JEE Main 2024 4 April Shift 1 Question Paper (Available)- Download Solution PDF with Answer Key

Let \( f(x) = \begin{cases} -2, & -2 \leq x \leq 0 \\ x - 2, & 0 < x \leq 2 \end{cases} \) and \( h(x) = f(|x|) + |f(x)| \). Then \(\int_{-2}^{2} h(x) \, dx\) is equal to:

• (1) 2
• (2) 4
• (3) 1
• (4) 6

One of the points of intersection of the curves \( y = 1 + 3x - 2x^2 \) and \( y = \frac{1}{x} \) is \( \left( \frac{1}{2}, 2 \right) \). Let the area of the region enclosed by these curves be \[\frac{1}{24} \left( \ell \sqrt{5} + m \right) - n \log_e \left( 1 + \sqrt{5} \right),\] where \( \ell, m, n \in \mathbb{N} \). Then \( \ell + m + n \) is equal to:

• (1) 32
• (2) 30
• (3) 29
• (4) 31

Let \( \alpha \in (0, \infty) \) and \( A = \begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \). If \( \det(\text{adj}(2A - A^\top) \cdot \text{adj}(A - 2A^\top)) = 2^8 \), then \( (\det(A))^2 \) is equal to:

• (1) 1
• (2) 49
• (3) 16
• (4) 36

There are 5 points \( P_1, P_2, P_3, P_4, P_5 \) on the side \( AB \), excluding \( A \) and \( B \), of a triangle \( \triangle ABC \). Similarly, there are 6 points \( P_6, P_7, \dots, P_{11} \) on the side \( BC \) and 7 points \( P_{12}, P_{13}, \dots, P_{18} \) on the side \( CA \) of the triangle.The number of triangles that can be formed using the points \( P_1, P_2, \dots, P_{18} \) as vertices, is:

• (1) 776
• (2) 751
• (3) 796
• (4) 771

A square is inscribed in the circle \( x^2 + y^2 - 10x - 6y + 30 = 0 \). One side of this square is parallel to \( y = x + 3 \). If \( (x_i, y_i) \) are the vertices of the square, then \( \sum (x_i^2 + y_i^2) \) is equal to:

• (1) 148
• (2) 156
• (3) 160
• (4) 152

Let \( a, b \in \mathbb{R} \). Let the mean and the variance of 6 observations \(-3, 4, 7, -6, a, b\) be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

• (1) \( \frac{13}{3} \)
• (2) \( \frac{16}{3} \)
• (3) \( \frac{11}{3} \)
• (4) \( \frac{14}{3} \)

Let \( f(x) = x^5 + 2e^{x/4} \) for all \( x \in \mathbb{R} \). Consider a function \( g(x) \) such that \( (g \circ f)(x) = x \) for all \( x \in \mathbb{R} \). Then the value of \( 8g'(2) \) is:

• (1) 16
• (2) 4
• (3) 8
• (4) 2

Let a unit vector which makes an angle of \(60^\circ\) with \( 2\hat{i} + 2\hat{j} - \hat{k} \) and an angle of \(45^\circ\) with \( \hat{i} - \hat{k} \) be \( \vec{C} \). Then \( \vec{C} + \left( -\frac{1}{2} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k} \right) \) is:

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

If the domain of the function \[\sin^{-1} \left( \frac{3x - 22}{2x - 19} \right) + \log_e \left( \frac{3x^2 - 8x + 5}{x^2 - 3x - 10} \right)\] is \( (\alpha, \beta] \), then \( 3\alpha + 10\beta \) is equal to:

• (1) 97
• (2) 100
• (3) 95
• (4) 98

Let the point, on the line passing through the points \( P(1, -2, 3) \) and \( Q(5, -4, 7) \), farther from the origin and at a distance of 9 units from the point \( P \), be \( (\alpha, \beta, \gamma) \). Then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

• (1) 155
• (2) 150
• (3) 160
• (4) 165

The vertices of a triangle are A(–1, 3), B(–2, 2) and C(3, –1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

• (1) \( x - y - \left( 2 + \sqrt{2} \right) = 0 \)
• (2) \( -x + y - \left( 2 - \sqrt{2} \right) = 0 \)
• (3) \( x + y - \left( 2 - \sqrt{2} \right) = 0 \)
• (4) \( x + y + \left( 2 + \sqrt{2} \right) = 0 \)

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then then probability that it is drawn from urn A is :

• (1) \(\frac{4}{17}\)
• (2) \(\frac{5}{18}\)
• (3) \(\frac{7}{18}\)
• (4) \(\frac{5}{16}\)

If the solution \( y = y(x) \) of the differential equation \( \left( x^4 + 2x^3 + 3x^2 + 2x + 2 \right) dy - \left( 2x^2 + 2x + 3 \right) dx = 0 \)
satisfies \( y(-1) = -\frac{\pi}{4} \), then \( y(0) \) is equal to:

• (1) \( -\frac{\pi}{12} \)
• (2) 0
• (3) \( \frac{\pi}{4} \)
• (4) \( \frac{\pi}{2} \)

Let the first three terms \(2\), \(p\), and \(q\), with \(q \neq 2\), of a G.P. be respectively the \(7^\text{th}\), \(8^\text{th}\), and \(13^\text{th}\) terms of an A.P. If the \(5^\text{th}\) term of the G.P. is the \(n^\text{th}\) term of the A.P., then \(n\) is equal to:

• (1) 151
• (2) 169
• (3) 177
• (4) 163

The sum of all rational terms in the expansion of \[ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} \] is equal to: 

• (1) 3133
• (2) 633
• (3) 931
• (4) 6131

If 2 and 6 are the roots of the equation \( ax^2 + bx + 1 = 0 \), then the quadratic equation, whose roots are \( \frac{1}{2a + b} \) and \( \frac{1}{6a + b} \), is:

• (1) \( 2x^2 + 11x + 12 = 0 \)
• (2) \( 4x^2 + 14x + 12 = 0 \)
• (3) \( x^2 + 10x + 16 = 0 \)
• (4) \( x^2 + 8x + 12 = 0 \)

Let the sum of the maximum and the minimum values of the function \( f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8} \) be \( \frac{m}{n} \), where \( \text{gcd}(m, n) = 1 \). Then \( m + n \) is equal to:

• (1) 182
• (2) 217
• (3) 195
• (4) 201

If the system of equations
\( x + \left( \sqrt{2} \sin \alpha \right) y + \left( \sqrt{2} \cos \alpha \right) z = 0 \)
\( x + \left( \cos \alpha \right) y + \left( \sin \alpha \right) z = 0 \)
\( x + \left( \sin \alpha \right) y - \left( \cos \alpha \right) z = 0 \)
has a non-trivial solution, then \( \alpha \in \left( 0, \frac{\pi}{2} \right) \) is equal to:

• (1) \( \frac{3\pi}{4} \)
• (2) \( \frac{7\pi}{24} \)
• (3) \( \frac{5\pi}{24} \)
• (4) \( \frac{11\pi}{24} \)

Let \( \alpha \) and \( \beta \) be the sum and the product of all the non-zero solutions of the equation \[(\bar{z})^2 + |z| = 0, \quad z \in \mathbb{C}.\] Then \( 4(\alpha^2 + \beta^2) \) is equal to:

• (1) 6
• (2) 4
• (3) 8
• (4) 2

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function given by
\[f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x < 0 \\\alpha, & x = 0, \text{ where } \alpha, \beta \in \mathbb{R}. \\\beta \sqrt{1 - \cos x} / x, & x > 0 \end{cases} \]
If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

• (1) 48
• (2) 12
• (3) 3
• (4) 6

Let the solution \( y = y(x) \) of the differential equation \[\frac{dy}{dx} - y = 1 + 4 \sin x\] satisfy \( y(\pi) = 1 \). Then \( y\left( \frac{\pi}{2} \right) + 10 \) is equal to _____

If the shortest distance between the lines \[ \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} \quad \text{and} \quad \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}\] is \(\frac{38}{3\sqrt{5}} k\) and \[\int_{0}^{k} \lfloor x^2 \rfloor dx = \alpha - \sqrt{\alpha}, \]where \([x]\) denotes the greatest integer function, then \( 6\alpha^3 \) is equal to _____

Let \( A \) be a square matrix of order 2 such that \( |A| = 2 \) and the sum of its diagonal elements is \(-3\). If the points \((x, y)\) satisfying \( A^2 + xA + yI = 0 \) lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is \( e \) and the length of the latus rectum is \( \ell \), then \( e^4 + \ell^4 \) is equal to _____

If \[ \lim_{x \to 1} \frac{(5x+1)^{1/3} - (x+5)^{1/3}}{(2x+3)^{1/2} - (x+4)^{1/2}} = \frac{m \sqrt{5}}{n (2n)^{2/3}}, \] where \( \text{gcd}(m, n) = 1 \), then \( 8m + 12n \) is equal to _____

Let \( A \) be a \( 3 \times 3 \) matrix of non-negative real elements such that \[A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = 3 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.\]Then the maximum value of \( \det(A) \) is _____

If \[\int_{0}^{\pi/4} \frac{\sin^2 x}{1 + \sin x \cos x} \, dx = \frac{1}{a} \log_e \left( \frac{a}{3} \right) + \frac{\pi}{b\sqrt{3}},\]where \( a, b \in \mathbb{N} \), then \( a + b \) is equal to _____

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _____

Let the length of the focal chord \( PQ \) of the parabola \( y^2 = 12x \) be 15 units. If the distance of \( PQ \) from the origin is \( p \), then \( 10p^2 \) is equal to _____

Let \( \triangle ABC \) be a triangle of area \( 15\sqrt{2} \) and the vectors \[ \overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}, \quad \overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}, \quad \text{and} \quad \overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}, \, d > 0.\]Then the square of the length of the largest side of the triangle \( \triangle ABC \) is

Let \[ a = 1 + \frac{{^2C_2}}{3!} + \frac{{^3C_2}}{4!} + \frac{{^4C_2}}{5!} + \dots,\]
\[ b = 1 + \frac{{^1C_0 + ^1C_1}}{1!} + \frac{{^2C_0 + ^2C_1 + ^2C_2}}{2!} + \frac{{^3C_0 + ^3C_1 + ^3C_2 + ^3C_3}}{3!} + \dots \]Then \( \frac{2b}{a^2} \) is equal to _____

An infinitely long positively charged straight thread has a linear charge density \( \lambda \, \text{Cm}^{-1} \). An electron revolves along a circular path having its axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of the electron as a function of the radius of the circular path from the wire is:

• (1)

• (2)

• (3)

• (4)

An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

• (1) the electron will be accelerated along the axis.
• (2) the electron will continue to move with uniform velocity along the axis of the solenoid.
• (3) the electron path will be circular about the axis.
• (4) the electron will experience a force at 45° to the axis and execute a helical path.

A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is:

• (1) \( \frac{GmM\pi}{2L^2} \)
• (2) 0
• (3) \( \frac{GmM\pi^2}{L^2} \)
• (4) \( \frac{2GmM\pi}{L^2} \)

An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D. Focal length of each of the convex lens is :

• (1) 20 cm
• (2) 50 cm
• (3) 500 cm
• (4) 25 cm

Given below are two statements:  
Statement I: When the speed of liquid is zero everywhere, the pressure difference at any two points depends on the equation \[ P_1 - P_2 = \rho g (h_2 - h_1). \] Statement II: In the ventury tube shown, \[ 2gh = v_1^2 - v_2^2. \]

• (1) Both Statement I and Statement II are correct.
• (2) Statement I is incorrect but Statement II is correct.
• (3) Both Statement I and Statement II are incorrect.
• (4) Statement I is correct but Statement II is incorrect.

If a rubber ball falls from a height h and rebounds upto the height of h/2. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

• (1) \( 50\%, \frac{\sqrt{gh}}{2} \)
• (2) \( 50\%, \sqrt{gh} \)
• (3) \( 40\%, \sqrt{2gh} \)
• (4) \( 50\%, \sqrt{2gh} \)

In an ac circuit, the instantaneous current is zero,when the instantaneous voltage is maximum. In this case, the source may be connected to :
A. pure inductor.
B. pure capacitor.
C. pure resistor.
D. combination of an inductor and capacitor.
Choose the correct answer from the options given below :

• (1) A, B and C only
• (2) B, C and D only
• (3) A and B only
• (4) A, B and D only

Which of the following nuclear fragments corresponding to nuclear fission between neutron \[\left( ^1_0 n \right) \, \text{and uranium isotope} \, \left( ^{235}_{92} \text{U} \right) \, \text{is correct:}\]

• (1) \( \frac{144}{56} \text{Ba} + \frac{89}{36} \text{Kr} + 4 \frac{1}{0} \text{n} \)
• (2) \( \frac{140}{56} \text{Xe} + \frac{94}{38} \text{Sr} + 3 \frac{1}{0} \text{n} \)
• (3) \( \frac{153}{51} \text{Sb} + \frac{99}{41} \text{Nb} + 3 \frac{1}{0} \text{n} \)
• (4) \( \frac{144}{56} \text{Ba} + \frac{89}{36} \text{Kr} + 3 \frac{1}{0} \text{n} \)

A body travels \( 102.5 \, \text{m} \) in \( n^\text{th} \) second and \( 115.0 \, \text{m} \) in \( (n+2)^\text{th} \) second. The acceleration is:

• (1) \( 9 \, \text{m/s}^2 \)
• (2) \( 6.25 \, \text{m/s}^2 \)
• (3) \( 12.5 \, \text{m/s}^2 \)
• (4) \( 5 \, \text{m/s}^2 \)

In an experiment to measure focal length (\(f\)) of a convex lens, the least counts of the measuring scales for the position of the object (\(u\)) and for the position of the image (\(v\)) are \( \Delta u \) and \( \Delta v \), respectively. The error in the measurement of the focal length of the convex lens will be:

• (1) \( \frac{\Delta u}{u} + \frac{\Delta v}{v} \)
• (2) \( f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right] \)
• (3) \( 2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)
• (4) \( f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right] \)

The coordinates of a particle moving in the \( x \)-\( y \) plane are given by: \[x = 2 + 4t, \quad y = 3t + 8t^2.\]The motion of the particle is:

• (1) non-uniformly accelerated.
• (2) uniformly accelerated having motion along a straight line.
• (3) uniform motion along a straight line.
• (4) uniformly accelerated having motion along a parabolic path.

The equation of a stationary wave is:
\[y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right)\]Which of the following is NOT correct:

• (1) The dimensions of \( nt \) is \([L]\)
• (2) The dimensions of \( n \) is \([LT^{-1}]\)
• (3) The dimensions of \( n / \lambda \) is \([T]\)
• (4) The dimensions of \( x \) is \([L]\)

Which figure shows the correct variation of applied potential difference (\(V\)) with photoelectric current (\(I\)) at two different intensities of light (\(I_1 < I_2\)) of same wavelengths:

• (1)

• (2)

• (3)

• (4)

P-T diagram of an ideal gas having three different densities \( \rho_1, \rho_2, \rho_3 \) (in three different cases) is shown in the figure. Which of the following is correct:

• (1) \( \rho_2 \lt \rho_3 \)
• (2) \( \rho_1 \gt \rho_2 \)
• (3) \( \rho_1 \lt \rho_2 \)
• (4) \( \rho_1 = \rho_2 = \rho_3 \)

The electric field in an electromagnetic wave is given by \[ \vec{E} = \hat{i} 40 \cos \omega \left( t - \frac{z}{c} \right) \, \text{NC}^{-1}.\]The magnetic field induction of this wave is (in SI unit):

• (1) \( \vec{B} = \hat{i} \frac{40}{c} \cos \omega \left( t - \frac{z}{c} \right) \)
• (2) \( \vec{B} = \hat{j} 40 \cos \omega \left( t - \frac{z}{c} \right) \)
• (3) \( \vec{B} = \hat{k} \frac{40}{c} \cos \omega \left( t - \frac{z}{c} \right) \)
• (4) \( \vec{B} = \hat{j} \frac{40}{c} \cos \omega \left( t - \frac{z}{c} \right) \)

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are \( 8 \, \Omega \) and \( 10 \, \Omega \) respectively. After inserting in a hot bath of temperature \( 400^\circ \text{C} \), the resistance of the platinum wire is:

• (1) \( 2 \, \Omega \)
• (2) \( 16 \, \Omega \)
• (3) \( 8 \, \Omega \)
• (4) \( 10 \, \Omega \)

The value of net resistance of the network as shown in the given figure is :

• (1) \( \frac{5}{2} \, \Omega \)
• (2) \( \frac{15}{4} \, \Omega \)
• (3) \( 6 \, \Omega \)
• (4) \( \frac{30}{11} \, \Omega \)

A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t. Which of the following curve best describes acceleration of the block with time :

• (1)

• (2)

• (3)

• (4)

To measure the internal resistance of a battery, a potentiometer is used. For \( R = 10 \, \Omega \), the balance point is observed at \( \ell = 500 \, \text{cm} \) and for \( R = 1 \, \Omega \), the balance point is observed at \( \ell = 400 \, \text{cm} \). The internal resistance of the battery is approximately:

• (1) \( 0.2 \, \Omega \)
• (2) \( 0.4 \, \Omega \)
• (3) \( 0.1 \, \Omega \)
• (4) \( 0.3 \, \Omega \)

On Celsius scale, the temperature of a body increases by \( 40^\circ \text{C} \). The increase in temperature on Fahrenheit scale is:

• (1) \( 70^\circ \text{F} \)
• (2) \( 68^\circ \text{F} \)
• (3) \( 72^\circ \text{F} \)
• (4) \( 75^\circ \text{F} \)

A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed \( v \). The sphere and the cylinder reach up to maximum heights \( h_1 \) and \( h_2 \), respectively, above the initial level. The ratio \( h_1 : h_2 \) is \( \frac{n}{10} \). The value of \( n \) is ______.

An infinite plane sheet of charge having uniform surface charge density \( +\sigma_s \, \text{C/m}^2 \) is placed on the \( x \)-\( y \) plane. Another infinitely long line charge having uniform linear charge density \( +\lambda_e \, \text{C/m} \) is placed at \( z = 4 \, \text{m} \) plane and parallel to the \( y \)-axis. If the magnitude values \( |\sigma_s| = 2 |\lambda_e| \), then at point \( (0, 0, 2) \), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is \( \pi \sqrt{n} : 1 \).The value of \( n \) is ______.

A hydrogen atom changes its state from \( n = 3 \) to \( n = 2 \). Due to recoil, the percentage change in the wavelength of emitted light is approximately \( 1 \times 10^{-n} \). The value of \( n \) is ______. \([ \text{Given: } Rhc = 13.6 \, \text{eV}, \, hc = 1242 \, \text{eV nm}, \, h = 6.6 \times 10^{-34} \, \text{J s}, \, \text{mass of the hydrogen atom} = 1.6 \times 10^{-27} \, \text{kg} ]\)

A soap bubble is blown to a diameter of \( 7 \, \text{cm} \). \( 36960 \, \text{erg} \) of work is done in blowing it further. If the surface tension of the soap solution is \( 40 \, \text{dyne/cm} \), then the new radius is ______ \( \text{cm} \). Take: \( \left( \pi = \frac{22}{7} \right) \).

An elastic spring under tension of 3 N has a length a. Its length is b under tension 2 N. For its length (3a – 2b), the value of tension will be_____ N.

The magnetic field existing in a region is given by \[\vec{B} = 0.2(1 + 2x)\hat{k} \, \text{T}.\]A square loop of edge \( 50 \, \text{cm} \) carrying \( 0.5 \, \text{A} \) current is placed in the \( x \)-\( y \) plane with its edges parallel to the \( x \)- and \( y \)-axes, as shown in the figure. The magnitude of the net magnetic force experienced by the loop is ______ \( \text{mN} \).

Two forces \( \vec{F}_1 \) and \( \vec{F}_2 \) are acting on a body. One force has magnitude thrice that of the other force, and the resultant of the two forces is equal to the force of larger magnitude. The angle between \( \vec{F}_1 \) and \( \vec{F}_2 \) is \( \cos^{-1}\left(\frac{1}{n}\right) \). The value of \( |n| \) is _____.

An alternating current at any instant is given by \[ i = \left[ 6 + \sqrt{56} \sin\left(100 \pi t + \frac{\pi}{3}\right) \right] \, \text{A}.\] The RMS value of the current is ______ A.

Twelve wires, each having resistance \( 2 \, \Omega \), are joined to form a cube. A battery of \( 6 \, \text{V} \) emf is joined across points \( a \) and \( c \). The voltage difference between \( e \) and \( f \) is ______ \( \text{V} \).

Two wavelengths \( \lambda_1 \) and \( \lambda_2 \) are used in Young's double slit experiment. \( \lambda_1 = 450 \, \text{nm} \) and \( \lambda_2 = 650 \, \text{nm} \). The minimum order of fringe produced by \( \lambda_2 \), which overlaps with the fringe produced by \( \lambda_1 \), is \( n \). The value of \( n \) is _____.

Number of complexes from the following with even number of unpaired "d" electrons is _____. \[ [\text{V}(\text{H}_2\text{O})_6]^{3+}, \, [\text{Cr}(\text{H}_2\text{O})_6]^{2+}, \, [\text{Fe}(\text{H}_2\text{O})_6]^{3+}, \, [\text{Ni}(\text{H}_2\text{O})_6]^{3+}, \, [\text{Cu}(\text{H}_2\text{O})_6]^{2+}\] [Given atomic numbers: \( \text{V} = 23, \, \text{Cr} = 24, \, \text{Fe} = 26, \, \text{Ni} = 28, \, \text{Cu} = 29 \)]

• (1) 2
• (2) 4
• (3) 5
• (4) 1

What will be the decreasing order of basic strength of the following conjugate bases?
\( \text{OH}^-, \, \text{R}^-, \, \text{CH}_3\text{COO}^-, \, \text{Cl}^- \)

• (1) \( \text{Cl}^- \gt \text{OH}^- \gt \text{R}^- \gt \text{CH}_3\text{COO}^- \)
• (2) \( \text{R}^- \gt \text{OH}^- \gt \text{CH}_3\text{COO}^- \gt \text{Cl}^- \)
• (3) \( \text{OH}^- \gt \text{R}^- \gt \text{CH}_3\text{COO}^- \gt \text{Cl}^- \)
• (4) \( \text{Cl}^- \gt \text{R}^- \gt \text{OH}^- \gt \text{CH}_3\text{COO}^- \)

Given below are two statements :
Statement I : Acidity of -hydrogens of aldehydes and ketones is responsible for Aldol reaction.
Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross – Aldol product.In the light of above statements, choose the most appropriate answer from the options given below.

• (1) Both Statement I and Statement II are correct.
• (2) Both Statement I and Statement II are incorrect.
• (3) Statement I is incorrect but Statement II is correct.
• (4) Statement I is correct but Statement II is incorrect.

Match List -I with List II:

Choose the correct answer from the options given below :

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

• (1)

• (2)

• (3)

• (4)

Identify the correct set of reagents or reaction conditions ‘X’ and ‘Y’ in the following set of transformation.

• (1) \( X = \text{conc. alc. NaOH, 80}^\circ \text{C}, \, Y = \text{Br}_2/\text{CHCl}_3 \)
• (2) \( X = \text{dil. aq. NaOH, 20}^\circ \text{C}, \, Y = \text{HBr/acetic acid} \)
• (3) \( X = \text{conc. alc. NaOH, 80}^\circ \text{C}, \, Y = \text{HBr/acetic acid} \)
• (4) \( X = \text{dil. aq. NaOH, 20}^\circ \text{C}, \, Y = \text{Br}_2/\text{CHCl}_3 \)

In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to :

• (1) Prevent interference by phosphate ions
• (2) Decrease concentration of \( \text{OH}^- \) ions
• (3) Increase concentration of \( \text{Cl}^- \) ions
• (4) Increase concentration of \( \text{NH}_4^+ \) ions

The element which shows only one oxidation state other than its elemental form is :

• (1) Cobalt
• (2) Scandium
• (3) Titanium
• (4) Nickel

Which of the following is the correct structure of L-Glucose ?

• (1)

• (2)

• (3)

• (4)

Identify the product in the following reaction :

• (1)

• (2)

• (3)

• (4)

Number of molecules/ions from the following in which the central atom is involved in sp3 hybridization is ________.\( \text{NO}_3^-, \, \text{BCl}_3, \, \text{ClO}_2^-, \, \text{ClO}_3 \)

• (1) 2
• (2) 4
• (3) 3
• (4) 1

The correct sequence of ligands in the order of decreasing field strength is :

• (1) \( \text{CO} \gt \text{H}_2\text{O} \gt \text{F}^- \gt \text{S}^{2-} \)
• (2) \( \text{OH}^- \gt \text{F}^- \gt \text{NH}_3 \gt \text{CN}^- \)
• (3) \( \text{NCS}^- \gt \text{EDTA}^{4-} \gt \text{CN}^- \gt \text{CO} \)
• (4) \( \text{S}^{2-} \gt \text{OH}^- \gt \text{EDTA}^{4-} \gt \text{CO} \)

The Molarity (M) of an aqueous solution containing \( 5.85 \, \text{g} \) of \( \text{NaCl} \) in \( 500 \, \text{mL} \) water is:
(Given: Molar Mass \( \text{Na} = 23 \) and \( \text{Cl} = 35.5 \, \text{g mol}^{-1} \))

• (1) 20
• (2) 0.2
• (3) 2
• (4) 4

Which among the following is incorrect statement?

• (1) Electromeric effect dominates over inductive effect
• (2) The electromeric effect is, temporary effect
• (3) The organic compound shows electromeric effect in the presence of the reagent only
• (4) Hydrogen ion (\( \text{H}^+ \)) shows negative electromeric effect.

One of the commonly used electrode is calomel electrode. Under which of the following categories calomel electrode comes ?

• (1) Metal – Insoluble Salt – Anion electrodes
• (2) Oxidation – Reduction electrodes
• (3) Gas – Ion electrodes
• (4) Metal ion – Metal electrodes

The correct order of first ionization enthalpy values of the following elements is :
(A) O (B) N (C) Be (D) F (E) B
Choose the correct answer from the options given below :

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

Which of the following nitrogen containing compound does not give Lassaigne’s test ?

• (1) Phenyl hydrazine
• (2) Glycene
• (3) Urea
• (4) Hydrazine

Which one of the following molecules has maximum dipole moment ?

• (1) \( \text{NF}_3 \)
• (2) \( \text{CH}_4 \)
• (3) \( \text{NH}_3 \)
• (4) \( \text{PF}_5 \)

What pressure (bar) of \( \text{H}_2 \) would be required to make the emf of a hydrogen electrode zero in pure water at \( 25^\circ \text{C} \)?

• (1) \( 10^{-14} \)
• (2) \( 10^{-7} \)
• (3) 1
• (4) 0.5

Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is ______.
B, C, N, S, O, F, P, Al, Si

• (1) 7
• (2) 5
• (3) 6
• (4) 3

\( 2.5 \, \text{g} \) of a non-volatile, non-electrolyte is dissolved in \( 100 \, \text{g} \) of water at \( 25^\circ \text{C} \). The solution showed a boiling point elevation by \( 2^\circ \text{C} \). Assuming the solute concentration is negligible with respect to the solvent concentration, the vapour pressure of the resulting aqueous solution is ______ mm of Hg (nearest integer).
(Given: Molal boiling point elevation constant of water (\( K_b \)) = \( 0.52 \, \text{K} \cdot \text{kg} \cdot \text{mol}^{-1} \),1 atm pressure = \( 760 \, \text{mm of Hg} \), molar mass of water = \( 18 \, \text{g mol}^{-1} \))

Only \( 2 \, \text{mL} \) of \( \text{KMnO}_4 \) solution of unknown molarity is required to reach the end point of a titration of \( 20 \, \text{mL} \) of oxalic acid (\( 2 \, \text{M} \)) in acidic medium. The molarity of \( \text{KMnO}_4 \) solution should be ______ \( \text{M} \).

Consider the following transformation involving first-order elementary reaction in each step at constant temperature as shown below: \[ \text{A} + \text{B} \xrightarrow{\text{Step 1}} \text{C} \xrightarrow{\text{Step 2}} \text{P}\] Some details of the above reaction are listed below:

The enthalpy of formation of ethane (\( \text{C}_2\text{H}_6 \)) from ethylene by addition of hydrogen,
where the bond energies of \( \text{C} - \text{H} \), \( \text{C} - \text{C} \), \( \text{H} - \text{H} \) are \( 414 \, \text{kJ} \), \( 347 \, \text{kJ} \), \( 615 \, \text{kJ} \), and \( 435 \, \text{kJ} \) respectively, is _________ \( \text{kJ} \).

\( X \, \text{g} \) of ethylamine is subjected to reaction with \( \text{NaNO}_2/\text{HCl} \) followed by water; evolved dinitrogen gas which occupied \( 2.24 \, \text{L} \) volume at STP. \( X \) is ______ \( \times 10^{-1} \, \text{g} \).

The number of correct reaction(s) among the following is ________.

The de-Broglie's wavelength of an electron in the \( 4^{\text{th}} \) orbit is ______ \( \pi a_0 \). (\( a_0 = \text{Bohr's radius} \))

The number of different chain isomers for \( \text{C}_7\text{H}_{16} \) is ______.

Number of molecules/species from the following having one unpaired electron is ________.
\( \text{O}_2, \, \text{O}_2^{1-}, \, \text{NO}, \, \text{CN}^{1-}, \, \text{O}_2^{2-} \)

Consider the following reaction: \( \text{MnO}_2 + \text{KOH} + \text{O}_2 \rightarrow \text{A} + \text{H}_2\text{O} \). Product ‘A’ in neutral or acidic medium disproportionates to give products ‘B’ and ‘C’ along with water. The sum of spin-only magnetic moment values of B and C is _________ BM (nearest integer).
(Given atomic number of Mn is 25)