JEE Main 2021 July 22 Shift 2 Question Paper with Solution PDF- Download Here

Question 1:

Let \(S_n\) denote the sum of first n-terms of an arithmetic progression. If \(S_{10} = 530\), \(S_5 = 140\), then \(S_{20} - S_6\) is equal to:

• (1) 1862
• (2) 1842
• (3) 1852
• (4) 1872

Question 2:

Let \(f : R \to R\) be defined as
\(f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x \gt 0 \\ 3xe^x, & x \le 0 \end{cases}\)
Then f is increasing function in the interval

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

Question 3:

Let \(y = y(x)\) be the solution of the differential equation \(\operatorname{cosec}^2 x\, dy + 2\, dx = (1 + y\cos 2x)\operatorname{cosec}^2 x\, dx\), with \(y\left(\frac{\pi}{4}\right) = 0\). Then, the value of \((y(0)+1)^2\) is equal to :

• (1) \(e^{1/2}\)
• (2) \(e^{-1/2}\)
• (3) \(e^{-1}\)
• (4) \(e\)

Question 4:

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in \(2 \times 2\) matrices. The probability that such formed matrix have all different entries and are non-singular, is :

• (1) \(\frac{45}{162}\)
• (2) \(\frac{21}{81}\)
• (3) \(\frac{22}{81}\)
• (4) \(\frac{43}{162}\)

Question 5:

If \(\int_0^{100\pi} \frac{\sin^2 x}{e^{\left(\frac{x}{\pi} - \left[\frac{x}{\pi}\right]\right)}}dx = \frac{\alpha\pi^3}{1 + 4\pi^2},\ \alpha \in R\) where \([x]\) is the greatest integer less than or equal to x, then the value of \(\alpha\) is :

• (1) \(200\,(1 - e^{-1})\)
• (2) \(100\,(1 - e)\)
• (3) \(50\,(e - 1)\)
• (4) \(150\,(e^{-1} - 1)\)

Question 6:

The values of \(\lambda\) and \(\mu\) such that the system of equations \(x + y + z = 6\), \(3x + 5y + 5z = 26\), \(x + 2y + \lambda z = \mu\) has no solution, are :

• (1) \(\lambda = 3,\ \mu = 5\)
• (2) \(\lambda = 3,\ \mu \ne 10\)
• (3) \(\lambda \ne 2,\ \mu = 10\)
• (4) \(\lambda = 2,\ \mu \ne 10\)

Question 7:

If the shortest distance between the straight lines \(3(x - 1) = 6(y - 2) = 2(z - 1)\) and \(4(x - 2) = 2(y - \lambda) = (z - 3),\ \lambda \in R\) is \(\frac{1}{\sqrt{38}}\), then the integral value of \(\lambda\) is equal to :

• (1) 3
• (2) 2
• (3) 5
• (4) \(-1\)

Question 8:

Let \(A = [a_{ij}]\) be a real matrix of order \(3 \times 3\), such that \(a_{i1} + a_{i2} + a_{i3} = 1\), for \(i = 1, 2, 3\). Then, the sum of all the entries of the matrix \(A^3\) is equal to :

• (1) 2
• (2) 1
• (3) 3
• (4) 9

Question 9:

Let \([x]\) denote the greatest integer less than or equal to x. Then, the values of \(x \in R\) satisfying the equation \([e^x]^2 + [e^x + 1] - 3 = 0\) lie in the interval :

• (1) \(\left[0, \frac{1}{e}\right)\)
• (2) \([\log_e 2, \log_e 3)\)
• (3) \([1, e)\)
• (4) \([0, \log_e 2)\)

Question 10:

Let the circle \(S : 36x^2 + 36y^2 - 108x + 120y + C = 0\) be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \(x - 2y = 4\) and \(2x - y = 5\) lies inside the circle S, then :

• (1) \(\frac{25}{9} \lt C \lt \frac{13}{3}\)
• (2) \(100 \lt C \lt 165\)
• (3) \(81 \lt C \lt 156\)
• (4) \(100 \lt C \lt 156\)

Question 11:

Let n denote the number of solutions of the equation \(z^2 + 3\overline{z} = 0\), where z is a complex number. Then the value of \(\sum_{k=0}^{\infty}\frac{1}{n^k}\) is equal to :

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

Question 12:

If the domain of the function \(f(x) = \cos^{-1}\left(\frac{2x-5}{11-3x}\right) + \sin^{-1}(2x^2-3x+1)\) is the interval \([\alpha, \beta]\), then \(\alpha + 2\beta\) is equal to:}

• (1) 3
• (2) 5
• (3) 1
• (4) 2

Question 13:

Let \(f : R \to R\) be defined as \(f(x) = \begin{cases} \frac{x^3}{(1 - \cos 2x)^2}\log_e\left(\frac{1 + 2xe^{-2x}}{(1 - xe^{-x})^2}\right), & x \ne 0 \\ \alpha, & x = 0 \end{cases}\)
If f is continuous at \(x = 0\), then \(\alpha\) is equal to :

• (1) 1
• (2) 3
• (3) 0
• (4) 2

Question 14:

Let \(E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\ a \gt b\). Let \(E_2\) be another ellipse such that it touches the end points of major axis of \(E_1\) and the foci of \(E_2\) are the end points of minor axis of \(E_1\). If \(E_1\) and \(E_2\) have same eccentricities, then its value is :

• (1) \(\frac{-1 + \sqrt{5}}{2}\)
• (2) \(\frac{-1 + \sqrt{8}}{2}\)
• (3) \(\frac{-1 + \sqrt{3}}{2}\)
• (4) \(\frac{-1 + \sqrt{6}}{2}\)

Question 15:

Let L be the line of intersection of planes \(\vec{r}\cdot(\hat{i} - \hat{j} + 2\hat{k}) = 2\) and \(\vec{r}\cdot(2\hat{i} + \hat{j} - \hat{k}) = 2\). If \(P(\alpha, \beta, \gamma)\) is the foot of perpendicular on L from the point \((1, 2, 0)\), then the value of \(35(\alpha + \beta + \gamma)\) is equal to :

• (1) 101
• (2) 119
• (3) 143
• (4) 134

Question 16:

• (1) 24
• (2) 29
• (3) 35
• (4) 42

Question 17:

Let three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be such that \(\vec{a}\times\vec{b} = \vec{c}\), \(\vec{b}\times\vec{c} = \vec{a}\) and \(|\vec{a}| = 2\). Then which one of the following is not true?

• (1) \(\vec{a}\times\left((\vec{b} + \vec{c})\times(\vec{b}\times\vec{c})\right) = \vec{0}\)
• (2) Projection of \(\vec{a}\) on \((\vec{b}\times\vec{c})\) is 2
• (3) \(\left[\vec{a}\ \vec{b}\ \vec{c}\right] + \left[\vec{c}\ \vec{a}\ \vec{b}\right] = 8\)
• (4) \(\left|3\vec{a} + \vec{b} - 2\vec{c}\right|^2 = 51\)

Question 18:

Which of the following Boolean expressions is not a tautology?

• (1) \((p \Rightarrow q) \vee (\sim q \Rightarrow p)\)
• (2) \((q \Rightarrow p) \vee (\sim q \Rightarrow p)\)
• (3) \((p \Rightarrow \sim q) \vee (\sim q \Rightarrow p)\)
• (4) \((\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)\)

Question 19:

The number of solutions of \(\sin^7 x + \cos^7 x = 1\), \(x \in [0, 4\pi]\) is equal to

• (1) \(11\)
• (2) \(7\)
• (3) \(5\)
• (4) \(9\)

Question 20:

Let a line \(L : 2x + y = k,\ k \gt 0\) be a tangent to the hyperbola \(x^2 - y^2 = 3\). If \(L\) is also a tangent to the parabola \(y^2 = \alpha x\), then \(\alpha\) is equal to :

• (1) \(12\)
• (2) \(-12\)
• (3) \(24\)
• (4) \(-24\)

Question 21:

Let \(A = \{0, 1, 2, 3, 4, 5, 6, 7\}\). Then the number of bijective functions \(f : A \to A\) such that \(f(1) + f(2) = 3 - f(3)\) is equal to

Question 22:

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________.

Question 23:

Let \(A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). Then the number of \(3 \times 3\) matrices \(B\) with entries from the set \(\{1, 2, 3, 4, 5\}\) and satisfying \(AB = BA\) is ____________.

Question 24: Consider the following frequency distribution :

Class :	0-6	6-12	12-18	18-24	24-30
Frequency :	\(a\)	\(b\)	12	9	5

If mean \(= \frac{309}{22}\) and median \(= 14\), then the value \((a - b)^2\) is equal to _____________.

Question 25:

The sum of all the elements of the set {α ∈ {1, 2, …, 100} : HCF(α, 24) = 1} is

Question 26:

The area (in sq. units) of the region bounded by the curves \(x^2 + 2y - 1 = 0\), \(y^2 + 4x - 4 = 0\) and \(y^2 - 4x - 4 = 0\), in the upper half plane is _______________.

Question 27:

Let \(f : \mathbb{R} \to \mathbb{R}\) be a function defined as
\(f(x) = \begin{cases} 3\left(1 - \frac{|x|}{2}\right) & \text{if } |x| \le 2 \\ 0 & \text{if } |x| \gt 2 \end{cases}\)
Let \(g : \mathbb{R} \to \mathbb{R}\) be given by
\(g(x) = f(x + 2) - f(x - 2)\).
If \(n\) and \(m\) denote the number of points in \(\mathbb{R}\) where \(g\) is not continuous and not differentiable, respectively, then \(n + m\) is equal to ______________.

Question 28:

If the constant term, in binomial expansion of \(\left(2x^r + \frac{1}{x^2}\right)^{10}\) is 180, then \(r\) is equal to __________________.

Question 29:

Let \(y = y(x)\) be the solution of the differential equation
\(\left((x + 2)e^{\left(\frac{y + 1}{x + 2}\right)} + (y + 1)\right)dx = (x + 2)\,dy\), \(y(1) = 1\).
If the domain of \(y = y(x)\) is an open interval \((\alpha, \beta)\), then \(|\alpha + \beta|\) is equal to _____________.

Question 30:

The number of elements in the set \(\{n \in \{1, 2, 3, \ldots\ldots, 100\} \mid (11)^n \gt (10)^n + (9)^n\}\) is ______________.

Question 31:

What will be the projection of vector
\(\vec{A} = \hat{i} + \hat{j} + \hat{k}\)
on vector
\(\vec{B} = \hat{i} + \hat{j}\) ?

• (1) \(\sqrt{2}\,(\hat{i} + \hat{j} + \hat{k})\)
• (2) \((\hat{i} + \hat{j})\)
• (3) \(\sqrt{2}\,(\hat{i} + \hat{j})\)
• (4) \(2(\hat{i} + \hat{j} + \hat{k})\)

Question 32:

A Copper (Cu) rod of length 25 cm and cross-sectional area 3 mm\(^2\) is joined with a similar Aluminium (Al) rod as shown in figure. Find the resistance of the combination between the ends A and B.
(Take Resistivity of Copper \(= 1.7 \times 10^{-8}\ \Omega\)m and Resistivity of Aluminium \(= 2.6 \times 10^{-8}\ \Omega\)m)

• (1) \(0.0858\ \mathrm{m}\Omega\)
• (2) \(1.420\ \mathrm{m}\Omega\)
• (3) \(0.858\ \mathrm{m}\Omega\)
• (4) \(2.170\ \mathrm{m}\Omega\)

Question 33:

In a circuit consisting of a capacitance and a generator with alternating emf \(E_g = E_{g_0}\sin\omega t\), \(V_C\) and \(I_C\) are the voltage and current. Correct phasor diagram for such circuit is

• (1)
• (2)
• (3)
• (4)

Question 34:

A porter lifts a heavy suitcase of mass 80 kg and at the destination lowers it down by a distance of 80 cm with a constant velocity. Calculate the work done by the porter in lowering the suitcase.

(take \(g = 9.8\ \mathrm{ms^{-2}}\))

• (1) \(+627.2\ \mathrm{J}\)
• (2) \(-62720.0\ \mathrm{J}\)
• (3) \(-627.2\ \mathrm{J}\)
• (4) \(784.0\ \mathrm{J}\)

Question 35:

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.

The correct statement for this situation is

• (1) All of them will have same velocity.
• (2) The ring has greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
• (3) The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.
• (4) The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.

Question 36:

Intensity of sunlight is observed as \(0.092\ \mathrm{Wm^{-2}}\) at a point in free space. What will be the peak value of magnetic field at the point?

(\(\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2N^{-1}m^{-2}}\))

• (1) \(2.77 \times 10^{-8}\ \mathrm{T}\)
• (2) \(1.96 \times 10^{-8}\ \mathrm{T}\)
• (3) \(8.31\ \mathrm{T}\)
• (4) \(5.88\ \mathrm{T}\)

Question 37:

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height \(h\) is ___________ s.

• (1) \(\sqrt{\dfrac{2R_e}{g}}\left[\left(1 + \dfrac{h}{R_e}\right)^{3/2} - 1\right]\)
• (2) \(\dfrac{1}{3}\sqrt{\dfrac{R_e}{2g}}\left[\left(1 + \dfrac{h}{R_e}\right)^{3/2} - 1\right]\)
• (3) \(\sqrt{\dfrac{R_e}{2g}}\left[\left(1 + \dfrac{h}{R_e}\right)^{3/2} - 1\right]\)
• (4) \(\dfrac{1}{3}\sqrt{\dfrac{2R_e}{g}}\left[\left(1 + \dfrac{h}{R_e}\right)^{3/2} - 1\right]\)

Question 38:

An electric dipole is placed on x-axis in proximity to a line charge of linear charge density \(3.0 \times 10^{-6}\ \mathrm{C/m}\). Line charge is placed on z-axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. If total force of 4N is exerted on the dipole, find out the amount of positive or negative charge of the dipole.

• (1) \(0.485\ \mathrm{mC}\)
• (2) \(815.1\ \mathrm{nC}\)
• (3) \(8.8\ \mu\mathrm{C}\)
• (4) \(4.44\ \mu\mathrm{C}\)

Question 39:

A nucleus with mass number 184 initially at rest emits an \(\alpha\)-particle. If the Q value of the reaction is 5.5 MeV, calculate the kinetic energy of the \(\alpha\)-particle.

• (1) \(5.5\ \mathrm{MeV}\)
• (2) \(5.0\ \mathrm{MeV}\)
• (3) \(5.38\ \mathrm{MeV}\)
• (4) \(0.12\ \mathrm{MeV}\)

Question 40:

A bullet of '4 g' mass is fired from a gun of mass 4 kg. If the bullet moves with the muzzle speed of \(50\ \mathrm{ms^{-1}}\), the impulse imparted to the gun and velocity of recoil of gun are :

• (1) \(0.2\ \mathrm{kg\,ms^{-1}},\ 0.1\ \mathrm{ms^{-1}}\)
• (2) \(0.4\ \mathrm{kg\,ms^{-1}},\ 0.05\ \mathrm{ms^{-1}}\)
• (3) \(0.2\ \mathrm{kg\,ms^{-1}},\ 0.05\ \mathrm{ms^{-1}}\)
• (4) \(0.4\ \mathrm{kg\,ms^{-1}},\ 0.1\ \mathrm{ms^{-1}}\)

Question 41:

Statement I : The ferromagnetic property depends on temperature. At high temperature, ferromagnet becomes paramagnet.

Statement II : At high temperature, the domain wall area of a ferromagnetic substance increases.

In the light of the above statements, choose the most appropriate answer from the options given below :

• (1) Statement I is false but Statement II is true
• (2) Statement I is true but Statement II is false
• (3) Both Statement I and Statement II are false
• (4) Both Statement I and Statement II are true

Question 42: Match List - I with List - II

	List - I		List - II
(a)	\(\omega L \gt \dfrac{1}{\omega C}\)	(i)	Current is in phase with emf
(b)	\(\omega L = \dfrac{1}{\omega C}\)	(ii)	Current lags behind the applied emf
(c)	\(\omega L \lt \dfrac{1}{\omega C}\)	(iii)	Maximum current occurs
(d)	Resonant frequency	(iv)	Current leads the emf

Choose the correct answer from the options given below

• (1) a(ii), b(i), c(iv), d(iii)
• (2) a(ii), b(i), c(iii), d(iv)
• (3) a(iii), b(i), c(iv), d(ii)
• (4) a(iv), b(iii), c(ii), d(i)

Question 43:

A ray of light passes from a denser medium to a rarer medium at an angle of incidence \(i\). The reflected and refracted rays make an angle of \(90^\circ\) with each other. The angle of reflection and refraction are respectively \(r\) and \(r'\). The critical angle is given by

• (1) \(\sin^{-1}(\tan r)\)
• (2) \(\sin^{-1}(\cot r)\)
• (3) \(\sin^{-1}(\tan r')\)
• (4) \(\tan^{-1}(\sin i)\)

Question 44:

An electron of mass \(m_e\) and a proton of mass \(m_p\) are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is

• (1) \(\dfrac{m_e}{m_p}\)
• (2) 1
• (3) \(\dfrac{m_p}{m_e}\)
• (4) \(\sqrt{\dfrac{m_p}{m_e}}\)

Question 45:

What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature \(T\)?

• (1) \(\dfrac{3}{2}k_BT\)
• (2) \(k_BT\)
• (3) \(\dfrac{2}{3}k_BT\)
• (4) \(\dfrac{1}{2}k_BT\)

Question 46:

\(T_0\) is the time period of a simple pendulum at a place. if the length of the pendulum is reduced to \(\dfrac{1}{16}\) times of its initial value, the modified time period is :

• (1) \(4\,T_0\)
• (2) \(\dfrac{1}{4}\,T_0\)
• (3) \(T_0\)
• (4) \(8\pi\,T_0\)

Question 47:

The motion of a mass on a spring, with spring constant \(K\) is as shown in figure.



The equation of motion is given by
\(x(t) = A\sin\omega t + B\cos\omega t\) with \(\omega = \sqrt{\dfrac{K}{m}}\)

Suppose that at time \(t = 0\), the position of mass is \(x(0)\) and velocity \(v(0)\), then its displacement can also be represented as \(x(t) = C\cos(\omega t - \phi)\), where \(C\) and \(\phi\) are :

• (1) \(C = \sqrt{\dfrac{2v(0)^2}{\omega^2} + x(0)^2},\ \phi = \tan^{-1}\left(\dfrac{x(0)\omega}{2v(0)}\right)\)
• (2) \(C = \sqrt{\dfrac{v(0)^2}{\omega^2} + x(0)^2},\ \phi = \tan^{-1}\left(\dfrac{x(0)\omega}{v(0)}\right)\)
• (3) \(C = \sqrt{\dfrac{v(0)^2}{\omega^2} + x(0)^2},\ \phi = \tan^{-1}\left(\dfrac{v(0)}{x(0)\omega}\right)\)
• (4) \(C = \sqrt{\dfrac{2v(0)^2}{\omega^2} + x(0)^2},\ \phi = \tan^{-1}\left(\dfrac{v(0)}{x(0)\omega}\right)\)

Question 48:

Consider a situation in which reserve biased current of a particular P-N junction increases when it is exposed to a light of wavelength \(\le 621\) nm. During this process, enhancement in carrier concentration takes place due to generation of hole-electron pairs. The value of band gap is nearly.

• (1) \(1\ \mathrm{eV}\)
• (2) \(4\ \mathrm{eV}\)
• (3) \(0.5\ \mathrm{eV}\)
• (4) \(2\ \mathrm{eV}\)

Question 49:

Choose the correct option

• (1) True dip is always equal to apparent dip.
• (2) True dip is not mathematically related to apparent dip.
• (3) True dip is less than the apparent dip.
• (4) True dip is always greater than the apparent dip.

Question 50:

The required height of a TV tower which can cover the population of \(603\) lakh is \(h\) If the average population density is \(100\) per square \(km\) and the radius of earth is \(6400\, km\), then the value of \(h\) will be ______\(m\)

Question 51:

A ray of light passing through a prism (\(\mu = \sqrt{3}\)) suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ________ (in degrees).

Question 52: Three students S₁, S₂ and S₃ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

Student No.	Length of Pendulum (cm)	No. of oscillations (n)	Total time for n oscillations	Time period (s)
1	64.0	8	128.0	16.0
2	64.0	4	64.0	16.0
3	20.0	4	36.0	9.0

(Least count of length = 0.1 cm and Least count for time = 0.1 s)
If \(E_1\), \(E_2\) and \(E_3\) are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. ________.

Question 53:

Three particles P, Q and R are moving along the vectors \(\vec{A} = \hat{i} + \hat{j}\), \(\vec{B} = \hat{j} + \hat{k}\) and \(\vec{C} = -\hat{i} + \hat{j}\) respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{B}\). Similarly particle Q is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{C}\). The angle between the direction of motion of P and Q is \(\cos^{-1}\left(\dfrac{1}{\sqrt{x}}\right)\). Then the value of x is ________.

Question 54:

Question 55:

The total charge enclosed in an incremental volume of \(2\times10^{-9}\ \mathrm{m^3}\) located at the origin is ________ nC, if electric flux density of its field is found as
\(\vec{D} = e^{-x}\sin y\,\hat{i} - e^{-x}\cos y\,\hat{j} + 2z\,\hat{k}\ \ \mathrm{C/m^2}\)

Question 56:

The position of the centre of mass of a uniform semi-circular wire of radius 'R' placed in x-y plane with its centre at the origin and the line joining its ends as x-axis is given by \(\left(0, \dfrac{xR}{\pi}\right)\). Then, the value of \(|x|\) is ________.

Question 57:

In an electric circuit, a cell of certain emf provides a potential difference of 1.25 V across a load resistance of \(5\ \Omega\). However, it provides a potential difference of 1 V across a load resistance of \(2\ \Omega\). The emf of the cell is given by \(\dfrac{x}{10}\) V. Then the value of x is ________.

Question 58:

The area of cross-section of a railway track is \(0.01\ \mathrm{m^2}\). The temperature variation is \(10^\circ\mathrm{C}\). Coefficient of liner expansion of material of track is \(10^{-5}/^\circ\mathrm{C}\). The energy stored per meter in the track is ________ J/m.
(Young's modulus of material of track is \(10^{11}\ \mathrm{Nm^{-2}}\))

Question 59:

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is 8 m/s. The speed of the particle on the rim of the wheel at the same level as the center of the wheel, will be:

• (1) \( 4\sqrt{2} \, \text{m/s} \)
• (2) 8 m/s
• (3) 4 m/s
• (4) \( 8\sqrt{2} \, \text{m/s} \)

Question 60:

In 5 minutes, a body cools from \(75^\circ\mathrm{C}\) to \(65^\circ\mathrm{C}\) at room temperature of \(25^\circ\mathrm{C}\). The temperature of body at the end of next 5 minutes is ________ \(^\circ\mathrm{C}\).

Question 61:

Which one of the following statements for D.I. Mendeleeff, is incorrect?

• (1) He authored the textbook - Principles of Chemistry
• (2) At the time, he proposed Periodic Table of elements structure of atom was known.
• (3) Element with atomic number 101 is named after him.
• (4) He invented accurate barometer.

Question 62:

Which purification technique is used for high boiling organic liquid compound (decomposes near its boiling point)?

• (1) Simple distillation
• (2) Steam distillation
• (3) Fractional distillation
• (4) Reduced pressure distillation

Question 63:

Which of the following compounds will provide a tertiary alcohol on reaction with excess of \(\mathrm{CH_3MgBr}\) followed by hydrolysis?

• (1)
• (2)
• (3)
• (4)

Question 64:

Which of the following compounds does not exhibit resonance?

• (1) \(\mathrm{CH_3CH_2OCH=CH_2}\)
• (2)
• (3) \(\mathrm{CH_3CH_2CH_2CONH_2}\)
• (4) \(\mathrm{CH_3CH_2CH=CHCH_2NH_2}\)

Question 65:

In the chemical reactions given above A and B respectively are :

• (1) \(\mathrm{H_3PO_2}\) and \(\mathrm{CH_3CH_2Cl}\)
• (2) \(\mathrm{CH_3CH_2OH}\) and \(\mathrm{H_3PO_2}\)
• (3) \(\mathrm{H_3PO_2}\) and \(\mathrm{CH_3CH_2OH}\)
• (4) \(\mathrm{CH_3CH_2Cl}\) and \(\mathrm{H_3PO_2}\)

Question 66:

Isotope(s) of hydrogen which emits low energy \(\beta^-\) particles with \(t_{1/2}\) value \(\gt\) 12 years is/are

• (1) Protium
• (2) Tritium
• (3) Deuterium
• (4) Deuterium and Tritium

Question 67: Match List-I with List-II :

List-I (Species)	List-II (Hybrid Orbitals)
(a) \(SF_4\)	(i) \(sp^3d^2\)
(b) \(IF_5\)	(ii) \(d^2sp^3\)
(c) \(NO_2^+\)	(iii) \(sp^3d\)
(d) \(NH_4^+\)	(iv) \(sp^3\)
	(v) \(sp\)

Choose the correct answer from the options given below :

• (1) (a)-(i), (b)-(ii), (c)-(v) and (d)-(iii)
• (2) (a)-(ii), (b)-(i), (c)-(iv) and (d)-(v)
• (3) (a)-(iii), (b)-(i), (c)-(v) and (d)-(iv)
• (4) (a)-(iv), (b)-(iii), (c)-(ii) and (d)-(v)

Question 68:

Which of the following molecules does not show stereo isomerism?

• (1) 3, 4-Dimethylhex-3-ene
• (2) 3-Methylhex-1-ene
• (3) 3-Ethylhex-3-ene
• (4) 4-Methylhex-1-ene

Question 69:

Which one of the following group-15 hydride is the strongest reducing agent?

• (1) \(\mathrm{AsH_3}\)
• (2) \(\mathrm{BiH_3}\)
• (3) \(\mathrm{PH_3}\)
• (4) \(\mathrm{SbH_3}\)

Question 70:

The set having ions which are coloured and paramagnetic both is :

• (1) \(\mathrm{Cu^{2+}, Cr^{3+}, Sc^{+}}\)
• (2) \(\mathrm{Cu^{2+}, Zn^{2+}, Mn^{4+}}\)
• (3) \(\mathrm{Sc^{3+}, V^{5+}, Ti^{4+}}\)
• (4) \(\mathrm{Ni^{2+}, Mn^{7+}, Hg^{2+}}\)

Question 71:

Thiamine and pyridoxine are also known respectively as :

• (1) Vitamin B₂ and Vitamin E
• (2) Vitamin E and Vitamin B₂
• (3) Vitamin B₆ and Vitamin B₂
• (4) Vitamin B₁ and Vitamin B₆

Question 72:

An organic compound A (\(\mathrm{C_6H_6O}\)) gives dark green colouration with ferric chloride. On treatment with \(\mathrm{CHCl_3}\) and KOH, followed by acidification gives compound B. Compound B can also be obtained from compound C on reaction with pyridinium chlorochromate (PCC). Identify A, B and C.

• (1)
• (2)
• (3)
• (4)

Question 73:

Which one of the following reactions does not occur?

• (1)
• (2)
• (3)
• (4)

Question 74:

Which one of the following 0.06 M aqueous solutions has lowest freezing point?

• (1) \(\mathrm{Al_2(SO_4)_3}\)
• (2) \(\mathrm{C_6H_{12}O_6}\)
• (3) KI
• (4) \(\mathrm{K_2SO_4}\)

Question 75:

The water having more dissolved \(\mathrm{O_2}\) is :

• (1) boiling water
• (2) water at \(80^\circ\mathrm{C}\)
• (3) polluted water
• (4) water at \(4^\circ\mathrm{C}\)

Question 76: Match List-I with List-II

List-I (Elements)	List-II (Properties)
(a) Ba	(i) Organic solvent soluble compounds
(b) Ca	(ii) Outer electronic configuration \(6s^2\)
(c) Li	(iii) Oxalate insoluble in water
(d) Na	(iv) Formation of very strong monoacidic base

Choose the correct answer from the options given below :

• (1) (a)-(ii), (b)-(iii), (c)-(i) and (d)-(iv)
• (2) (a)-(iv), (b)-(i), (c)-(ii) and (d)-(iii)
• (3) (a)-(iii), (b)-(ii), (c)-(iv) and (d)-(i)
• (4) (a)-(i), (b)-(iv), (c)-(ii) and (d)-(iii)

Question 77:

25 mL of silver nitrate solution (1M) is added dropwise to 25 mL of potassium iodide (1.05 M) solution. The ions(s) present in very small quantity in the solution is/are:

• (1) NO₃⁻ only
  
• (2) Ag⁺ and I⁻ both
  
• (3) Ag⁺ only
  
• (4) I⁻ only

Question 78:

Given below are the statements about diborane
(a) Diborane is prepared by the oxidation of \(\mathrm{NaBH_4}\) with \(\mathrm{I_2}\)
(b) Each boron atom is in \(sp^2\) hybridized state
(c) Diborane has one bridged 3 centre-2-electron bond
(d) Diborane is a planar molecule
The option with correct statements is :

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

Question 79: Match List-I with List-II :

List-I	List-II
(a) Chloroprene	(i), (ii), (iii): structures as shown in the figure
(b) Neoprene
(c) Acrylonitrile	(iv) \(\mathrm{CH_2=CH-CN}\)
(d) Isoprene

Choose the correct answer from the options given below :

• (1) (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
• (2) (a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)
• (3) (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
• (4) (a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)

Question 80:

Sulphide ion is soft base and its ores are common for metals.
(a) Pb (b) Al (c) Ag (d) Mg

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

Question 81:

The total number of unpaired electrons present in \(\mathrm{[Co(NH_3)_6]Cl_2}\) and \(\mathrm{[Co(NH_3)_6]Cl_3}\) is :

Question 82:

Methylation of 10 g of benzene gave 9.2 g of toluene. Calculate the percentage yield of toluene ________. (Nearest integer)

Question 83:

The number of acyclic structural isomers (including geometrical isomers) for pentene are ___________

Question 84:

Assume a cell with the following reaction

\(\mathrm{Cu_{(s)} + 2Ag^{+}(1\times 10^{-3}\,M) \rightarrow Cu^{2+}(0.250\,M) + 2Ag_{(s)}}\)

\(E^{\Theta}_{cell} = 2.97\) V

\(E_{cell}\) for the above reaction is ______________ V. (Nearest integer)

[Given : log 2.5 = 0.3979, T = 298 K]

Question 85:

Value of \(K_P\) for the equilibrium reaction

\(\mathrm{N_2O_{4(g)} \rightleftharpoons 2NO_{2(g)}}\) at 288 K is 47.9. The \(K_C\) for this reaction at same temperature is ____________. (Nearest integer)

(R = 0.083 L bar \(\mathrm{K^{-1}\,mol^{-1}}\))

Question 86:

If the standard molar enthalpy change for combustion of graphite powder is \(-2.48\times 10^{2}\) kJ \(\mathrm{mol^{-1}}\), the amount of heat generated on combustion of 1 g of graphite powder is ___________ kJ. (Nearest integer)

Question 87:

Number of electrons that Vanadium (Z = 23) has in p-orbitals is equal to _____________

Question 88:

\(\mathrm{N_2O_{5(g)} \rightarrow 2NO_{2(g)} + \frac{1}{2}O_{2(g)}}\)

In the above first order reaction the initial concentration of \(\mathrm{N_2O_5}\) is \(2.40\times 10^{-2}\) mol \(\mathrm{L^{-1}}\) at 318 K. The concentration of \(\mathrm{N_2O_5}\) after 1 hour was \(1.60\times 10^{-2}\) mol \(\mathrm{L^{-1}}\). The rate constant of the reaction at 318 K is ______________ \(\times 10^{-3}\) \(\mathrm{min^{-1}}\). (Nearest integer)

[Given : log 3 = 0.477, log 5 = 0.699]

Question 89:

The osmotic pressure of blood is 7.47 bar at 300 K. To inject glucose to a patient intravenously, it has to be isotonic with blood. The concentration of glucose solution in \(\mathrm{gL^{-1}}\) is ____________. (Molar mass of glucose = 180 g \(\mathrm{mol^{-1}}\), R = 0.083 L bar \(\mathrm{K^{-1}\,mol^{-1}}\)) (Nearest integer)

Question 90:

A copper complex crystallising in a CCP lattice with a cell edge of 0.4518 nm has been revealed by employing X-ray diffraction studies. The density of a copper complex is found to be 7.62 g \(\mathrm{cm^{-3}}\). The molar mass of copper complex is ____________ g \(\mathrm{mol^{-1}}\). (Nearest integer)

[Given : \(N_A = 6.022\times 10^{23}\ \mathrm{mol^{-1}}\)]