JEE Main 2019 B.E./B.Tech Question Paper with Answer Keys January 10 Forenoon Session

If the system of equations
\(x+y+z=5\)
\(x+2y+3z=9\)
\(x+3y+\alpha z=\beta\)
has infinitely many solutions, then \(\beta-\alpha\) equals -

• (1) 8
• (2) 21
• (3) 18
• (4) 5

Let \(\vec{a}=2\hat{i}+\lambda_1\hat{j}+3\hat{k}\), \(\vec{b}=4\hat{i}+(3-\lambda_2)\hat{j}+6\hat{k}\) and \(\vec{c}=3\hat{i}+6\hat{j}+(\lambda_3-1)\hat{k}\) be three vectors such that \(\vec{b}=2\vec{a}\) and \(\vec{a}\) is perpendicular to \(\vec{c}\). Then a possible value of \((\lambda_1,\lambda_2,\lambda_3)\) is :

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

If \(\frac{dy}{dx}+\frac{3}{\cos^2 x}\,y=\frac{1}{\cos^2 x}\), \(x\in\left(-\frac{\pi}{3},\frac{\pi}{3}\right)\) and \(y\left(\frac{\pi}{4}\right)=\frac43\), then \(y\left(-\frac{\pi}{4}\right)\) equals -

• (1) \(\frac13+e^6\)
• (2) \(\frac13\)
• (3) \(\frac13+e^3\)
• (4) \(-\frac43\)

For each \(t\in\mathbb{R}\), let \([t]\) be the greatest integer less than or equal to \(t\). Then \(\displaystyle\lim_{x\to1^+}\frac{(1-|x|+\sin|1-x|)\,\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|\,[1-x]}\)

• (1) equals \(-1\)
• (2) equals 1
• (3) equals 0
• (4) does not exist

Let \(f(x)=\begin{cases}\max\{|x|,x^2\} & |x|\le2\\ 8-2|x| & 2<|x|\le4\end{cases}\)
Let \(S\) be the set of points in the interval \((-4,4)\) at which \(f\) is not differentiable. Then \(S\)

• (1) equals \(\{-2,-1,1,2\}\)
• (2) equals \(\{-2,-1,0,1,2\}\)
• (3) equals \(\{-2,2\}\)
• (4) is an empty set

The area enclosed between the curves \(y = x|x|\) and \(y = x - |x|\) is:

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

A point \(P\) moves on the line \(2x-3y+4=0\). If \(Q(1,4)\) and \(R(3,-2)\) are fixed points, then the locus of the centroid of \(\triangle PQR\) is a line :

• (1) parallel to y-axis
• (2) with slope \(\frac23\)
• (3) parallel to x-axis
• (4) with slope \(\frac32\)

Let \(z_1\) and \(z_2\) be any two non-zero complex numbers such that \(3|z_1| = 4 |z_2|\). If \(z = \frac{3z_{1}}{2z_{2}} + \frac{2z_{2}}{3z_{1}}\) then :

• (1) \(\left|z\right| = \frac{1}{2} \sqrt{\frac{17}{2}}\)
• (2) \(Re(z) = 0\)
• (3) \(\left|z\right| = \sqrt{\frac{5}{2}}\)
• (4) \(Im(z) = 0\)

If \(5,\ 5r,\ 5r^2\) are the lengths of the sides of a triangle, then \(r\) cannot be equal to :

• (1) \(\frac74\)
• (2) \(\frac54\)
• (3) \(\frac34\)
• (4) \(\frac32\)

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered \(1,2,3,\ldots,9\) is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :

• (1) \(\frac{19}{36}\)
• (2) \(\frac{15}{72}\)
• (3) \(\frac{13}{36}\)
• (4) \(\frac{19}{72}\)

If the line \(3x+4y-24=0\) intersects the x-axis at the point \(A\) and the y-axis at the point \(B\), then the incentre of the triangle \(OAB\), where \(O\) is the origin, is :

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

If the third term in the binomial expansion of \(\left(1+x^{\log_2x}\right)^5\) equals 2560, then a possible value of \(x\) is -

• (1) \(2\sqrt2\)
• (2) \(4\sqrt2\)
• (3) \(\frac18\)
• (4) \(\frac14\)

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -

• (1) 1356
• (2) 1256
• (3) 1365
• (4) 1465

Let \(I=\displaystyle\int_a^b\left(x^4-2x^2\right)dx\). If \(I\) is minimum then the ordered pair \((a,b)\) is -

• (1) \(\left(\sqrt2,-\sqrt2\right)\)
• (2) \(\left(0,\sqrt2\right)\)
• (3) \(\left(-\sqrt2,\sqrt2\right)\)
• (4) \(\left(-\sqrt2,0\right)\)

Let \(d\in\mathbb{R}\), and \(A=\begin{bmatrix}-2&4+d&(\sin\theta)-2\\1&(\sin\theta)+2&d\\5&(2\sin\theta)-d&(-\sin\theta)+2+2d\end{bmatrix}\), \(\theta\in[0,2\pi]\). If the minimum value of \(\det(A)\) is 8, then a value of \(d\) is -

• (1) \(-7\)
• (2) \(2(\sqrt2+2)\)
• (3) \(-5\)
• (4) \(2(\sqrt2+1)\)

The shortest distance between the point \(\left(\frac32,0\right)\) and the curve \(y=\sqrt{x}\), \((x>0)\), is -

• (1) \(\frac{\sqrt3}{2}\)
• (2) \(\frac54\)
• (3) \(\frac32\)
• (4) \(\frac{\sqrt5}{2}\)

Let \(f:\mathbb{R}\to\mathbb{R}\) be a function such that \(f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)\), \(x\in\mathbb{R}\). Then \(f(2)\) equals -

• (1) 30
• (2) \(-2\)
• (3) \(-4\)
• (4) 8

The number of integral values of \(k\), for which one root of the equation \[2x^2 - 8x + k = 0\] lies in the interval \((1, 2)\) and its other root lies in the interval \((2, 3)\), is:

• (1) 2
• (2) 0
• (3) 1
• (4) 2

In a class of \(140\) students numbered \(1\) to \(140\), all even numbered students opted mathematics course, those whose number is divisible by \(3\) opted Physics course and those whose number is divisible by \(5\) opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

• (1) 102
• (2) 42
• (3) 1
• (4) 38

The mean of five observations is \(5\) and their variance is \(9.20.\) If three of the given five observations are \(1, 3\) and \(8\), then a ratio of other two observations is :

• (1) 4:09
• (2) 6:07
• (3) 5:08
• (4) 10:03

Let \(n \ge 2\) be a natural number and \(0 \lt \theta \lt \frac{\pi}{2}\). Then
\(\int \frac{(\sin^n\theta - \sin\theta)^{1/n}\cos\theta}{\sin^{n+1}\theta}\,d\theta\)
is equal to - (where C is a constant of integration)

• (1) \(\frac{n}{n^2-1}\left(1+\frac{1}{\sin^{n-1}\theta}\right)^{\frac{n+1}{n}}+C\)
• (2) \(\frac{n}{n^2-1}\left(1-\frac{1}{\sin^{n+1}\theta}\right)^{\frac{n+1}{n}}+C\)
• (3) \(\frac{n}{n^2-1}\left(1-\frac{1}{\sin^{n-1}\theta}\right)^{\frac{n+1}{n}}+C\)
• (4) \(\frac{n}{n^2+1}\left(1-\frac{1}{\sin^{n-1}\theta}\right)^{\frac{n+1}{n}}+C\)

The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:

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

If a circle \(C\) passing through the point \((4,0)\) touches the circle \(x^2 + y^2 + 4x - 6y = 12\) externally at the point \((1, -1)\), then the radius of \(C\) is :

• (1) \(\sqrt{57}\)
• (2) \(4\)
• (3) \(2 \sqrt{5}\)
• (4) \(5\)

The equation of a tangent to the hyperbola \(4x^2 - 5y^2 = 20\) parallel to the line \(x - y = 2\) is :

• (1) \(x - y + 9 = 0\)
• (2) \(x - y - 3 = 0\)
• (3) \(x - y + 1 = 0\)
• (4) \(x - y + 7 = 0\)

Consider the statement : "P(n) : \(n^2 - n + 41\) is prime". Then which one of the following is true ?

• (1) P(5) is false but P(3) is true
• (2) Both P(3) and P(5) are true
• (3) P(3) is false but P(5) is true
• (4) Both P(3) and P(5) are false

Let A be a point on the line \(\vec{r} = (1-3\mu)\hat{i} + (\mu-1)\hat{j} + (2+5\mu)\hat{k}\) and B(3, 2, 6) be a point in the space. Then the value of \(\mu\) for which the vector \(\overrightarrow{AB}\) is parallel to the plane \(x - 4y + 3z = 1\) is -

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

Consider a triangular plot \(ABC\) with sides \(AB = 7m, BC = 5m\) and \(CA = 6m\). A vertical lamp-post at the mid point \(D\) of \(AC\) subtends an angle \(30^{\circ}\) at \(B\). The height (in \(m\)) of the lamp-post is:

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

If the parabolas \(y^2 = 4b(x - c)\) and \(y^2 = 8ax\) have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

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

The plane passing through the point \((4, -1, 2)\) and parallel to the lines
\(\frac{x+2}{3} = \frac{y-2}{-1} = \frac{z+1}{2}\) and \(\frac{x-2}{1} = \frac{y-3}{2} = \frac{z-4}{3}\)
also passes through the point -

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

If \(\sum_{i=1}^{20}\left(\frac{{}^{20}C_{i-1}}{{}^{20}C_i + {}^{20}C_{i-1}}\right)^3 = \frac{k}{21}\) then k is equal to

• (1) 100
• (2) 200
• (3) 50
• (4) 400

A block of mass m is kept on a platform which starts from rest with constant acceleration g/2 upward, as shown in figure. Work done by normal reaction on block in time \(t\) is -

• (1) \(\frac{mg^2t^2}{8}\)
• (2) \(\frac{3mg^2t^2}{8}\)
• (3) \(-\frac{mg^2t^2}{8}\)
• (4) 0

A uniform metallic wire has a resistance of \(18 \; \Omega\) and is bent into an equilateral triangle. Then, the resistance between any two vertices of the triangle is :

• (1) \( 8 \; \Omega \)
• (2) \( 12 \; \Omega \)
• (3) \( 4 \; \Omega \)
• (4) \( 2 \; \Omega \)

An insulating thin rod of length \(\ell\) has a x linear charge density \(p(x) = \rho_0 \frac{x}{\ell}\) on it . The rod is rotated about an axis passing through the origin \((x = 0)\) and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is :

• (1) \(\frac{\pi }{4} n \rho \ell^3\)
• (2) \( n \rho \ell^3\)
• (3) \( \pi n \rho \ell^3\)
• (4) \(\frac{\pi }{3} n \rho \ell^3\)

In Young's double-slit experiment with slit separation $0.1$ mm, one observes a bright fringe at angle $\frac{1}{4^{\circ }}$ rad by using the light of wavelength ${\lambda }_1$. When the light of wavelength ${\lambda }_2$ is used a bright fringe is seen at the same angle in the same setup. Given that $1$ and $2$ are in the visible range $(380nm$ to $740nm)$, their values are

• (1) (A) $400nm,500nm$
  
• (2) (B) $2500nm,1250nm$
  
• (3) (C) $380nm,525nm$
  
• (4) (D) $380nm,500nm$
  

A homogeneous solid cylindrical roller of radius \(R\) and mass \(M\) is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is :

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

In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons used. To resolve a width of \(7.5\times10^{-12}\) m, the minimum electron energy required is close to -

• (1) 25 keV
• (2) 500 keV
• (3) 100 keV
• (4) 1 keV

A plano convex lens of refractive index \(\mu_1\) and focal length \(f_1\) is kept in contact with another plano concave lens of refractive index \(\mu_2\) and focal length \(f_2\). If the radius of curvature of their spherical faces is R each and \(f_1 = 2f_2\), then \(\mu_1\) and \(\mu_2\) are related as :

• (1) \(\mu_1 + \mu_2 = 3\)
• (2) \(2 \mu_1 - \mu_2 = 1\)
• (3) \(2\mu_2 - \mu_1 = 1\)
• (4) \(3 \mu_2 - 2\mu_1 = 1\)

In the given circuit the cells have zero internal resistance. The currents (in Amperes) passing through resistance \(R_1\), and \(R_2\) respectively, are:

• (1) 44594
• (2) 0, 1
• (3) 44563
• (4) 0.5 , 0

A string of length \(1\, m\) and mass \(5\, g\) is fixed at both ends. The tension in the string is \(8.0\, N\). The siring is set into vibration using an external vibrator of frequency \(100\, Hz\). The separation between successive nodes on the string is close to :

• (1) 16.6 cm
• (2) 20.0 cm
• (3) 10.0 cm
• (4) 33.3 cm

A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass 'm' is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic energy of the object is -

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

If the magnetic field of a plane electromagnetic wave is given by (the speed of light = \(3\times10^{8}\) B = \(100\times10^{-6}\) sin
\(\left[2\pi\times2\times10^{15}\left(t-\frac{x}{c}\right)\right]\)
then the maximum electric field associated with it is -

• (1) \(4.5\times10^4\) N/C
• (2) \(4\times10^4\) N/C
• (3) \(6\times10^4\) N/C
• (4) \(3\times10^4\) N/C

A \(2\, W\) carbon resistor is color coded with green, black, red and brown respectively. The maximum current which can be passed through this resistor is :

• (1) 63 mA
• (2) 0.4 mA
• (3) 100 mA
• (4) 20 mA

Two guns \(A\) and \(B\) can fire bullets at speeds \(1\, km/s\) and \(2\, km/s\) respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is :

• (1) 1:02
• (2) 1:04
• (3) 1:08
• (4) 1:16

A charge \(Q\) is distributed over two concentric conducting thin spherical shells radii \(r\) and \(R\) \(( R > r )\). If the surface charge densities on the two shells are equal, the electric potential at the common centre is :

• (1) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{( R +2 r ) Q }{2\left( R ^{2}+ r ^{2}\right)}\)
• (2) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{( R + r )}{2\left( R ^{2}+ r ^{2}\right)} Q\)
• (3) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{( R + r )}{\left( R ^{2}+ r ^{2}\right)} Q\)
• (4) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{(2 R+r)}{\left(R^{2}+r^{2}\right)} Q\)

To mop-clean a floor, a cleaning machine presses a circular mop of radius \(R\) vertically down with a total force \(F\) and rotates it with a constant angular speed about its axis. If the force \(F\) is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is \(\mu\), the torque, applied by the machine on the mop is :

• (1) \(\frac{2}{3} \mu FR\)
• (2) \( \mu FR /3\)
• (3) \( \mu FR / 2\)
• (4) \( \mu FR / 6\)

A solid metal cube of edge length 2 cm is moving in a positive y-direction at a constant speed of 6 m/s. There is a uniform magnetic field of 0.1 T in the positive z-direction. The potential difference between the two faces of the cube perpendicular to the x-axis, is -

• (1) 2 mV
• (2) 12 mV
• (3) 6 mV
• (4) 1 mV

Water flows into a large tank with flat bottom at the rate of \(10^{-4} \; m^3 s^{-1}\). Water is also leaking out of a hole of area \(1\, cm^2\) at its bottom. If the height of the water in the tank remains steady, then this height is :

• (1) 4 cm
• (2) 2.9 cm
• (3) 1.7 cm
• (4) 5.1 cm

To get output 1 at R, for the given logic gate circuit the input values must be

• (1) x = 0, y = 0
• (2) x = 1, y = 0
• (3) x = 0, y = 1
• (4) x = 1, y = 1

In the cube of side \('a'\) shown in the figure, the vector from the central point of the face \(ABOD\) to the central point of the face \(BEFO\) will be:

• (1) \(\frac{1}{2} a \left(\hat{i} -\hat{k}\right) \)
• (2) \(\frac{1}{2} a \left(\hat{j} -\hat{i}\right) \)
• (3) \(\frac{1}{2} a \left(\hat{k} -\hat{i}\right) \)
• (4) \(\frac{1}{2} a \left(\hat{j} -\hat{k}\right) \)

Two metallic blocks M₁ and M₂ of the same cross-section area are connected (as shown in the figure). If the thermal conductivity of M₂ is K then the thermal conductivity of M₁ will be : [Assume steady state heat conduction]

• (1) 10K
• (2) 8K
• (3) 12.5K
• (4) 2K

The density of a material in SI units is \(128 \; kg \; m^{-3}\). In certain units in which the unit of length is \(25\, cm\) and the unit of mass is \(50\, g\), the numerical value of density of the material is :

• (1) 410
• (2) 640
• (3) 16
• (4) 40

A magnet of total magnetic moment \(10^{-2} \hat{i} \; A-m^2\) is placed in a time varying magnetic field, \(B \hat{i} (\cos t\omega t)\) where \(B = 1 \)Tesla and \(\omega = 0.125 \; rad/s\). The work done for reversing the direction of the magnetic moment at \(t = 1\) second, is :

• (1) 0.007 J
• (2) 0.02 J
• (3) 0.01 J
• (4) 0.028 J

A parallel plate capacitor is of area \(6 \; cm^2\) and a separation \(3\, mm\). The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants \(K_1, = 10, K_2 = 12 \) and \(K_3 = 14\). The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be :

• (1) 12
• (2) 4
• (3) 36
• (4) 14

Two electric dipoles, \(A, B\) with respective dipole moments \(\vec{d}_A = - 4 \,qa \,\hat{i}\) and \(\vec{d}_B = - 2 \,qa\, \hat{i}\) placed on the x-axis with a separation \(R\), as shown in the figure The distance from \(A\) at which both of them produce the same potential is :

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

A piece of wood of mass \(0.03\, kg\) is dropped from the top of a \(100 \,m\) height building. At the same time, a bullet of mass \(0.02\, kg\) is fired vertically upward, with a velocity \(100 \; ms^{-1}\), from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is : \((g =10\,ms^{-2})\)

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

A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its frequency registered by the observer is \(f_1\). If the speed of the train is reduced to 17 m/s, the frequency registered is \(f_2\). If speed of sound is 340 m/s, then the ratio \(f_1/f_2\) is -

• (1) 19/18
• (2) 20/19
• (3) 21/20
• (4) 18/17

Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At t = 0 it was 1600 counts per second and t = 8 seconds it was 100 counts per second. The count rate observed, as counts per second, at t = 6 seconds is close to -

• (1) 200
• (2) 150
• (3) 400
• (4) 360

Three Carnot engines operate in series between a heat source at a temperature \(T_1\) and a heat sink at temperature \(T_4\) (see figure). There are two other reservoirs at temperature \(T_2\), and \(T_3\), as shown, with \(T_2 > T_2 > T_3 > T_4\) . The three engines are equally efficient if:

• (1) \(T_{2} =\left(T_{1}^{2} T_{4}\right)^{1/3} ; T_{3} =\left(T_{1} T_{4}^{2}\right)^{1/3} \)
• (2) \(T_{2} =\left(T_{1} T_{4}^{2} \right)^{1/3} ; T_{3} =\left(T_{1}^{2} T_{4}\right)^{1/3} \)
• (3) \(T_{2} =\left(T_{1}^{3} T_{4}\right)^{1/4} ; T_{3} =\left(T_{1} T_{4}^{3}\right)^{1/4} \)
• (4) \(T_{2} =\left(T_{1} T_{4}\right)^{1/2} ; T_{3} =\left(T_{1}^{2} T_{4}\right)^{1/3} \)

A potentiometer wire \(AB\) having length \(L\) and resistance \(12\, r\) is joined to a cell \(D\) of emf \(\varepsilon\) and internal resistance \(r\). A cell \(C\) having \(emf \, \varepsilon /2\) and internal resistance \(3r\) is connected. The length \(AJ\) at which the galvanometer as shown in fig. shows no deflection is :

• (1) \(\frac{5}{12}\) L
• (2) \(\frac{11}{24}\) L
• (3) \(\frac{11}{12}\) L
• (4) \(\frac{13}{24}\) L

A TV transmission tower has a height of 140 m and the height of the receiving antenna is 40 m. What is the maximum distance upto which signals can be broadcasted from this tower is LOS (Line of Sight) mode ? (Given : radius of earth = \(6.4 \times 10^{6}\) m).

• (1) 40 km
• (2) 65 km
• (3) 48 km
• (4) 80 km

Which dicarboxylic acid in presence of a dehydrating agent is least reactive to give an anhydride?

• (1)
• (2)
• (3)
• (4)

Liquids \(A\) and \(B\) form an ideal solution in the entire composition range. At \(350\, K\), the vapor pressures of pure \(A\) and pure \(B\) are \(7 \times 10^3\) Pa and \(12 \times 10^3\, Pa\), respectively. The composition of the vapor in equilibrium with a solution containing \(40\) mole percent of \(A\) at this temperature is :

• (1) \(x_A = 0.37; x_B = 0.63\)
• (2) \(x_A = 0.28; x_B = 0.72\)
• (3) \(x_A = 0.76; x_B = 0.24\)
• (4) \(x_A = 0.4; x_B = 0.6\)

Wilkinson catalyst is :

• (1) \({[(Ph3P)3RhCl] }\)
• (2) \({[Et_3P)_3IrCl]}\)
• (3) \({[Et_3P)_3RhCl]}\)
• (4) \({[Ph_3P]_3IrCl]}\)

The increasing order of the pKa values of the following compounds is :

(A) phenol; (B) 4-nitrophenol; (C) 3-nitrophenol; (D) 4-methoxyphenol

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

A mixture of 100 m mol of Ca(OH)\(_2\) and 2 g of sodium sulphate was dissolved in water and the volume was made up to 100 mL. The mass of calcium sulphate formed and the concentration of OH\(^-\) in resulting solution, respectively, are : (Molar mass of Ca (OH)\(_2\), Na\(_2\)SO\(_4\) and CaSO\(_4\) are 74, 143 and 136 g mol\(^{-1}\), respectively; K\(_{sp}\) of Ca(OH)\(_2\) is \(5.5 \times 10^{-6}\) )

• (1) 13.6g, 0.28 mol L\(^{-1}\)
• (2) 13.6g, 0.14 mol L\(^{-1}\)
• (3) 1.9g, 0.28 mol L\(^{-1}\)
• (4) 1.9g, 0.14 mol L\(^{-1}\)

The type of hybridisation and number of lone pair (s) of electrons of Xe in XeOF\(_4\), respectively, are:

• (1) sp\(^3\)d and 2
• (2) sp\(^3\)d\(^2\) and 2
• (3) sp\(^3\)d and 1
• (4) sp\(^3\)d\(^2\) and 1

Consider the following reduction processes: \[ \text{Al}^{3+} + 3e^- \longrightarrow \text{Al(s)}, \quad E^\circ = -1.66\,\text{V} \] \[ \text{Fe}^{3+} + e^- \longrightarrow \text{Fe}^{2+}, \quad E^\circ = +0.77\,\text{V} \] \[ \text{Co}^{3+} + e^- \longrightarrow \text{Co}^{2+}, \quad E^\circ = +1.81\,\text{V} \] \[ \text{Cr}^{3+} + 3e^- \longrightarrow \text{Cr(s)}, \quad E^\circ = -0.74\,\text{V} \] The tendency to act as reducing agent decreases in the order:

• (1) \( \text{Cr}>\text{Fe}^{2+}>\text{Al}>\text{Co}^{2+} \)
• (2) \( \text{Al}>\text{Cr}>\text{Co}^{2+}>\text{Fe}^{2+} \)
• (3) \( \text{Al}>\text{Cr}>\text{Fe}^{2+}>\text{Co}^{2+} \)
• (4) \( \text{Al}>\text{Fe}^{2+}>\text{Cr}>\text{Co}^{2+} \)
  

The electronegativity of aluminium is similar to :

• (1) Beryllium
• (2) Carbon
• (3) Boron
• (4) Lithium

The major product formed in the reaction given below will be :

(Reactant: a hydrindane (cyclohexane fused to cyclopentane) carrying a \(-\mathrm{CH_2NH_2}\) group on the five-membered ring; reagent: NaNO\(_2\), aq. HCl, 0-5\(^\circ\)C.)

• (1)
• (2)
• (3)
• (4)

Which of the graphs shown below does not represent the relationship between incident light and the electron ejected from metal surface ?

• (1)
• (2)
• (3)
• (4)

The total number of isomers for a square planar complex [M(F)(Cl)(SCN)(NO\(_2\))] is

• (1) 16
• (2) 8
• (3) 12
• (4) 4

The correct structure of product 'P' in the following reactions is :
\(\mathrm{Asn - Ser + (CH_3CO)_2O\ (excess)\ \xrightarrow{NEt_3}\ P}\)

• (1)
• (2)
• (3)
• (4)

The effect of lanthanoid contraction in the lanthanoid series of elements by and large means :

• (1) decrease in both atomic and ionic radii
• (2) increase in atomic radii and decrease in ionic radii
• (3) increase in both atomic and ionic radii
• (4) decrease in atomic radii and increase in ionic radii

A process has \(\Delta\)H = 200 J mol\(^{-1}\) and \(\Delta\)S = 40 JK\(^{-1}\) mol\(^{-1}\). Out of the values given below, choose the minimum temperature above which the process will be spontaneous :

• (1) 4 K
• (2) 20 K
• (3) 5 K
• (4) 12 K

The decreasing order of ease of alkaline hydrolysis for the following esters in

(I) ethyl benzoate; (II) ethyl 4-chlorobenzoate; (III) ethyl 4-nitrobenzoate; (IV) ethyl 4-methoxybenzoate

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

Two pi and half sigma bonds are present in:

• (1) \({N_{2}^{+}}\)
• (2) \({N_{2} }\)
• (3) \({O_{2}^{+}}\)
• (4) \({O_2}\)

The values of K\(_p\)/K\(_c\) for the following reactions at 300 K are, respectively : (At 300 K, RT = 24.62 dm\(^3\) atm mol\(^{-1}\))
\(\mathrm{N_2(g) + O_2(g) \rightleftharpoons 2\,NO(g)}\)
\(\mathrm{N_2O_4(g) \rightleftharpoons 2\,NO(g)}\)
\(\mathrm{N_2(g) + 3H_2(g) \rightleftharpoons 2\,NH_3(g)}\)

• (1) \(24.62\ \mathrm{dm^3\,atm\,mol^{-1}},\ 606.0\ \mathrm{dm^6\,atm^2\,mol^{-2}},\ 1.65\times10^{-3}\ \mathrm{dm^{-6}\,atm^{-2}\,mol^{2}}\)
• (2) \(1,\ 4.1\times10^{-2}\ \mathrm{dm^{-3}\,atm^{-1}\,mol},\ 606\ \mathrm{dm^6\,atm^2\,mol^{-2}}\)
• (3) \(1,\ 24.62\ \mathrm{dm^3\,atm\,mol^{-1}},\ 1.65\times10^{-3}\ \mathrm{dm^{-6}\,atm^{-2}\,mol^{2}}\)
• (4) \(1,\ 24.62\ \mathrm{dm^3\,atm\,mol^{-1}},\ 606.0\ \mathrm{dm^6\,atm^2\,mol^{-2}}\)

Consider the given plots for a reaction obeying Arrhenius equation \((0^{\circ}C < T < 300^{\circ}C)\) : (\(k\) and \(E_a\) are rate constant and activation energy, respectively) Choose the correct option :

• (1) Both I and II are wrong
• (2) I is wrong but II is right
• (3) Both I and II are correct
• (4) I is right but II is wrong

If dichloromethane (DCM) and water \({(H_2O)}\) are used for differential extraction, which one of the following statements is correct ?

• (1) DCM and \({(H_2O)}\) would stay as lower and upper layer respectively in the S.F.
• (2) DCM and \({(H_2O)}\) will be miscible clearly
• (3) DCM and \({(H_2O)}\) would stay as upper and lower layer respectively in the separating funnel (S.F.)
• (4) DCM and \({(H_2O)}\) will make trubid/colloidal mixture

The major product 'X' formed in the following reaction is :

(Reactant: a cyclohexenone carrying a \(-\mathrm{CH_2COOCH_3}\) group on the carbon next to C=O; reagent: NaBH\(_4\), MeOH.)

• (1)
• (2)
• (3)
• (4)

The major product of the following reaction is :

(Reactant: \(\mathrm{3\text{-}CH_3O\text{-}C_6H_4\text{-}CH_2CH_2CH(CH_3)CH_2Cl}\); reagents: (i) AlCl\(_3\) (anhyd.), (ii) H\(_2\)O.)

• (1)
• (2)
• (3)
• (4)

The major product of the following reaction is

• (1)
• (2)
• (3)
• (4)

Which hydrogen in compound (E) is easily replaceable during bromination reaction in presence of light ?
\(\mathrm{CH_3-CH_2-CH=CH_2}\) (E)
Labels: \(\mathrm{CH_3}\) = \(\delta\), \(\mathrm{CH_2}\) = \(\gamma\), \(\mathrm{CH}\) = \(\beta\), terminal \(\mathrm{=CH_2}\) = \(\alpha\).

• (1) \(\beta\) - hydrogen
• (2) \(\alpha\) - hydrogen
• (3) \(\gamma\) - hydrogen
• (4) \(\delta\) - hydrogen

The total number of isotopes of hydrogen and number of radioactive isotopes among them, respectively, are :

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

Hall-Heroult's process is given by :

• (1) \({Cr2O3 + 2Al -> Al2O3 + 2Cr}\)
• (2) \({Cu^{2+} (a) + H_2(g) \to Cu(s) + 2H^{+} (aq)}\)
• (3) \({ZnO + C ->[Coke, 1673 K] Zn + CO}\)
• (4) \({2 Al_2 O_3 + 3C \to 4Al + 3CO_2 }\)

The metal used for making \(X\)-ray tube window is :

• (1) \(Mg\)
• (2) \(Na\)
• (3) \(Ca\)
• (4) \(Be\)

Which of the following is not an example of heterogenous catalytic reaction?

• (1) Ostwald's process
• (2) Combustion of coal
• (3) Hydrogenation of vegetable oils
• (4) Haber's process

The chemical nature of hydrogen peroxide is :

• (1) Oxidising agent in acidic medium, but not in basic medium
• (2) Oxidising and reducing agent in both acidic and basic medium
• (3) Reducing agent in basic medium, but not in acidic medium
• (4) Oxidising and reducing agent in acidic medium, but not in basic medium.

Which premitive unit cell has unequal edge lengths (\(a \ne b \ne c\)) and all axial angles different from \(90^\circ\)?

• (1) Hexagonal
• (2) Tetragonal
• (3) Triclinic
• (4) Monoclinic

Water filled in two glasses A and B have BOD values of 10 and 20, respectively. The correct statement regarding them, is :

• (1) B is more polluted than A.
• (2) A is suitable for drinking, whereas B is not .
• (3) Both A and B are suitable for dirnking
• (4) A is more polluted than B.