AP EAPCET (AP EAMCET) 2025 Question Paper May 26 Shift 2(Available): Download Solutions with Answer Key

Question 1:

Let \([x]\) represent the greatest integer less than or equal to \(x\), \(\{x\} = x - [x]\). Given \[ \sqrt{2} = 1.414 \quad and \quad \sqrt{3} = 1.732. \]
If \[ f(x) = x + \frac{x}{1 + x^2} \]
is a real valued function, then find \[ f(\sqrt{2}) + f(-\sqrt{3}). \]

• (1) 0.682
• (2) 0.318
• (3) 0.146
• (4) 1.146

If the range of the function \(f(x) = -3x - 3\) is \(\{3, -6, -9, -18\}\), then which one of the following is not in the domain of \(f\)?

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

Evaluate \[ \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \frac{1}{7 \cdot 9} + \cdots up to 24 terms. \]

• (1) \(\frac{23}{147}\)
• (2) \(\frac{6}{35}\)
• (3) \(\frac{6}{37}\)
• (4) \(\frac{8}{51}\)

If B is the inverse of a third order matrix A and det B = k, then \((adj(adj A))^{-1}=\)

• (1) \(kB\)
• (2) \(\frac{1}{k}B\)
• (3) \(kB^{-1}\)
• (4) \(B + kI\)

If \(A = \begin{bmatrix} 2 & 2 & 1
1 & 3 & 1
1 & 2 & 2 \end{bmatrix}\) and \(\alpha, \beta, \gamma\) are the roots of the equation represented by \(|A - xI| = 0\), then \(\alpha^2 + \beta^2 + \gamma^2 =\)

• (1) 50
• (2) 29
• (3) 17
• (4) 27

If the values of \(x, y,\) and \(z\) satisfy the equations \[ 2x - 3y + 2z + 15 = 0, \quad 3x + y - z + 2 = 0, \quad x - 3y - 3z + 8 = 0 \]
simultaneously are \(\alpha, \beta,\) and \(\gamma\) respectively, then

• (1) \(\beta + \gamma = \alpha\)
• (2) \(\alpha + \beta = 2\gamma\)
• (3) \(2\alpha + \beta = \gamma\)
• (4) \(2\beta + \gamma = 2\alpha\)

If \(x = 3 - 2\sqrt{3}i\), then evaluate \(x^4 - 12x^3 + 54x^2 - 108x - 54 = \)

• (1) 0
• (2) 6
• (3) -6
• (4) 9

\(z_1, z_2, z_3\) represent the vertices A, B, C of a triangle ABC respectively in Argand plane. If \[ |z_1 - z_2| = \sqrt{25 - 12 \sqrt{3}}, \quad \left|\frac{z_1 - z_3}{z_2 - z_3}\right| = \frac{3}{4}, \quad and \angle ACB = 30^\circ, \]
then the area (in sq. units) of that triangle is

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

The product of the four values of the complex number \((1+i)^{3/4}\) is

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

If the difference of the roots of the equation \(x^2 - 7x + 10 = 0\) is same as the difference of the roots of the equation \(x^2 - 17x + k = 0\), then a divisor of \(k\) is

• (1) 14
• (2) 17
• (3) 6
• (4) 15

The product of all the real roots of the equation \(|x^2 - 5||x| + 6 = 0\) is

• (1) 25
• (2) 36
• (3) 4
• (4) 16

If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(5x^3 - 4x^2 + 3x - 2 = 0\), then \(\alpha^3 + \beta^3 + \gamma^3\) equals

• (1) \(\dfrac{17}{25}\)
• (2) \(\dfrac{394}{125}\)
• (3) \(\dfrac{34}{125}\)
• (4) \(\dfrac{34}{25}\)

After the roots of the equation \(6x^3 + 7x^2 - 4x - 2 = 0\) are diminished by \(h\), if the transformed equation does not contain \(x\) term, then the product of all possible values of \(h\) is

• (1) \(\dfrac{1}{3}\)
• (2) 2
• (3) \(-\dfrac{2}{9}\)
• (4) \(\dfrac{7}{3}\)

The number of integers greater than 6000 that can be formed by using the digits 0, 5, 6, 7, 8 and 9 without repetition is

• (1) 240
• (2) 840
• (3) 1440
• (4) 1680

The number of distinct quadratic equations \(ax^2 + bx + c = 0\) with unequal real roots that can be formed by choosing the coefficients \(a, b, c\) (with \(a \ne 0\)) from the set \(\{0,1,2,4\}\) is

• (1) 4
• (2) 6
• (3) 5
• (4) 12

The number of ways of dividing 15 persons into 3 groups containing 3, 5 and 7 persons so that two particular persons are not included into the 5 persons group is

• (1) \(\frac{117(11!)}{3!(7!)}\)
• (2) \({15 \choose 5} {10 \choose 3}\)
• (3) \(90 \times \frac{13!}{7!}\)
• (4) \({15 \choose 5} {8 \choose 3}\)

The coefficient of \(x^{10}\) in the expansion of \(\left(x + \frac{2}{x} - 5 \right)^{12}\) is

• (1) 1674
• (2) 2132
• (3) 1892
• (4) 862

Assertion (A): \(S_3 = 55 \times 2^9\)

Reason (R): \(S_1 = 90 \times 2^8\) and \(S_2 = 10 \times 2^8\)

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

If \[ \frac{2x^4 - 3x^2 + 4}{(x^2 + 1)(x^2 + 2)} = a + \frac{px + q}{x^2 + 1} + \frac{mx + n}{x^2 + 2}, \]
then \(\frac{n}{q} =\)

• (1) \(p + m - a\)
• (2) \(\frac{p + m}{a}\)
• (3) \(\frac{a}{p + m}\)
• (4) \(p + m + a\)

Evaluate \[ (4 \cos^2 \frac{\pi}{20} - 1)(4 \cos^2 \frac{3\pi}{20} - 1)(4 \cos^2 \frac{5\pi}{20} - 1)(4 \cos^2 \frac{7\pi}{20} - 1)(4 \cos^2 \frac{9\pi}{20} - 1). \]
 

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

If A and B are the values such that \((A + B)\) and \((A - B)\) are not odd multiples of \(\frac{\pi}{2}\) and \(2\tan(A+B) = 3 \tan(A-B)\), then \(\sin A \cos A =\)

• (1) \(\sin B \cos B\)
• (2) \(5 \sin B \cos B\)
• (3) \(\sin 2B\)
• (4) \(\cos 2B\)

If \(\cos 80^\circ + \cos 40^\circ - \cos 20^\circ = k\), then \(\frac{4k}{3} =\)

• (1) \(\sin \left(\frac{4\pi}{3}\right)\)
• (2) \(\cos \left(\frac{2\pi}{3}\right)\)
• (3) \(\tan \left(\frac{\pi}{3}\right)\)
• (4) \(\sec \left(\frac{2\pi}{3}\right)\)

The number of solutions of the equation \(4 \cos 2\theta \cos 3\theta = \sec \theta\) in the interval \([0, 2\pi]\) is

• (1) 12
• (2) 8
• (3) 16
• (4) 4

Evaluate: \(\tan\left(2\tan^{-1}\left(-\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right)\right) =\)

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

Evaluate: \(\tan^{-1}\left(\frac{1}{3}\right) + \cot^{-1}(3) =\)

• (1) \(\sec^{-1}\left(\frac{1}{3}\right)\)
• (2) \(cosec^{-1}\left(\frac{1}{3}\right)\)
• (3) \(\cos^{-1}\left(\frac{4}{3}\right)\)
• (4) \(\sin^{-1}\left(\frac{3}{4}\right)\)

In \(\triangle ABC\), if \(A = 30^\circ\) and \[ \frac{b}{(\sqrt{3}+1)^2 + 2(\sqrt{2} - 1)}, \quad \frac{c}{(\sqrt{3}+1)^2 - 2(\sqrt{2} - 1)}, \]
then find the angle \(B\).

• (1) \(60^\circ\)
• (2) \(97.5^\circ\)
• (3) \(75^\circ\)
• (4) \(52.5^\circ\)

In \(\triangle ABC\), if the line joining the circumcentre and incentre is parallel to \(BC\), then find \( \cos B + \cos C \).

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

In a triangle \(ABC\), if \(r_1 : r_2 = 3 : 4\) and \(r_1 : r_3 = 2 : 3\), then find the ratio \(a : b : c\).

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

Let \(\mathbf{a} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and \(\mathbf{b} = 6\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) be two vectors. If \[ |\mathbf{a} \times \mathbf{b}|^2 + |\mathbf{a} \cdot \mathbf{b}|^2 = f(x,y)(x+y) - 46 = 0, \]
then what does this represent?

• (1) A pair of lines
• (2) An ellipse
• (3) A hyperbola
• (4) A circle

Line \(L_1\) passes through the points \(\mathbf{i} + \mathbf{j}\) and \(\mathbf{k} - \mathbf{i}\). Line \(L_2\) passes through the point \(\mathbf{j} + 2\mathbf{k}\) and is parallel to the vector \(\mathbf{i} + \mathbf{j} + \mathbf{k}\). If \(\mathbf{x}i + \mathbf{y}j + \mathbf{z}k\) is the point of intersection of the lines \(L_1\) and \(L_2\), then find \((y - x) =\)

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

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be position vectors of three non-collinear points on a plane. If \[ \alpha = \left[\mathbf{a} \quad \mathbf{b} \quad \mathbf{c}\right] and \mathbf{r} = \mathbf{a} \times \mathbf{b} - \mathbf{c} \times \mathbf{b} - \mathbf{a} \times \mathbf{c}, \]
then \(\frac{|\alpha|}{|\mathbf{r}|}\) represents:

• (1) Ratio of areas of the triangles formed by \(\mathbf{0}, \mathbf{a}, \mathbf{b}\) to \(\mathbf{0}, \mathbf{b}, \mathbf{c}\)
• (2) Ratio of the numerical values of volume of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\) and its height
• (3) Ratio of lengths of the diagonals of the parallelepiped formed with \(\mathbf{0}, \mathbf{a}, \mathbf{b}, \mathbf{c}\)
• (4) Length of the perpendicular from origin to the plane

If \[ P = (a x \mathbf{i})^2 + (a x \mathbf{j})^2 + (a x \mathbf{k})^2 \quad and \quad Q = (a \cdot \mathbf{i})^2 + (a \cdot \mathbf{j})^2 + (a \cdot \mathbf{k})^2, \]
then find the relation between \(P\) and \(Q\).

• (1) \(P = Q\)
• (2) \(P = 2Q\)
• (3) \(P = 3Q\)
• (4) \(P = 4Q\)

Given vectors \(\mathbf{a} = \mathbf{i} + \mathbf{j} - 2\mathbf{k}\), \(\mathbf{b} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\), \(\mathbf{c} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}\), and \(\mathbf{r}\) such that \[ \mathbf{r} \cdot \mathbf{a} = 0, \quad \mathbf{r} \cdot \mathbf{c} = 3, \quad [\mathbf{r} \quad \mathbf{a} \quad \mathbf{b}] = 0, \]
then find \(|\mathbf{r}|\).

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

The mean deviation from the median for the following data is:
\[ \begin{array}{c|ccccc} x_i & 9 & 3 & 7 & 2 & 5
f_i & 1 & 6 & 2 & 8 & 4
\end{array} \]

• (1) \(\frac{94}{21}\)
• (2) \(\frac{12}{7}\)
• (3) \(\frac{10}{7}\)
• (4) \(\frac{100}{21}\)

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive houses get one sample is:

• (1) \(\frac{7}{99}\)
• (2) \(\frac{5}{12}\)
• (3) \(\frac{4}{13}\)
• (4) \(\frac{5}{31}\)

A and B are two independent events of a random experiment and \(P(A) > P(B)\). If the probability that both A and B occur is \(\frac{1}{6}\) and neither of them occurs is \(\frac{1}{3}\), then the probability of the occurrence of B is:

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

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If A is the event of getting a prime number as their sum and B is the event of getting a number greater than 8 as their sum, then find \(P(A \cap \overline{B})\).

• (1) \(\frac{1}{4}\)
• (2) \(\frac{13}{36}\)
• (3) \(\frac{2}{9}\)
• (4) \(\frac{5}{18}\)

A family consists of 8 persons. If 4 persons are chosen at random and they are found to be 2 men and 2 women, then the probability that there are equal numbers of men and women in that family is:

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

The number of trials conducted in a binomial distribution is 6. If the difference between the mean and variance of this variate is \(\frac{27}{8}\), then the probability of getting at most 2 successes is:

• (1) \(\frac{106}{4^6}\)
• (2) \(\frac{144}{4^6}\)
• (3) \(\frac{126}{4^6}\)
• (4) \(\frac{154}{4^6}\)

Let \(X \sim B(n, p)\) with mean \(\mu\) and variance \(\sigma^2\). If \(\mu = 2\sigma^2\) and \(\mu + \sigma^2 = 3\), then find \(P(X \leq 3)\).

• (1) \(\frac{40}{49}\)
• (2) \(\frac{40}{43}\)
• (3) \(\frac{100}{101}\)
• (4) \(\frac{15}{16}\)

If \(A(\cos \alpha, \sin \alpha)\), \(B(\sin \alpha, -\cos \alpha)\), and \(C(1, 2)\) are the vertices of \(\triangle ABC\), then find the locus of its centroid.

• (1) \(3(x^2 + y^2) - 2x - 4y + 1 = 0\)
• (2) \(x^2 + y^2 - 2x - 4y + 1 = 0\)
• (3) \(x^2 + y^2 - 2x - 4y + 3 = 0\)
• (4) \(2(x^2 + y^2) - 2x - 4y + 5 = 0\)

If the axes are translated to the orthocentre of the triangle formed by points \(A(7,5), B(-5,-7), C(7,-7)\), then the coordinates of the incentre of the triangle in the new system are?

• (1) \((-6,6)\)
• (2) \(\left(\frac{5}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\)
• (3) \(\left(\frac{-12}{2+\sqrt{2}}, \frac{12}{2+\sqrt{2}}\right)\)
• (4) \(\left(-5\sqrt{2}, -7\sqrt{2}\right)\)

The angle made by a line \(L\) with positive X-axis measured in the positive direction is \(\frac{\pi}{6}\) and the intercept made by \(L\) on Y-axis is negative. If \(L\) is at a distance 5 units from the origin, then the perpendicular distance from the point \(\left(1,-\sqrt{3}\right)\) to the line \(L\) is?

• (1) 2
• (2) 1
• (3) 4
• (4) 3

Lines \(L_1\) and \(L_2\) have slopes 2 and \(-\frac{1}{2}\) respectively. If both \(L_1\) and \(L_2\) are concurrent with the lines \(x - y + 2 = 0\) and \(2x + y + 3 = 0\), then the sum of the absolute values of the intercepts made by the lines \(L_1\) and \(L_2\) on the coordinate axes is?

• (1) 2
• (2) 7
• (3) 12
• (4) 9

The lines \(L_1: y - x = 0\) and \(L_2: 2x + y = 0\) intersect the line \(L_3: y + 2 = 0\) at points \(P\) and \(Q\) respectively. The bisector of the angle between \(L_1\) and \(L_2\) divides the segment \(PQ\) internally at \(R\). Consider:

Statement-I: \(PR : RQ = 2\sqrt{2} : \sqrt{5}\).

Statement-II: In any triangle, bisector of an angle divides that triangle into two similar triangles.


Which statement(s) is/are correct?

• (1) Statement-I is true, Statement-II is false
• (2) Statement-I is false, Statement-II is true
• (3) Statement-I and Statement-II are true, but Statement-II is not a correct explanation for Statement-I

If \[ 2x^2 + 3xy - 2y^2 - 5x + 2fy - 3 = 0 \]
represents a pair of straight lines, then one of the possible values of \(f\) is?

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

A circle passing through origin cuts the coordinate axes at \(A\) and \(B\). If the straight line \(AB\) passes through a fixed point \((x_1,y_1)\), then the locus of the centre of the circle is?

• (1) \(\frac{x_1}{x} + \frac{y_1}{y} = 1\)
• (2) \(xy = x_1 y_1\)
• (3) \(x y_1 + y x_1 = 2\)
• (4) \(\frac{x_1}{x} + \frac{y_1}{y} = 2\)

If \((\alpha, \beta)\) is the external centre of similitude of the circles \[ x^2 + y^2 = 3 \]
and \[ x^2 + y^2 - 2x + 4y + 4 = 0, \]
then find \(\frac{\beta}{\alpha}\).

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

The equation of the circle touching the lines \(|x-2| + |y-3| = 4\) is?

• (1) \(x^2 + y^2 - 6x - 4y + 5 = 0\)
• (2) \(x^2 + y^2 - 4x - 6y + 5 = 0\)
• (3) \(x^2 + y^2 - x - 2y - 5 = 0\)
• (4) \(x^2 + y^2 - 2x - y - 5 = 0\)

If the chord joining points \((1,2)\) and \((2,-1)\) on a circle subtends an angle \(\frac{\pi}{4}\) at any point on its circumference, then the equation of such a circle is?

• (1) \(x^2 + y^2 + 6x - 2y + 5 = 0\)
• (2) \(x^2 + y^2 - 6x - 2y + 5 = 0\)
• (3) \(x^2 + y^2 - 6x + 2y + 5 = 0\)
• (4) \(x^2 + y^2 + 6x + 2y + 5 = 0\)

The equation of the circle which cuts all the three circles \[ 4(x-1)^2 + 4(y-1)^2 = 1, \quad 4(x+1)^2 + 4(y-1)^2 = 1, \quad 4(x+1)^2 + 4(y+1)^2 = 1, \]
orthogonally is?

• (1) \(4x^2 + 4y^2 = 49\)
• (2) \(4(x-1)^2 + 4(y+1)^2 = 1\)
• (3) \((x-1)^2 + (y+1)^2 = 4\)
• (4) \(4x^2 + 4y^2 = 7\)

If the normal chord drawn at the point \(\left(\frac{15}{2\sqrt{2}}, \frac{15}{2\sqrt{2}}\right)\) to the parabola \(y^2 = 15x\) subtends an angle \(\theta\) at the vertex of the parabola, then \(\sin \frac{\theta}{3} + \cos \frac{2\theta}{3} - \sec \frac{4\theta}{3} =\) ?

• (1) 0
• (2) 3
• (3) 1
• (4) 2

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b)\) is normal to the circle \((x+1)^2 + (y+1)^2 = 1\), then \(a^2\) lies in the interval?

• (1) \(\left(\sqrt{\frac{2}{5}}, 2\right)\)
• (2) \(\left(\frac{2}{5}, 4\right)\)
• (3) \(\left(1, \frac{10}{9}\right)\)
• (4) \((3,5)\)

Let \(P(a \sec \theta, b \tan \theta)\) and \(Q(a \sec \phi, b \tan \phi)\) where \(\theta + \phi = \frac{\pi}{2}\) be two points on the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). If \((h,k)\) is the point of intersection of the normals drawn at \(P\) and \(Q\), then find \(k\).

• (1) \(\frac{a^2 + b^2}{a}\)
• (2) \(-\frac{a^2 + b^2}{b}\)
• (3) \(-\frac{a^2 + b^2}{a}\)
• (4) \(\frac{a^2 + b^2}{b}\)

If the angle between the asymptotes of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is \(2 \tan^{-1} \left(\frac{1}{3}\right)\) and \(a^2 - b^2 = 45\), then find \(ab\).

• (1) 20
• (2) 24
• (3) 45
• (4) 54

The point in the \(xy\)-plane which is equidistant from the points \(A(2,0,3), B(0,3,2)\) and \(C(0,0,1)\) has the coordinates?

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

If the direction ratios of two lines \(L_1\) and \(L_2\) are \((1,-2,2)\) and \((-2,3,-6)\) respectively, then the direction ratios of the line which is perpendicular to both \(L_1\) and \(L_2\) are?

• (1) \((1,-2,3)\)
• (2) \((-2,3,5)\)
• (3) \((6,2,-1)\)
• (4) \((2,-1,3)\)

If the image of the point \(A(1,1,1)\) with respect to the plane \(4x + 2y + 4z + 1 = 0\) is \(B(\alpha, \beta, \gamma)\), then find \(\alpha + \beta + \gamma\).

• (1) \(-2\)
• (2) \(\frac{28}{9}\)
• (3) \(\frac{55}{36}\)
• (4) \(\frac{35}{16}\)

Evaluate \[ \lim_{x \to 0} \sqrt{\frac{x + 2 \sin x + 3 \tan x - \tan^3 x}{x^2 + 2 \sin x + \tan x + 3 - \sqrt{\sin^2 x - 2 \tan x - x + 3}}} = ? \]

• (1) \(2 \sqrt{3}\)
• (2) 10
• (3) 25
• (4) \(\sqrt{17}\)

Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]

• (1) \(3^{60}\)
• (2) \(-1\)
• (3) 1
• (4) 0

If a real valued function \[ f(x) = \begin{cases} \log(1 + [x]), & x \geq 0
\sin^{-1}[x], & -1 \leq x < 0
k([x] + |x|), & x < -1 \end{cases} \]
is continuous at \(x = -1\), then find \(k\).

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

If \(y = \sin^{-1} \left(\frac{2x}{1 + x^2}\right)\) and \(\left(\frac{d^2 y}{dx^2}\right)_{x=2} = k\), then find \(25k\).

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

If \(f(x) = x^{\sec^{-1} x}\), then find \(f'(2)\).

• (1) \(\frac{2^{\pi/3}}{6} \left(\pi - \sqrt{3} \log 2\right)\)
• (2) \(\frac{2^{\pi/6}}{6} \left(\pi + \sqrt{3} \log 2\right)\)
• (3) \(\frac{2^{\pi/3}}{6} \left(\pi + \sqrt{3} \log 2\right)\)
• (4) \(\frac{2^{\pi/6}}{6} \left(\pi - \sqrt{3} \log 2\right)\)

If \(f(x) = \sec^{-1} \left(\frac{1}{2x^2 -1}\right)\) and \(g(x) = \tan^{-1} \left(\frac{\sqrt{1 + x^2} - 1}{x}\right)\), then the derivative of \(f(x)\) with respect to \(g(x)\) is?

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

If the tangent to the curve \(xy + ax + by = 0\) at \((1,1)\) makes an angle \(\tan^{-1} 2\) with X-axis, then find \(\frac{ab}{a+b}\).

• (1) 1
• (2) 2
• (3) 3
• (4) 4

If the displacement \(S\) of a particle travelling along a straight line in \(t\) seconds is given by \[ S = 2t^3 + 2t^2 - 2t - 3, \]
then the time taken (in seconds) by the particle to change its direction is?

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

If the function \(f(x) = x^3 + b x^2 + c x - 6\) satisfies all conditions of Rolle's theorem in \([1,3]\) and \[ f'\left(\frac{2\sqrt{3} + 1}{\sqrt{3}}\right) = 0, \]
then find \(bc\).

• (1) 18
• (2) -66
• (3) 38
• (4) -46

If \(P(\alpha, \beta)\) is a point on the curve \(9x^2 + 4 y^2 = 144\) in the first quadrant and the minimum area of the triangle formed by the tangent of the curve at \(P\) with the coordinate axes is \(S\), then find \(S\).

• (1) \(S = \sqrt{\alpha \beta}\)
• (2) \(S = \alpha \beta\)
• (3) \(S = 2 \sqrt{\alpha \beta}\)
• (4) \(S = 2 \alpha \beta\)

Evaluate \[ \int (\log 2x)^3 \, dx = ? \]

• (1) \(x \left[(\log 2x)^3 - 3 (\log 2x)^2 + 6 (\log 2x) - 6 \right] + c\)
• (2) \(\frac{x}{4} \left[4(\log 2x)^3 - 6 (\log 2x)^2 + 6 (\log 2x) - 3 \right] + c\)
• (3) \(\frac{x}{2} \left[(\log 2x)^3 - 3 (\log 2x)^2 + 3 (\log 2x) - 6\right] + c\)
• (4) \(x \left[(\log 2x)^3 - 6 (\log 2x)^2 + 18 (\log 2x) - 54\right] + c\)

Evaluate \[ \int \frac{x + 1}{(x - 2) \sqrt{1 - x}} \, dx = ? \]

• (1) \(\log(x + 1) - \log (x - 2) \sqrt{1 - x} + c\)
• (2) \(\log(x - 2) \sqrt{1 - x} + c\)
• (3) \(6 \tan^{-1} \sqrt{1 - x} - 2 \sqrt{1 - x} + c\)
• (4) \(4 \tan^{-1} \sqrt{1 - x} - 2 \sqrt{1 - x} + c\)

Evaluate \[ \int \frac{1}{1 + x^2} \, dx = ? \]

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

If \[ \int \frac{dx}{(x \tan x + 1)^2} = f(x) + c, \]
then \(\lim_{x \to \frac{\pi}{2}} f(x)\) is?

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

Evaluate \[ \int \sin^3 x \cos^2 x \, dx = ? \]

• (1) \(\frac{\sin^4 x}{5} - \frac{\sin^2 x \cos^2 x}{15} + 2 \cos x + c\)
• (2) \(\frac{\sin^4 x}{5} - \frac{\sin^2 x \cos^2 x}{15} + 2 \cos x + c\)
• (3) \(\frac{\sin^4 x}{5} - \frac{\sin^2 x \cos^2 x}{15} + 2 \cos x + c\)
• (4) \(\frac{\sin^4 x}{5} - \frac{\sin^2 x \cos^2 x}{15} + 2 \cos x + c\)

Evaluate \[ \lim_{n \to \infty} \frac{1}{2n} \left( \sin \frac{\pi}{2n} + \sin \frac{\pi}{n} + \sin \frac{2\pi}{2n} + \dots \right) = ? \]

• (1) 1
• (2) 0
• (3) 4
• (4) 3

Evaluate \[ \int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ? \]

• (1) \(\frac{873}{2240}\)
• (2) \(\frac{3\pi}{12}\)
• (3) \(\frac{1641}{4480}\)
• (4) \(\frac{3\pi}{128}\)

Evaluate \[ \int \frac{1}{x^4 + 1} \, dx = ? \]

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

The area of the region (in sq.units) bounded by the curves \(x^2 + y^2 = 16\) and \(x^2 + y^2 = 6x\) is?

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

If \(a\) and \(b\) are arbitrary constants, then the differential equation corresponding to the family of curves \(y = \tan (ax + b)\) is?

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

The general solution of the differential equation \(xy(y + 2y') + (y^2 - y) \, dx = 0\) is?

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

The general solution of the differential equation \((1 + \sin^2 x) \, \frac{dy}{dx} + \sin 2x = 0\) is?

• (1) \((\sin 2x) y = \sin x + \sin^2 x + c\)
• (2) \((\sin 2x) y = \sin x + \sin^2 x + c\)
• (3) \((1 + \sin^2 x) y = \sin x + \sin^2 x + c\)
• (4) \((\sin 2x) y = \sin 2x \)

If force \( F = \frac{\alpha}{\beta} \), then the dimensional formulae of \( \alpha \) and \( \beta \) are respectively:

• (1) \([M L^2 T^{-3}]\), \([M L^{-1} T^{-1}]\)
• (2) \([M L^2 T^{-1}]\), \([M^{1/3} L^1 T^{-1}]\)
• (3) \([M^2 L^{-2} T^{-3}]\), \([M^{1/3} L^{-1} T^3]\)
• (4) \([M^2 L T^{-2}]\), \([M L^3 T^{-1}]\)

The displacement (\(x\)) and time (\(t\)) graph of a particle moving along a straight line is shown in the figure. The average velocity of the particle in the time of 10s is:

• (1) 2 m s\(^{-1}\)
• (2) 4 m s\(^{-1}\)
• (3) 6 m s\(^{-1}\)
• (4) 8 m s\(^{-1}\)

If the horizontal range of a body projected with a velocity \(u\) is 3 times the maximum height reached by it, then the range of the body is:

• (1) \(\frac{2u^2}{3g}\)
• (2) \(\frac{4u^2}{5g}\)
• (3) \(\frac{12u^2}{13g}\)
• (4) \(\frac{24u^2}{25g}\)

If the velocity at the maximum height of a projectile projected at an angle of \(45^\circ\) is \(20 \, m/s\), then the maximum height reached by the projectile is:

• (1) 10 m
• (2) 20 m
• (3) 30 m
• (4) 40 m

A body of mass \(m\) moving along a straight line collides with a stationary body of mass \(2m\). After collision if the two bodies move together with the same velocity, then the fraction of kinetic energy lost in the process is:

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

If a body of mass \(2 \, kg\) moving with initial velocity of \(4 \, m/s\) is subjected to a force of \(3 \, N\) for a time of 2 seconds normal to the direction of its initial velocity, then the resultant velocity of the body is:

• (1) 7 m s\(^{-1}\)
• (2) 5 m s\(^{-1}\)
• (3) 2 m s\(^{-1}\)
• (4) 7.5 m s\(^{-1}\)

If a constant force of \((2i + 3j + 4k) \, N\) acting on a body of mass \(5 \, kg\) displaces it from \((3i - 4k)\, m\) to \((2i + 2j + 3k)\, m\), then the work done by the force on the body is:

• (1) 32 J
• (2) 28 J
• (3) 36 J
• (4) 44 J

A motor can pump 7560 kg of water per hour from a well of depth 100 m. If the efficiency of the pump is 70%, then the power of the pump is:

• (1) 4 kW
• (2) 6 kW
• (3) 3 kW
• (4) 7 kW

A circular disc of diameter 0.8 m and mass 4 kg is rolling on a smooth horizontal plane. If 2.56 N torque is acting on the disc, then its angular acceleration is:

• (1) 8 rad s\(^{-2}\)
• (2) 4 rad s\(^{-2}\)
• (3) 2 rad s\(^{-2}\)
• (4) 16 rad s\(^{-2}\)

A solid sphere and a solid cylinder have same mass and same radius. The ratio of the moment of inertia of the solid sphere about its diameter and the moment of inertia of the solid cylinder about its axis is:

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

A particle is executing simple harmonic motion with amplitude \( A \). The ratio of the kinetic energies of the particle when it is at displacements of \( \frac{A}{4} \) and \( \frac{A}{2} \) from the mean position is:

• (1) 4 : 1
• (2) 2 : 1
• (3) 5 : 4
• (4) 9 : 16

If the potential energy of a particle of mass \(0.1 \, kg\) moving along the x-axis is \(5x(x - 4)\), then the speed of the particle is maximum at a position of:

• (1) \( x = 2 \, m \)
• (2) \( x = 3 \, m \)
• (3) \( x = 0.5 \, m \)
• (4) \( x = 5 \, m \)

The potential energy of a satellite of mass \( m \) revolving around the earth at a height of \( R \) from the surface of the earth is:

• (1) \( -0.5mgR_e \)
• (2) \( -mgR_e \)
• (3) \( -2mgR_e \)
• (4) \( -4mgR_e \)

The elastic potential energy stored in a copper rod of length one meter and area of cross-section \(1 \, mm^2\) when stretched by 1 mm is:

• (1) \( 6 \times 10^{-2} \, J \)
• (2) \( 3 \times 10^{-2} \, J \)
• (3) \( 60 \, J \)
• (4) \( 3 \, J \)

When the temperature increases, the viscosity of

• (1) gases decreases but liquids increases.
• (2) gases increases but liquids decreases.
• (3) both gases and liquids increase.
• (4) both gases and liquids decrease.

If a body cools from a temperature of 62°C to 50°C in 10 minutes and to 42°C in the next 10 minutes, then the temperature of the surroundings is

• (1) 12°C
• (2) 26°C
• (3) 36°C
• (4) 21°C

If the ratio of universal gas constant and specific heat capacity at constant volume of a gas is given by 0.67, then the gas is

• (1) monoatomic
• (2) diatomic
• (3) polyatomic
• (4) a mixture of diatomic and polyatomic gases

The internal energy of 4 moles of a monoatomic gas at a temperature of 77°C is

• (1) 1500R
• (2) 1800R
• (3) 2100R
• (4) 3500R

If 5.6 liters of a monoatomic gas at STP is adiabatically compressed to 0.7 liters, then the work done on the gas is nearly

• (1) 307R
• (2) 357R
• (3) 367R
• (4) 407R

If the rms speed of the molecules of a diatomic gas at a temperature of 322K is 2000 m/s, then the gas is

• (1) hydrogen
• (2) nitrogen
• (3) oxygen
• (4) chlorine

The equation of a transverse wave propagating along a stretched string of length 80 cm is \(y = 1.5 \sin ( (5 \times 10^3)x + 20t )\), here 'x' and 'y' are in cm and the time 't' is in seconds. If the mass of the string is 3 g, then the tension in the string is

• (1) 12 N
• (2) 4 N
• (3) 6 N
• (4) 8 N

When an object is placed in front of a convex mirror at a distance 'u' from the pole of the mirror such that the size of the image is 'n' times that of the object, then the object distance 'u' is

• (1) \(\frac{f}{n^2}\)
• (2) \(nf - f\)
• (3) \(\frac{f}{n}\)
• (4) \(\frac{f}{n^2}\)

A narrow slit of width 2 mm is illuminated with monochromatic light of wavelength 500 nm. If the distance between the slit and the screen is 1 m, then first minima are separated by a distance of

• (1) 5 mm
• (2) 0.5 mm
• (3) 1 mm
• (4) 10 mm

The force between two conducting spheres of same radius having charges +8 μC and -4 μC separated by some distance in air is F. If the spheres are connected by a conducting wire and after some time the wire is removed, then the magnitude of the force between the two conducting spheres is

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

In space, the electric potential varies as \(V = 20|r|\) volt, where \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) is the position vector. Then electric field in (N/C) at the point (4 m, 3 m, -5 m) is

• (1) \(-\sqrt{2} (4i + 3j - 10k)\)
• (2) \(-\sqrt{2} (8i + 6j - 10k)\)
• (3) \(-(8i + 6j - 10k)\)
• (4) \(4i + 3j - 5k\)

A capacitor of capacitance 2 μF is charged with the help of a 60 V battery. After disconnecting the battery, if this capacitor is connected in parallel with another uncharged capacitor of capacitance 1 μF, then the potential difference across the plates of the 2 μF capacitor is

• (1) 30 V
• (2) 60 V
• (3) 40 V
• (4) 20 V

The readings of the voltmeter and ammeter in the circuit shown in the diagram are respectively

• (1) 5 V, 3 A
• (2) 7 V, 3 A
• (3) 5 V, 1 A
• (4) 7 V, 1 A

When two identical batteries of internal resistance 1 Ω each are connected in series across a resistor R, the rate of heat produced in R is P1. When the same batteries are connected in parallel across R, the rate of heat produced is P2. If P1 = 2P2, then the value of R is

• (1) 4 Ω
• (2) 10 Ω
• (3) 5 Ω
• (4) 2 Ω

The magnetic field at the center of a long solenoid having 400 turns per unit length and carrying a current \(I = 6.24 \times 10^{-7}\) A is

• (1) \(1.56 \times 10^{-3}\) T
• (2) \(2.4 \times 10^{-3}\) T
• (3) \(2.6 \times 10^{-3}\) T
• (4) \(1.6 \times 10^{-3}\) T

If a proton of kinetic energy 8.35 MeV enters a uniform magnetic field of 10 T at right angles to the direction of the field, then the force acting on the proton is

• (1) \(48 \times 10^{-12}\) N
• (2) \(16 \times 10^{-12}\) N
• (3) \(64 \times 10^{-12}\) N
• (4) \(32 \times 10^{-12}\) N

A sample of a ferromagnetic iron in the shape of a cube of side 1.0 μm contains \(8.7 \times 10^{29}\) atoms per cubic meter and the magnetic dipole moment of each iron atom is \(9.3 \times 10^{-29}\) Am². Then the maximum possible magnetic dipole moment (in Am²) of the sample is nearly

• (1) \(8.1 \times 10^{-12}\)
• (2) \(8.1 \times 10^{-14}\)
• (3) \(8.1 \times 10^{-16}\)
• (4) \(8.1 \times 10^{-18}\)

When current in a coil changes from 2 A to 5 A in a time of 0.3 s, if the emf induced in the coil is 40 mV, then the self-inductance of the coil is

• (1) 4 H
• (2) 4 mH
• (3) 40 mH
• (4) 4 μH

In a series LCR circuit, the voltages across the capacitor, resistor, and inductor are in the ratio 2:3:6. If the voltage of the source in the circuit is 240 V, then the voltage across the inductor is

• (1) 240 V
• (2) 144 V
• (3) 96 V
• (4) 288 V

If a 10 W bulb emits electromagnetic waves uniformly in all directions, then the intensity of light at a distance 0.5 m from the source is nearly

• (1) 3.18 W/m²
• (2) 0.31 W/m²
• (3) 0.62 W/m²
• (4) 5 W/m²

The ratio of de Broglie wavelengths associated with thermal neutrons at temperatures 127°C and 352°C is

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

The ratio of the time periods of the revolution of the electrons in the second and third excited states of hydrogen atom is

• (1) 9 : 16
• (2) 27 : 64
• (3) 4 : 9
• (4) 8 : 27

If the surface areas of two nuclei are in the ratio 9 : 49, then the ratio of their mass numbers is

• (1) 27 : 343
• (2) 9 : 49
• (3) 3 : 7
• (4) 49 : 81

In the given options, the diode that is forward biased is

In a common emitter transistor amplifier the resistance of collector is 3 kΩ. If the current amplification factor is 100 and the base resistance is 2 kΩ, then the power gain of the transistor is

• (1) 150
• (2) 10000
• (3) 1500
• (4) 15000

The layer of the atmosphere that reflects low frequency (LF) electromagnetic waves during daytime only is

• (1) D
• (2) E
• (3) F1
• (4) F2

a, b, c, d are electromagnetic radiations. Frequencies of a, b, c are \(3 \times 10^{15}\) Hz, respectively, whereas the wavelength of c, d are 400 nm, 750 nm, respectively. The increasing order of their energies is

• (1) b, d, c, a
• (2) a, b, c, d
• (3) a, c, b, d
• (4) b, c, d, a

The number of electrons with magnetic quantum number \(m = 0\) in the elements with atomic numbers \(Z = 24\) and \(Z = 29\) are respectively

• (1) 12, 13
• (2) 12, 12
• (3) 13, 12
• (4) 14, 15

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

• (1) Li < Na < K - metallic radius
• (2) Br < F < Cl - electron gain enthalpy
• (3) C < N < O - first ionization enthalpy
• (4) Mg\(^{2+}\) < Na\(^+\) < F\(^-\) - ionic radius

Match the following:

List-I (Molecule)   List-II (Dipole moment in D)
A) HCCl                 I) 1.85
B) NH₃                   II) 1.07
C) H₂O                  III) 0.23
D) NF₃                  IV) 1.47

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

Which of the following sets are correctly matched?

List-I                 List-II
Molecule      Number of lone pairs of electrons on central atom & Hybridization
I) PCF₃          1 & sp³
II) SO₂           2 & sp²
III) SF₄          2 & sp³d²
IV) CF₄         2 & sp³

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

The correct equation for one mole of a real gas is (a, b are constants)

• (1) \( p + \frac{a}{v^2} \) \( V - b \) = RT
• (2) \( p + \frac{a}{v^2} \) (V + b) = RT
• (3) \( p - \frac{a}{v^2} \) (V - b) = RT
• (4) \( p + \frac{a}{v^2} \) (V - b) = RT

A and B are ideal gases. At T(K), 2 L of A with a pressure of 1 bar is mixed with 4 L of B with a pressure of P\(_B\) bar in a 100 L flask. The pressure exerted by gaseous mixture is 0.1 bar. What is the value of P\(_B\) in bar?

• (1) 0.04
• (2) 0.02
• (3) 0.10
• (4) 0.05

The mass of a mixture containing NaCl and NaBr is 4.0 g. If Na is 20% of the total mixture, the composition of NaCl in the mixture is

• (1) 48%
• (2) 55%
• (3) 45%
• (4) 52%

The number of extensive and intensive properties in the list given below is respectively:
% List
- Density, enthalpy, mass, temperature, volume, pressure

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

One mole of ethanol (C₂H₅OH) was completely burnt in oxygen to form CO₂(g) and H₂O(l). What is the \(\Delta H_f^{\circ}\) (in kJ/mol) for this reaction?

• (1) 1921
• (2) -1921
• (3) 1367
• (4) -1367

For the following given equilibrium reaction, K\(_p\) is equal to 1076 at T(K). What is the value of T (in K)? \[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]
Given that \( R = 0.082 \, L atm K^{-1} mol^{-1} \), \( K_p = 1076 \), the equation is: \[ K_p = 1076, \, R = 0.082 \, L atm K^{-1} mol^{-1} \]

• (1) 500
• (2) 600
• (3) 400
• (4) 450

The molar solubility of PbI₂ in 0.2 M Pb(NO₃)₂ solution in terms of \(K_{sp}\) (solubility product) is

• (1) \( K_{sp}^{1/2} \, 0.2 \)
• (2) \( K_{sp}^{1/2} \, 0.8 \)
• (3) \( K_{sp}^{1/4} \, 0.4 \)
• (4) \( K_{sp}^{1/3} \, 0.8 \)

Which of the following property is less for D₂O than H₂O?

• (1) Dielectric constant
• (2) Viscosity
• (3) Density
• (4) Melting point

Identify the correct statements from the following:
A) Among alkali metal ions, Li⁺ has the highest hydration enthalpy.

B) Boiling point of alkali metals increases from Li to Cs.

C) Density of K is less than that of Na and Rb.

D) Li has strong tendency to form superoxide.

• (1) A & B
• (2) B & C
• (3) A & C
• (4) A & D

The correct order of electronegativity of group 13 elements is

• (1) B \(>\) Ga \(>\) Al \(>\) Tl \(>\) In
• (2) B \(>\) Al \(>\) Tl \(>\) Ga \(>\) In
• (3) B \(>\) Al \(>\) Ga \(>\) In \(>\) Tl
• (4) B \(>\) Tl \(>\) In \(>\) Ga \(>\) Al

Identify the correct statements from the following:
I) CO reduces the oxygen carrying ability of blood.
II) Producer gas contains CO and N₂.
III) C-O bond length in CO₂ is 115 pm.

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

The incorrect statement from the following is

• (1) Classical smog is also called reducing smog.
• (2) Common components of classical smog are O₃, NO, HCHO.
• (3) Photochemical smog leads to cracking of rubber and corrosion of metals.
• (4) Photochemical smog occurs in warm, dry, and sunny climate.

IUPAC names of the given compounds (I) and (II) are respectively

• (1) 5-Phenyl-3-nitrobutane; 2-ethyl-2-methyl-1-propylcyclohexane
• (2) 2-Nitro-1-phenylbutane; 2-ethyl-2-methyl-1-propylcyclohexane
• (3) 2-Nitro-1-phenylbutane; 1-ethyl-1-methyl-2-propylcyclohexane
• (4) 3-Nitro-5-phenylbutane; 2-ethyl-2-methyl-1-propylcyclohexane

Identify the most stable carbocation from the following

A metal crystallizes in simple cubic lattice. The volume of one unit cell is \( 6.4 \times 10^{-7} \, pm^3 \). What is the radius of the metal atom in pm?

• (1) 100
• (2) 200
• (3) 300
• (4) 400

What is the approximate molality of 10% (w/w) aqueous glucose solution?
(Molar mass of glucose = 180 g mol\(^{-1}\))

• (1) 0.31 m
• (2) 0.62 m
• (3) 0.93 m
• (4) 1.24 m

The van't Hoff factor for 0.5 m aqueous CH₃COOH solution is 1.075. What is the experimentally observed \(\Delta T_f\) (in K) for this solution? \( K_f = 1.86 \, K kg mol^{-1} \)

• (1) 1.156
• (2) 1.075
• (3) 1.0
• (4) 0.95

Match the following:

List-I (symbol of electrical property)   List-II (units)
A) \( \Omega \)                                        I) S cm\(^{-1}\)
B) G                                                        II) m\(^{-1}\)
C) \( \kappa \)                                         III) S cm\(^2\) mol\(^{-1}\)
D) G*                                                      IV) S

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

The following graph is obtained for a first-order reaction (A → P). The activation energy (E\(_a\) in kJ/mol\(^{-1}\)) and heat of reaction (ΔH in kJ/mol\(^{-1}\)) for this reaction are respectively

• (1) 5, 15
• (2) 15, 5
• (3) 25, 5
• (4) 10, 25

Match the following:

List-I (Sol)   List-II (Method of preparation)
A) As₂S₃       I) Bredig's arc method
B) Au           II) Oxidation
C) S             III) Hydrolysis
D) Fe(OH)₃  IV) Double decomposition

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

Which of the following enzymatic reaction is not correctly matched with enzyme shown against it in brackets?

• (1) Proteins → Peptides (Pepsin)
• (2) Starch → Maltose (Zymase)
• (3) Sucrose → Glucose and fructose (Invertase)
• (4) Maltose → Glucose (Maltase)

Which of the following methods is useful for producing semiconductor grade metals of high purity?

• (1) Liquation
• (2) Vapour phase refining
• (3) Electrolytic refining
• (4) Zone refining

Observe the following (Products = .........):

P₄ + SOCl₂ → Products

P₄ + SO₂Cl₂ → Products


In both the reactions, a common product 'X' is obtained. The number of lone pair of electrons on the central atom of 'X' is:

• (1) 1
• (2) 2
• (3) 3
• (4) zero

The IUPAC name of the complex shown below is:

K₃[Co(COₓ)₃]

• (1) Tripotassium trioxalatecobaltate (III)
• (2) Potassium trioxalatecobaltate (III)
• (3) Potassium trioxalatecobaltat (III)
• (4) Potassium trioxalatecobaltate (III)

Identify the ion (hydrated in solution) which is not correctly matched with its spin only magnetic moment (in BM) given in brackets:

• (1) Cr³⁺ (4.90)
• (2) Cu²⁺ (1.73)
• (3) Co³⁺ (4.90)
• (4) Fe³⁺ (4.90)

Which one of the statements regarding X is not correct?

3-Hydroxybutanoic acid → 3-Hydroxypentanoic acid → X
X is:

• (1) It is a condensation polymer
• (2) It is not biodegradable
• (3) It is used in orthopedic devices
• (4) It is known as PHBV

Identify the essential amino acids from the following:

A) Leucine
B) Tyrosine
C) Cysteine
D) Histidine

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

Which of the following represents nucleoside of RNA?

Which of the following is not an antibiotic?

• (1) Chloramphenicol
• (2) Ofloxacin
• (3) Penicillin
• (4) Novestrol

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

Which of the following will undergo methylation with CH₃Cl / anhyd. AlCl₃?

• (1) Aniline and Anisole
• (2) Chlorobenzene and Benzoic acid
• (3) Benzoic acid and Anisole
• (4) Aniline and Chlorobenzene

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

Match the following:

List-I (Compound)   List-II (pKa)
A) p-Nitrophenol     I) 15.9
B) Phenol                II) 7.1
C) Ethanol               III) 10.0
D) p-Cresol              IV) 10.2
 

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

The structures of succinic acid (X) and malonic acid (Y), respectively, are:

Benzyl amine can be prepared from which of the following reactions?