AP EAPCET (AP EAMCET) 2025 Question Paper May 23 Shift 2: Download Solutions with Answer Key

Let \( [t] \) denote the greatest integer function and \( [t - m] = [t] - m \) when \( m \in \mathbb{Z} \). If \( k = 2[2x - 1] - 1 \) and \( 3[2x - 2] + 1 = 2[2x - 1] - 1 \), then the range of \( f(x) = [k + 5x] \) is:

• (1) \(\{7, 8, 9\}\)
• (2) \(\{4, 5, 6\}\)
• (3) \(\{5, 6, 7\}\)
• (4) \(\{6, 7, 8\}\)

If \( f(x) = (x+1)^2 - 1, x \ge -1 \), then \( \{x \mid f(x) = f^{-1}(x)\} \) is:

1. \( \{0, -1\} \) \\
2. \( \{-1, 0, 1\} \) \\
3. \( \left\{-1, 0, \frac{-3+\sqrt{3}i}{2}, \frac{-3-\sqrt{3}i}{2}\right\} \) \\
4. an empty set

If \( 11^{12} - 11^2 = k(5 \times 10^9 + 6 \times 10^9 + 33 \times 10^8 + 110 \times 10^7 + \ldots + 33) \), then find the value of \( k \).

• (1) 20
• (2) 50
• (3) 100
• (4) 200

If \( P = \begin{bmatrix} 1 & \alpha & 3
1 & 3 & 3
2 & 4 & 4 \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( \det(A) = 4 \), then the value of \( \alpha \) is:

• (1) \(3\)
• (2) \(22\)
• (3) \(11\)
• (4) \(4\)

If \( \alpha \) is a real root of the equation \( x^3 + 6x^2 + 5x - 42 = 0 \), then the determinant of the matrix
\[ \begin{bmatrix} \alpha - 1 & \alpha + 1 & \alpha + 2
\alpha - 2 & \alpha + 3 & \alpha - 3
\alpha + 4 & \alpha - 4 & \alpha + 5 \end{bmatrix} \]

is
 

• (1) 90
• (2) 120
• (3) -105
• (4) -135

The rank of the matrix \( \begin{bmatrix} 2 & -3 & 4 & 0
5 & -4 & 2 & 1
1 & -3 & 5 & -4 \end{bmatrix} \) is

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

If \( z \) is a complex number such that \( \frac{z-1}{z-i} \) is purely imaginary and the locus of \( z \) represents a circle with center \( (\alpha, \beta) \) and radius \( r \), then the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) is:

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

If the least positive integer n satisfying the equation \(\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n = -1\) is p and the least positive integer m satisfying the equation \(\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^m = cis\left(\frac{2\pi}{3}\right)\) is q, then \(\sqrt{p^2 + q^2}\) is equal to:

• (1) \(5\)
• (2) \(10\)
• (3) \( \sqrt{13} \)
• (4) \( \sqrt{17} \)

The sum of the squares of the imaginary roots of the equation \( z^8 - 20z^4 + 64 = 0 \) is:

• (1) \(0\)
• (2) \(-12\)
• (3) \(-4\)
• (4) \(-16\)

Let \((a-3)x^2 + 12x + (a+6) > 0, \forall x \in R and a \in (t, \infty)\). If \(\alpha\) is the least positive integral value of \(a\), then the roots of \((\alpha-3)x^2 + 12x + (\alpha+2) = 0\) are:

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

If the roots of the equation \(x^2 + 2ax + b = 0\) are real, distinct and differ utmost by \(2m\), then b lies in the interval

• (1) \((a^2, a^2 + m^2]\)
• (2) \((a^2 + m^2, a^2)\)
• (3) \([a^2, a^2 + 2m^2]\)
• (4) \([a^2 - m^2, a^2)\)

The cubic equation whose roots are the squares of the roots of the equation \( x^3 - 2x^2 + 3x - 4 = 0 \) is

• (1) \( x^3 + 2x^2 + 7x - 16 = 0 \)
• (2) \( x^3 + 2x^2 - 7x - 16 = 0 \)
• (3) \( x^3 - 2x^2 - 7x + 16 = 0 \)
• (4) \( x^3 - 2x^2 + 7x + 16 = 0 \)

If \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + px^2 + qx + r = 0 \), then \( \alpha^3 + \beta^3 + \gamma^3 = \)

• (1) \(p^3 - 3pq + r\)
• (2) \(p^2 - 2pq + r\)
• (3) \(3pq - 3r - p^3\)
• (4) \(3pq + 3r + p^3\)

If all possible 4-digit numbers are formed by choosing 4 different digits from the given digits \( 1, 2, 3, 5, 8 \), then the sum of all such 4-digit numbers is:

• (1) \(199980\)
• (2) \(999990\)
• (3) \(506616\)
• (4) \(479952\)

The number of ways of forming the ordered pairs (p, q) such that p > q by choosing p and q from the first 50 natural numbers is:

• (1) \(1275 \)
• (2) \(1250 \)
• (3) \(1225 \)
• (4) \(1200 \)

The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is:

• (1) \(1865 \)
• (2) \(2370 \)
• (3) \(3050 \)
• (4) \(4380 \)

If \(C_0, C_1, C_2, \dots, C_n\) are the binomial coefficients in the expansion of \((1 + x)^n\), then \((C_0 + C_1) - (C_2 + C_3) + (C_4 + C_5) - (C_6 + C_7) + \dots = \)

• (1) \(2^{n/2} \left(\cos\left(\frac{n\pi}{4}\right) + i\sin\left(\frac{n\pi}{4}\right)\right)\)
• (2) \(2^{n/2} \left(\cos\left(\frac{n\pi}{3}\right) + \sin\left(\frac{n\pi}{3}\right)\right)\)
• (3) \(2^{n/2} \left(\cos\left(\frac{n\pi}{3}\right) + i\sin\left(\frac{n\pi}{3}\right)\right)\)
• (4) \(2^{n/2} \left(\cos\left(\frac{n\pi}{4}\right) + \sin\left(\frac{n\pi}{4}\right)\right)\)

\( 1 + \frac{4}{15} + \frac{4 \cdot 10}{15 \cdot 30} + \frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45} + \cdots \infty \)

• (1) \( \left( \frac{3}{5} \right)^{2/3} \)
• (2) \( \left( \frac{5}{3} \right)^{2/3} \)
• (3) \( \left( \frac{3}{5} \right)^{3/2} \)
• (4) \( \left( \frac{5}{3} \right)^{3/2} \)

If \( \frac{3x+1}{(x-1)(x^2+2)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+2} \), then \( 5(A-B) = \)

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

\(cosec 48^\circ + cosec 96^\circ + cosec 192^\circ + cosec 384^\circ =\)

Options :

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

If \( \sqrt{3}\cos\theta + \sin\theta > 0 \), then the range of \( \theta \) is:

• (1) \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \)
• (2) \( -\frac{\pi}{3} < \theta < \frac{2\pi}{3} \)
• (3) \( -\frac{2\pi}{3} < \theta < \frac{\pi}{3} \)
• (4) \( -\frac{\pi}{6} < \theta < \frac{5\pi}{6} \)

If \(\cos\theta = -\frac{3}{5}\) and \(\theta\) does not lie in second quadrant, then \(\tan\frac{\theta}{2} =\)

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

The general solution satisfying both the equations \(\sin x = -\frac{3}{5}\) and \(\cos x = -\frac{4}{5}\) is:

• (1) \(x = (2n+1)\pi + \operatorname{Tan}^{-1}\left(\frac{3}{4}\right), n \in Z \)
• (2) \(x = 2n\pi + \operatorname{Tan}^{-1}\left(\frac{3}{4}\right), n \in Z \)
• (3) \(x = n\pi + \operatorname{Tan}^{-1}\left(\frac{3}{4}\right), n \in Z \)
• (4) \(x = n\pi \pm \operatorname{Tan}^{-1}\left(\frac{3}{4}\right), n \in Z \)

The number of solutions of \( \tan^{-1} 1 + \frac{1}{2} \cos^{-1} x^2 - \tan^{-1}\left(\frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}}\right) = 0 \) is

• (1) \( 3 \)
• (2) \( 0 \)
• (3) \( 1 \)
• (4) infinitely many ( \(\infty\) )

\(\operatorname{Tanh}^{-1}(\sin\theta) =\)

• (1) \(\operatorname{Sinh}^{-1}(\operatorname{cosec}\theta) \)
• (2) \(\operatorname{Sinh}^{-1}(\sec\theta) \)
• (3) \(\operatorname{Cosh}^{-1}(\operatorname{cosec}\theta) \)
• (4) \(\operatorname{Cosh}^{-1}(\sec\theta) \)

In \( \triangle ABC \), if \( a = 8 \), \( b = 10 \), \( c = 12 \), then \( \frac{r}{R} = \)

• (1) \( \frac{8}{15} \)
• (2) \( \frac{7}{16} \)
• (3) \( \frac{3}{5} \)
• (4) \( \frac{5}{8} \)

In triangle \( ABC \), if \( a = 13 \), \( b = 8 \), \( c = 7 \), then \( \cos(B+C) = \)

• (1) \( \frac{11}{13} \)
• (2) \( \frac{23}{26} \)
• (3) \( \frac{3}{4} \)
• (4) \( \frac{1}{2} \)

In a triangle ABC, if \((r_1 - r_3)(r_1 - r_2) - 2r_2r_3 = 0\), then \(a^2 - b^2 =\)

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

If the median AD of the triangle ABC is bisected at E and BE meets AC in F, then AF : AC =

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

If \( \overline{a} = 2\overline{i} - 3\overline{j} + 5\overline{k} \) and \( \overline{b} = -\overline{i} + 3\overline{j} + 3\overline{k} \) are two vectors, then the vector of magnitude 28 units in the direction of the vector \( \overline{a} - \overline{b} \) is:

• (1) \( 3\overline{i} + 6\overline{j} - 2\overline{k} \)
• (2) \( 12\overline{i} - 24\overline{j} + 8\overline{k} \)
• (3) \( 3\overline{i} - 6\overline{j} - 2\overline{k} \)
• (4) \( 12\overline{i} + 24\overline{j} - 8\overline{k} \)

If \( \overline{a} \) is a unit vector, then \( |\overline{a} \times \overline{i}|^2 + |\overline{a} \times \overline{j}|^2 + |\overline{a} \times \overline{k}|^2 = \)

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

If \( \overline{a} = \overline{i} - 2\overline{j} - 3\overline{k} \), \( \overline{b} = -2\overline{i} + 3\overline{j} + 4\overline{k} \), \( \overline{c} = 5\overline{i} - 4\overline{j} + 3\overline{k} \), and \( \overline{d} = 3\overline{i} + \overline{j} + 5\overline{k} \) are four vectors, then \( (\overline{a} \times \overline{b}) \cdot (\overline{c} \times \overline{d}) = \)

• (1) \( 18\overline{i} + 6\overline{j} + 30\overline{k} \)
• (2) \( 8\overline{i} - 3\overline{j} + 8\overline{k} \)
• (3) \( 19\overline{i} - 5\overline{j} + 21\overline{k} \)
• (4) \( 27\overline{i} - 8\overline{j} + 29\overline{k} \)

If \( 3\overline{i} + \overline{j} + \overline{k} \), \( 2\overline{i} + \overline{k} \), and \( \overline{i} + 5\overline{j} \) are the position vectors of three non-collinear points A, B, C respectively. If the perpendicular drawn from C onto \( \overline{AB} \) meets \( \overline{AB} \) at the point \( a\overline{i} + b\overline{j} + c\overline{k} \), then \( a + b + c = \)

• (1) \( 5 \)
• (2) \( 3 \)
• (3) \( 7 \)
• (4) \( 9 \)

Let \(x_1, x_2, \ldots, x_{11}\) be the observations satisfying \(\sum_{i=1}^{11} (x_i - 4) = 22\) and \(\sum_{i=1}^{11} (x_i - 4)^2 = 154\). If the mean and variance of the observations are \(\alpha\) and \(\beta\), then the quadratic equation having the roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is:

• (1) \(15x^2 - 16x + 15 = 0 \)
• (2) \(15x^2 - 34x + 15 = 0 \)
• (3) \(x^2 - 16x + 60 = 0 \)
• (4) \(12x^2 - 25x + 20 = 0 \)

There are 8 boys and 7 girls in a class room. If the names of all those children are written on paper slips and 3 slips are drawn at random from them, then the probability of getting the names of one boy and two girls or one girl and two boys is

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

A four member committee is to be formed from a group containing 9 men and 5 women. If a committee is formed randomly, then the probability that it contains atleast one woman is

• (1) \(\frac{125}{143} \)
• (2) \(\frac{18}{143} \)
• (3) \(\frac{60}{143} \)
• (4) \(\frac{65}{143} \)

A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and B be the event of getting an even number when the die is thrown second time. Then \(P(A/B) =\)

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

A bag contains 5 balls of unknown colors. There are equal chances that out of these five balls, there may be 0 or 1 or 2 or 3 or 4 or 5 red balls. A ball is taken out from the bag at random and is found to be red. The probability that it is the only red ball in the bag is:

• (1) \( \frac{1}{5} \)
• (2) \( \frac{1}{6} \)
• (3) \( \frac{1}{15} \)
• (4) \( \frac{1}{30} \)

If \( X \sim B(9, p) \) is a binomial variate satisfying the equation \( P(X = 3) = P(X = 6) \), then \( P(X < 3) = \)

• (1) \( \frac{23}{256} \)
• (2) \( \frac{65}{256} \)
• (3) \( \frac{5}{256} \)
• (4) \( \frac{45}{512} \)

The mean and variance of a binomial distribution are \(x\) and \(5\) respectively. If \(x\) is an integer, then the possible values for \(x\) are

• (1) \(6, 10, 30 \)
• (2) \(8, 12, 28 \)
• (3) \(10, 15, 25 \)
• (4) \(9, 18, 24 \)

If the locus of a point which is equidistant from the coordinate axes forms a triangle with the line \(y = 3\), then the area of the triangle is

• (1) \(18 \)
• (2) \(9 \)
• (3) \(6 \)
• (4) \(3 \)

After the coordinate axes are rotated through an angle \(\frac{\pi}{4}\) in the anti clockwise direction without shifting the origin, if the equation \(x^2 + y^2 - 2x - 4y - 20 = 0\) transforms to \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) in the new coordinate system, then \(\begin{vmatrix} a & h & g
h & b & f
g & f & c \end{vmatrix} =\)

• (1) \(-20\)
• (2) \(-25\)
• (3) \(-30\)
• (4) \(-35\)

A\((-2, 3)\) is a point on the line \(4x + 3y - 1 = 0\). If the points on the line that are 10 units away from the point A are \((x_1, y_1)\) and \((x_2, y_2)\), then \((x_1 + y_1)^2 + (x_2 + y_2)^2 =\ ?\)

• (1) \( 10 \)
• (2) \( 90 \)
• (3) \( 180 \)
• (4) \( 405 \)

If \(\alpha\) is the angle made by the perpendicular drawn from origin to the line \(12x - 5y + 13 = 0\) with the positive X-axis in anti-clockwise direction, then \(\alpha =\)

• (1) \(\operatorname{Tan}^{-1} \frac{5}{12} \)
• (2) \(2\pi - \operatorname{Tan}^{-1} \frac{5}{12} \)
• (3) \(\pi - \operatorname{Tan}^{-1} \frac{5}{12} \)
• (4) \(\pi + \operatorname{Tan}^{-1} \frac{5}{12} \)

If the equation of the pair of lines passing through (1, 1) and perpendicular to the pair of lines \(2x^2 + xy - y^2 - x + 2y - 1 = 0\) is \(ax^2 + 2hxy + by^2 + 2gx + 3y = 0\). then \(\frac{b}{a} =\)

• (1) \(g/h\)
• (2) \(2(g+h)\)
• (3) \(2(g-h)\)
• (4) \(gh\)

If the combined equation of the lines joining the origin to the points of intersection of the curve \( x^2 + y^2 - 2x - 4y + 2 = 0 \) and the line \( x + y - 2 = 0 \) is \( (l_1x + m_1y)(l_2x + m_2y) = 0 \), then \( l_1 + l_2 + m_1 + m_2 = \)

• (1) \( 16 \)
• (2) \( -6 \)
• (3) \( -2 \)
• (4) \( 10 \)

The slope of one of the direct common tangents drawn to the circles \(x^2 + y^2 - 2x + 4y + 1 = 0\) and \(x^2 + y^2 - 4x - 2y + 4 = 0\) is

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

If (1, a), (b, 2) are conjugate points with respect to the circle \(x^2 + y^2 = 25\), then \(4a + 2b =\)

• (1) \(25\)
• (2) \(50\)
• (3) \(100\)
• (4) \(150\)

If the pole of the line \(x + 2by - 5 = 0\) with respect to the circle \(S = x^2 + y^2 - 4x - 6y + 4 = 0\) lies on the line \(x + by + 1 = 0\), then the polar of the point \((b, -b)\) with respect to the circle \(S = 0\) is

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

If P(\(\alpha, \beta\)) is the radical centre of the circles \(S=x^2+y^2+4x+7=0\), \(S'=2x^2+2y^2+3x+5y+9=0\) and \(S''=x^2+y^2+y=0\), then the length of the tangent drawn from P to S' = 0 is

• (1) \(5\)
• (2) \(8\)
• (3) \(4\)
• (4) \(2\)

If the tangents of the parabola \( y^2 = 8x \) passing through the point \( P(1, 3) \) touch the parabola at points \( A \) and \( B \), then the area (in sq. units) of \( \triangle ABC \) is

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

The equation of the normal drawn at the point \((\sqrt{2}+1, -1)\) to the ellipse \(x^2 + 2y^2 - 2x + 8y + 5 = 0\) is

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

If \(3x+2\sqrt{2}y+k=0\) is a normal to the hyperbola \(4x^2-9y^2-36=0\) making positive intercepts on both the axes, then \(k=\)

Options :

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

If a hyperbola has asymptotes \(3x-4y-1=0\) and \(4x-3y-6=0\), then the transverse and conjugate axes of that hyperbola are

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

If \( A(0, 1, 2) \), \( B(2, -1, 3) \), and \( C(1, -3, 1) \) are the vertices of a triangle, then the distance between its circumcentre and orthocentre is

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

If the direction cosines of two lines satisfy the equations \( l - 2m + n = 0 \) and \( lm + 10mn - 2nl = 0 \), and \( \theta \) is the angle between the lines, then \( \cos \theta = \)

• (1) \( \frac{\pi}{6} \)
• (2) \( \frac{8}{\sqrt{70}} \)
• (3) \( \frac{\pi}{3} \)
• (4) \( \frac{20}{3\sqrt{70}} \)

If \((2, -1, 3)\) is the foot of the perpendicular drawn from the origin \((0, 0, 0)\) to a plane then the equation of that plane is

• (1) \(2x + y - 3z + 6 = 0 \)
• (2) \(2x - y + 3z - 14 = 0 \)
• (3) \(2x - y + 3z - 13 = 0 \)
• (4) \(2x + y + 3z - 10 = 0 \)

Evaluate the limit: \(\lim_{x \to 0} \frac{x^2 \sin^2(3x) + \sin^4(6x)}{(1 - \cos 3x)^2}\)

• (1) \(\frac{580}{9}\)
• (2) \(\frac{145}{3}\)
• (3) \(\frac{580}{3}\)
• (4) \(\frac{145}{9}\)

If a real valued function \(f(x) = \begin{cases} (1 + \sin x)^{\operatorname{cosec} x} & , -\pi/2 < x < 0
a & , x = 0
\frac{e^{2/x} + e^{3/x}}{ae^{2/x} + be^{3/x}} & , 0 < x < \pi/2 \end{cases}\) is continuous at \(x = 0\), then \(ab =\)

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

Evaluate the limit: \(\lim_{x \to 0} \frac{(\operatorname{cosec} x - \cot x)(e^x - e^{-x})}{\sqrt{3} - \sqrt{{2} + \cos x}}\)

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

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

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

If \( y = \tan^{-1}\sqrt{x^2 - 1} + \sinh^{-1}\sqrt{x^2 - 1} \), \( x > 1 \), then \( \frac{dy}{dx} = \)

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

If \( y = (\log x)^{1/x} + x^{\log x} \), then at \( x = e \), \( \frac{dy}{dx} \) equals:

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

The interval in which the function \( f(x) = \tan^{-1}(\sin x + \cos x) \) is an increasing function, is:

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

The slope of a tangent drawn at the point \( P(\alpha, \beta) \) lying on the curve \( y = \frac{1}{2x - 5} \) is \(-2\). If \( P \) lies in the fourth quadrant, then \( \alpha - \beta = \)

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

The function \(f(x) = xe^{-x} \forall x \in \mathbb{R}\) attains a maximum value at \(x = k\), then \(k =\)

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

If \( m \) and \( M \) are the absolute minimum and absolute maximum values of the function \( f(x) = 2\sqrt{2} \sin x - \tan x \) in the interval \( \left[0, \frac{\pi}{3} \right] \), then \( m + M = \)

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

Evaluate \( \int \frac{\sec^2 x}{\sin^7 x} \, dx - \int \frac{7}{\sin^7 x} \, dx \):

• (1) \( \frac{1}{\sin^6 x \cos x} + c \)
• (2) \( \frac{\tan x}{\sin^8 x} + c \)
• (3) \( \sin^8 x \cos x + c \)
• (4) \( \sec x \tan^7 x + c \)

If \( \int \left( x^6 + x^4 + x^2 \right) \sqrt{2x^4 + 3x^2 + 6} \, dx = f(x) + c \), then \( f(3) = \)

• (1) \( \dfrac{3}{2} (95)^{3/2} \)
• (2) \( \dfrac{3}{2} (195)^{3/2} \)
• (3) \( \dfrac{3}{2} (265)^{3/2} \)
• (4) \( \dfrac{3}{2} (175)^{3/2} \)

Evaluate: \[ \int \frac{dx}{(x+1)\sqrt{x^2+1}} \]

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

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

• (1) \( \frac{\pi}{12} \)
• (2) \( \frac{\pi}{8} \)
• (3) \( \frac{5\pi}{12} \)
• (4) \( \frac{5\pi}{8} \)

If \(\int \frac{1}{((x+4)^3 (x+1)^5)^{1/4}} dx = A \cdot \left(\frac{x+4}{x+1}\right)^n + c\), then

• (1) \(n \cdot A = 3 \)
• (2) \(n + \frac{1}{A} = -\frac{1}{2} \)
• (3) \(A + n = 1 \)
• (4) \(A = n \)

\( \int_{-\pi/2}^{\pi/2} \sin^2 x \cos^2 x (\sin x + \cos x) \, dx = \)

• (1) \( 0 \)
• (2) \( \frac{2}{15} \)
• (3) \( \frac{4}{15} \)
• (4) \( \frac{2}{5} \)

Evaluate the integral \( \displaystyle \int_{1/5}^{1/2} \frac{\sqrt{x - x^2}}{x^3} \, dx \):

• (1) \( \frac{21}{2} \)
• (2) \( \frac{14}{3} \)
• (3) \( \frac{7}{3} \)
• (4) \( \frac{7}{2} \)

Evaluate: \( \int_0^{400\pi} \sqrt{1 - \cos 2x} \, dx \)

• (1) \( 100\sqrt{2} \)
• (2) \( 200\sqrt{2} \)
• (3) \( 400\sqrt{2} \)
• (4) \( 800\sqrt{2} \)

Area of the region (in sq. units) bounded by the curve \( y = x^2 - 5x + 4 \), \( x = 0 \), \( x = 2 \), and the X-axis is

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

If the order and degree of the differential equation \(x \frac{d^2 y}{dx^2} = \left(1 + \left(\frac{d^2 y}{dx^2}\right)^2\right)^{-1/2}\) are \(k\) and \(l\) respectively, then \(k, l\) are the roots of

• (1) \(x^2 - 5x + 6 = 0 \)
• (2) \(x^2 - 3x + 2 = 0 \)
• (3) \(x^2 - 7x + 12 = 0 \)
• (4) \(x^2 - 6x + 8 = 0 \)

The equation of the curve passing through the point \( (0, \pi) \) and satisfying the differential equation \( ydx = (x + y^3 \cos y)dy \) is

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

The general solution of the differential equation \( \left(x - (x + y)\log(x + y)\right) dx + x\,dy = 0 \) is:

• (1) \( y \log(x + y) = cx \)
• (2) \( x \log(x + y) = cy \)
• (3) \( \log(x + y) = cy \)
• (4) \( \log(x + y) = cx \)

If the equation for the velocity of a particle at time 't' is \(v = at + \frac{b}{t+c}\), then the dimensions of a, b, c are respectively

• (1) \(LT^{-2}, L, T \)
• (2) \(L^2, L, T \)
• (3) \(LT^{-2}, LT, L \)
• (4) \(L, LT, L^2 \)

If a stone thrown vertically upwards from a bridge with an initial velocity of \(5 \operatorname{ms}^{-1}\), strikes the water below the bridge in a time of 3 s, then the height of the bridge above the water surface is (Acceleration due to gravity = \(10 \operatorname{ms}^{-2}\))

• (1) \(10 \operatorname{m} \)
• (2) \(26 \operatorname{m} \)
• (3) \(30 \operatorname{m} \)
• (4) \(18 \operatorname{m} \)

If \( \alpha \), \( \beta \), and \( \gamma \) are the angles made by a vector with the \( x \)-, \( y \)-, and \( z \)-axes respectively, then find the value of \( \sin^2\alpha + \sin^2\beta \).

• (1) \( \sin^2\gamma \)
• (2) \( \cos^2\gamma \)
• (3) \( 1 + \cos^2\gamma \)
• (4) \( 1 + \sin^2\gamma \)

A particle moving along a straight line covers the first half of the distance with a speed of \( 3 m s^{-1} \), the other half of the distance is covered in two equal time intervals with speeds of \( 4.5 m s^{-1} \) and \( 7.5 m s^{-1} \) respectively, then the average speed of particle during the motion is

• (1) \( 4.0 m s^{-1} \)
• (2) \( 5.0 m s^{-1} \)
• (3) \( 5.5 m s^{-1} \)
• (4) \( 4.8 m s^{-1} \)

Water flowing through a pipe of area of cross-section \( 2 \times 10^{-3} m^2 \) hits a vertical wall horizontally with a velocity of \( 12 m s^{-1} \). If the water does not rebound after hitting the wall, then the force acting on the wall due to water is

• (1) \( 24 N \)
• (2) \( 144 N \)
• (3) \( 288 N \)
• (4) \( 72 N \)

Two blocks A and B of masses 2 kg and 4 kg respectively are kept on a rough horizontal surface. If same force of 20 N is applied on each block, then the ratio of the accelerations of the blocks A and B is (Coefficient of kinetic friction between the surface and the blocks is 0.3 and acceleration due to gravity = \(10 \operatorname{ms}^{-2}\))

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

If a force of \((6x^2 - 4x) \operatorname{N}\) acts on a body of mass 10 kg, then work to be done by the force in displacing the body from \(x = 2 \operatorname{m}\) to \(x = 4 \operatorname{m}\) is

• (1) \(22 \operatorname{J} \)
• (2) \(44 \operatorname{J} \)
• (3) \(66 \operatorname{J} \)
• (4) \(88 \operatorname{J} \)

A circular well of diameter \(2 \, m\) has water up to the ground level. If the bottom of the well is at a depth of \(14 \, m\), the time taken in seconds to empty the well using a \(1.4 \, kW\) motor is (Acceleration due to gravity = \(10 \, m/s^2\))

• (1) 1860
• (2) 2200
• (3) 2660
• (4) 3300

The coordinates of the centre of mass of a uniform L-shaped plate of mass 3 kg shown in the figure is:

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

A force \( F \) is applied on a body of mass \( m \) so that the body starts moving from rest. The power delivered by the force at time \( t \) is proportional to:

• (1) \( t \)
• (2) \( t^2 \)
• (3) \( t^3 \)
• (4) \( \sqrt{t} \)

The equations for the displacements of two particles in simple harmonic motion are \( y_1 = 0.1\sin\left(100\pi t + \frac{\pi}{3}\right) \) and \( y_2 = 0.1\cos(\pi t) \) respectively. The phase difference between the velocities of the two particles at a time \( t = 0 \) is

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

A spring is stretched by \( 0.2 m \) when a mass of \( 0.5 kg \) is suspended to it. The time period of the spring when \( 0.5 kg \) mass is replaced with a mass of \( 0.25 kg \) is
(Acceleration due to gravity \( = 10 m s^{-2} \))

• (1) \( 0.628 s \)
• (2) \( 6.28 s \)
• (3) \( 62.8 s \)
• (4) \( 0.0628 s \)

An artificial satellite is revolving around a planet of radius \( R \) in a circular orbit of radius \( a \). If the time period of revolution of the satellite, \( T \propto a^{3/2}g^xR^y \), then the values of \( x \) and \( y \) are respectively:

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

If the longitudinal strain of a stretched wire is 0.2% and the Poisson's ratio of the material of the wire is 0.3, then the volume strain of the wire is

• (1) \(0.12% \)
• (2) \(0.08% \)
• (3) \(0.14% \)
• (4) \(0.26% \)

If two soap bubbles A and B of radii \(r_1\) and \(r_2\) respectively are kept in vaccum at constant temperature, then the ratio of masses of air inside the bubbles A and B is

• (1) \(r_2^2 : r_1^2 \)
• (2) \(r_1^2 : r_2^2 \)
• (3) \(r_1 : r_2 \)
• (4) \(r_2 : r_1 \)

A small quantity of water of mass 'm' at temperature \( \theta ^\circC \) is mixed with a large mass 'M' of ice which is at its melting point. If 's' is specific heat capacity of water and 'L' is the Latent heat of fusion of ice, then the mass of ice melted is

• (1) \( \frac{ML}{ms\theta} \)
• (2) \( \frac{ms\theta}{ML} \)
• (3) \( \frac{Ms\theta}{L} \)
• (4) \( \frac{ms\theta}{L} \)

In a Carnot engine, if the absolute temperature of the source is \( 25% \) more than the absolute temperature of the sink, then the efficiency of the engine is

• (1) \( 25% \)
• (2) \( 50% \)
• (3) \( 20% \)
• (4) \( 40% \)

The work done by 6 moles of helium gas when its temperature increases by \( 20^\circ C \) at constant pressure is (Universal gas constant = \( 8.31 \, J mol^{-1} \, K^{-1} \))

• (1) \( 807.2 \, J \)
• (2) \( 887.2 \, J \)
• (3) \( 997.2 \, J \)
• (4) \( 1007.2 \, J \)

If a heat engine and a refrigerator are working between the same two temperatures \(T_1\) and \(T_2\) (\(T_1 > T_2\)), then the ratio of efficiency of heat engine to coefficient of performance of refrigerator is

• (1) \(\frac{(T_1 - T_2)}{T_1 T_2} \)
• (2) \(\frac{(T_1 + T_2)}{T_1 T_2} \)
• (3) \(\frac{(T_1 - T_2)^2}{T_1 T_2} \)
• (4) \(\frac{(T_1 + T_2)^2}{T_1 T_2} \)

If the internal energy of 3 moles of a gas at a temperature of 27 °C is 2250R, then the number of degrees of freedom of the gas is (R - Universal gas constant)

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

If two progressive sound waves represented by \( y_1 = 3 \sin 250 \pi t \) and \( y_2 = 2 \sin 260 \pi t \) (where displacement is in metre and time is in second) superimpose, then the time interval between two successive maximum intensities is

• (1) \( 0.1 s \)
• (2) \( 0.4 s \)
• (3) \( 0.5 s \)
• (4) \( 0.2 s \)

If the least distance of distinct vision for a boy is \( 35 cm \), then the lens to be used by the boy for correcting the defect of his eye is

• (1) convex lens of focal length \( 35 cm \)
• (2) concave lens of focal length \( 35 cm \)
• (3) convex lens of focal length \( 87.5 cm \)
• (4) concave lens of focal length \( 87.5 cm \)

In Young's double-slit experiment, if the distance between the slits is increased to 3 times its initial distance, then the ratio of initial and final fringe widths is.

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

A solid of mass 1 kg has \(6 \times 10^{24}\) atoms. If one electron is removed from every one atom of 0.005% of the atoms, then the charge gained by the solid is

• (1) \(+24 \operatorname{C} \)
• (2) \(+48 \operatorname{C} \)
• (3) \(+96 \operatorname{C} \)
• (4) \(+60 \operatorname{C} \)

One of the two identical capacitors having the same capacitance C, is charged to a potential \(V_1\) and the other is charged to a potential \(V_2\). If they are connected with their like plates together, then the decrease in the electrostatic potential energy of the combined system is

• (1) \(\frac{C}{4}(V_1^2 - V_2^2) \)
• (2) \(\frac{C}{4}(V_1^2 + V_2^2) \)
• (3) \(\frac{C}{4}(V_1 - V_2)^2 \)
• (4) \(\frac{C}{4}(V_1 + V_2)^2 \)

If the energy stored in a spherical conductor having a charge of \( 12 \, \muC \) is \( 6 \, J \), then the radius of the spherical conductor is.

• (1) \( 10.8 \, cm \)
• (2) \( 0.108 \, cm \)
• (3) \( 1.08 \, cm \)
• (4) \( 108 \, cm \)

A part of a circuit is shown in the figure. The ratio of the potential differences between the points A and C, and the points D and E is.

• (1) \( 4 : 5 \)
• (2) \( 2 : 3 \)
• (3) \( 8 : 15 \)
• (4) \( 11 : 15 \)

A DC supply of \(160\,V\) is used to charge a battery of EMF \(10\,V\) and internal resistance \(1\,\Omega\) by connecting a series resistance of \(24\,\Omega\). The terminal voltage of the battery during charging is:

• (1) \(8 \operatorname{V} \)
• (2) \(12 \operatorname{V} \)
• (3) \(16 \operatorname{V} \)
• (4) \(4 \operatorname{V} \)

The magnetic moment of an electron moving in a circular orbit of radius R with a time period T is

• (1) \(\frac{2\pi Re}{T} \)
• (2) \(\frac{\pi eR}{T} \)
• (3) \(\frac{\pi eR^2}{T} \)
• (4) \(\pi R^2 eT \)

A solenoid of one meter length and \( 3.55 cm \) inner diameter carries a current of \( 5 A \). If the solenoid consists of five closely packed layers each with \( 700 \) turns along its length, then the magnetic field at its centre is

• (1) \( 22 mT \)
• (2) \( 35 mT \)
• (3) \( 44 mT \)
• (4) \( 15 mT \)

The work done in rotating a bar magnet which is initially in the direction of a uniform magnetic field through \( 45^\circ \) is \( W \). The additional work to be done to rotate the magnet further through \( 15^\circ \) is.

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

When a current of \( 4 \, mA \) passes through an inductor, if the flux linked with it is \( 32 \times 10^{-6} \, Tm^2 \), then the energy stored in the inductor is.

• (1) \( 64 \times 10^{-9} \, J \)
• (2) \( 32 \times 10^{-9} \, J \)
• (3) \( 128 \times 10^{-9} \, J \)
• (4) \( 96 \times 10^{-9} \, J \)

In a series resonant LCR circuit, for the power dissipated to become half of the maximum power dissipated, the current amplitude is

• (1) \(\frac{1}{\sqrt{2}}\) times its maximum value.
• (2) \(\frac{1}{2}\) times its maximum value.
• (3) twice its maximum value.
• (4) \(\sqrt{2}\) times its maximum value.

The waves having maximum wavelength among the following electromagnetic waves is

• (1) X-rays
• (2) Radio waves
• (3) UV waves
• (4) Visible rays

If the de Broglie wavelength of an electron is \( 2 nm \), then its kinetic energy is nearly
(Planck's constant \( = 6.6 \times 10^{-34} J s \) and mass of electron \( = 9 \times 10^{-31} kg \))

• (1) \( 0.48 eV \)
• (2) \( 0.68 eV \)
• (3) \( 0.38 eV \)
• (4) \( 0.25 eV \)

The ratio of the wavelengths of the spectral lines emitted due to transitions \( 3 \rightarrow 2 \) and \( 2 \rightarrow 1 \) orbits in the hydrogen atom is

• (1) \( 3:1 \)
• (2) \( 9:17 \)
• (3) \( 27:5 \)
• (4) \( 25:9 \)

The density (in \(kg m^{-3}\)) of nuclear matter is of the order of

• (1) \( 10^{21} \)
• (2) \( 10^{17} \)
• (3) \( 10^{12} \)
• (4) \( 10^{8} \)

In a common emitter amplifier of a transistor, if the ratio of the voltage gain and current amplification factor is 4, then the ratio of the collector and base resistances is.

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

If three logic gates are connected as shown in the figure, then the correct truth table of the circuit is

• (1)
• (2)
• (3)
• (4)

Ionosphere acts as a reflector for the frequency range of

• (1) \(3 - 30 \operatorname{kHz} \)
• (2) \(3 - 30 \operatorname{MHz} \)
• (3) \(3 - 30 \operatorname{Hz} \)
• (4) \(3 - 30 \operatorname{GHz} \)

The uncertainty in the velocities of two particles \( A \) and \( B \) are \( 0.03 \) and \( 0.01 \, m/s \), respectively. The mass of \( B \) is four times the mass of \( A \). The ratio of uncertainties in their positions is.

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

The total maximum number of electrons possible in 3d, 6d, 5s and 4f orbitals with \(m_l\) (magnetic quantum number) value -2 is

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

The period and group numbers of the element having maximum electronegativity in the long form of periodic table, respectively, are

• (1) \(2, 17 \)
• (2) \(3, 17 \)
• (3) \(1, 18 \)
• (4) \(2, 16 \)

Identify the pair of molecules which have the same hybridisation as the hybridisation in Xenon (II) fluoride.

• (1) \( XeO_3 \), \( SF_4 \)
• (2) \( BrF_5 \), \( PF_5 \)
• (3) \( ClF_3 \), \( SF_4 \)
• (4) \( PCl_3 \), \( NH_3 \)

Identify the set containing isoelectronic species.

• (1) \( N_2 \), \( O_2^- \), \( NO^+ \)
• (2) \( N_2 \), \( CO \), \( NO^+ \)
• (3) \( F_2 \), \( O_2^- \), \( N_2 \)
• (4) \( N_2 \), \( O_2^+ \), \( C_2 \)

Choose the incorrect statement from the following

• (1) At Boyle temperature a real gas obeys ideal gas law over an appreciable range of pressure
• (2) Critical temperature of CO\(_2\) is 27.5°C
• (3) Above critical temperature, a real gas behaves like an ideal gas
• (4) At room temperature and 1 atm pressure the compressibility factor (Z) for H\(_2\) gas is greater than 1

An ideal gas mixture of C\(_2\)H\(_6\) and C\(_2\)H\(_4\) occupies a volume of 28 L at 1 atm and 273 K. This mixture reacts completely with 128 g of O\(_2\) to produce CO\(_2\) and H\(_2\)O(\(l\)). What is the mole fraction of C\(_2\)H\(_4\) in the mixture ?

• (1) \(0.4 \)
• (2) \(0.8 \)
• (3) \(0.5 \)
• (4) \(0.6 \)

Identify the incorrect statements from the following.

I. For an adiabatic process, \( \Delta U = w_{ad} \).


II. Enthalpy is an intensive property.


III. For the process \( H_2O(\ell) \rightarrow H_2O(s) \), the entropy increases.

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

The enthalpies of formation of \( CO_2(g) \), \( H_2O(l) \), and \( C_6H_{12}O_6(s) \) are \( -393 \), \( -286 \), and \( -1170 \, kJ mol^{-1} \), respectively. The quantity of heat liberated when \( 18 \, g \) of \( C_6H_{12}O_6(s) \) is burnt completely in oxygen is.

• (1) \( 520 \, kJ \)
• (2) \( 145 \, kJ \)
• (3) \( 290 \, kJ \)
• (4) \( 420 \, kJ \)

The percentage of ionization of \( 1 \, L \) of \( x \, M \) acetic acid is \( 4.242% \) and is called solution "A". The percentage of ionization of \( 1 \, L \) of \( y \, M \) acetic acid is \( 3% \) and is called solution "B". Solution "A" is mixed with solution "B". What is the concentration of acetic acid in the resultant solution? (\( K_a \) of acetic acid = \( 1.8 \times 10^{-5} \))

• (1) \( 0.05 \, M \)
• (2) \( 0.015 \, M \)
• (3) \( 0.02 \, M \)
• (4) \( 0.15 \, M \)

At 298 K, the value of K\(_p\) for N\(_2\)O\(_4\)(g) \(\rightleftharpoons\) 2NO\(_2\)(g) is 0.113 atm. The partial pressure of N\(_2\)O\(_4\) at equilibrium is 0.2 atm. What is the partial pressure (in atm) of NO\(_2\) at equilibrium ?

• (1) \(0.05 \)
• (2) \(0.075 \)
• (3) \(0.30 \)
• (4) \(0.15 \)

H\(_2\)O\(_2\) reduces KMnO\(_4\) in acidic medium to 'x' and in basic medium to 'y'. What are x and y?

• (1) x = MnO\(_2\), y = Mn\(^{2+}\)
• (2) x = Mn\(^{2+}\), y = MnO\(_2\)
• (3) x = MnO\(_4^{2-}\), y = Mn\(^{2+}\)
• (4) x = MnO\(_2\), y = MnO\(_4^{2-}\)

Which chloride does not exist as hydrate ?

• (1) MgCl\(_2\)
• (2) CaCl\(_2\)
• (3) LiCl
• (4) KCl

Identify the incorrect statement about the group 13 elements

• (1) Nature of aqueous solution of borax is alkaline
• (2) Orthoboric acid is a weak tribasic acid
• (3) Metaboric acid on heating gives an acidic oxide
• (4) LiBH\(_4\) acts as a reducing agent

Which of the following statements are correct ?

I) SnF\(_4\) is ionic in nature

II) Stability of dihalides of group 14 elements increases down the group

III) GeCl\(_2\) is more stable than GeCl\(_4\)

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

Which of the following when present in excess in drinking water causes the disease methemoglobinemia ?

• (1) SO\(_4^{2-}\)
• (2) NO\(_3^-\)
• (3) F\(^-\)
• (4) Pb

IUPAC name of the following compound is

• (1) \( 2-Methyl-4-ethylhexane \)
• (2) \( 4-Ethyl-2-methylhexane \)
• (3) \( 5-Methyl-3-ethylhexane \)
• (4) \( 3-Ethyl-5-methylhexane \)

The empirical formula weight of 'Z' in the given reaction sequence is
n-propyl bromide \(\xrightarrow{Na}\) X \(\xrightarrow{V_2O_5 773 K}\) Y \(\xrightarrow{Cl_2 UV 500 K}\) Z
Dry ether 20 atm

• (1) \(47.5 \)
• (2) \(54.5 \)
• (3) \(84.5 \)
• (4) \(48.5 \)

If AgCl is doped with \(1 \times 10^{-4}\) mole percent of CdCl\(_2\), the number of cation vacancies (in mol\(^{-1}\)) is

• (1) \(6.023 \times 10^{19} \)
• (2) \(6.023 \times 10^{21} \)
• (3) \(6.023 \times 10^{17} \)
• (4) \(6.023 \times 10^{23} \)

In an aqueous glucose solution, the mole fraction of water is 40 times the mole fraction of glucose. What is the weight percentage (w/w) of glucose in the solution?

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

Benzoic acid molecules undergo dimerisation in benzene. 2.44 g of benzoic acid when dissolved in 30 g of benzene caused depression in freezing point of 2 K. What is the percentage of association of it ?

(Given \(K_f (C_6H_6) = 5\operatorname{K kg mol^{-1}}\); molar mass of benzoic acid \(= 122\operatorname{g mol^{-1}}\))

When the lead storage battery is in use (during discharge) the reaction that occurs at the anode is

The following equation is obtained for a first order reaction at 300 K.
\(\log_{10} \frac{k}{A} = 0.00174\)
What is the activation energy (in J mol\(^{-1}\)) of the reaction ?

(R = 8.314 J mol\(^{-1}\) K\(^{-1}\))

Match the following

\begin{tabular{ll
List-I (colloidal solution) & List-II (use)

\addlinespace
A) Colloidal antimony & I) Eye lotion

B) Argyrol & II) Intramuscular injection

C) Colloidal gold & III) Kalaazar

D) Milk of magnesia & IV) Stomach disorders

\end{tabular


The correct answer is

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

Adsorption of a gas on solids follows Freundlich adsorption isotherm. The graph drawn between log \(\frac{x}{m}\) (on y-axis) and log p (on x-axis) is a straight line with slope equal to 3 and intercept equal to 0.30. What is the value of \(\frac{x}{m}\) at a pressure of 2 atm ?

(Given; log 2 = 0.3)

• (1) 48
• (2) 32
• (3) 16
• (4) 8

Nature of two oxides of nitrogen X and Y formed in the reaction of sodium nitrite with hydrochloric acid is

• (1) Both X and Y are acidic in nature
• (2) X is acidic and Y is neutral in nature
• (3) Both X and Y are neutral in nature
• (4) X is amphoteric and Y is neutral in nature
• (1) A-IV, B-I, C-III, D-II
• (2) A-III, B-I, C-IV, D-II
• (3) A-I, B-II, C-III, D-IV
• (4) A-II, B-IV, C-I, D-III

Identify the complex ion with spin only magnetic moment of 4.90 BM.

What are \( X \) and \( Y \) in the following reaction?
\(\)
\text{nCl/CH_2 \xrightarrow{X Y \(\)

• (1) \( Na / NH_3(l) \) - thermosetting polymer
• (2) \( (C_6H_5COO)_2 \) - thermoplastic polymer
• (3) \( Na / NH_3(l) \) - condensation polymer
• (4) \( (C_6H_5COO)_2 \) - Network polymer

Consider the following

Statement-I: Cane sugar is a disaccharide of \(\alpha\)-D-glucose and \(\beta\)-D-fructose

Statement-II: Milk sugar is a disaccharide of \(\alpha\)-D-glucose and \(\beta\)-D-galactose


Options:

• (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

The deficiency of vitamin (X) causes convulsions. Source of X is Y. What are X and Y ?

Which of the following is not an example of a synthetic detergent?

• (1) Cetyltrimethylammonium bromide
• (2) Sodium stearate
• (3) Sodium laurylsulfate
• (4) Sodium dodecylbenzenesulfonate

The most reactive compound towards nucleophilic substitution with an aqueous \( NaOH \) is:

• (1)
• (2)
• (3)
• (4)

An alkyl bromide \( X (C_5H_{11}Br) \) undergoes hydrolysis in a two-step mechanism. \( X \) is converted to a Grignard reagent and then reacted with \( CO_2 \) in dry ether followed by acidification gave \( Y \). What is \( Y \)?

• (1)
• (2)
• (3)
• (4)

Consider the following sequence of reactions.
\(\operatorname{C}_6\operatorname{H}_5\operatorname{COONa} \xrightarrow{NaOH/CaO, \Delta}\) X \(\xrightarrow{CO+HCl, Anhy. AlCl_3}\) Y \(\xrightarrow{NaOH}\) A + B

If A is the reduction product of Y, what is B ?

• (1) Sodium formate
• (2) Sodium phenoxide
• (3) Sodium salt of benzoic acid
• (4) Sodium salt of salicylic acid

What is \( A \) in the following reaction? \(\)
\text{CH_3-\text{CH=\text{CH-\text{CH_2-\text{CH_2-\text{CN \xrightarrow{(1)\,A(\text{i-Bu)_2{(2)\,\text{H_2\text{O \rightarrow A \(\)

• (1) \( CH_3-CH=CH-CH_2-CH_2-NH_2 \)
• (2) \( CH_3-CH_2-CH_2-CH_2-CH_2-CH_2-NH_2 \)
• (3) \( CH_3-CH=CH-CH_2-CH_2-CHO \)
• (4) \( CH_3-CH_2-CH_2-CH_2-CH_2-CHO \)

The correct statement regarding X and Y formed in the following reaction is
\((\operatorname{CH}_3)_3\operatorname{COC}_2\operatorname{H}_5 \xrightarrow{HI, \Delta}\) halide(X) + alcohol(Y)

Options:

• (1) X undergoes substitution by \(\operatorname{S}_{N}2\) mechanism
• (2) X undergoes substitution with water in two steps
• (3) Y gets converted to corresponding chloride with conc.HCl at room temperature
• (4) Reaction of Y with Cu / 573 K gives ketone

Consider the following

Statement-I: In the nitration of aniline, more amount of m-nitroaniline is formed than expected.

Statement-II: In the presence of a strongly acidic medium, aniline is protonated to form anilinium ion, which is meta directing.

• (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