AP EAPCET 2026 May 13 Shift 1 Engineering Question Paper Available- Download Solution PDF

A \(\subseteq \mathbb{R}\) and \(f:A \to \mathbb{R}\) is a function defined by \(f(x)=\dfrac{x-1}{x+1}\). If the domain of the function \(f(2x)\) is \(B\), then \(A \cap B =\)

• (A) \(\mathbb{R} – \{-1,-2\}\)
• (B) \(\mathbb{R} – \left\{-1,-\dfrac12\right\}\)
• (C) \(\mathbb{R} – \left\{-1,\dfrac13\right\}\)
• (D) \(\mathbb{R} – \{-1,1\}\)

Let \(f\) be a function with domain \([0,7]\) and \(g\) be a function defined by \(g(x)=|2x+1|\). Then the domain of \((f\circ g)(x)\) is

• (A) \([0,7]\)
• (B) \([-7,0]\)
• (C) \([-4,3]\)
• (D) \([-3,4]\)

If \[ \frac34+\frac{15}{16}+\frac{63}{64}+\cdots+n terms =\frac{939}{256}, \]
then \(5n=\)

• (A) \(55\)
• (B) \(20\)
• (C) \(15\)
• (D) \(35\)

If \(\alpha,\beta,\gamma\) are the roots of the equation \(x^3+bx+c=0\), then \[ \begin{vmatrix} \alpha & \beta & \gamma \\
\beta & \gamma & \alpha \\
\gamma & \alpha & \beta \end{vmatrix} = \]

• (A) \(-b^3\)
• (B) \(b^3-3c\)
• (C) \(b^2-3c\)
• (D) \(0\)

For a fixed positive integer \(n\), if \[ D= \begin{vmatrix} n! & (n+1)! & (n+2)! \\
(n+1)! & (n+2)! & (n+3)! \\
(n+2)! & (n+3)! & (n+4)! \end{vmatrix}, \]
then \[ \frac{D}{n!(n+1)!(n+2)!} = \]

• (A) \(-4\)
• (B) \(-2\)
• (C) \(2\)
• (D) \(4\)

If \[ y= \begin{vmatrix} \sin x & \cos x & \sin x \\
\cos x & -\sin x & \cos x \\
x & 1 & 1 \end{vmatrix}, \]
then \[ \frac{dy}{dx}= \]

• (A) \(x\cos x\)
• (B) \(x\sin x\)
• (C) \(\sin x+\cos x\)
• (D) \(1\)

If \[ \frac{3}{2+\cos\theta+i\sin\theta}=x+iy, \]
then \[ (x-1)(x-3)= \]

• (A) \(y^2\)
• (B) \(-y^2\)
• (C) \(0\)
• (D) \(1\)

If \(z_1,z_2\) are two roots of the equation \[ z^2+az+b=0 \]
and on the Argand plane the points represented by \(z_1,z_2\) and the origin form an equilateral triangle, then \(a^2=\)

• (A) \(b\)
• (B) \(2b\)
• (C) \(3b\)
• (D) \(4b\)

If \[ (1+\sin\alpha+i\cos\alpha)^8 = A(\cos\theta+i\sin\theta), \]
then \(A\) and \(\theta\) are respectively

• (A) \[ 2^8\cos^8\left(\frac{\pi}{4}-\frac{\alpha}{2}\right),\;2\alpha \]
• (B) \[ 2^8\cos^8\left(\frac{\pi}{4}-\frac{\alpha}{2}\right),\;-4\alpha \]
• (C) \[ 2^8\cos^8\left(\frac{\pi}{4}+\frac{\alpha}{2}\right),\;2\alpha \]
• (D) \[ 2^8\cos^8\left(\frac{\pi}{4}+\frac{\alpha}{2}\right),\;-4\alpha \]

If \(\alpha,\beta\) are the roots of the equation \[ x^2-p(x+1)-c=0, \]
then \[ \frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+c} + \frac{\beta^2+2\beta+1}{\beta^2+2\beta+c} = \]

• (A) \(3\)
• (B) \(2\)
• (C) \(1\)
• (D) \(0\)

If \(\alpha \neq \beta\) and \(\alpha^2=5\alpha-3,\ \beta^2=5\beta-3\), then the equation whose roots are \(\dfrac{\alpha}{\beta}\) and \(\dfrac{\beta}{\alpha}\) is

• (A) \(3x^2+19x+3=0\)
• (B) \(3x^2+19x-3=0\)
• (C) \(3x^2-19x+3=0\)
• (D) \(3x^2-19x-3=0\)

If the roots of the equation \[ x^3-px^2+qx-s=0 \]
are in geometric progression, then

• (A) \(q^2=ps^2\)
• (B) \(q^2=p^2s\)
• (C) \(q^3=ps^3\)
• (D) \(q^3=p^3s\)

If \(\alpha,\beta,\gamma\) are the roots of \[ x^3-10x^2+7x+8=0, \]
match List-I with List-II and choose the correct option.

• (A) A-V, B-III, C-I, D-II
• (B) A-V, B-III, C-II, D-IV
• (C) A-V, B-III, C-II, D-I
• (D) A-V, B-II, C-III, D-I

A student is allowed to select at most \(n\) books from a collection of \(2n+1\) books. If the total number of ways in which he can select at least one book is \(255\), then the value of \(n\) is

• (A) \(6\)
• (B) \(5\)
• (C) \(4\)
• (D) \(3\)

The number of different nine-digit numbers that can be formed by rearranging all the digits of the number \(223355888\) so that odd digits always occupy even positions is

• (A) \(180\)
• (B) \(120\)
• (C) \(60\)
• (D) \(36\)

If all the letters of the word RANKS are permutated in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word RANKS is

• (A) \(74\)
• (B) \(76\)
• (C) \(75\)
• (D) \(77\)

The value of \(n\) for which the sum \[ \frac{{}^nC_0}{2^n} + 2\frac{{}^nC_1}{2^n} + 3\frac{{}^nC_2}{2^n} +\cdots+ (n+1)\frac{{}^nC_n}{2^n} =16 \]
is

• (A) \(10\)
• (B) \(20\)
• (C) \(25\)
• (D) \(30\)

In the binomial expansion of \((x+a)^{15}\), if the eleventh term is the geometric mean of the eighth and twelfth terms, then the numerically greatest term in its expansion is

• (A) \(8\)
• (B) \(9\)
• (C) \(10\)
• (D) \(11\)

If \[ \frac{x}{(x^2+1)^2(x-1)} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} + \frac{E}{x-1}, \]
then \[ A+B-C+2D= \]

• (A) \(\dfrac{1}{2}\)
• (B) \(1\)
• (C) \(\dfrac{3}{2}\)
• (D) \(2\)

Evaluate \[ \cos\frac{6\pi}{17}\cos\frac{10\pi}{17}\cos\frac{12\pi}{17}\cos\frac{14\pi}{17}. \]

• (A) \(-\dfrac{1}{16}\)
• (B) \(\dfrac{1}{16}\)
• (C) \(-16\)
• (D) \(\dfrac{1}{4}\)

The number of distinct real roots of \[ \begin{vmatrix} \sin x & \cos x & \cos x \\
\cos x & \sin x & \cos x \\
\cos x & \cos x & \sin x \end{vmatrix}=0 \]
in the interval \[ \left(-\frac{\pi}{4},\frac{\pi}{4}\right) \]
is

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(3\)

The value of \[ \tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ \]
is

• (A) \(2\)
• (B) \(3\)
• (C) \(4\)
• (D) \(5\)

For \(n\in\mathbb Z\), a set of values of \(\theta\) satisfying \[ \sec\theta-1=(\sqrt2-1)\tan\theta \]
is

• (A) \[ 2n\pi-\frac{\pi}{4} \]
• (B) \[ 2n\pi+\frac{\pi}{2} \]
• (C) \[ (2n+1)\pi+\frac{\pi}{4} \]
• (D) \[ 2n\pi+\frac{\pi}{4} \]

The number of real solutions of the equation \[ \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \frac{\pi}{2} \]
is

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) Infinitely many

If \[ u=\log\tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right), \]
then \[ \tanh\frac{u}{2} = \]

• (A) \[ \tan\frac{\theta}{2} \]
• (B) \[ \cot\frac{\theta}{2} \]
• (C) \[ \sec\frac{\theta}{2} \]
• (D) \[ \sin\frac{\theta}{2} \]

If \(\Delta\) denotes the area of triangle \(ABC\) and \(s\) is its semi-perimeter, then

• (A) \[ \Delta=\frac{s^2}{2} \]
• (B) \[ \Delta\le\frac{s^2}{3\sqrt3} \]
• (C) \[ \Delta>\frac{s^2}{\sqrt3} \]
• (D) \[ \Delta=\frac{s^2}{2\sqrt3} \]

In a triangle \(ABC\), \(C=90^\circ\). If \(r\) is the inradius and \(R\) is the circumradius of the triangle, then \[ 2(r+R)= \]

• (A) \(a+b\)
• (B) \(b+c\)
• (C) \(a+c\)
• (D) \(a+b+c\)

The sides of a triangle are in the ratio \[ 1:\sqrt3:2. \]
Then the angles opposite to these sides are in the ratio

• (A) \(1:2:3\)
• (B) \(1:3:2\)
• (C) \(2:3:1\)
• (D) \(1:2:2\)

Assertion (A): If each of the angles \(A,B,C\) is not a multiple of \(\pi\), then the vectors \(\vec r_1=(\sec^2A)\hat i+\hat j+\hat k,\) \(\vec r_2=\hat i+(\sec^2B)\hat j+\hat k,\) \(\vec r_3=\hat i+\hat j+(\sec^2C)\hat k\) are coplanar.

Reason (R): The three vectors \(\vec a=a_1\hat i+a_2\hat j+a_3\hat k,\) \(\vec b=b_1\hat i+b_2\hat j+b_3\hat k,\) \() \vec c=c_1\hat i+c_2\hat j+c_3\hat k \) are coplanar if and only if \[ \begin{vmatrix} a_1&a_2&a_3
b_1&b_2&b_3
c_1&c_2&c_3 \end{vmatrix}=0.\] Which of the following is true?

• (A) (A) is true, (R) is true and (R) is a correct explanation of (A)
• (B) (A) is true, (R) is true but (R) is not a correct explanation of (A)
• (C) (A) is true but (R) is false
• (D) (A) is false but (R) is true

Let \(\vec a,\vec b\) and \(\vec c\) be three non-zero vectors such that no two of them are collinear. If the vector \(\vec a+\vec b\) is collinear with \(\vec c\) and \(\vec b+\vec c\) is collinear with \(\vec a\), then \[ \vec a+\vec b+\vec c= \]

• (A) \(\vec a\)
• (B) \(\vec b\)
• (C) \(\vec c\)
• (D) \(\vec 0\)

Let \[ \vec a=4\hat i-\hat j+\alpha\hat k \]
and \[ \vec b=\hat i+\alpha\hat j-4\hat k \]
be two vectors. If \(\alpha_1,\alpha_2\) (\(\alpha_1<\alpha_2\)) are two different values of \(\alpha\) such that \[ (\vec a,\vec b)=\cos^{-1}\left(-\frac{2}{7}\right), \]
then \[ \alpha_1+2\alpha_2= \]

• (A) \(15\)
• (B) \(24\)
• (C) \(33\)
• (D) \(52\)

The shortest distance between the two lines \[ \vec r=(\hat i-\hat j)+s(\hat j+2\hat k) \]
and \[ \vec r=(2\hat i+\hat k)+t(\hat i-\hat j+\hat k) \]
is

• (A) \(4\)
• (B) \(5\)
• (C) \[ \sqrt{\frac{6}{5}} \]
• (D) \[ \sqrt{\frac{8}{7}} \]

If \(\vec a,\vec b,\vec c\) are \(3\) vectors such that \[ \vec b=2\hat i-\hat j,\qquad \vec c=\hat j+2\hat k, \] \[ |\vec a+\vec b|=3,\qquad |\vec a\times(\vec b\times\vec c)|=3\sqrt2 \]
and \[ (\vec a,\vec b\times\vec c)=\frac{\pi}{3}, \]
then \(\vec a\cdot\vec b=\)

• (A) \(\dfrac{3}{2}\)
• (B) \(\sqrt6\)
• (C) \(\dfrac{\sqrt3}{\sqrt2}\)
• (D) \(6\)

The mean deviation about the mean for the following data is
\[ \begin{array}{c|cccc} x_i & 1 & 2 & 4 & 7 \\
\hline f_i & 3 & 2 & 4 & 1 \end{array} \]

• (A) \(3\)
• (B) \(2\)
• (C) \(1.5\)
• (D) \(1.6\)

If five unit squares are selected at random from a chess board, then the probability that they all lie on a diagonal is

• (A) \(\dfrac{112}{{}^{64}C_5}\)
• (B) \(\dfrac{56}{{}^{64}C_5}\)
• (C) \(\dfrac{448}{{}^{64}C_5}\)
• (D) \(\dfrac{224}{{}^{64}C_5}\)

Let \(A\) and \(B\) be two non-empty sets with \(n(A)=4\) and \(n(B)=5\). If a mapping is selected at random from the set of all mappings from \(A\) to \(B\), then the probability of getting a many-one mapping is

• (A) \(\dfrac{29}{125}\)
• (B) \(\dfrac{24}{125}\)
• (C) \(\dfrac{96}{125}\)
• (D) \(\dfrac{101}{125}\)

Out of the first \(20\) consecutive natural numbers, \(3\) numbers are chosen at random. If these \(3\) numbers are in arithmetic progression with common difference \(d\in\mathbb N\), then the probability of getting those \(3\) numbers whose common difference is a prime number is

• (A) \(\dfrac{4}{9}\)
• (B) \(\dfrac{1}{3}\)
• (C) \(\dfrac{23}{45}\)
• (D) \(\dfrac{29}{90}\)

Bag \(A\) contains \(3\) white and \(4\) black balls. Bag \(B\) contains \(4\) white and \(3\) black balls. Bag \(C\) contains \(2\) white and \(5\) black balls. A bag is randomly selected and then a ball is randomly drawn from that bag. If the ball drawn was found to be white, then the probability that the ball is drawn from bag \(C\) is

• (A) \(\dfrac{1}{6}\)
• (B) \(\dfrac{2}{9}\)
• (C) \(\dfrac{1}{4}\)
• (D) \(\dfrac{2}{13}\)

Consider the random experiment of throwing a die and tossing a coin. Let \[ \{(a,b)\mid a\in\{H,T\},\ b\in\{1,2,\ldots,6\}\} \]
denote the outcomes of the experiment. If \(X\) is a random variable defined by \[ X(a,b)=b, \]
then the variance of \(X\) is

• (A) \(\dfrac{49}{4}\)
• (B) \(\dfrac{35}{3}\)
• (C) \(\dfrac{49}{2}\)
• (D) \(\dfrac{35}{12}\)

If \(X\sim B(n,p)\) is a binomial variate, \(8p^2+15p-2=0\) and the product of the mean and variance of \(X\) is \(\dfrac78\), then \(P(X=4)=\)

• (A) \({}^{6}C_{4}\dfrac{5^2}{6^6}\)
• (B) \({}^{7}C_{4}\dfrac{4^3}{5^7}\)
• (C) \({}^{5}C_{4}\dfrac{3}{4^5}\)
• (D) \({}^{8}C_{4}\dfrac{7^4}{8^8}\)

If the distance of a variable point \(P\) from the fixed line \[ 2x-y+1=0 \]
is twice the distance of \(P\) from another fixed line \[ 2x+y-2=0, \]
then a point on the locus of \(P\) is

• (A) \(\left(\dfrac14,\dfrac34\right)\)
• (B) \((2,3)\)
• (C) \((1,1)\)
• (D) \(\left(\dfrac32,\dfrac14\right)\)

When the origin is shifted to the point \[ \left(\frac74,-\frac14\right) \]
by translation of axes, the transformed equation of \[ x^2-2xy+3y^2+2gx+2fy-6=0 \]
is \[ 8X^2-16XY+24Y^2+k=0. \]
Then \[ -27(2f+g)= \]

• (A) \(4k\)
• (B) \(k\)
• (C) \(3k\)
• (D) \(2k\)

Let \(A(1,2)\) and \(C(3,4)\) be the end points of one of the diagonals of a square \(ABCD\). If \(B(\alpha,\beta)\) and \(D(\gamma,\delta)\) are the end points of another diagonal of this square, then \(\alpha+\beta-\gamma+\delta=\)

• (A) 0
• (B) 6
• (C) 8
• (D) 4

The equation of a line \(L_{1}\) passing through the point \((2, 4)\) and making an angle \(\tan^{-1}(2)\) with another line \(x+2y=4\) is \(ax+by+c=0\). If this line \(L_{1}\) is neither horizontal nor vertical, then \(\frac{b+c}{a}=\)

• (A) 1
• (B) \(\frac{14}{3}\)
• (C) 2
• (D) 0

The product of the lengths of the perpendiculars drawn from the point \((1, 2)\) to the pair of lines \(2x^{2}-3xy-2y^{2}=0\) is

• (A) \(\frac{12}{5}\)
• (B) 2
• (C) \(\frac{4}{5}\)
• (D) 6

If the equation \(ax^{2}+2hxy+by^{2}+2gx+2fy+c=0\) represents a pair of parallel lines, then \(g^{2}h^{2}=\)

• (A) \(a^{2}b^{2}\)
• (B) \(a^{2}\)
• (C) \(f^{2}\)
• (D) \(a^{2}f^{2}\)

If a circle passing through the points \((1, 5)\) and \((4,0)\) makes equal intercepts on coordinate axes and if its centre lies in the first quadrant, then \(\sqrt{4g^{2}-c^{2}}=\)

• (A) 2
• (B) 4
• (C) 3
• (D) 5

If a tangent drawn to the circle \(x^{2}+y^{2}-6x-8y-11=0\) is perpendicular to the line \(3x + 4y + k = 0\), then the distance from the origin to this tangent is

• (A) 4
• (B) 5
• (C) 6
• (D) 2

If \(L\) represents a normal drawn at the point \(P\left(\frac{\pi}{4}\right)\) on the circle \(x^{2}+y^{2}+6x-6y-14=0\), then the equation of the diameter of this circle which is perpendicular to \(L\) is

• (A) \(x-y+6=0\)
• (B) \(2x+y+3=0\)
• (C) \(3x+2y+3=0\)
• (D) \(x+y=0\)

If \((h, k)\) is the pole of the line \(2x-3y+4=0\) with respect to the circle \(x^{2}+y^{2}-4x+6y-3=0\), then \(10h+k=\)

• (A) 0
• (B) 1
• (C) 4
• (D) 3

If the equation of a circle passing through the point (2, 1) and the points of intersection of the circles \(x^{2}+y^{2}+4x-6y-3=0\) and \(x^{2}+y^{2}-2x+2y-2=0\) is \(x^{2}+y^{2}+2gx+2fy+c=0\), then \(2g+f=\)

• (A) -2c
• (B) 2c
• (C) c
• (D) -c

If \(y=mx+c\), \(m>0\) is a common tangent to the parabolas \(y^{2}=8x\) and \(y^{2}=1+4x\), then \(m+c=\)

• (A) \(\frac{\sqrt{5}}{2}+\frac{9}{2\sqrt{5}}\)
• (B) 5
• (C) \(\frac{\sqrt{5}}{8}+\frac{2}{\sqrt{5}}\)
• (D) 3

Through the focus of the parabola \(x^{2}-4x-8y+44=0\), if tangents are drawn to another parabola \(y^{2}=20x\), then the sum of the Y - coordinates of the points of contact of these tangents is

• (A) 20
• (B) 8
• (C) 14
• (D) 32

If the major axis of an ellipse subtends an angle of \(120^{\circ}\) at one end of its minor axis and the length of its semi latus rectum is \(\frac{4}{\sqrt{3}}\), then the sum of the lengths of its axes is

• (A) 12
• (B) 24
• (C) \(8(\sqrt{3}+1)\)
• (D) \(4(\sqrt{3}+1)\)

The equation of the conjugate hyperbola of the hyperbola \(x^{2}-4y^{2}-2x-8y-19=0\) is

• (A) \(x^{2}-4y^{2}-2x-8y-33=0\)
• (B) \(x^{2}-4y^{2}-2x-8y+33=0\)
• (C) \(x^{2}-4y^{2}-2x-8y+13=0\)
• (D) \(x^{2}-4y^{2}-2x-8y-13=0\)

The points (0, \(\lambda\), 1), (\(\mu\), 3, -1), (\(\lambda\), 5, 0), (\(\mu\), 6, \(\mu\)) taken in that order, form a square. If \(\lambda\), \(\mu\) are positive real numbers, then the length of its side is

• (A) 1
• (B) 2
• (C) 3
• (D) 4

If \(\theta\) is the acute angle between a line \(\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{1}\) and a normal to the plane \(2x+3y+4z=0\) then \(\tan^{2}\theta+sec^{2}\theta=\)

• (A) 1
• (B) \(\frac{16}{13}\)
• (C) \(\frac{55}{3}\)
• (D) \(\frac{22}{7}\)

If the point P which divides the line segment joining \(A(1,1,1)\) and \(B(2,2,2)\) in the ratio 1: m lies on the plane \(x+2y+3z-1=0\), then \(m=\)

• (A) \(-\frac{3}{2}\)
• (B) \(\frac{4}{3}\)
• (C) \(-\frac{11}{5}\)
• (D) \(-\frac{1}{2}\)

If \(\lim_{n\rightarrow5}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=k\), then \(\lim_{n\rightarrow k^{+}}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=\)

• (A) k+1
• (B) k-1
• (C) k+2
• (D) k

\(\lim_{x\rightarrow1}\frac{(9x-1)(\sqrt{x}-1)}{3x^{2}+2x-5}=\)

• (A) \(\frac{1}{2}\)
• (B) \(\frac{3}{5}\)
• (C) 2
• (D) \(-\frac{1}{2}\)

If the function \(f(x)=\begin{cases}\frac{2h(x)-g(x)}{(h(x)+7)^{2/3}}, & x\ne0
\frac{7}{4}, & x=0\end{cases}\) is continuous at \(x=0\) and \(\lim_{x\rightarrow0}h(x)=1\), then \(\lim_{x\rightarrow0}g(x)=\)

• (A) 3
• (B) 4
• (C) -5
• (D) 7

Let \([y]\) represent the greatest integer less than or equal to \(y\). Then the set of all \(x\) at which \(f(x)=\cos^{-1}[4x+3]\) is differentiable is

• (A) \(\mathbb{R}\)
• (B) \([-1,1]\)
• (C) \([-1,-\frac{1}{4})-\{-\frac{3}{4},-\frac{1}{2}\}\)
• (D) \((-\frac{3}{4},\infty)\)

If \(\tan^{-1}x^{2}+\tan^{-1}y^{2}=\frac{\pi}{2}\), then \(\left(\frac{dy}{dx}\right)_{(-1,2)}=\)

• (A) 0
• (B) 1
• (C) \(\frac{1}{2}\)
• (D) \(-\frac{1}{2}\)

If \(f(x)=\begin{cases}\frac{1-\sin^{3}x}{3\cos^{2}x}, & x\ne\frac{\pi}{2}
\frac{1}{2}, & x=\frac{\pi}{2}\end{cases}\), then \(f^{\prime}\left(\frac{\pi}{2}\right)=\)

• (A) 1
• (B) \(\frac{1}{2}\)
• (C) -1
• (D) 0

If the length of the tangent at a point on the parabola \(y^{2}=4ax\) is \(4a\sqrt{5}\), then the length of the sub-normal at that point is

• (A) 4a
• (B) a
• (C) 8a
• (D) 2a

If the acute angle between the parabolas \(y=8x-x^{2}\) and \(y=x^{2}-4x\) is \(\theta\), then \(\tan\theta=\)

• (A) 1
• (B) \(\sqrt{3}\)
• (C) \(\frac{15}{17}\)
• (D) \(\frac{12}{31}\)

Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be such that \(f(2+x)=f(2-x)\,\,\forall x\in\mathbb{R}\). If \(f(x)\) is twice differentiable such that \(f^{\prime}(1)=0\), then which one of the following is true?

• (A) there exist at least one \(c\) in \((0, 1)\) such that \(f^{\prime}(c)=0\)
• (B) there exist at least one \(c in (1, 2)\) such that \(f^{\prime\prime}(c)=0\)
• (C) there exist at least one \(c\) in \((0, 1)\) such that \(f^{\prime\prime}(c)=0\)
• (D) there exist at least one \(c\) in \((1, 2)\) such that \(f^{\prime}(c)=0\)

If the local maximum 'M' and local minimum 'm' of the function \(f(x)=x-\frac{x^{2}}{2}-xe^{2-x}\) exist at \(x=\alpha\) and \(x=\beta\) respectively, then \(2\alpha m+\beta M=\)

• (A) -2e
• (B) \(\frac{1}{e}\)
• (C) -4e
• (D) \(\frac{1}{e^{2}}\)

Let \(f:\mathbb{R}^{+}\rightarrow\mathbb{R}\) be such that \(f(e)=\frac{7}{4}\) and \(f^{\prime}(x^{2})=\frac{3\log x}{x^{2}}\), then \(f(e^{2})=\)

• (A) \(\frac{5}{4}\)
• (B) 4
• (C) 10
• (D) \(\frac{49}{16}\)

If \(\int\frac{1}{x^{2}+4x+\alpha}dx=\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x+2}{2\sqrt{2}}\right)+c\), then \(\int\frac{1}{x^{2}+4x-\alpha}dx=\)

• (A) \(\frac{1}{4}\log\left|\frac{x-12}{x+2}\right|+k\)
• (B) \(\frac{1}{8}\log\left|\frac{x-2}{x+6}\right|+k\)
• (C) \(\frac{1}{8}\log\left|\frac{x+6}{x+8}\right|+k\)
• (D) \(\frac{1}{4}\log\left|\frac{x-12}{x+16}\right|+k\)

If \(\int\frac{2\sin x+a\cos x}{b\sin x+4\cos x}dx=\frac{2}{5}x-\frac{1}{5}\log(b\sin x+4\cos x)+c\), then \(a+b=\)

• (A) 2
• (B) 3
• (C) 4
• (D) 5

If \(\int(4\sin\theta+\cos\theta)\cot\theta\cos\theta d\theta=2\theta+\sin2\theta+\log(\sin\theta)+f(2\theta)+c\) and \(f(0)=\frac{1}{2}\), then \(f(x)=\)

• (A) \(\frac{1}{2}\cos x\)
• (B) \(\frac{1}{2}\sin x\)
• (C) \(\frac{1}{2}\sin x\cos x\)
• (D) \(\frac{\sin^{2}x}{2}\)

If \(I_{n}=\int\sin^{n}x dx\), then \(I_{6}-\frac{5}{6}I_{4}=\)

• (A) \(\frac{-\sin^{5}x\cos x}{6}\)
• (B) \(\frac{\sin^{5}x\cos x}{5}\)
• (C) \(\frac{-\sin^{5}x\cos^{2}x}{6}\)
• (D) \(\frac{\sin^{5}x\cos^{2}x}{5}\)

\[ \sum_{n=1}^{\infty}\int_{0}^{1}x^{3}(1-x^{2})^{n}dx =\]

• (A) \(\frac{1}{4}\)
• (B) \(\frac{1}{2}\)
• (C) 1
• (D) 2

If \(f(x)\) is a twice differentiable function and \(f^{\prime}(0)=0\), then \(\int_{0}^{\pi/2}(f(x)+f^{\prime\prime}(x))\cos x dx=\)

• (A) \(f\left(\frac{\pi}{2}\right)\)
• (B) \(f^{\prime}\left(\frac{\pi}{2}\right)\)
• (C) 1
• (D) 0

If \(f(x)=\left|\frac{x}{3}-3^{-x}\right|\) and \(\int_{0}^{5}f(x)dx=K+\frac{3^{n}+1}{3^{5}\log 3}\), then \(nK=\)

• (A) \(\frac{46}{3}\)
• (B) \(\frac{23}{6}\)
• (C) \(\frac{54}{5}\)
• (D) \(\frac{28}{9}\)

The area of the region bounded by the curve \(y=x^{2}+x\), the lines \(y=x\), \(x=1\) and \(y=2\) is

• (A) \(\frac{12}{5}\)
• (B) \(\frac{7}{2}\)
• (C) \(\frac{4}{5}\)
• (D) \(\frac{1}{3}\)

If the degree of the differential equation \(\left(\frac{d^{2}y}{dx^{2}}\right)^{3/2}+5\left(\frac{d^{2}y}{dx^{2}}\right)^{5/2}=7y\) is \(m\) and its order is \(n\), then \(y=Ae^{mx}+Be^{nx}\) is solution of the differential equation

• (A) \(\frac{d^{2}y}{dx^{2}}-12\frac{dy}{dx}+20y=0\)
• (B) \(\frac{d^{2}y}{dx^{2}}-7\frac{dy}{dx}+14y=0\)
• (C) \(\frac{d^{2}y}{dx^{2}}-10\frac{dy}{dx}+16y=0\)
• (D) \(\frac{d^{2}y}{dx^{2}}-8\frac{dy}{dx}+12y=0\)

If \(y=f(x)\) is a solution of \(\frac{dy}{dx}=(y-kx)^{2}+K\) when \(x=0\) and \(y=1\), then \(f(2)=\)

• (A) \(2K+1\)
• (B) \(2K-1\)
• (C) \(2K+5\)
• (D) \(2K-5\)

At a point \(P(x,y)\) on a curve \(x=f(y)\), the x-intercept of the tangent is always equal to the y-coordinate of the point of contact, then \(f(y)=\)

• (A) \(e^{cy^{2}}\)
• (B) \(y\log\left(\frac{c}{y}\right)\)
• (C) \(cy^{2}\)
• (D) \(\sin(c+y)\)

\(\beta=\frac{F}{v^{2}}\cos(\alpha t)\), if \(F\) is force, \(v\) is velocity, \(t\) is time, then the dimensional formulae of \(\alpha\), \(\beta\) are respectively

• (A) \(M^{0}L^{0}T^{0}\), \(ML^{-1}T^{0}\)
• (B) \(M^{0}L^{0}T^{-1}\), \(MLT^{0}\)
• (C) \(M^{0}L^{0}T^{-1}\), \(ML^{-1}T^{0}\)
• (D) \(ML^{0}T^{-1}\), \(ML^{-1}T\)

Cars \(X\) and \(Y\) begin the race simultaneously with velocities \(8 ms^{-1}\) and \(4 ms^{-1}\), moving in a straight line with uniform accelerations \(2 ms^{-2}\) and \(4 ms^{-2}\) respectively. If they reach final point at the same instant, then the length of the path is

• (A) \(24 m\)
• (B) \(48 m\)
• (C) \(32 m\)
• (D) \(16 m\)

A projectile is thrown into space so that it attains a maximum possible range of \(200 m\). Taking the point of projection as origin, the coordinates of the point where the velocity of projectile is minimum is

• (A) \((100,50)\)
• (B) \((200, 100)\)
• (C) \((100, 200)\)
• (D) \((50,100)\)

A bike race is conducted on a circular track of radius \(60 m\). If the slowest bike, weighing \(120 kg\), moved around the track at a constant speed of \(108 kmph\), then the time taken by it to complete one lap and its acceleration are respectively

• (A) \(3.49 s\), \(1.11 ms^{-2}\)
• (B) \(12.56 s\), \(15 ms^{-2}\)
• (C) \(3.49 s\), \(0 ms^{-2}\)
• (D) \(2 s\), \(9.8 ms^{-2}\)

A block is kept at the top of a rough inclined plane of length \(2.8 m\) and angle of inclination \(\cos^{-1}(0.6)\). If the coefficient of kinetic friction between the block and the upper half of the plane is \(0.3\) and between the block and the lower half of the plane is \(0.5\), then the velocity with which the block reaches the bottom of the plane is (acceleration due to gravity \(=10 ms^{-2}\))

• (A) \(1.4 ms^{-1}\)
• (B) \(5.6 ms^{-1}\)
• (C) \(4.2 ms^{-1}\)
• (D) \(2.8 ms^{-1}\)

A stone of mass \(400 g\) tied to one end of a string is rotated in a horizontal circle of radius \(125 m\). If the string can withstand a maximum tension of \(50\pi^{2} N\) then the minimum time period with which the stone can be rotated is

• (A) \(3 s\)
• (B) \(2 s\)
• (C) \(4 s\)
• (D) \(6 s\)

A body of mass \(5 kg\) is projected vertically upwards from a point \(X\) from the ground with an initial speed of \(20 ms^{-1}\). It rises to a point \(Y\) where its kinetic energy is reduced to \(400 J\). If the body experiences a constant air resistance of \(10 N\) throughout its motion, then the vertical distance between \(X\) and \(Y\) is (Acceleration due to gravity \(=10 ms^{-2}\))

• (A) \(5 m\)
• (B) \(10 m\)
• (C) \(8 m\)
• (D) \(16 m\)

Three balls \(A\), \(B\) and \(C\) of equal mass are moving in the same order along the same straight line on a smooth horizontal surface. Initially, ball \(A\) moves towards right with a velocity of \(6 ms^{-1}\), ball \(B\) moves towards left with a velocity of \(4 ms^{-1}\) and ball \(C\) moves towards right with a velocity of \(3 ms^{-1}\). If the coefficient of restitution between any two balls is \(0.8\), then the relative velocity of balls \(B\) and \(C\) after collision is

• (A) \(1.6 ms^{-1}\)
• (B) \(0.8 ms^{-1}\)
• (C) \(3.2 ms^{-1}\)
• (D) \(2.4 ms^{-1}\)

If three particles of masses \(2m\), \(m\) and \(4m\) are moving in three mutually perpendicular directions with velocities \(3 ms^{-1}\), \(4 ms^{-1}\) and \(3 ms^{-1}\) respectively, then the magnitude of the velocity of the center of mass of the system of three particles is

• (A) \(3.5 ms^{-1}\)
• (B) \(2 ms^{-1}\)
• (C) \(2.5 ms^{-1}\)
• (D) \(3 ms^{-1}\)

The acceleration of a uniform disc rolling down an inclined plane of length \(1.75 m\) is \(1/3\) times the acceleration due to gravity. If a solid sphere is rolling down from the top of the same inclined plane, then the velocity with which it reaches the bottom of the plane is

• (A) \(2.5 ms^{-1}\)
• (B) \(5 ms^{-1}\)
• (C) \(3.5 ms^{-1}\)
• (D) \(7 ms^{-1}\)

In simple harmonic motion, at mean position

• (A) KE maximum
• (B) PE maximum
• (C) KE zero
• (D) Acceleration maximum

A particle starting from the origin executes simple harmonic motion along x-axis. Its velocity at any instant t is given by \(v_{x}=22 \cos(\frac{\pi t}{2}) cm s^{-1}\). The total distance covered by the particle in time \(t=4.5 sec\) is

• (A) 74.5 cm
• (B) 51.4 cm
• (C) 65.9 cm
• (D) 49.8 cm

A 1200 kg artificial satellite is in an orbit of radius \(2R_{E}\) about the earth. The energy required to transfer it to an orbit of radius \(3R_{E}\) is (\(g=10ms^{-2}\), \(R_{E}=6400 km\))

• (A) \(1.2\times10^{9} J\)
• (B) \(6.4\times10^{9} J\)
• (C) \(6400\times10^{3} J\)
• (D) \(3.2\times10^{9} J\)

A copper wire of negligible mass, length 1 m and area of cross-section \(10^{-6}m^{2}\) is kept on a smooth horizontal table. One end of the wire is fixed and other end of the wire attached with a ball of mass 1 kg. If the ball and wire are rotating with 20 rad \(s^{-1}\), an elongation of \(10^{-3}m\) is observed in the wire, find its Young's modulus

• (A) \(4\times10^{11}Nm^{-2}\)
• (B) \(8\times10^{11}Nm^{-2}\)
• (C) \(4\times10^{8}Nm^{-2}\)
• (D) \(400 Nm^{-2}\)

Calculate the amount of work done in spraying a drop of a liquid of radius 1 mm into million identical droplets under isothermal conditions. [surface tension of liquid \(=550\times10^{-3}Nm^{-1}\)]

• (A) \(6.84\times10^{-4} J\)
• (B) \(5.50\times10^{-4} J\)
• (C) \(7.25\times10^{-4} J\)
• (D) \(3.42\times10^{-4} J\)

A blackened platinum wire of surface area \(10^{-5}m^{2}\) is maintained at temperature of 3000K. At what rate the wire is loosing energy [Stefan-Boltzmann constant \(\sigma=5.67\times10^{-8}W m^{-2}K^{-4}\)]

• (A) 50 W
• (B) 46 W
• (C) 76 W
• (D) 38 W

An earthen pitcher containing 9.5 kg of water loses one gram of water per minute due to evaporation. If water equivalent of pitcher is 0.5 kg. The time required to cool the water in pitcher from \(30^{\circ}C\) to \(28^{\circ}C\) is (Neglect radiation effect and Take latent heat of vaporization 500 cal \(g^{-1}\))

• (A) 30 min
• (B) 60 min
• (C) 40 min
• (D) 20 min

A Carnot engine of 50 % efficiency takes heat from a source of 400 K. To change efficiency to 70 % without changing the sink temperature, the new temperature of source should be (nearer to)

• (A) 765 K
• (B) 525 K
• (C) 800 K
• (D) 667 K

In which of the following thermodynamic process, the internal energy of system remains constant

• (A) Isobaric
• (B) Isochoric
• (C) Adiabatic
• (D) Isothermal

The rms speed of oxygen molecule at some temperature is \(150 ms^{-1}\). Then the rms speed of hydrogen molecule at the same temperature is

• (A) \(400 ms^{-1}\)
• (B) \(600 ms^{-1}\)
• (C) \(200 ms^{-1}\)
• (D) \(800 ms^{-1}\)

Two identical strings A and B of same material are having tensions \(T_{A}\) and \(T_{B}\) respectively. If their fundamental frequencies are 450 Hz and 300 Hz, then \(T_{A}/T_{B}\) is

• (A) \(\frac{4}{9}\)
• (B) \(\frac{2}{9}\)
• (C) \(\frac{6}{4}\)
• (D) \(\frac{9}{4}\)

The refractive index of water is \(\frac{4}{3}\) and that of glass is \(\frac{3}{2}\). The refractive index of glass with respect to water is

• (A) \(9/8\)
• (B) \(8/9\)
• (C) \(2/3\)
• (D) \(3/4\)

For the same angle of incidence, the angles of refraction of light ray in different media A, B, C are \(35^{\circ}\), \(25^{\circ}\), \(15^{\circ}\). If \(V_{A}, V_{B}, V_{C}\) are velocities of light in A, B, C media respectively, then

• (A) \(V_{A} = V_{B} = V_{C} = 0\)
• (B) \(V_{A} = V_{B} = V_{C}\)
• (C) \(V_{A} > V_{B} > V_{C}\)
• (D) \(V_{A} < V_{B} < V_{C}\)

In a Young's double slit experiment, the intensity at a point where the path difference is \(\frac{\lambda}{6}\) (\(\lambda\) being the wavelength of the light used) is \(I\). If \(I_{0}\) denotes the maximum intensity, \(\frac{I}{I_{0}}\) is

• (A) \(\frac{1}{\sqrt{2}}\)
• (B) \(\frac{\sqrt{3}}{2}\)
• (C) \(\frac{1}{2}\)
• (D) \(\frac{3}{4}\)

Two identical balls each of mass \(\frac{40}{3}\) g carry equal and opposite charge. They are suspended from a horizontal plate by silk threads each of length 1 m with a separation of 1.5 m between points of suspension. At equilibrium, if the distance between the balls is 30 cm then magnitude of charge on each ball is (Acceleration due to gravity \(= 10 ms^{-2}\))

• (A) \(10^{-5} C\)
• (B) \(10^{-6} C\)
• (C) \(10^{-3} C\)
• (D) \(2\times 10^{-6} C\)

A conducting sphere of radius 4 cm is charged such that it has a potential of 5V on its surface. Then the potential at a point which is at a depth of 1 cm from its surface is

• (A) 3 V
• (B) 4 V
• (C) 0 V
• (D) 5 V

A parallel plate capacitor with dielectric is charged completely and the battery is then disconnected. Now if the dielectric is slowly pulled out of the capacitor then the variation of the potential (V) of the capacitor with respect to the length (x) of the dielectric pulled out is represented by

• (A)   
• (B)   
• (C)   
• (D)
  

The potential difference between points A and B of adjoining figure is


 

• (A) \(\frac{2}{3} V\)
• (B) \(\frac{8}{9} V\)
• (C) \(\frac{4}{3} V\)
• (D) \(\frac{5}{3} V\)

A potentiometer wire is 10m long and has a resistance of \(18\,\Omega\). It is connected to a battery of emf 5V and internal resistance \(2\,\Omega\). Calculate the potential gradient along the wire.

• (A) \(0.45 Vm^{-1}\)
• (B) \(0.5 Vm^{-1}\)
• (C) \(0.2 Vm^{-1}\)
• (D) \(0.1 Vm^{-1}\)

A moving coil galvanometer has a resistance of \(50\,\Omega\) and gives a full-scale deflection for a current of \(2 mA\). To convert it into a voltmeter reading up to 10 V, the required series resistance to be connected is

• (A) \(4950\,\Omega\)
• (B) \(5000\,\Omega\)
• (C) \(4450\,\Omega\)
• (D) \(5050\,\Omega\)

A wire carrying current I and other parallel wire carrying current 2I in the same direction produces a magnetic field B at the midpoint between them. Then the magnitude of field at the same point, when the 2I wire is switched off

• (A) B/2
• (B) 2 B
• (C) B
• (D) 4 B

The earth's magnetic field at a certain place has a total strength of 0.5 Gauss and the horizontal component of 0.3 Gauss. Then the angle of dip at that place is

• (A) \(\tan^{-1}\frac{3}{4}\)
• (B) \(\tan^{-1}\frac{4}{3}\)
• (C) \(\sin^{-1}\frac{3}{4}\)
• (1) \(\sin^{-1}\frac{3}{5}\)

A 2 m long solenoid with diameter 2 cm and 2000 turns has a secondary coil of 1000 turns wound closely near its midpoint. The mutual inductance between the two coils is

• (A) \(2.4\times10^{-4} H\)
• (B) \(3.9\times10^{-4} H\)
• (C) \(1.28\times10^{-3} H\)
• (D) \(3.14\times10^{-3} H\)

In the given circuit, the readings of voltmeters \(V_{1}\) and \(V_{2}\) are 300 V each. The reading of voltmeter \(V_{3}\) and ammeter A are respectively

• (A) 100 V, 2.0 A
• (B) 150 V, 2.2 A
• (C) 220 V, 2.0 A
• (D) 220 V, 2.2 A

Electromagnetic waves do not transport

• (A) Charge
• (B) Energy
• (C) Momentum
• (D) Information

Work function of a metallic surface is 5.01 eV. When a light of wavelength 2000Å falls on the metallic surface, it emits photo electrons. The potential difference required to stop the fastest photo electron is? (in Volt)

• (A) 1.2
• (B) 1.6
• (C) 2.4
• (D) 2.8

The atomic number (Z) of a hydrogen like atom whose shortest wavelength of Brackett Series is same as the shortest wavelength of Balmer series of hydrogen atom is

• (A) Z = 1
• (B) Z = 2
• (C) Z = 3
• (D) Z = 4

Binding energy per nucleon of deuteron and Helium nuclei are 1.1 MeV and 7 MeV respectively. If a single Helium nucleus was formed by adding two Deuterons, the energy released is

• (A) 23.6 MeV
• (B) 32.4 MeV
• (C) 28.6 MeV
• (D) 13.6 MeV

The minimum energy gap of a semi-conductor used in manufacturing LED is

• (A) 3 eV
• (B) 3.6 eV
• (C) 2.8 eV
• (D) 1.8 eV

If \(f_{m}\) is a modulating frequency and \(f_{c}\) is carrier wave frequency, then Bandwidth in Amplitude Modulated wave is

• (A) \(2f_{c}\)
• (B) \(f_{c} + f_{m}\)
• (C) \(2f_{m}\)
• (D) \(\frac{(f_{c} + f_{m})}{2}\)

The energy of spectral line of lowest frequency in Lyman series of \(Li^{2+}\) spectrum is x J. The energy of second spectral line in Balmer series of \(He^{+}\) spectrum is y J. The ratio of x and y is

• (A) 3:1
• (B) 1:3
• (C) 1:9
• (D) 9:1

Quantum number sets of four electrons I, II, III, IV are given below.
I. \(n=3\), \(l=1\), \(m_{l}=-1\), \(m_{s}=+\frac{1}{2}\)
II. \(n=4\), \(l=1\), \(m_{l}=0\), \(m_{s}=+\frac{1}{2}\)
III. \(n=4\), \(l=2\), \(m_{l}=-2\), \(m_{s}=+\frac{1}{2}\)
IV. \(n=3\), \(l=2\), \(m_{l}=-1\), \(m_{s}=-\frac{1}{2}\) The correct order of the energy of these electrons is

• (A) \(I > IV > II > III\)
• (B) \(III > II > IV > I\)
• (C) \(III > IV > II > I\)
• (D) \(III > IV > I > II\)

Which of the following statement is not correct?

• (A) The increasing order of first ionization enthalpy of Be, B, C is \(B < Be < C\)
• (B) The IUPAC name of element Livermorium is ununhexium
• (C) The element Tennessene belongs to group 16 in periodic table
• (D) \(Al_{2}O_{3}\), \(As_{2}O_{3}\) are amphoteric oxides

Identify the correct pair of species having same number of valency electrons

• (A) \(SF_{4}, ClO_{3}^{-}\)
• (B) \(PO_{4}^{3-}, SF_{4}\)
• (C) \(ClF_{3}, SO_{4}^{2-}\)
• (D) \(CCl_{4}, PO_{4}^{3-}\)

Which of the following orders is not correct for the property mentioned against them?

• (A) \(H_{2}O > HF > NH_{3} > H_{2}S\)  – (boiling point)
• (B) \(H_{2}O > NH_{3} > NF_{3} > CF_{4}\) – (dipole moment)
• (C) \(C-C > N-O > C-O > C-H\) – (bond length)
• (D) \(O_{2}^{+} > O_{2} > O_{2}^{-} > O_{2}^{2-}\) – (bond order)

An ideal gas expands from volume \(V_{1}\) to \(V_{2}\) through an isothermal reversible process. The work done by the system is given by

• (A) \(W = -nRT \ln\left(\frac{V_{2}}{V_{1}}\right)\)
• (B) \(W = nRT \ln\left(\frac{V_{2}}{V_{1}}\right)\)
• (C) \(W = zero\)
• (D) \(W = -P\Delta V\)

For a reaction, the rate constant double when temperature is increased from 300 K to 310 K. The activation energy (\(E_{a}\)) of this reaction is (Take \(R = 8.314 J K^{-1} mol^{-1}\))

• (A) \(53.6 kJ mol^{-1}\)
• (B) \(43.6 kJ mol^{-1}\)
• (C) \(58.5 kJ mol^{-1}\)
• (D) \(62.4 kJ mol^{-1}\)

The conjugate base of \(H_{2}PO_{4}^{-}\) is

• (A) \(HPO_{4}^{2-}\)
• (B) \(PO_{4}^{3-}\)
• (C) \(H_{3}PO_{4}\)
• (D) \(H_{2}PO_{4}^{2-}\)

Which of the following compounds will exhibit geometrical isomerism?

• (A) 1-Butene
• (B) 2-Butene
• (C) Propene
• (D) 2-Methylpropene

The temporary hardness of water is due to the presence of

• (A) Calcium and Magnesium chlorides
• (B) Calcium and Magnesium sulfates
• (C) Calcium and Magnesium bicarbonates
• (D) Sodium and Potassium carbonates

For the reaction \(2HI(g)\rightleftharpoons H_{2}(g)+I_{2}(g)\), the correct expression relating the degree of dissociation (\(\alpha\)) of HI and equilibrium constant \(K_{p}\) is

• (A) \(\alpha=\frac{1+2\sqrt{K_{p}}}{2}\)
• (B) \(\alpha=\sqrt{1+\frac{2K_{p}}{2}}\)
• (C) \(\alpha=\frac{2\sqrt{K_{p}}}{1+2\sqrt{K_{p}}}\)
• (D) \(\alpha=\sqrt{\frac{2K_{p}}{1+2K_{p}}}\)

In gas phase structure of \(H_{2}O_{2}\) the dihedral angle is

• (A) 94.5\(^{\circ}\)
• (B) 101.9\(^{\circ}\)
• (C) 111.5\(^{\circ}\)
• (D) 90.2\(^{\circ}\)

Which one of the carbonates is unstable in air and is kept in \(CO_{2}\) atmosphere to avoid decomposition?

• (A) \(MgCO_{3}\)
• (B) \(CaCO_{3}\)
• (C) \(BaCO_{3}\)
• (D) \(BeCO_{3}\)

Total number of electrons involved in the formation of bridge bonds in the structure of diborane is

• (A) 6
• (B) 2
• (C) 4
• (D) 3

Consider the following statements:

Statement - I: Silica in its normal form is almost non-reactive because of very high Si-O bond enthalpy.

Statement - II: Silica has no reaction with HF.

• (A) Both the statements I & II are correct.
• (B) Statement I is correct but statement - II is NOT correct.
• (C) Statement I is NOT correct but statement - II is correct.
• (D) Both the statements I & II are NOT correct.

Identify the sets which contain greenhouse gases only:

I. \(CO_{2}\), \(SO_{3}\), \(N_{2}\)

II. \(CO_{2}\), \(CH_{4}\), \(O_{3}\)

III. \(O_{2}\), \(NO_{2}\), \(SO_{2}\)

IV. \(O_{3}\), CFCs, \(N_{2}O\)

• (A) I, II only
• (B) II, III only
• (C) II, IV only
• (D) I, IV only

An organic compound containing an extra element E on reaction with \(Na_{2}O_{2}\) followed by boiling with \(HNO_{3}\) gives a compound. This on treatment with ammonium molybdate solution gives yellow precipitate. What is E?

• (A) S
• (B) N
• (C) P
• (D) I

The number of hyperconjugative hydrogens present in the product Y is
\(CH_{2}=CH_{2} \xrightarrow{HBr} X \xrightarrow[Anhy. AlCl_{3}]{C_{6}H_{6}} Y\)

• (A) 2
• (B) 3
• (C) 5
• (D) 1

Which of the following compounds will show geometrical isomerism?

I. \((CH_{3})_{2}C=CH~C_{2}H_{5}\)

II. \(C_{6}H_{5}CH=CH~C_{2}H_{5}\)

III. \(C_{6}H_{5}CH=CH_{2}\)

IV. \(CH_{3}CH=C(Cl)CH_{3}\)

• (A) I & II only
• (B) II & III only
• (C) I & III only
• (D) II & IV only

A metal X of atomic mass 75 u forms a cubic lattice of edge length 5 Å. If the density of the lattice is 2 g cm\(^{-3}\), the radius (in Å) of the metal atom is (\(N=6\times10^{23} mol^{-1}\))

• (A) 1.083
• (B) 4.330
• (C) 2.165
• (D) 6.495

An aqueous solution of a non-volatile and non-electrolytic solute boils at \(100.5^{\circ}C.\) What will be the freezing point of the same solution? (Given \(K_{b}=0.512~K\) kg \(mol^{-1}\) and \(K_{f}=1.86~K\) kg mol\(^{-1}\))

• (A) \(-2.816^{\circ}C\)
• (B) \(-1.816^{\circ}C\)
• (C) \(-0.908^{\circ}C\)
• (D) \(-3.632^{\circ}C\)

X, Y and Z represent three electrodes \(Al^{3+}/Al\), \(Cu^{2+}/Cu\) and \(Ag^{+}/Ag\) with \(E^{\circ}\) values -1.66, 0.34 and 0.80 V respectively. The correct order of oxidising power of these three electrodes is

• (A) \(X>Y>Z\)
• (B) \(Z>Y>X\)
• (C) \(X=Y=Z\)
• (D) \(Y>Z>X\)

Given below are two statements: \ Statement I: The electrical conductivity of a solution decreases with dilution. \ Statement II: The overall reaction in \(H_{2}-O_{2}\) fuel cell is: \(2H_{2}O(l)\longrightarrow2H_{2}(g)+O_{2}(g)\).

• (A) Both statements I and statement II are correct.
• (B) Both statements I and statement II are not correct.
• (C) Statement I is correct but statement II is not correct.
• (D) Statement I is not correct but statement II is correct.

In the Arrhenius equation, \(\exp\left(-\frac{E_{a}}{RT}\right)\), is equal to

• (A) Frequency factor
• (B) Fraction of molecules that have energy higher than \(E_{a}\)
• (C) Rate of the reaction
• (D) Rate constant of the reaction

At \(T(K)\) adsorption of acetic acid on 1g of charcoal gave a Freundlich adsorption isotherm with slope of 0.5 and intercept of 1. What is the value of x, when the concentration of acetic acid is 0.1 mol \(L^{-1}\)? (Given: \(antilog(0.5) [cite_start]= 3.162\); \(antilog(0.301) = 2.0\))

• (A) 0.3162
• (B) 3.162
• (C) 0.5
• (D) 0.2

Match List - I with List - II:



The correct match is:

• (A) A-III, B-I, C-IV, D-II
• (B) A-III, B-I, C-II, D-IV
• (C) A-IV, B-II, C-I, D-III
• (D) A-II, B-I, C-III, D-IV

The ore calamine on calcination gives its oxide X. The common name of X is

• (A) Cuprite
• (B) Bauxite
• (C) Zincite
• (D) Siderite

The reduction products formed when copper and zinc metals are separately oxidised with dilute \(HNO_{3}\) respectively are

• (A) NO, NO\({2}\)
• (B) N\({2}\)O, NO
  [cite: 372]
• (C) [cite_start]NO, N\({2}O\)
• (D) NO\({2}\), NO

Consider the following compounds:

\(H_{3}PO_{2}, Se_{2}Cl_{2}, HNO_{2}, HNO_{3}, H_{2}SO_{4}, H_{2}O_{2}\)
How many of the above compounds undergo disproportionation reaction?

• (A) 2
• (B) 3
• (C) 4
• (D) 5

Which of the following oxoacids of phosphorus has a peroxy (\(O-O\)) linkage present within its molecular structure?

• (A) Orthophosphoric acid
• (B) Pyrophosphoric acid
• (C) Peroxodiphosphoric acid
• (D) Hypophosphoric acid

The metal ion, ligands present in Wilkinson catalyst are respectively

• (A) \(Rh^{3+}, Cl^{-}, PPh_{3}\)
• (B) \(Rh^{+}, Cl^{-}, PPh_{3}\)
• (C) \(Rh^{2+}, Br^{-}, PPh_{3}\)
• (D) \(Re^{+}, I^{-}, P(CH_{3})_{3}\)

Consider the following statements: Statement-I: Polystyrene is a thermosetting polymer. Statement-II: Bakelite is a thermoplastic polymer.

• (A) Both Statement-I and statement-II are correct
• (B) Statement-I is correct but statement-II is not correct
• (C) Statement-I is not correct but statement-II is correct
• (D) Both Statement-I and statement-II are not correct

The incorrect statement in the following is:

• (A) Amylopectin has \(C_{1}-C_{4}\) and \(C_{1}-C_{6}\) glycosidic linkages
• (B) Cellulose is a straight chain polysaccharide of \(\alpha-D\)-glucose units through \(C_{1}-C_{4}\) glycosidic linkage
• (C) Glycogen is found in Yeast and fungi
• (D) Glycogen is also known as animal starch

Which of the following structures represent an example of a tranquilizer?

• (A)
• (B)
• (C)
• (D)

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

• (A) KCl; KF
• (B) HCl; KF
• (C) HCl; (i) \(HBF_{4}\) (ii) \(\Delta\)
• (D) Cu | HCl; (i) \(HBF_{4}\) (ii) \(\Delta\)

Consider the following reactions I, II and III. The correct order of reactivity of X, Y and Z towards \(S_{N}1\) reaction is:

• (A) \(Z > Y > X\)
• (B) \(X > Z > Y\)
• (C) \(X > Y > Z\)
• (D) \(Z > X > Y\)

Consider the reaction of Cumene: (i) \(O_{2}\) (ii) \(H_{3}O^{+} \rightarrow A + B\). A is acidic and forms a salt with NaOH. X and Y are reagents to convert A and B to hydrocarbons. What are X and Y?

• (A) \(X=Zn; Y=NaBH_{4}\)
• (B) \(X=NaBH_{4}; Y=Zn, \Delta\)
• (C) \(X=Zn; Y=Zn(Hg), HCl\)
• (D) \(X=Zn(Hg), HCl; Y=Zn\)

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

• (A) \(H_{2}/cat; (CH_{3})_{2}Cd\)
• (B) \(Na/C_{2}H_{5}OH; CH_{3}MgBr, H_{2}O\)
• (C) \(DIBAL-H, H_{2}O; CH_{3}MgBr, H_{2}O\)
• (D) \(LiAlH_{4}, H_{2}O; (CH_{3})_{2}Cd\)

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

• (A)
• (B)
• (C)
• (D)

In which of the following set/s, reactant and reagent are correctly matched to get ethyl isonitrile as major product?

• (A) I, III
• (B) II, III
• (C) I only
• (D) II only