JEE Main 2017 B.E./B.Tech Set A Question Paper with Answer Key April 2 Forenoon Session

Let \(a, b, c \, \in \, R\). If \(f(x) = ax^2 + bx + c\) is such that \(a + b + c = 3\) and \(f (x + y) = f (x) + f (y) + xy, \forall \, x, y \, \in \, R,\) then \(\displaystyle\sum^{10}_{n = 1} f(n)\) is equal to :

• (1) 165
• (2) 190
• (3) 255
• (4) 330

The function \(f:R\to\left[-\frac{1}{2},\frac{1}{2}\right]\) defined as \(f(x)=\frac{x}{1+x^2}\), is

• (1) invertible
• (2) injective but not surjective.
• (3) surjective but not injective
• (4) neither injective nor surjective.

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is:

• (1) 468
• (2) 469
• (3) 484
• (4) 485

If S is the set of distinct values of 'b' for which the following system of linear equations
\(x+y+z=1\)
\(x+ay+z=1\)
\(ax+by+z=0\)
has no solution, then S is :

• (1) an empty set
• (2) an infinite set
• (3) a finite set containing two or more elements
• (4) a singleton

If \(A=\begin{bmatrix}2&-3\\-4&1\end{bmatrix}\), then adj\((3A^2+12A)\) is equal to

• (1) \(\begin{bmatrix}51&63\\84&72\end{bmatrix}\)
• (2) \(\begin{bmatrix}51&84\\63&72\end{bmatrix}\)
• (3) \(\begin{bmatrix}72&-63\\-84&51\end{bmatrix}\)
• (4) \(\begin{bmatrix}72&-84\\-63&51\end{bmatrix}\)

If, for a positive integer n, the quadratic equation, \(x(x+1)+(x+1)(x+2)+....+(x + \overline{ n - 1}) (x+ n)=10n\) has two consecutive integral solutions, then \(n\) is equal to :

• (1) 9
• (2) 10
• (3) 11
• (4) 12

If for \(x \epsilon \left(0, \frac{1}{4}\right) ,\) the derivative of \( \tan^{-1} \left(\frac{6x \sqrt{x}}{1-9x^{3}}\right) \) is \(\sqrt{x} . g(x)\), then \(g(x)\) equals :

• (1) \(\frac{3x \sqrt{x}}{1 - 9x^3}\)
• (2) \(\frac{3x }{1 - 9x^3}\)
• (3) \(\frac{3 }{1 + 9x^3}\)
• (4) \(\frac{9 }{1 + 9x^3}\)

Let \(l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C\), where \(C\) is a constant of integration, then the ordered pair \((a, b)\) is equal to :

• (1) \(\left(\frac{1}{5} , 0\right) \)
• (2) \(\left(\frac{1}{5} , - 1 \right) \)
• (3) \(\left( - \frac{1}{5} , 0\right) \)
• (4) \(\left( - \frac{1}{5} , 1 \right) \)

If \((2 + \sin \, x ) \frac{dy}{dx} + (y + 1) \cos \, x = 0\) and \(y(0) = 1,\) then \(y \left( \frac{\pi}{2} \right)\) is equal to :

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

Let \(\vec{a}=2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\). Let \(\vec{c}\) be a vector such that \(|\vec{c}-\vec{a}|=3\), \(|(\vec{a}\times\vec{b})\times\vec{c}|=3\) and the angle between \(\vec{c}\) and \(\vec{a}\times\vec{b}\) is \(30^\circ\). Then \(\vec{a}\cdot\vec{c}\) is equal to :

• (1) 2
• (2) 5
• (3) \(\frac{1}{8}\)
• (4) \(\frac{25}{8}\)

If two different numbers are taken from the set \(\{0,1,2,3, \ldots \ldots, 10\}\)then the probability that their sum as well as absolute difference are both multiple of \(4\), is :

• (1) \(\frac{12}{55}\)
• (2) \(\frac{14}{45}\)
• (3) \(\frac{7}{55}\)
• (4) \(\frac{6}{55}\)

The integral \(\int\limits^{\frac{3\, \pi}{4}}_{\frac{\pi}{4}} \frac{dx}{ 1 + \cos \, x}\) is equal to :

• (1) 2
• (2) 4
• (3) -1
• (4) -2

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in \(s m\)) of the flower-bed, is :

• (1) 10
• (2) 25
• (3) 30
• (4) 12.5

If \(5\left(\tan^2x-\cos^2x\right)=2\cos2x+9\), then the value of \(\cos4x\) is :

• (1) \(\frac{1}{3}\)
• (2) \(\frac{2}{9}\)
• (3) \(-\frac{7}{9}\)
• (4) \(-\frac{3}{5}\)

Let k be an integer such that the triangle with vertices \((k,-3k)\), \((5,k)\) and \((-k,2)\) has area 28 sq. units. Then the orthocentre of this triangle is at the point :

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

\(\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{\left(\pi - 2x\right)^{3}} \) equals :

• (1) \(\frac{1}{16}\)
• (2) \(\frac{1}{8}\)
• (3) \(\frac{1}{4}\)
• (4) \(\frac{1}{24}\)

For three events A, B and C,
P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = \(\frac{1}{4}\)
and P(All the three events occur simultaneously) = \(\frac{1}{16}\).
Then the probability that at least one of the events occurs, is :

• (1) \(\frac{7}{16}\)
• (2) \(\frac{7}{64}\)
• (3) \(\frac{3}{16}\)
• (4) \(\frac{7}{32}\)

The area (in s units) of the region \(\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y\) and \(y \leq 1 + \sqrt{x} \}\) is

• (1) \(\frac{3}{2}\)
• (2) \(\frac{7}{3}\)
• (3) \(\frac{5}{2}\)
• (4) \(\frac{59}{12}\)

Let \(\omega\) be a complex number such that \(2 \omega + 1 = z\) where \(z = \sqrt{-3}\) ,If \(\begin{vmatrix}1&1&1\\ 1&-\omega^{2} - 1 &\omega^{2}\\ 1&\omega^{2}& \omega^{7}\end{vmatrix} = 3 k , \) then \(k\) is equal to :

• (1) z
• (2) -1
• (3) 1
• (4) -z

Let \( C \) be the circle of minimum area touching the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3} |x| \). Then, which one of the following points lies on the circle \( C \)?

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

The normal to the curve \(y(x-2)(x-3)=x+6\) at the point where the curve intersects the y-axis passes through the point :

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

Let a vertical tower \(AB\) have its end \(A\) on the level ground. Let \(C\) be the mid-point of \(AB\) and \(P\) be a point on the ground such that \(AP = 2AB\). If \(\angle BPC = \beta \), then tan \(\beta\) is equal to :

• (1) \(\frac{1}{4}\)
• (2) \(\frac{2}{9}\)
• (3) \(\frac{4}{9}\)
• (4) \(\frac{6}{7}\)

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is \(\frac{1}{2}\). If P(1, \(\beta\)), \(\beta \gt 0\) is a point on this ellipse, then the equation of the normal to it at P is :

• (1) \(4x - 3y = 2\)
• (2) \(8x - 2y = 5\)
• (3) \(7x - 4y = 1\)
• (4) \(4x - 2y = 1\)

The following statement \((p\to q)\to\left[(\sim p\to q)\to q\right]\) is :

• (1) equivalent to \(\sim p\to q\)
• (2) equivalent to \(p\to\sim q\)
• (3) a fallacy
• (4) a tautology

A box contains \(15\) green and \(10\) yellow balls. If \(10\) balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :

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

The distance of the point \( Q(0, 2, -2) \) from the line passing through the point \( P(5, -4, 3) \) and perpendicular to the lines \[ \vec{r} = (-3\hat{i} + 2\hat{k}) + \lambda (2\hat{i} + 3\hat{j} + 5\hat{k}), \quad \lambda \in \mathbb{R} \] and \[ \vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu (-\hat{i} + 3\hat{j} + 2\hat{k}), \quad \mu \in \mathbb{R} \] is

• (1) \( \sqrt{86} \)
• (2) \( \sqrt{20} \)
• (3) \( \sqrt{54} \)
• (4) \( \sqrt{74} \)

A hyperbola passes through the point \(P\left(\sqrt{2},\sqrt{3}\right)\) and has foci at \((\pm2,0)\). Then the tangent to this hyperbola at P also passes through the point :

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

The value of \((^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + (^{21}C_{3} - ^{10}C_{3}) +(^{21}C_{4} - ^{10}C_{4}) +....+(^{21}C_{10} - ^{10}C_{10})\) is :

• (1) \(2^{21} - 2^{10}\)
• (2) \(2^{20} - 2^{9}\)
• (3) \(2^{20} - 2^{10}\)
• (4) \(2^{21} - 2^{11}\)

If the image of the point \(P(1,-2,3)\) in the plane \(2x+3y-4z+22=0\) measured parallel to the line \(\frac{x}{1}=\frac{y}{4}=\frac{z}{5}\) is Q, then PQ is equal to:

• (1) \(2\sqrt{42}\)
• (2) \(\sqrt{42}\)
• (3) \(6\sqrt{5}\)
• (4) \(3\sqrt{5}\)

For any three positive real numbers a, b and c, \(9(25a^2 + b^2) + 25 (c^2 - 3ac) = 15b (3a + c)\). Then :

• (1) \(b, c\) and \(a\) are in \(A.P\)
• (2) \(a, b\) and \(c\) are in \(A.P\)
• (3) \(a, b\) and \(c\) are in \(G.P\)
• (4) \(b, c\) and \(a\) are in \(G.P\)

\(C_p\) and \(C_v\) are specific heats at constant pressure and constant volume respectively. It is observed that \(C_p-C_v\) = a for hydrogen gas \(C_p-C_v\) = b for nitrogen gas The correct relation between \(a\) and \(b\) is :

• (1) \(a = \frac{1}{14} b \)
• (2) \(a = b \)
• (3) \(a = 14 \, b \)
• (4) \(a = 28 \, b \)

A diverging lens with magnitude of focal length \(25\, cm\) is placed at a distance of \(15\, cm\) from a converging lens of magnitude of focal length \(20\, cm\). A beam of parallel light falls on the diverging lens. The final image formed is :

• (1) real and at a distance of 40 cm from convergent lens
• (2) virtual and at a distance of 40 cm from convergent lens
• (3) real and at a distance of 40 cm from the divergent lens
• (4) real and at a distance of 6 cm from the convergent lens

As shown in the figure, the ratio of \(T_1\) and \(T_2\) is 

• (1) \(\frac{5}{3}\)
  
• (2) \(\frac{4}{3}\)
  
• (3) \(\frac{10}{3}\)
  
• (4) \(\frac{6}{3}\)

In a coil of resistance \(100 \, \Omega\), a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is :

• (1) 200 Wb
• (2) 225 Wb
• (3) 250 Wb
• (4) 275 Wb

In a Young's double slit experiment, slits are separated by 0.5 mm, and the screen is placed 150 cm away. A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes on the screen. The least distance from the common central maximum to the point where the bright fringes due to both the wavelengths coincide is

• (1) 15.6 mm
• (2) 1.56 mm
• (3) 7.8 mm
• (4) 9.75 mm

Which of the following statements is false?

• (1) Kirchhoff's second law represents energy conservation.
• (2) Wheatstone bridge is the most sensitive when all the four resistances are of the same order of magnitude.
• (3) In a balanced wheatstone bridge if the cell and the galvanometer are exchanged, the null point is disturbed.
• (4) A rheostat can be used as a potential divider.

The following observations were taken for determining surface tension T of water by capillary method: diameter of capillary, \(D = 1.25 \times 10^{-2} \, m\)rise of water, \(h = 1.45 \times 10^{-2} m\). Using \(g = 9.80 \, m/s^2\) and the simplified relation \(T = \frac{rhg}{2} \times 10^3 \, N/m\) , the possible error in surface tension is closest to:

• (1) \(0.15\%\)
• (2) \(1.5\%\)
• (3) \(2.4\%\)
• (4) \(10\%\)

A particle is executing simple harmonic motion with a time period \(T\). At time \(t = 0\), it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like :

• (1)
• (2)
• (3)
• (4)

An electron beam is accelerated by a potential difference \(V\) to hit a metallic target to produce \(X-\)rays. It produces continuous as well as characteristic \(X\)-rays. If \(\lambda_{\text{min}}\) is the smallest possible wavelength of \(X\)-ray in the spectrum, the variation of \(\log \, \lambda_{\text{min}}\) with log \(V\) is correctly represented in :

• (1)
• (2)
• (3)
• (4)

When a current of \(5 \,mA\) is passed through a galvanometer having a coil of resistance \(15 \, \Omega\), it shows full scale deflection. The value of the resistance to be put in series with the galvanometer to convert it into a voltmeter of range \(0-10 \,V\) is :

• (1) \(1.985 \times 10^3 \, \Omega\)
• (2) \(2.045 \times 10^3 \, \Omega\)
• (3) \(2.535 \times 10^3 \, \Omega\)
• (4) \(4.005 \times 10^3 \, \Omega\)

A capacitance of \(2 \, \mu F\) is required in an electrical circuit across a potential difference of \(1.0\, kV\). A large number of \(1 \, \mu F\) capacitors are available which can withstand a potential difference of not more than \(300\, V\). The minimum number of capacitors required to achieve this is :

• (1) 2
• (2) 16
• (3) 24
• (4) 32

A man grows into a giant such that his linear dimensions increase by a factor of \(9\). Assuming that his density remains same, the stress in the leg will change by a factor of :

• (1) 9
• (2) \(\frac{1}{9}\)
• (3) 81
• (4) \(\frac{1}{81}\)

A body of mass \(m=10^{-2} \, kg\) is moving in a medium and experiences a frictional force \(F=-kv^2\). Its initial speed is \(v_0=10 \, ms^{-1}\). If, after \(10\,s\), its energy is \(\frac{1}{8} mv_0{^{2}}\), the value of \(k\) will be :

• (1) \(10^{-3} kg \, m^{-1}\)
• (2) \(10^{-3} kg \, s^{-1}\)
• (3) \(10^{-4} kg \, m^{-1}\)
• (4) \(10^{-1} kg \, m^{-1} s^{-1}\)

A copper ball of mass \(100\, gm\) is at a temperature \(T\). It is dropped in a copper calorimeter of mass \(100\, gm\), filled with \(170\, gm\) of water at room temperature. Subsequently, the temperature of the system is found to be \(75^{\circ} C\). \(T\) is given by : (Given : room temperature = \(30^{\circ}C\), specific heat of copper = \(0.1\, cal\,g^{-1}\,^{\circ} C^{-1}\))

• (1) \(800^{\circ}C\)
• (2) \(885^{\circ}C\)
• (3) \(1250^{\circ}C\)
• (4) \(825^{\circ}C\)

The variation of acceleration due to gravity \(g\) with distance \(d\) from centre of the earth is best represented by (R = Earth?s radius) :

• (1)
• (2)
• (3)
• (4)

The temperature of an open room of volume \(30 \, m^3\) increases from \(17^{\circ}C\) to \(27^{\circ}C\) due to the sunshine. The atmospheric pressure in the room remains \(1 \times 10^5 \, Pa\). If \(n_i\) and \(n_f\) are the number of molecules in the room before and after heating, then \(n_f-n_i\) will be :

• (1) \( - 1.61 \times 10^{23}\)
• (2) \(1.38 \times 10^{23}\)
• (3) \(2.5 \times 10^{25}\)
• (4) \( - 2.5 \times 10^{25}\)

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time?

• (1)
• (2)
• (3)
• (4)

A time dependent force \(F = 6t\) acts on a particle of mass \(1\,kg\). If the particle starts from rest, the work done by the force during the first \(1\,sec\). will be :

• (1) 4.5 J
• (2) 22 J
• (3) 9 J
• (4) 18 J

The moment of inertia of a uniform cylinder of length \(l\) and radius \(R\) about its perpendicular bisector is \(l\). What is the ratio \(\frac{l}{R}\) such that the moment of inertia is minimum ?

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

A slender uniform rod of mass \(M\) and length \(l\) is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle \(\theta\) with the vertical is :

• (1) \(\frac{3g}{2 l} \sin \, \theta\)
• (2) \(\frac{2g}{3l} \sin \, \theta\)
• (3) \(\frac{3g}{2l} \cos\, \theta\)
• (4) \(\frac{2g}{2l} \sin\, \theta\)

An external pressure \(P\) is applied on a cube at \(0^{\circ}C\) so that it is equally compressed from all sides.\(K\) is the bulk modulus of the material of the cube and \(\alpha\) is its coefficient of linear expansion.Suppose we want to bring the cube to its original size by heating it. The temperature should be raised by :

• (1) \(\frac{P}{3 \alpha K}\)
• (2) \(\frac{P}{ \alpha K}\)
• (3) \(\frac{3 \alpha}{P K}\)
• (4) \(3 P K \alpha \)

A particle A of mass \(m\) and initial velocity \(v\) collides with a particle B of mass \(m/2\) which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths \(\lambda_A\) to \(\lambda_B\) after the collision is:

• (1) \(\frac{\lambda_A}{\lambda_B}=\frac{1}{3}\)
• (2) \(\frac{\lambda_A}{\lambda_B}=2\)
• (3) \(\frac{\lambda_A}{\lambda_B}=\frac{2}{3}\)
• (4) \(\frac{\lambda_A}{\lambda_B}=\frac{1}{2}\)

In the above circuit the current in each resistance is

• (1) 1 A
• (2) 0.25 A
• (3) 0.5 A
• (4) 0 A

In the given circuit diagram when the current reaches steady state in the circuit, the charge on the capacitor of capacitance \(C\) will be:

• (1) \(CE\,\frac{r_1}{(r_1 + r)}\)
• (2) \(CE\)
• (3) \(CE\,\frac{r_1}{(r_2 + r)}\)
• (4) \(CE\,\frac{r_2}{(r + r_2)}\)

A magnetic needle of magnetic moment \(6.7 \times 10^{-2} \, Am^2\) and moment of inertia \(7.5 \times 10^{-6} \, kg \, m^2\) is performing simple harmonic oscillations in a magnetic field of \(0.01\, T\). Time taken for \(10\) complete oscillations is :

• (1) 6.65 s
• (2) 8.89 s
• (3) 6.98 s
• (4) 8.76 s

An electric dipole has a fixed dipole moment \(\vec{p}\), which makes angles \(\theta\) with respect to x-axis. When subjected to an electric field \(\vec{E}_1 = E \hat{i}\), it experiences a torque \(\vec{T}_1 = \tau \hat{k}\). When subjected to another electric field \(\vec{E}_2 = \sqrt{3 E_1} \hat{j}\) it experiences a torque \(\vec{T}_2 = - \vec{T}_1\). The angle \(\theta\) is

• (1) \(30^{\circ}\)
• (2) \(45^{\circ}\)
• (3) \(60^{\circ}\)
• (4) \(90^{\circ}\)

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency \(10\, GHz\). What is the frequency of the microwave measured by the observer ? (speed of light \(= 3 \times 10^8 \, ms^{-1}\))

• (1) 10.1 GHz
• (2) 12.1 GHz
• (3) 17.3 GHz
• (4) 15.3 GHz

A radioactive nucleus \(A\) with a half-life \(T\), decays into a nucleus \(B\). At \(t = 0\), there is no nucleus \(B\). At sometime t, the ratio of the number of \(B\) to that of \(A\) is \(0.3\). Then, \(t\) is given by :

• (1) \(t = \frac{T}{2} \frac{\log \, 2}{\log \, (1.3)}\)
• (2) \(t = T \frac{\log \, (1.3)}{\log \, 2}\)
• (3) \(t = T \, \log (1.3)\)
• (4) \(t = \frac{T }{\log \, (1.3)}\)

In a common emitter amplifier circuit using an n-p-n transistor, the phase difference between the input and the output voltages will be:

• (1) \(180^\circ\)
• (2) \(45^\circ\)
• (3) \(90^\circ\)
• (4) \(135^\circ\)

In amplitude modulation, sinusoidal carrier frequency used is denoted by \(\omega_c\) and the signal frequency is denoted by \(\omega_m\). The bandwidth (\(\Delta\omega_m\)) of the signal is such that \(\Delta\omega_m \lt\lt \omega_c\). Which of the following frequencies is not contained in the modulated wave?

• (1) \(\omega_m\)
• (2) \(\omega_c\)
• (3) \(\omega_m + \omega_c\)
• (4) \(\omega_c - \omega_m\)

The major product obtained in the following reaction is
 

• (1)
• (2)
• (3)
• (4)

Which of the following reactions is an example of a redox reaction?

• (1) \(XeF _{6}+ H _{2} O \rightarrow XeOF _{4}+2 HF\)
• (2) \(XeF _{6}+2 H _{2} O \rightarrow XeO _{2} F _{2}+4 HF\)
• (3) \(XeF _{4}+ O _{2} F _{2} \rightarrow XeF _{6}+ O _{2}\)
• (4) \(XeF _{2}+ PF _{5} \rightarrow[ XeF ]^{+} PF _{6}^{-}\)

The correct sequence of reagents for the following conversion will be :

• (1) \(\mathrm{CH_3MgBr,\ H^+/\,CH_3OH,\ [Ag(NH_3)_2]^+OH^-}\)
• (2) \(\mathrm{[Ag(NH_3)_2]^+OH^-,\ CH_3MgBr,\ H^+/\,CH_3OH}\)
• (3) \(\mathrm{[Ag(NH_3)_2]^+OH^-,\ H^+/\,CH_3OH,\ CH_3MgBr}\)
• (4) \(\mathrm{CH_3MgBr,\ [Ag(NH_3)_2]^+OH^-,\ H^+/\,CH_3OH}\)

On treatment of \(100\, mL\) of \(0.1\, M\) solution of \(CoCl _{3} \cdot 6 H _{2} O\) with excess \(AgNO _{3} ; 1.2 \times 10^{22}\) ions are precipitated. The complex is

• (1) \(\left[ Co \left( H _{2} O \right)_{6}\right] Cl _{3}\)
• (2) \(\left[ Co \left( H _{2} O \right)_{5} Cl \right] Cl _{2} \cdot H _{2} O\)
• (3) \(\left[ Co \left( H _{2} O \right)_{4} Cl _{2}\right] Cl \cdot 2 H _{2} O\)
• (4) \(\left[ Co \left( H _{2} O \right)_{3} Cl _{3}\right] \cdot 3 H _{2} O\)

The major product obtained in the following reaction is :

• (1) \(\mathrm{C_6H_5CH=CHC_6H_5}\)
• (2) \((+)\text{-}\mathrm{C_6H_5CH(O\text{-}t\text{-}Bu)CH_2C_6H_5}\)
• (3) \((-)\text{-}\mathrm{C_6H_5CH(O\text{-}t\text{-}Bu)CH_2C_6H_5}\)
• (4) \((\pm)\text{-}\mathrm{C_6H_5CH(O\text{-}t\text{-}Bu)CH_2C_6H_5}\)

The increasing order of the reactivity of the following compounds towards electrophilic aromatic substitution reactions is :

• (1) \(\mathrm{III} \lt \mathrm{II} \lt \mathrm{I}\)
• (2) \(\mathrm{III} \lt \mathrm{I} \lt \mathrm{II}\)
• (3) \(\mathrm{II} \lt \mathrm{I} \lt \mathrm{III}\)
• (4) \(\mathrm{I} \lt \mathrm{III} \lt \mathrm{II}\)

Given \(E _{ Cl _{2} / Cl ^{-}}^{\circ}=1 . 36\, V,\, E _{ Cr ^{3+} / Cr }^{\circ}=-7 \cdot 74\, V\) \(E _{ Cr _{2} O _{7}^{2-} / Cr ^{3+}}^{\circ}=1 .33\, V,\, E _{ MnO _{4}^{-} / Mn ^{2}+}^{\circ}=1. 51\, V\) Among the following, the strongest reducing agent is:

• (1) \(Cr^{3+}\)
• (2) \(Cl^{-}\)
• (3) \(Cr\)
• (4) \(Mn^{2+}\)

The freezing point of benzene decreases by \(0.45^\circ\mathrm{C}\) when \(0.2\ \mathrm{g}\) of acetic acid is added to \(20\ \mathrm{g}\) of benzene. If acetic acid associates to form a dimer in benzene, percentage association of acetic acid in benzene will be:
(\(K_f\) for benzene \(= 5.12\ \mathrm{K\ kg\ mol^{-1}}\))

• (1) \(80.4\ \%\)
• (2) \(74.6\ \%\)
• (3) \(94.6\ \%\)
• (4) \(64.6\ \%\)

Which of the following molecules is least resonance stabilized?

• (1)
• (2)
• (3)
• (4)

The group having isoelectronic species is:

• (1) \(\mathrm{O^{2-},\ F^-,\ Na^+,\ Mg^{2+}}\)
• (2) \(\mathrm{O^{2-},\ F^-,\ Na,\ Mg^{2+}}\)
• (3) \(\mathrm{O^-,\ F^-,\ Na^+,\ Mg^{2+}}\)
• (4) \(\mathrm{O^-,\ F^-,\ Na,\ Mg^+}\)

\(3\)-Methylpent-\(2\)-ene on reaction with \(HBr\) in presence of peroxide forms an addition product. The number of possible stereoisomers for the product is :

• (1) Two
• (2) Four
• (3) Six
• (4) Zero

Sodium salt of an organic acid 'X' produces effervescence with conc. \(\mathrm{H_2SO_4}\). 'X' reacts with the acidified aqueous \(\mathrm{CaCl_2}\) solution to give a white precipitate which decolourises acidic solution of \(\mathrm{KMnO_4}\). 'X' is :

• (1) \(\mathrm{CH_3COONa}\)
• (2) \(\mathrm{Na_2C_2O_4}\)
• (3) \(\mathrm{C_6H_5COONa}\)
• (4) \(\mathrm{HCOONa}\)

Pre-exponential factors of two different reactions of same order are identical. Let activation energy of first reaction exceed the activation energy of second reaction by 20 kJ mol\(^{-1}\). If \(k_1\) and \(k_2\) are the rate constants of first and second reaction respectively at 300 K, then \(\ln \frac{k_{2}}{k_{1}}\) will be___. (nearest integer) [\(R=8.3 \text{ J K}^{-1} \text{ mol}^{-1}\)]

Given,
\(\mathrm{C_{(graphite)} + O_2 \rightarrow CO_2(g)}\); \(\Delta_r H^\circ = -393.5\ \mathrm{kJ\ mol^{-1}}\)
\(\mathrm{H_2(g) + \frac{1}{2}O_2(g) \rightarrow H_2O(l)}\); \(\Delta_r H^\circ = -285.8\ \mathrm{kJ\ mol^{-1}}\)
\(\mathrm{CO_2(g) + 2H_2O(l) \rightarrow CH_4(g) + 2O_2(g)}\); \(\Delta_r H^\circ = +890.3\ \mathrm{kJ\ mol^{-1}}\)
Based on the above thermochemical equations, the value of \(\Delta_r H^\circ\) at 298 K for the reaction \(\mathrm{C_{(graphite)} + 2H_2(g) \rightarrow CH_4(g)}\) will be :

• (1) \(+144.0\ \mathrm{kJ\ mol^{-1}}\)
• (2) \(-74.8\ \mathrm{kJ\ mol^{-1}}\)
• (3) \(-144.0\ \mathrm{kJ\ mol^{-1}}\)
• (4) \(+74.8\ \mathrm{kJ\ mol^{-1}}\)

\(\Delta U \) is equal to :

• (1) Adiabatic work
• (2) Isothermal work
• (3) Isochoric work
• (4) Isobaric work

Which of the following, upon treatment with tert-BuONa followed by addition of bromine water, fails to decolorize the colour of bromine?

• (1)
• (2)
• (3)
• (4)

Which of the following compounds will form significant amount of meta product during mono-nitration reaction?

• (1)
• (2)
• (3)
• (4)

In the following reactions, \(\mathrm{ZnO}\) is respectively acting as a/an :
(a) \(\mathrm{ZnO + Na_2O \rightarrow Na_2ZnO_2}\)
(b) \(\mathrm{ZnO + CO_2 \rightarrow ZnCO_3}\)

• (1) base and base
• (2) acid and acid
• (3) acid and base
• (4) base and acid

\(pK_a\) of a weak acid (HA) and \(pK_b\) of a weak base (BOH) are \(3.2\) and \(3.4,\) respectively. .The pH of their salt (AB) solution is :

• (1) 7
• (2) 1
• (3) 7.2
• (4) 6.9

Which of the following compounds will behave as a reducing sugar in an aqueous KOH solution?

• (1)
• (2)
• (3)
• (4)

The most abundant elements by mass in the body of a healthy human adult are: Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%); and Nitrogen (2.6%). The weight which a 75 kg person would gain if all \(^{1}\mathrm{H}\) atoms are replaced by \(^{2}\mathrm{H}\) atoms is:

• (1) 37.5 kg
• (2) 7.5 kg
• (3) 10 kg
• (4) 15 kg

Which of the following species is not paramagnetic?

• (1) \(O_2\)
• (2) \(B_2\)
• (3) \(NO\)
• (4) \(CO\)

1 gram of a carbonate \((M_2CO_3)\) on treatment with excess \(HCl\) produces \(0.01186\) mole of \(CO_2\). The molar mass of \(M_2CO_3\) in g \(mol^{-1}\) is :

• (1) \(118.6\)
• (2) \(11.86\)
• (3) \(1186\)
• (4) \(84.3\)

The radius of the second Bohr orbit for hydrogen atom is: (Planck's Const. \(h =6 . 6262 \times 10^{-34}\, Js\); mass of electron \(=9 . 1091 \times 10^{-31}\, Kg\) charge of electron \(e =1 .60210 \times 10^{-19} C\); permittivity of vacuum \(\left.\left(\epsilon_{0}\right)=8 \cdot 854185 \times 10^{-12}\, Kg ^{-1}\, m ^{-3} A ^{2}\right)\)

• (1) 0.529\, ??
• (2) 2.12\, ??
• (3) 1.65\, ??
• (4) 4.76\, ??

Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect, is :

• (1) Both form nitrides
• (2) Nitrates of both \(Li\) and \(Mg\) yield \(NO_2\) and \(O_2\) on heating
• (3) Both form basic carbonates
• (4) Both form soluble bicarbonates

The formation of which of the following polymers involves hydrolysis reaction?

• (1) Bakelite
• (2) Nylon 6, 6
• (3) Terylene
• (4) Nylon 6

A water sample has ppm level concentration of following anions \(F^{-} = 10 ;\, SO^{2-}_4 = 100 ;\, NO^{-}_3 = 50\) The anion/anions that make/makes the water sample unsuitable for drinking is / are :

• (1) only \(F^{-}\)
• (2) only \(SO_4^{2-}\)
• (3) only \(NO_4^{2-}\)
• (4) only \(SO_4^{2-}\) and \(NO^{-}_3\)

The products obtained when chlorine gas reacts with cold and dilute aqueous $NaOH$ are:

• (1) (A) $C{l}^{-}$ and $Cl{O}^{-}$
• (2) (B) $C{l}^{-}$ and $Cl{O}_2^{-}$
  
• (3) (C) $C{l}^{-}$ and $Cl{O}_3^{-}$
  
• (4) (D) $C{l}^{-}$ and $Cl{O}_4^{-}$
  

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is \('a'\), the closest approach between two atoms in metallic crystal will be :

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

The Tyndall effect is observed only when following conditions are satisfied (a) The diameter of the dispersed particles is much smaller than the wavelength of the light used. (b) The diameter of the dispersed particle is not much smaller than the wavelength of the light used (c) The refractive indices of the dispersed phase and dispersion medium are almost similar in magnitude (d) The refractive indices of the dispersed phase and dispersion medium differ greatly in magnitude

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