AP EAPCET (AP EAMCET) 2025 Question Paper May 24 Shift 2 (Available): Download Solutions with Answer Key

If \(A = \{x \in \mathbb{R} \mid \sin^{-1}(\sqrt{x^2+x+1}) \in [-\frac{\pi}{2}, \frac{\pi}{2}]\}\) and \(B = \{y \in \mathbb{R} \mid y = \sin^{-1}(\sqrt{x^2+x+1}), x \in A\}\), then

• (1) \(A \cap B \neq \emptyset\)
• (2) \(A \cap B = [0,1]\)
• (3) \(A \cap B = [\frac{\pi}{3}, \frac{\pi}{2}]\)
• (4) \(A \cap B = \mathbb{R} - [-1,0] \cup [\frac{\pi}{3},\frac{\pi}{2}]\)

The domain of the function \(f(x) = \ln\left(\frac{1}{\sqrt{x^2-4x+4}}\right) + \sin^{-1}(x^2-2)\) is

• (1) \([1,3]\)
• (2) \([1,3)\)
• (3) \([1,\sqrt{3}]\)
• (4) \([1,\sqrt{3})\)

For all \(n \in \mathbb{N}\), if \(n(n^2+3)\) is divisible by \(k\), then the maximum value of \(k\) is

• (1) 4
• (2) 6
• (3) 8
• (4) 2

If \(a\) is the determinant of the adjoint of the matrix \(\begin{bmatrix} 1 & 1 & 2
1 & 2 & 3
2 & 3 & 3 \end{bmatrix}\) and \(b\) is the determinant of the inverse of the matrix \(\begin{bmatrix} 2 & 1 & 3
1 & -4 & -1
2 & 1 & 4 \end{bmatrix}\), then \(\frac{b+1}{18b} = \)

• (1) \(a\)
• (2) \(10a\)
• (3) \(2 + a\)
• (4) \(2a\)

Consider two systems of 3 linear equations in 3 unknowns \(AX = B\) and \(CX = D\). If \(AX = B\) has the unique solution \(X = D\) and \(CX = D\) has the unique solution \(X = B\), then the solution of \((A - C^{-1})X = 0\) is

• (1) \(B\)
• (2) \(D\)
• (3) \(B + D\)
• (4) \(B - D\)

\(f(x)\) is an \(n^{th}\) degree polynomial satisfying \(f(x) = \frac{1}{2}\left[f(x)f\left(\frac{1}{x}\right) + f\left(\frac{f(x)}{x}\right)\right]\). If \(f(2) = 33\), then the value of \(f(3)\) is

• (1) 126
• (2) 214
• (3) 244
• (4) -124

If the point \(P\) denotes the complex number \(z = x + iy\) in the Argand plane and \(\frac{z - (2 - i)}{z + (1 + 2i)}\) is purely imaginary, then the locus of \(P\) is

• (1) a hyperbola not containing the point \((-1, -2)\)
• (2) an ellipse not containing the point \((-1, -2)\)
• (3) a parabola not containing the point \((-1, -2)\)
• (4) a circle not containing the point \((-1, -2)\) and having its centre on the line \(x + y + 1 = 0\)

If \((\sqrt{3} - i)^n = 2^n\), \(n \in \mathbb{N}\), then the least possible value of \(n\) is

• (1) 3
• (2) 4
• (3) 6
• (4) 12

\((1 + \sqrt{5} + i \sqrt{10 - 2\sqrt{5}})^3 = \)

• (1) 1024
• (2) -1024
• (3) 512
• (4) -512

The number of solutions of the equation \(\sqrt{3x^2 + x + 5} = x - 3\) is

• (1) 2
• (2) 1
• (3) 0
• (4) 4

The set of all real values of \(x\) for which \(\frac{x^2-1}{(x-4)(x-3)} \ge 1\) is

• (1) \([-1,1] \cup (3,4)\)
• (2) \([1,\frac{7}{3}] \cup (4,\infty)\)
• (3) \((-\infty, \frac{-13}{7}] \cup (3,4)\)
• (4) \(\mathbb{R} - [3,4]\)

If \(\alpha\), \(\beta\), and \(\gamma\) are the roots of the equation \(2x^3 + 3x^2 - 5x - 7 = 0\), then \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} =\)

• (1) \(\frac{17}{49}\)
• (2) \(-\frac{23}{49}\)
• (3) \(\frac{55}{49}\)
• (4) \(\frac{67}{49}\)

Two roots of the equation \(ax^4 + bx^3 + cx^2 + dx + e = 0\) are positive and equal. If the product of the other two real roots is 1, then

• (1) \(be^2 = a^2d\)
• (2) \(\frac{3e + 2b\sqrt{e} + c}{\sqrt{a}} = a\)
• (3) \(e + 2b\sqrt{e} + 3c = a\sqrt{a}\)
• (4) \(b^2e = ad^2\)

The number of integers between 10 and 10,000 such that in every integer every digit is greater than its immediate preceding digit, is

• (1) 1112
• (2) 437
• (3) 216
• (4) 182

All letters of the word `AGAIN' are permuted in all possible ways, and the words so formed (with or without meaning) are written as in a dictionary. Then the \(50^{th}\) word is

• (1) IAANG
• (2) INAGA
• (3) NAAIG
• (4) NAAGI

The number of ways in which a cricket team of 11 members can be formed out of 6 batsmen, 6 bowlers, 4 all-rounders, and 4 wicket-keepers by selecting at least 4 batsmen, at least 3 bowlers, at least 2 all-rounders, and only one wicket-keeper is

• (1) 11560
• (2) 6480
• (3) 7680
• (4) 13080

If \(y = \frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \dots \infty\), then

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

Sum of the coefficients of \(x^4\) and \(x^6\) in the expansion of \((1 + x - x^2)^6\) is

• (1) 121
• (2) -91
• (3) 11
• (4) 31

If \(\frac{3x^2 - 7x + 1}{(x - 2)^3} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3}\), then \(A(B + C + D + E) =\)

• (1) 0
• (2) 64
• (3) 348
• (4) 256

\(\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{4\pi}{7}\right) + \tan\left(\frac{4\pi}{7}\right)\tan\left(\frac{\pi}{7}\right) + \tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right) =\)

• (1) 7
• (2) -7
• (3) 3
• (4) -3

\(\cos(13^\circ)\sin(17^\circ)\sin(21^\circ)\cos(47^\circ) =\)

• (1) \(\frac{1}{32}\)
• (2) \(\frac{1}{16}\)
• (3) \(\frac{1}{32}(1 + 2\sqrt{3} - \sqrt{5})\)
• (4) \(\frac{1}{16}(1 + \sqrt{3} + \sqrt{5})\)

\(\sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) + \sin\left(\frac{3\pi}{5}\right) + \sin\left(\frac{4\pi}{5}\right) =\)

• (1) 1
• (2) \(\sqrt{5}\)
• (3) \(\frac{1}{4}(\sqrt{5} + 1)(\sqrt{10 + 2\sqrt{5}})\)
• (4) \(\frac{1}{4}(\sqrt{5} - 1)(\sqrt{10 - 2\sqrt{5}})\)

The sum of the solutions of \(\cos x \sqrt{16 \sin^2 x} = 1\) in \((0, 2\pi)\) is

• (1) \(2\pi\)
• (2) \(\frac{13\pi}{2}\)
• (3) \(\frac{17\pi}{4}\)
• (4) \(4\pi\)

If \(\cot(\cos^{-1} x) = \sec\left(\tan^{-1}\left(\frac{a}{\sqrt{b^2 - a^2}}\right)\right)\), \(b > a\), then \(x =\)

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

If \(\sinh^{-1}(x) = \log 3\) and \(\cosh^{-1}(y) = \log\left(\frac{3}{2}\right)\), then \(\tanh^{-1}(x - y) =\)

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

In a triangle ABC, if \(a, b, c\) are in arithmetic progression and the angle \(A\) is twice the angle \(C\), then \(\cos A : \cos B : \cos C =\)

• (1) \(2 : 3 : 4\)
• (2) \(3 : 4 : 8\)
• (3) \(2 : 9 : 12\)
• (4) \(1 : 9 : 6\)

In a triangle ABC, if A, B, and C are in arithmetic progression, \(r_3 = r_1 r_2\), and \(c = 10\), then \(a^2 + b^2 + c^2 =\)

• (1) 128
• (2) 288
• (3) 392
• (4) 200

In a \(\triangle ABC\), \(\frac{2(r_1 + r_3)}{a c (1 + \cos B)} =\)

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

In a right-angled triangle, if the position vector of the vertex having the right angle is \(-3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}\) and the position vector of the midpoint of its hypotenuse is \(6\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}\), then the position vector of its centroid is

• (1) \(3\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\)
• (2) \(3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}\)
• (3) \(\frac{3\mathbf{i} + 7\mathbf{j} + 7\mathbf{k}}{2}\)
• (4) \(4\mathbf{j} + 3\mathbf{k}\)

If the position vectors of the vertices A, B, C of a triangle are \(3\mathbf{i} + 4\mathbf{j} - \mathbf{k}\), \(\mathbf{i} + 3\mathbf{j} + \mathbf{k}\), and \(5(\mathbf{i} + \mathbf{j} + \mathbf{k})\) respectively, then the magnitude of the altitude drawn from A onto the side BC is

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

If the vectors \(2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\), \(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\), and \(p\mathbf{i} - 2\mathbf{j} + \mathbf{k}\) are coplanar, then the unit vector in the direction of the vector \(9p\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}\) is

• (1) \(\frac{1}{6}(2\mathbf{i} - 4\mathbf{j} + 4\mathbf{k})\)
• (2) \(\frac{1}{\sqrt{57}}(5\mathbf{i} - 4\mathbf{j} + 4\mathbf{k})\)
• (3) \(\frac{1}{\sqrt{68}}(6\mathbf{i} - 4\mathbf{j} + 4\mathbf{k})\)
• (4) \(\frac{1}{9}(-7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k})\)

Assertion (A): For the lines \(\mathbf{r} = \mathbf{a} + t \mathbf{b}\) and \(\mathbf{r} = \mathbf{p} + s \mathbf{q}\), if \((\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) \neq 0\), then the two lines are coplanar. Reason (R): \(|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})|\) is \(|\mathbf{b} \times \mathbf{q}|\) times the shortest distance between the lines \(\mathbf{r} = \mathbf{a} + t \mathbf{b}\) and \(\mathbf{r} = \mathbf{p} + s \mathbf{q}\).

• (1) (A) is true, (R) is true, and (R) is the correct explanation to (A)
• (2) (A) is true, (R) is true, and (R) is not the correct explanation to (A)
• (3) (A) is true, (R) is false
• (4) (A) is false, (R) is true

Let \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{b}\) be two perpendicular vectors in the XOY-plane. A vector \(\mathbf{c}\) in the same plane and having projections 1 and 2 respectively on \(\mathbf{a}\) and \(\mathbf{b}\) is

• (1) \(\mathbf{i} + 2\mathbf{j}\)
• (2) \(2\mathbf{i} + \mathbf{j}\)
• (3) \(\mathbf{i} - 2\mathbf{j}\)
• (4) \(2\mathbf{i} - \mathbf{j}\)

The mean deviation about the mean for the following data is

• (1) \(\frac{20}{11}\)
• (2) \(\frac{40}{11}\)
• (3) \(\frac{11}{40}\)
• (4) 2

A basket contains 5 apples and 7 oranges, and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is

• (1) \(\frac{1}{6}\)
• (2) \(\frac{7}{18}\)
• (3) \(\frac{17}{36}\)
• (4) \(\frac{19}{36}\)

Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is

• (1) \(\frac{11}{663}\)
• (2) \(\frac{11}{1326}\)
• (3) \(\frac{11}{312}\)
• (4) \(\frac{11}{156}\)

An item is tested on a device for its defectiveness. The probability that such an item is defective is 0.3. The device gives an accurate result in 8 out of 10 such tests. If the device reports that an item tested is not defective, then the probability that it is actually defective is

• (1) \(\frac{2}{15}\)
• (2) \(\frac{3}{29}\)
• (3) \(\frac{3}{31}\)
• (4) \(\frac{4}{51}\)

In a school there are 3 sections A, B, and C. Section A contains 20 girls and 30 boys, section B contains 40 girls and 20 boys, and section C contains 10 girls and 30 boys. The probabilities of selecting section A, B, and C are 0.2, 0.3, and 0.5, respectively. If a student selected at random from the school is a girl, then the probability that she belongs to section A is

• (1) \(\frac{121}{200}\)
• (2) \(\frac{16}{121}\)
• (3) \(\frac{14}{81}\)
• (4) \(\frac{16}{81}\)

If the probability distribution of a random variable \(X\) is as follows, then the mean of \(X\) is

• (1) \(\frac{193}{27}\)
• (2) \(\frac{25}{27}\)
• (3) \(\frac{23}{27}\)
• (4) \(\frac{83}{27}\)

If \(X\) is a binomial variate with mean \(\frac{16}{5}\) and variance \(\frac{48}{25}\), then \(P(X \leq 2) =\)

• (1) \(\frac{3^6 (169)}{5^8}\)
• (2) \(\frac{3^6 (71)}{5^8}\)
• (3) \(\frac{3^8 (43)}{5^8}\)
• (4) \(\frac{3^6 (158)}{5^8}\)

A(\(a\), 0) is a fixed point, and \(\theta\) is a parameter such that \(0 < \theta < 2\pi\). If P(\(a \cos \theta\), \(a \sin \theta\)) is a point on the circle \(x^2 + y^2 = a^2\) and Q(\(b \sin \theta\), \(-b \cos \theta\)) is a point on the circle \(x^2 + y^2 = b^2\), then the locus of the centroid of the triangle APQ is

• (1) a circle with centre at \(\left( \frac{a}{3}, 0 \right)\) and radius \(\frac{\sqrt{a^2 + b^2}}{3}\)
• (2) a circle with centre at \((a, 0)\) and radius \(\frac{\sqrt{a^2 + b^2}}{3}\)
• (3) a parabola with focus at \(\left( \frac{a}{3}, 0 \right)\)
• (4) a parabola with focus at \((a, 0)\)

The point P(4, 1) undergoes the following transformations in succession: (i) origin is shifted to the point (1, 6) by translation of axes, (ii) translation through a distance of 2 units along the positive direction of the x-axis, (iii) rotation of axes through an angle of \(90^\circ\) in the positive direction. Then the coordinates of the point P in its final position are

• (1) (3, 4)
• (2) (4, 3)
• (3) (-5, -5)
• (4) (1, 0)

\(L_1 = ax - 3y + 5 = 0\) and \(L_2 = 4x - 6y + 8 = 0\) are two parallel lines. If \(p, q\) are the intercepts made by \(L_1 = 0\) and \(m, n\) are the intercepts made by \(L_2 = 0\) on the X and Y coordinate axes, respectively, then the equation of the line passing through the points \((p, q)\) and \((m, n)\) is

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

If \((h, k)\) is the image of the point \((2, -3)\) with respect to the line \(5x - 3y = 2\), then \(h + k =\)

• (1) \(-3\)
• (2) \(\frac{3}{34}\)
• (3) \(-\frac{1}{34}\)
• (4) \(5\)

If the pair of lines \(ax^2 - 7xy - 3y^2 = 0\) and \(2x^2 + xy - 6y^2 = 0\) have exactly one line in common and '\(a\)' is an integer, then the equation of the pair of bisectors of the angles between the lines \(ax^2 - 7xy - 3y^2 = 0\) is

• (1) \(7x^2 + 18xy - 7y^2 = 0\)
• (2) \(x^2 - 16xy - y^2 = 0\)
• (3) \(7x^2 - 9xy - 7y^2 = 0\)
• (4) \(x^2 - 8xy - y^2 = 0\)

If the angle between the pair of lines \(2x^2 + 2hxy + 2y^2 - x + y - 1 = 0\) is \(\tan^{-1}\left(\frac{3}{4}\right)\) and \(h\) is a positive rational number, then the point of intersection of these two lines is

• (1) \((1, -1)\)
• (2) \(\left(-\frac{1}{9}, \frac{1}{9}\right)\)
• (3) \((-1, 1)\)
• (4) \((3, 1)\)

If the equation of the circle passing through the point \((8, 8)\) and having the lines \(x + 2y - 2 = 0\) and \(2x + 3y - 1 = 0\) as its diameters is \(x^2 + y^2 + px + qy + r = 0\), then \(p^2 + q^2 + r =\)

• (1) \(244\)
• (2) \(100\)
• (3) \(-44\)
• (4) \(44\)

If \(2x - 3y + 1 = 0\) is the equation of the polar of a point \(P(x_1, y_1)\) with respect to the circle \(x^2 + y^2 - 2x + 4y + 3 = 0\), then \(3x_1 - y_1 =\)

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

If a unit circle \(S = x^2 + y^2 + 2gx + 2fy + c = 0\) touches the circle \(S' = x^2 + y^2 - 6x + 6y + 2 = 0\) externally at the point \((-1, -3)\), then \(g + f + c =\)

• (1) \(0\)
• (2) \(1\)
• (3) \(15\)
• (4) \(25\)

\(3x+4y-43=0\) is a tangent to the circle \(S = x^2+y^2-6x+8y+k=0\) at a point P. If C is the center of the circle and Q is a point which divides CP in the ratio -1:2, then the power of the point Q with respect to the circle S=0 is

• (1) 50
• (2) 21
• (3) 0
• (4) 5

If the radical axis of the circles \(x^2+y^2+2gx+2fy+c=0\) and \(2x^2 + 2y^2 + 3x + 8y + 2c = 0\) touches the circle \(x^2 + y^2 + 2x + 2y + 1 = 0\), then

• (1) either \(g=\frac{3}{2}\) or \(f=2\)
• (2) either \(g=\frac{3}{4}\) or \(f=\frac{1}{2}\)
• (3) either \(g=\frac{3}{4}\) or \(f=2\)
• (4) either \(g=\frac{1}{2}\) or \(f=\frac{3}{4}\)

Tangents are drawn at three points P(\(t_1\)), Q(\(t_2\)), R(\(t_3\)) on the parabola \(y^2 = x\). Let these tangents intersect each other at the points L, M, N. If \(t_1 = 2\), \(t_2 = -4\), \(t_3 = 6\), then the area of the triangle LMN is

• (1) 24
• (2) 18.5
• (3) 7.5
• (4) 12

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse \( 9x^2 + 4y^2 = 72 \) at the point (2, 3) with the X-axis is

• (1) \(\frac{25}{2}\)
• (2) \(\frac{39}{4}\)
• (3) \(\frac{35}{4}\)
• (4) \(\frac{45}{4}\)

If \( 3\sqrt{2}x - 4y = 12 \) is a tangent to the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and \(\frac{5}{4}\) is its eccentricity, then \( a^2 - b^2 = \)

• (1)
• (2) 7
• (3) 9
• (4) 11

If the normal drawn to the hyperbola \( xy = 16 \) at (8, 2) meets the hyperbola again at a point \((\alpha, \beta)\), then \( |\beta| + \frac{1}{|\alpha|} = \)

• (1) 40 
• (2) 34
• (3) 28
• (4) 54

The locus of a point at which the line joining the points (-3, 1, 2), (1, -2, 4) subtends a right angle, is

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

If A(1, 2, 3), B(2, 3, -1), C(3, -1, -2) are the vertices of a triangle ABC, then the direction ratios of the bisector of \(\angle\)ABC are

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

Let A = (2, 0, -1), B = (1, -2, 0), C = (1, 2, -1), and D = (0, -1, -2) be four points. If \(\theta\) is the acute angle between the plane determined by A, B, C and the plane determined by A, C, D, then \(\tan\theta =\)

• (1) \(\frac{\sqrt{14}}{3}\)
• (2) \(\sqrt{14}\)
• (3) \(\frac{3}{\sqrt{5}}\)
• (4) \(\frac{\sqrt{5}}{3}\)

Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)

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

\(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)

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

If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & if x > 0
1 & if x = 0
\frac{\cos 4x - \cos bx}{\tan^2 x} & if x < 0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)

• (1) 4
• (2) 5
• (3) 3
• (4) 7

If \(y = \tan^{-1}(\frac{3x - x^3}{1-3x^2}) + \tan^{-1}(\frac{7x}{1-12x^2})\), then at \(x=0\), \(\frac{dy}{dx} =\)

• (1) 6
• (2) 7
• (3) 9
• (4) 10

If \(y = \frac{x^4\sqrt{3x-5}}{\sqrt{(x^2-3)(2x-3)}}\), then \(\frac{dy}{dx}|_{x=2} =\)

• (1) 5
• (2) 0
• (3) 1
• (4) -5

If \(x^2 + y^2 + \sin y = 4\), then the value of \(\frac{d^2y}{dx^2}\) at \(x=-2\) is

• (1) -30
• (2) -34
• (3) -32
• (4) -18

If the surface area of a spherical bubble is increasing at the rate of 4 sq.cm/sec, then the rate of change in its volume (in cubic cm/sec) when its radius is 8 cms is

• (1) 8
• (2) 12
• (3) 15
• (4) 16

The number of turning points of the curve \(f(x) = 2\cos x - \sin 2x\) in the interval \([-\pi, \pi]\) is

• (1) 4
• (2) 3
• (3) 1
• (4) 2

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft) is

• (1) \((0.088)(\sqrt{2}+1)\)
• (2) \((0.616)(\sqrt{2}+1)\)
• (3) \((0.616)(\sqrt{2})\)
• (4) \((0.088)(\sqrt{2})\)

If the slope of the tangent drawn at any point \((x, y)\) on a curve is \(x + y\), then the equation of that curve is

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

\(\int (\sqrt{\tan x} + \sqrt{\cot x}) \, dx =\)

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

\(\int \frac{\sqrt{x - 2}}{2x + 4} \, dx =\)

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

If \(\int \left( \frac{x^{49} \tan^{-1}(x^{50})}{1 + x^{100}} + \frac{x^{50}}{1 + x^{100}} \right) dx = k f(x) + c\) where \(k\) is a constant, then \(f(x) - f\left( \frac{1}{x^{49}} \right) =\)

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

\(\int \frac{x}{\sqrt{x^2 - 2x + 5}} \, dx =\)

• (1) \(\sqrt{x^2 - 2x + 5} + \sinh^{-1}\left( \frac{x - 1}{2} \right) + c\)
• (2) \(\frac{1}{2} \sqrt{x^2 - 2x + 5} + \sin^{-1}\left( \frac{x - 1}{2} \right) + c\)
• (3) \(2 \sqrt{x^2 - 2x + 5} + \cosh^{-1}\left( \frac{x - 1}{2} \right) + c\)
• (4) \(\sqrt{x^2 - 2x + 5} + \cos^{-1}\left( \frac{x - 1}{2} \right) + c\)

For \( 0 < x < 1 \), \(\int_0^1 \left( \tan^{-1}\left( \frac{1 + x^2 - x}{x} \right) + \tan^{-1}(1 - x + x^2) \right) dx =\)

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

\(\int_{-2\pi}^{2\pi} \sin^2(2x) \cos^4(2x) \, dx =\)

• (1) \(\frac{3\pi}{64}\) 
• (2) \(\frac{9\pi}{64}\)
• (3) \(\frac{9\pi}{35}\)
• (4) \(\frac{9\pi}{280}\)

If \( f(t) = \int_0^t \tan^{2n-1}(x) \, dx \), \( n \in \mathbb{N} \), then \( f(t + \pi) =\)

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

\(\int_0^2 \frac{x^{\frac{8}{3}}}{|x - 1|^{\frac{5}{2}}} \, dx =\)

• (1) \(\frac{215}{63}\)
• (2) \(\frac{216}{315}\)
• (3) \(\frac{216}{189}\)
• (4) \(\frac{210}{63}\)

The area (in sq. units) of the region bounded by the curves \( y = x^2 \) and \( y = 8 - x^2 \) is

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

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

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

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

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

If \( x \log_{10} x \frac{dy}{dx} + y = 2 \log_{10} x \) and \( y(e) = 0 \), then \( y(e^2) = \)

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

If the error in the measurement of the surface area of a sphere is 1.2%, then the error in the determination of the volume of the sphere is

• (1) 2.4%
• (2) 1.8%
• (3) 1.2%
• (4) 0.6%

A body starts from rest with uniform acceleration and its velocity at a time of \( n \) seconds is \( v \). The total displacement of the body in the \( n \)-th and \( (n - 1) \)-th seconds of its motion is

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

If the range of a body projected with a velocity of 60 m/s is \( 180\sqrt{3} \) m, then the angle of projection of the body is (Acceleration due to gravity = 10 m/s\(^2\))

• (1) 30° or 60°
• (2) 37° or 53°
• (3) 20° or 70°
• (4) 15° or 75°

If the height of a projectile at a time of 2 s from the beginning of motion is 60 m, then the time of flight of the projectile is (Acceleration due to gravity = 10 m/s\(^2\))

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

A disc of mass 0.2 kg is kept floating in air without falling by vertically firing bullets each of mass 0.05 kg on the disc at the rate of 10 bullets per every second. If the bullets rebound with the same speed, then the speed of each bullet is (Acceleration due to gravity = 10 m/s\(^2\))

• (1) 2 m/s
• (2) 10 m/s
• (3) 20 m/s
• (4) 1 m/s

Two bodies A and B of masses 1.5 kg and 3 kg are moving with velocities 20 m/s and 15 m/s respectively. If the same retarding force is applied on the two bodies, then the ratio of the distances travelled by the bodies A and B before they come to rest is

• (1) 1:1
• (2) 8:9
• (3) 2:3
• (4) 3:8

If a force F = (3i - 2j) N acting on a body displaces it from point (1 m, 2 m) to point (2 m, 0 m), then work done by the force is

• (1) 5 J
• (2) 6 J
• (3) 4 J
• (4) 7 J

A body moving along a straight line collides another body of same mass moving in the same direction with half of the velocity of the first body. If the coefficient of restitution between the two bodies is 0.5, then the ratio of the velocities of the two bodies after collision is (treat the collision as one dimensional)

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

If a solid sphere is rolling without slipping on a horizontal plane, then the ratio of its rotational and total kinetic energies is

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

As shown in the figure, two thin coplanar circular discs A and B each of mass 'M' and radius 'r' are attached to form a rigid body. The moment of inertia of this system about an axis perpendicular to the plane of disc B and passing through its centre is

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

The time period of a simple pendulum on the surface of the earth is T. If the pendulum is taken to a height equal to half of the radius of the earth, then its time period is

• (1) T/2
• (2) 3T/2
• (3) 2T
• (4) 3T

A particle is executing simple harmonic motion starting from its mean position. If the time period of the particle is 1.5 s, then the minimum time at which the ratio of the kinetic and total energies of the particle becomes 3:4 is

• (1) 1/4 s
• (2) 1/12 s
• (3) 1/8 s
• (4) 1/6 s

If the escape velocity of a body from the surface of the earth is 11.2 km/s, then the orbital velocity of a satellite in an orbit which is at a height equal to the radius of the earth is

• (1) 11.2 km/s
• (2) 2.8 km/s
• (3) 22.4 km/s
• (4) 5.6 km/s

A wire is stretched 1 mm by a force F. If a second wire of same material, same length and 4 times the diameter of the first wire is stretched by the same force F, then the elongation of the second wire is

• (1) 1/8 mm
• (2) 8 mm
• (3) 16 mm
• (4) 1/16 mm

In a water tank, an air bubble rises from the bottom to the top surface of the water. If the depth of the water in the tank is 7.28 m and atmospheric pressure is 10 m of water, then the ratio of the radii of the bubble at the bottom of the tank and at the top surface of the water is (Temperature of the water in the tank is constant)

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

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6}\) m\(^2\) at a temperature of 100°C is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to 0°C, but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is [Young's modulus of material of the wire = \(10^{11}\) N/m\(^2\); coefficient of linear expansion of the material of the wire = \(10^{-5}\) K\(^{-1}\) and acceleration due to gravity = 10 m/s\(^2\)]

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

The temperature of water of mass 100 g is raised from 24°C to 90°C by adding steam to it. The mass of the steam added is (Latent heat of steam = 540 cal/g and specific heat capacity of water = 1 cal/g°C)

• (1) 10 g
• (2) 12 g
• (3) 8 g
• (4) 16 g

When 80 J of heat is supplied to a gas at constant pressure, if the work done by the gas is 20 J, then the ratio of the specific heat capacities of the gas is

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

A refrigerator of coefficient of performance 5 that extracts heat from the cooling compartment at the rate of 250 J per cycle is placed in a room. The heat released per cycle to the room by the refrigerator is

• (1) 250 J
• (2) 50 J
• (3) 200 J
• (4) 300 J

In a container of volume 16.62 m\(^3\) at 0°C temperature, 2 moles of oxygen, 5 moles of nitrogen and 3 moles of hydrogen are present, then the pressure in the container is (Universal gas constant = 8.31 J/mol K)

• (1) 1570 Pa
• (2) 1270 Pa
• (3) 1365 Pa
• (4) 2270 Pa

If a travelling wave is given by \( y(x, t) = 0.5 \sin(70.1x - 10\pi t) \), where \( x \) and \( y \) are in metres, the time \( t \) is in seconds, then the frequency of the wave is

• (1) 6 Hz
• (2) 7 Hz
• (3) 4 Hz
• (4) 5 Hz

The ratio of the focal lengths of a convex lens when kept in air and when it is immersed in a liquid is 1:2. If the refractive index of the material of the lens is 1.5, then the refractive index of the liquid is

• (1) 1.20
• (2) 1.30
• (3) 1.25
• (4) 1.35

The path difference between two waves given by the equations \( y_1 = a_1 \sin\left(\omega t - \frac{2\pi x}{\lambda}\right) \) and \( y_2 = a_2 \sin\left(\omega t - \frac{2\pi x}{\lambda} + \phi\right) \) is

• (1) \( \frac{\lambda}{2\pi} \left| \phi \right| \)
• (2) \( \frac{\lambda}{2\pi} \left( \frac{\pi}{2} - \phi \right) \)
• (3) \( \frac{\lambda}{2\pi} \phi \)
• (4) \( \frac{\lambda}{2\pi} \left( \frac{\pi}{2} - \frac{\phi}{2} \right) \)

The sum of two point positive charges separated by a distance of 1.5 m in air is \( 25 \, \mu C \). If the electrostatic force between the two charges is 0.6 N, then the difference between the two charges is

• (1) \( 5 \, \mu C \)
• (2) \( 8 \, \mu C \)
• (3) \( 3 \, \mu C \)
• (4) \( 6 \, \mu C \)

The energy stored in a capacitor of capacitance \( 10 \, \mu F \) when charged to a potential of 6 kV is

• (1) 100 J
• (2) 200 J
• (3) 180 J
• (4) 160 J

A parallel plate capacitor has plates of area 0.4 m\(^2\) and spacing of 0.5 mm. If a slab of thickness 0.5 mm and dielectric constant 4.5 is introduced between the plates of the capacitor, then the capacitance of the capacitor is

• (1) 100 nF
• (2) 60 pF
• (3) 100 pF
• (4) 60 nF

In the given circuit, the potential difference across the plates of the capacitor \( C \) in steady state is


 

• (1) 6.5 V
• (2) 6 V
• (3) 9 V
• (4) 7.5 V

The potential difference across a conducting wire of length 20 cm is 30 V. If the electron mobility is \(2 \times 10^{-6}\) m\(^2\)V\(^{-1}\)s\(^{-1}\), then the drift velocity of the electrons is

• (1) \(3 \times 10^{-3}\) m/s
• (2) \(1.5 \times 10^{-3}\) m/s
• (3) \(1.5 \times 10^{-4}\) m/s
• (4) \(3 \times 10^{-4}\) m/s

A maximum current of 0.5 mA can pass through a galvanometer of resistance 15 \(\Omega\). The resistance to be connected in series to the galvanometer to convert it into a voltmeter of range 0–10 V is

• (1) 9985 \(\Omega\)
• (2) 20015 \(\Omega\)
• (3) 20000 \(\Omega\)
• (4) 19985 \(\Omega\)

Two charged particles of specific charges in the ratio 2:1 and masses in the ratio 1:4 moving with same kinetic energy enter a uniform magnetic field at right angles to the direction of the field. The ratio of the radii of the circular paths in which the particles move under the influence of the magnetic field is

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

A sample of paramagnetic salt contains \(2 \times 10^{24}\) atomic dipoles each of dipole moment \(1.5 \times 10^{-23}\) JT\(^{-1}\). The sample is placed under homogeneous magnetic field of 0.6 T and cooled to a temperature 4.2 K. The degree of magnetic saturation achieved is 20%. Then total dipole moment of the sample for a magnetic field of 0.9 T and a temperature of 2.8 K is

• (1) 4.5 JT\(^{-1}\)
• (2) 13.5 JT\(^{-1}\)
• (3) 0.64 JT\(^{-1}\)
• (4) 7 JT\(^{-1}\)

A sample of paramagnetic salt contains \(2 \times 10^{24}\) atomic dipoles each of dipole moment \(1.5 \times 10^{-23} \, JT^{-1}\). The sample is placed under homogeneous magnetic field of 0.6 T and cooled to a temperature 4.2 K. The degree of magnetic saturation achieved is 20%. Then total dipole moment of the sample for a magnetic field of 0.9 T and a temperature of 2.8 K is

• (1) 4.5 JT\(^{-1}\)
• (2) 13.5 JT\(^{-1}\)
• (3) 0.64 JT\(^{-1}\)
• (4) 7 JT\(^{-1}\)

A coil of resistance 200 \(\Omega\) is placed in a magnetic field. If the magnetic flux \(\phi\) (in weber) linked with the coil varies with time 't' (in second) as per the equation \(\phi = 50t^2 + 4\), then the current induced in the coil at a time t = 2 s is

• (1) 2 A
• (2) 1 A
• (3) 0.5 A
• (4) 0.1 A

The oscillating electric and magnetic field vectors of an electromagnetic wave are along

• (1) the same direction and in same phase.
• (2) the same direction but have a phase difference of 90\(^{\circ}\).
• (3) mutually perpendicular directions and are in same phase.
• (4) mutually perpendicular directions but have a phase difference of 90\(^{\circ}\).

A laser produces a beam of light of frequency \(5 \times 10^{14}\) Hz with an output power of 33 mW. The average number of photons emitted by the laser per second is (Planck's constant = \(6.6 \times 10^{-34}\) Js)

• (1) \(40 \times 10^{16}\)
• (2) \(10 \times 10^{16}\)
• (3) \(30 \times 10^{16}\)
• (4) \(20 \times 10^{16}\)

The ratio of energies of photons produced due to transition of an electron in hydrogen atom from second energy level to first energy level and fifth energy level to second energy level is

• (1) 2:1
• (2) 1:4
• (3) 3:2
• (4) 25:7

The half life of a radioactive substance is 10 minutes. If \(n_1\) and \(n_2\) are the number of atoms decayed in 20 and 30 minutes respectively, then \(n_1 : n_2 = \)

• (1) 7:8
• (2) 1:2
• (3) 6:7
• (4) 3:4

If X, Y and Z are the sizes of the emitter, base and collector of a transistor respectively, then

• (1) X \(>\) Z \(>\) Y
• (2) X \(>\) Y \(>\) Z
• (3) Z \(>\) X \(>\) Y
• (4) Z \(>\) Y \(>\) X

The logic gate equivalent to the circuit given in the figure is


 

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

If the ratio of the maximum and minimum amplitudes of an amplitude modulated wave is 7:3, then the modulation index is

• (1) 0.6
• (2) 0.7
• (3) 0.4
• (4) 0.3

Which of the following represents the wavelength of spectral line of Balmer series of He\(^+\) ion? (R = Rydberg constant, n \(>\) 2)

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

The work functions (in eV) of Mg, Cu, Ag, Na respectively are 3.7, 4.8, 4.3, 2.3. From how many metals, the electrons will be ejected if their surfaces are irradiated with an electromagnetic radiation of wavelength 300 nm? (h = \(6.6 \times 10^{-34}\) Js, 1 eV = \(1.6 \times 10^{-19}\) J)

• (1) 1
• (2) 2
• (3) 3
• (4) 4

The order of negative electron gain enthalpy of Li, Na, S, Cl is

• (1) Na \(>\) S \(>\) Cl \(>\) Li
• (2) Cl \(>\) S \(>\) Li \(>\) Na
• (3) Cl \(>\) Li \(>\) S \(>\) Na
• (4) Li \(>\) Na \(>\) S \(>\) Cl

The number of molecules having lone pair of electrons on central atom in the following is: BF\(_3\), SF\(_4\), SiCl\(_4\), XeF\(_4\), NCl\(_3\), XeF\(_6\), PCl\(_5\), HgCl\(_2\), SnCl\(_2\)

• (1) 6
• (2) 3
• (3) 4
• (4) 5

Observe the following substances. Ethanol, acetic acid, ethylamine, trimethylamine, salicylic acid, ethanal. In the above list, the number of substances with H-bonding is

• (1) 4
• (2) 3
• (3) 5
• (4) 2

Consider the following:

Statement-I: If thermal energy is stronger than intermolecular forces, the substance prefers to be in gaseous state.

Statement-II: At constant temperature, the density of an ideal gas is proportional to its pressure.

The correct answer is

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

At 27\(^{\circ}\)C, 1 L of H\(_2\) with a pressure of 1 bar is mixed with 2 L of O\(_2\) with a pressure of 2 bar in a 10 L flask. What is the pressure exerted by gaseous mixture in bar? (Assume H\(_2\) and O\(_2\) as ideal gases)

• (1) 4
• (2) 0.05
• (3) 1
• (4) 0.5

Two acids A and B are titrated separately. 25 mL of 0.5 M Na\(_2\)CO\(_3\) solution requires 10 mL of A and 40 mL of B for complete neutralisation. The volume (in L) of A and B required to produce 1 L of 1 N acid solution respectively are

• (1) 0.2, 0.8
• (2) 0.8, 0.2
• (3) 0.3, 0.7
• (4) 0.7, 0.3

If \(\Delta_r H^{\ominus}\) and \(\Delta_r S^{\ominus}\) are standard enthalpy change and standard entropy change respectively for a reaction, the incorrect option is

• (1) \(\Delta_r H^{\ominus}\) = negative; \(\Delta_r S^{\ominus}\) = positive; spontaneous at all temperatures
• (2) \(\Delta_r H^{\ominus}\) = negative; \(\Delta_r S^{\ominus}\) = negative; non-spontaneous at low temperatures
• (3) \(\Delta_r H^{\ominus}\) = positive; \(\Delta_r S^{\ominus}\) = positive; non-spontaneous at low temperatures
• (4) \(\Delta_r H^{\ominus}\) = negative; \(\Delta_r S^{\ominus}\) = negative; spontaneous at low temperatures

The C\(_p\) of H\(_2\)O(l) is 75.3 J mol\(^{-1}\) K\(^{-1}\). What is the energy (in J) required to raise 180 g of liquid water from 10\(^{\circ}\)C to 15\(^{\circ}\)C? (H\(_2\)O = 18 u)

• (1) 3.765
• (2) 3765
• (3) 753
• (4) 376.5

At T(K), consider the following gaseous reaction, which is in equilibrium: N\(_2\)O\(_5 \rightleftharpoons 2\)NO\(_2 + \frac{1}{2}\)O\(_2\). What is the fraction of N\(_2\)O\(_5\) decomposed at constant volume and temperature, if the initial pressure is 300 mm Hg and pressure at equilibrium is 480 mm Hg? (Assume all gases as ideal)

• (1) 0.2
• (2) 0.6
• (3) 0.4
• (4) 0.8

Observe the following molecules/ions. NH\(_4^+\), NH\(_3\), BF\(_3\), OH\(^-\), CH\(_3^-\), H\(^+\), CO, C\(_2\)H\(_4\). The number of Lewis bases in the above list is

• (1) 2
• (2) 3
• (3) 4
• (4) 5

Observe the following reactions (g = gas):

I. N\(_2\)(g) + 3H\(_2\)(g) \(\xrightarrow[200 atm]{773K, X}\) 2NH\(_3\)(g)

II. CO(g) + H\(_2\)O(g) \(\xrightarrow[673 K]{Y}\) CO\(_2\)(g) + H\(_2\)(g)

III. CH\(_4\)(g) + H\(_2\)O(g) \(\xrightarrow[1270 K]{Z}\) CO(g) + 3H\(_2\)(g)

Catalysts X, Y, Z respectively are

• (1) Iron, sodium arsenite, cobalt
• (2) Iron, zinc, cobalt
• (3) Cobalt, zinc, nickel
• (4) Iron, iron chromate, nickel

Consider the following:

Statement-I: Both BeSO\(_4\) and MgSO\(_4\) are readily soluble in water.

Statement-II: Among the nitrates of alkaline earth metals, only Be(NO\(_3\))\(_2\) on strong heating gives its oxide, NO\(_2\), and O\(_2\).

The correct answer is

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

Which of the following is not associated with water molecules?

• (1) cryolite
• (2) bauxite
• (3) kernite
• (4) borax

Identify the incorrect statement about silica.

• (1) It is acidic in nature.
• (2) It has no reaction with most of acids except HF.
• (3) With NaOH it forms sodium silicate.
• (4) Like graphite, it has two dimensional structure.

Which one of the following statements related to photochemical smog is not correct?

• (1) It is controlled by the use of catalytic converters in automobiles.
• (2) It causes corrosion of metals.
• (3) It is a mixture of SO\(_2\), smoke, and fog.
• (4) It causes extensive damage to plant life.

In compound (X), hyperconjugation is present and in (Y), resonance effect is present. What are X and Y, respectively?

• (1) Toluene, prop-2-en-1-ol
• (2) Aniline, 2-propenal
• (3) Toluene, nitrobenzene
• (4) 1-Bromopropane, phenol

An alcohol X(C\(_4\)H\(_{10}\)O) on dehydration gave alkene (C\(_4\)H\(_8\)) as major product, which on bromination followed by treatment with Y gave alkyne C\(_4\)H\(_6\). Alkyne C\(_4\)H\(_6\) does not react with sodium metal. What are X and Y?

An element occurs in the body-centred cubic structure with an edge length of 288 pm. The density of the element is 7.2 g cm\(^{-3}\). The number of atoms present in 208 g of the element is nearly

• (1) \(24.2 \times 10^{23}\)
• (2) \(12.1 \times 10^{23}\)
• (3) \(24.2 \times 10^{24}\)
• (4) \(36.3 \times 10^{23}\)

An aqueous solution containing 0.2 g of a non-volatile solute 'A' in 21.5 g of water freezes at 272.814 K. If the freezing point of water is 273.16 K, the molar mass (in g mol\(^{-1}\)) of solute A is [K\(_f\)(H\(_2\)O) = 1.86 K kg mol\(^{-1}\)]

• (1) 80
• (2) 75
• (3) 100
• (4) 50

At T(K), the vapor pressure of x molal aqueous solution containing a non-volatile solute is 12.078 kPa. The vapor pressure of pure water at T(K) is 12.3 kPa. What is the value of x?

• (1) 10
• (2) 1.018
• (3) 0.1018
• (4) 0.018

Consider the following cell reaction: 2Fe\(^{3+}\)(aq) + 2I\(^-\) (aq) \(\rightarrow\) 2Fe\(^{2+}\)(aq) + I\(_2\)(s). At 298 K, the cell emf is 0.237 V. The equilibrium constant for the reaction is 10\(^x\). The value of x is (F = 96500 C mol\(^{-1}\); R = 8.3 J K\(^{-1}\) mol\(^{-1}\))

• (1) 8
• (2) 7
• (3) 6
• (4) 9

For a first-order reaction, the ratio between the time taken to complete \(\frac{3}{4}\)th of the reaction and time taken to complete half of the reaction is

• (1) 2
• (2) 3
• (3) 1.5
• (4) 2.5

What is the indicator used in Argentometric titrations?

• (1) Starch solution
• (2) Eosin dye
• (3) KMnO\(_4\) solution
• (4) Phenolphthalein

In a Freundlich adsorption isotherm, if the slope is unity and k is 0.1, the extent of adsorption at 2 atm is (log 2 = 0.30)

• (1) 0.6
• (2) 0.4
• (3) 0.2
• (4) 0.8

Match the following:


List-I (Process)

A. Hall-Heroult process

B. Mond process

C. Van-Arkel process

D. Zone refining process



List-II (Metal)

I. Ti

II. In

III. Al

IV. Ni

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

The number of P=O and P-O-P bonds present in the oxoacid of phosphorus, prepared by treating red P\(_4\) with alkali, are respectively

• (2) 1, 1
• (3) 1, 2
• (4) 2, 2

Which one of the following statements is not correct?

• (1) CrO is basic but Cr\(_2\)O\(_3\) is amphoteric.
• (2) Nitrite is oxidized to nitrate in acidic medium by KMnO\(_4\).
• (3) PdCl\(_2\) is the catalyst in Wacker process.
• (4) The reactivity of the earlier members of lanthanide series is similar to that of aluminum.

The co-ordination number of chromium in K[Cr(H\(_2\)O)\(_2\)(C\(_2\)O\(_4\))\(_2\)] is

• (1) 5
• (2) 4
• (3) 6
• (4) 3

Consider the following:

Statement-I: Nylon 6 is a condensation copolymer.

Statement-II: Nylon 6,6 is a condensation polymer of adipic acid and tetramethylene diamine.

The correct answer is

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

Match the following:

List-I (Glycosidic linkage)
A. \(\alpha\)-1,4
B. \(\beta\)-1,4
C. \(\alpha\)-1,4, \(\alpha\)-1,6

List-II (Polysaccharide)
I. Amylose
II. Amylopectin
III. Cellulose

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

The list given below contains essential amino acids that are basic (X) and also non-essential amino acids that are neutral (Y). X and Y, respectively are

a) Lysine

b) Alanine

c) Serine

d) Arginine

e) Tyrosine

• (1) X = b, c, e; Y = a, d
• (2) X = a, d; Y = b, c, e
• (3) X = a, c; Y = b, d, e
• (4) X = a, b, c; Y = d, e

The artificial sweetener X contains glycosidic linkage and Y contains amide and ester linkages. X and Y respectively are

• (1) Sucralose, Alitame
• (2) Sucralose, Aspartame
• (3) Saccharin, Alitame
• (4) Saccharin, Aspartame

Which one of the following halogen compounds is least reactive towards hydrolysis by the S\(_N\)1 mechanism?

• (1) Tertiary butyl chloride
• (2) Isopropyl chloride
• (3) Allyl chloride
• (4) Ethyl chloride

Which one of the following halogen compounds is least reactive towards hydrolysis by S\(_N\)1 mechanism?

• (2) Isopropyl chloride
• (3) Allyl chloride
• (4) Ethyl chloride

p-Chlorotoluene is the major product in which of the following reactions?

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

Arrange the following in decreasing order of electrophilicity of carbonyl carbon.

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

What is the ratio of sp\(^3\) carbons to sp\(^2\) carbons in the product 'P' of the given sequence of reactions?

• (2) 2 : 1
• (3) 1 : 2
• (4) 1 : 3

The final product (C) in the given reaction sequence is

• (2) Diphenyl methane
• (3) Diphenylmethanol
• (4) Benzoic acid

What are X and Y in the following reaction sequence?

• (2) KCN; C\(_6\)H\(_5\)C(OH)(CH\(_3\))\(_2\)
• (3) CuCN | KCN; C\(_6\)H\(_5\)CH(OH)CH\(_3\)
• (4) CuCN | KCN; C\(_6\)H\(_5\)COCH\(_3\)