 \times 10^{-7}\)
• (3) 2300
• (4) 230

For the reaction: A + 2B \(\rightarrow\) 3C + 2D, if rate of disappearance of B is \(x \times 10^{-2}\) mol L\(^{-1}\) s\(^{-1}\), the ratio of rate of reaction to rate of appearance of C is:

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

Identify the catalytic reaction in which both reactants are in different phases.

• (1) Ammonia synthesis by Haber process
• (2) Synthesis of sulphur trioxide by lead chamber process
• (3) Hydrogenation of vegetable oils
• (4) Hydrolysis of methyl acetate

Consider the following.

Statement-I: Gold sol is prepared by Bredig’s arc method.

Statement-II: Bredig’s arc method involves only dispersion but not condensation.

• (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 are correctly matched?

I) Hg — distillation
II) Cu — poling
III) B — zone refining
IV) Ti — liquation

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

The oxides of nitrogen obtained by the reaction of nitric acid with

(i) P\(_4\)O\(_{10}\) and (ii) P\(_4\) respectively are:

• (1) NO, N\(_2\)O
• (2) N\(_2\)O\(_3\), NO
• (3) N\(_2\)O\(_5\), NO\(_2\)
• (4) NO\(_2\), N\(_2\)O

Match the following:
 

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

The ion with 4f\(^7\) configuration is:

• (1) Pr\(^{3+}\)
• (2) Lu\(^{3+}\)
• (3) Eu\(^{2+}\)
• (4) Ce\(^{4+}\)

Which of the following is the common monomer for the polymers Bakelite and Melamine?

Activation energy for the hydrolysis of sucrose by acid is X kJ mol\(^{-1}\) and by sucrase is Y kJ mol\(^{-1}\). X and Y respectively are

• (1) 6.22, 2.15
• (2) 2.15, 6.22
• (3) 6.22, 6.22
• (4) 2.15, 2.15

The structure of the nitrogen-containing heterocyclic base shown below represents:
Structure of Uracil shown

• (1) Adenine
• (2) Thymine
• (3) Uracil
• (4) Cytosine

What is the drug used to control depression and hypertension?

• (1) Bithionol
• (2) Equanil
• (3) Dimetapp
• (4) Prontosil

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

In the following sequence of reactions, what is the end product (D)?

C\(_2\)H\(_5\)Br \(\xrightarrow{KCN}\) A \(\xrightarrow{H_3O^+}\) B \(\xrightarrow{LiAlH_4}\) C \(\xrightarrow{Cu, 573 K}\) D

• (1) Acetaldehyde
• (2) Acetone
• (3) Propionaldehyde
• (4) Propanol-1

The most acidic carboxylic acid is:

• (1) Benzoic acid
• (2) Phenylacetic acid
• (3) Formic acid (HCOOH)
• (4) Acetic acid (CH\(_3\)COOH)

A carbonyl compound X(C\(_5\)H\(_8\)O) gives yellow precipitate with NaOI. Hemiacetal of X with methanol/dry HCl is:

Which of the following does not involve in Friedel-Crafts reaction?

Consider the following statements:

Statement-I: CH\(_3\)NH\(_2\) is more basic than NH\(_3\), but C\(_6\)H\(_5\)NH\(_2\) is less basic than NH\(_3\).

Statement-II: The order of basic strength of amines in aqueous phase follows

(C\(_2\)H\(_5\))\(_2\)NH \(>\) C\(_2\)H\(_5\)NH\(_2\) \(>\) C\(_6\)H\(_5\)NH\(_2\)

• (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