JEE Main 2023 Jan 25 Shift 1 Question Paper with Solution PDF- Download Here

The vector \(\vec{a}=-\hat{i}+2 \hat{j}+\hat{k}\) is rotated through a right angle, passing through the y-axis in its way and the resulting vector is \(\vec{b}\). Then the projection of \(3 \vec{a}+\sqrt{2} \vec{b}\) on \(\vec{c}=5 \hat{i}+4 \hat{j}+3 \hat{k}\) is :

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

The minimum value of the function \(f(x)=\int\limits_0^2 e^{|x-t|} d t\) is :

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

Let \(x=2\) be a local minima of the function \(f(x)=2 x^4-18 x^2+8 x+12\), \(x \in(-4,4)\) If \(M\) is local maximum value of the function \(f\) in \((-4,4)\), then \(M =\)

• (1) \(18 \sqrt{6}-\frac{33}{2}\)
• (2) \(12 \sqrt{6}-\frac{33}{2}\)
• (3) \(12 \sqrt{6}-\frac{31}{2}\)
• (4) \(18 \sqrt{6}-\frac{31}{2}\)

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively Later, the marks of one of the students is increased from 8 to 12 If the new mean of the marks is \(10.2\), then their new variance is equal to :

• (1) \(3.92\)
• (2) \(3.96\)
• (3) \(4.04\)
• (4) \(4.08\)

The value of \(\displaystyle\lim _{n \rightarrow \infty} \frac{1+2-3+4+5-6+\ldots +(3 n-2)+(3 n-1)-3 n}{\sqrt{2 n^4+4 n+3-} \sqrt{n^4+5 n+4}}\) is :

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

Let \(M\) be the maximum value of the product of two positive integers when their sum is 66. Let the sample space \(S =\left\{x \in Z : x(66-x) \geq \frac{5}{9} M\right\}\) and the event \(A =\{x \in S : x\) is a multiple of 3\(\}\). Then \(P ( A )\) is equal to

• (1) \(\frac{15}{44}\)
• (2) \(\frac{1}{3}\)
• (3) \(\frac{7}{22}\)
• (4) \(\frac{1}{5}\)

Let \(f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x\) If \(f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)\), then \(f(4)\) is equal to

• (1) \(\log _{ e } 17-\log _{ e } 18\)
• (2) \(\log _e 19-\log _e 20\)
• (3) \(\frac{1}{2}\left(\log _e 19-\log _e 17\right)\)
• (4) \(\frac{1}{2}\left(\log _e 17-\log _e 19\right)\)

The distance of the point \(P (4,6,-2)\) from the line passing through the point \((-3,2,3)\) and parallel to a line with direction ratios \(3,3,-1\) is equal to :

• (1) \(2 \sqrt{3}\)
• (2) \(\sqrt{14}\)
• (3) 3
• (4) \(\sqrt{6}\)

Let \(y (x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)\). Then \(y^{\prime}-y^{\prime \prime}\) at \(x=-1\) is equal to :

• (1) 976
• (2) 944
• (3) 496
• (4) 464

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

The points of intersection of the line \(a x+b y=0,(a \neq b)\) and the circle \(x^2+y^2-2 x=0\) are \(A (\alpha, 0)\) and \(B (1, \beta)\) The image of the circle with \(AB\) as a diameter in the line \(x+y+2=0\) is :

• (1) \(x^2+y^2+3 x+3 y+4=0\)
• (2) \(x^2+y^2+5 x+5 y+12=0\)
• (3) \(x^2+y^2+3 x+5 y+8=0\)
• (4) \(x^2+y^2-5 x-5 y+12=0\)

Let \(z_1=2+3 i\) and \(z_2=3+4 i\). The set \(S=\left\{z \in C:\left|z-z_1\right|^2-\left|z-z_2\right|^2=\left|z_1-z_2\right|^2\right\}\) represents a

• (1) straight line with the sum of its intercepts on the coordinate axes equals \(-18\)
• (2) hyperbola with eccentricity 2
• (3) hyperbola with the length of the transverse axis 7
• (4) straight line with the sum of its intercepts on the coordinate axes equals 14

Let \(S_1\) and \(S_2\) be respectively the sets of all \(a \in R -\{0\}\) for which the system of linear equations \( a x+2 a y-3 a z=1\) \( (2 a+1) x+(2 a+3) y+(a+1) z=2\) \((3 a+5) x+(a+5) y+(a+2) z=3\) has unique solution and infinitely many solutions Then

• (1) \(S_1=\Phi\) and \(S_2=R-\{0\}\)
• (2) \(S_1\) is an infinite set and \(n\left(S_2\right)=2\)
• (3) \(S_1=R-\{0\}\) and \(S_2=\Phi\)
• (4) \(n \left( S _1\right)=2\) and \(S _2\) is an infinite set

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}=\frac{y}{x}\left(1+x y^2\left(1+\log _e x\right)\right), x>0, y(1)=3\) Then \(\frac{y^2(x)}{9}\) is equal to :

• (1) \(\frac{x^2}{5-2 x^3\left(2+\log _e x^3\right)}\)
• (2) \(\frac{x^2}{3 x^3\left(1+\log _e x^2\right)-2}\)
• (3) \(\frac{x^2}{7-3 x^3\left(2+\log _e x^2\right)}\)
• (4) \(\frac{x^2}{2 x^3\left(2+\log _e x^3\right)-3}\)

Let \(f:(0,1) \rightarrow R\) be a function defined by

\(f(x)=\frac{1}{1-e^{-x}}\), and \(g(x)=(f(-x)-f(x))\) Consider two statements

(I) \(g\) is an increasing function in \((0,1)\)

(II) \(g\) is one-one in \((0,1)\)Then,

• (1) Only (I) is true
• (2) Both (I) and (II) are true
• (3) Neither (I) nor (II) is true
• (4) Only (II) is true

The statement \((p \wedge(\sim q)) \Rightarrow(p \Rightarrow(\sim q))\) is

• (1) equivalent to \(p \vee q\)
• (2) equivalent to \((-p) \vee(-q)\)
• (3) a contradiction
• (4) a tautology

If \(a_r\) is the coefficient of \(x^{10-r}\) in the Binomial expansion of \((1+x)^{10}\), then \(\displaystyle\sum_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2\) is equal to

• (1) 3025
• (2) 1210
• (3) 5445
• (4) 4895

Let \(x, y, z>1\) and \(A=\begin{bmatrix}1 & \log _x y & \log _x z \\ \log _y x & 2 & \log _y z \\ \log _z x & \log _z y & 3\end{bmatrix}\) Then ∣adj(adjA2)∣ is equal to

• (1) \(4^8\)
• (2) \(2^8\)
• (3) \(2^4\)
• (4) \(6^4\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non zero vectors such that \(\vec{b} \cdot \vec{c}=0\) and \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}\) If \(\vec{d}\) be a vector such that \(\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}\), then \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})\) is equal to

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

The distance of the point \((6,-2 \sqrt{2})\) from the common tangent \(y=m x+c, m>\), of the curves \(x=2 y^2\) and \(x=1+y^2\) is :

• (1) \(5 \sqrt{3}\)
• (2) \(\frac{14}{3}\)
• (3) \(\frac{1}{3}\)
• (4) 5

The constant term in the expansion of \(\left(2 x+\frac{1}{x^7}+3 x^2\right)^5\) is_____

Let \(S =\{1,2,3,5,7,10,11\}\) The number of non-empty subsets of \(S\) that have the sum of all elements a multiple of 3 , is _____

Let \(x\) and \(y\) be distinct integers where \(1 \leq x \leq 25\) and \(1 \leq y \leq 25\) Then, the number of ways of choosing \(x\) and \(y\), such that \(x+y\) is divisible by 5 , is ____

for some a,b,c ∈ \(\N\), let f(x) = ax-3 and g(x)=xb+c, x ∈ \(\R\). If (fog)^-1 (x) = \((\frac{x-7}{2})^{\frac{1}{3}}\) then (fog) (ac) + (gof) (b) is equal to _________ .

Let \(S=\left\{\alpha: \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2\right\}\) Then the maximum value of \(\beta\) for which the equation \(x^2-2\left(\displaystyle\sum_{\alpha \in s} \alpha\right)^2 x+\displaystyle\sum_{\alpha \in s}(\alpha+1)^2 \beta=0\) has real roots, is _____

Let \(A_1, A_2, A_3\) be the three \(A P\) with the same common difference \(d\) and having their first terms as \(A , A +1, A +2\), respectively, Let \(a, b, c\) be the \(7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}\) terms of \(A_1, A_2, A_3\), respectively such that \(\begin{vmatrix}a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{vmatrix}+70=0\)If \(a=29\), then the sum of first 20 terms of an AP whose first term is \(c-a-b\) and common difference is \(\frac{d}{12}\), is equal to

It the area enclosed by the parabolas \(P_1: 2 y=5 x^2\) and \(P_2: x^2-y+6=0\) is equal to the area enclosed by \(P_1\) and \(y=\alpha x, \alpha>0\), then \(\alpha^3\) is equal to ____ .

If the sum of all the solutions of \(\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3},-1 < x < 1, x \neq 0\), is \(\alpha-\frac{4}{\sqrt{3}}\), then \(\alpha\) is equal to ___

The vertices of a hyperbola \(H\) are \((\pm 6,0)\) and its eccentricity is \(\frac{\sqrt{5}}{2}\) Let \(N\) be the normal to \(H\) at a point in the first quadrant and parallel to the line \(\sqrt{2} x+y=2 \sqrt{2}\)If \(d\) is the length of the line segment of \(N\) between \(H\) and the \(y\)-axis then \(d^2\) is equal to

Let the equation of the plane passing through the line \(x-2 y-z-5=0=x+y+3 z-5\) and parallel to the line \(x+y+2 z-7=0=2 x+3 y+z-2\) be \(a x+b y+c z=65\) Then the distance of the point \((a, b, c)\) from the plane \(2 x+2 y-z+16=0\) is _______

A parallel plate capacitor has plate area \(40\, cm ^2\) and plates separation \(2\, mm\) The space between the plates is filled with a dielectric medium of a thickness \(1 \,mm\) and dielectric constant \(5\) The capacitance of the system is :

• (1) \(\frac{10}{3} \varepsilon_0 F\)
• (2) \(\frac{3}{10} \varepsilon_0 F\)
• (3) \(24 \varepsilon_0 F\)
• (4) \(10 \varepsilon_0 F\)

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

The root mean square velocity of molecules of gas is

• (1) Inversely proportional to square root of temperature \(\left(\sqrt{\frac{1}{T}}\right)\)
• (2) Proportional to square root of temperature \((\sqrt{T})\)
• (3) Proportional to square of temperature \(\left(T^2\right)\)
• (4) Proportional to temperature (T)

Given below are two statements ; one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\)

Assertion A: Photodiodes are used in forward bias usually for measuring the light intensity

Reason R: For a p-n junction diode at applied voltage \(V\) the current in the forward bias is more than the current in the reverse bias for \(\left|V_2\right|>\pm V \geq\left|V_0\right|\) where \(V_0\) is the threshold voltage and \(V _2\) is the breakdown voltage

In the light of the above statements, choose the correct answer from the options given below

• (1) A is true but \(R\) is false
• (2) A is false but \(R\) is true
• (3) Both \(A\) and \(R\) are true but \(R\) is NOT the correct explanation \(A\)
• (4) Both \(A\) and \(R\) are true and \(R\) is correct explanation \(A\)

Match List I with List II 

List I (Current configuration)		List II(Magnetic field at point O)
A		i	\(B_0=\frac{\mu_0I}{4\pi r}[\pi+2]\)
B		ii	\(B_0=\frac{\mu_0}{4}\frac{I}{r}\)
C		iii	\(B_0=\frac{\mu_0I}{2\pi r}[\pi-1]\)
D		iv	\(B_0=\frac{\mu_0I}{4\pi r}[\pi+1]\)

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

In Young's double slits experiment, the position of \(5^{\text {th }}\) bright fringe from the central maximum is \(5 \,cm\) The distance between slits and screen is \(1 \,m\) and wavelength of used monochromatic light is \(600 \,mm\) The separation between the slits is:

• (1) \(12\, \mu m\)
• (2) \(60\, \mu m\)
• (3) \(36\, \mu m\)
• (4) \(48 \,\mu m\)

\(T\) is the time period of simple pendulum on the earth's surface Its time period becomes \(x T\) when taken to a height \(R\) (equal to earth's radius) above the earth's surface Then, the value of \(x\) will be:

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

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth It is found that when a particle is released in this tunnel, it executes a simple harmonic motion The mass of the particle is \(100 \,g\) The time period of the motion of the particle will be (approximately)(Take \(g=10 \, m s ^{-2}\), radius of earth \(=6400 \,km\) )

• (1) 1 hour 24 minutes
• (2) 1 hour 40 minutes
• (3) 12 hours
• (4) 24 hours

A car travels a distance of ' \(x\) ' with speed \(v_1\) and then same distance ' \(x\) ' with speed \(v_2\) in the same direction The average speed of the car is:

• (1) \(\frac{v_1 v_2}{2\left(v_1+v_2\right)}\)
• (2) \(\frac{2 x}{v_1+v_2}\)
• (3) \(\frac{2 v_1 v_2}{v_1+v_2}\)
• (4) \(\frac{v_1+v_2}{2}\)

Electron beam used in an electron microscope, when accelerated by a voltage of \(20 kV\), has a de-Broglie wavelength of \(\lambda_0\). If the voltage is increased to \(40 kV\), then the de-Broglie wavelength associated with the electron beam would be:

• (1) \(\frac{\lambda_0}{2}\)
• (2) \(3 \lambda_0\)
• (3) \(\frac{\lambda_0}{\sqrt{2}}\)
• (4) \(9 \lambda_0\)

A uniform metallic wire carries a current \(2 A\), when \(34 V\) battery is connected across it The mass of uniform metallic wire is \(892 \times 10^{-3} kg\), density is \(892 \times 10^3 kg / m ^3\) and resistivity is \(17 \times 10^{-8} \Omega- m\) The length of wire is :

• (1) \(l=10\, m\)
• (2) \(l=100\, m\)
• (3) \(l=5\, m\)
• (4) \(l=6.8 \, m\)

A solenoid of \(1200\) tums is wound uniformly in a single layer on a glass tube \(2\, m\) long and \(02\, m\) in diameter The magnetic intensity at the center of the solenoid when a current of \(2 A\) flows through it is:

• (1) \(2.4 \times 10^3 A m ^{-1}\)
• (2) \(1.2 \times 10^3 A m ^{-1}\)
• (3) \(1 Am ^{-1}\)
• (4) \(2.4 \times 10^{-3} A m ^{-1}\)

An electromagnetic wave is transporting energy in the negative \(z\) direction At a certain point and certain time the direction of electric field of the wave is along positive \(y\) direction What will be the direction of the magnetic field of the wave at that point and instant?

• (1) Positive direction of \(z\)
• (2) Negative direction of \(y\)
• (3) Positive direction of \(x\)
• (4) Negative direction of \(x\)

An object of mass \(8 kg\) is hanging from one end of a uniform rod CD of mass \(2 kg\) and length \(1 m\) pivoted at its end \(C\) on a vertical wall as shown in figure It is supported by a cable \(AB\) such that the system is in equilibrium The tension in the cable is : (Take \(g =10 m / s ^2\) )

• (1) \(240 N\)
• (2) \(30 N\)
• (3) \(300 N\)
• (4) \(90 N\)

In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes \(x\) times its initial resonant frequency \(\omega_0\) The value of \(x\) is:

• (1) \(1 / 4\)
• (2) 4
• (3) 16
• (4) 1/16

A car is moving with a constant speed of \(20\, m / s\) in a circular horizontal track of radius \(40\, m\) A bob is suspended from the roof of the car by a massless string The angle made by the string with the vertical will be : (Take \(g=10 \, m / s ^2\) )

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

A Carnot engine with efficiency \(50 \%\) takes heat from a source at \(600 \,K\) In order to increase the efficiency to \(70 \%\), keeping the temperature of sink same, the new temperature of the source will be :

• (1) \(900 K\)
• (2) \(300 K\)
• (3) \(1000 K\)
• (4) \(360 K\)

The ratio of the density of oxygen nucleus \(\left({ }_8^{16} O \right)\) and helium nucleus \(\left({ }_2^4 He \right)\) is

• (1) \(4: 1\)
• (2) \(1: 1\)
• (3) \(8: 1\)
• (4) \(2: 1\)

A message signal of frequency \(5 \,kHz\) is used to modulate a carrier signal of frequency \(2\, MHz\) The bandwidth for amplitude modulation is:

• (1) \(10 \,kHz\)
• (2) \(2.5 \,kHz\)
• (3) \(20\, kHz\)
• (4) \(5\, kHz\)

A bowl filled with very hot soup cools from \(98^{\circ} C\) to \(86^{\circ} C\) in 2 minutes when the room temperature is \(22^{\circ} C\). How long it will take to cool from \(75^{\circ} C\) to \(69^{\circ} C\) ?

• (1) 2 minutes
• (2) 1 minute
• (3) \(1.4\) minutes
• (4) \(0.5\) minute

If \(\vec{P}=3 \hat{i}+\sqrt{3} \hat{j}+2 \hat{k}\) and \(\vec{Q}=4 \hat{i}+\sqrt{3} \hat{j}+25 \hat{k}\) then, The unit vector in the direction of \(\vec{P} \times \vec{Q}\) is \(\frac{1}{x}(\sqrt{3} i+\hat{j}-2 \sqrt{3} \hat{k})\) The value of \(x\) is

\(I _{ CM }\) is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc \(I_{A B}\) is it's moment of inertia about an axis \(AB\) perpendicular to plane and parallel to axis \(CM\) at a distance \(\frac{2}{3} R\) from center Where \(R\) is the radius of the dise The ratio of \(I _{ AB }\) and \(I _{ CM }\) is \(x: 9\) The value of \(x\) is______

A ray of light is incident from air on a glass plate having thickness \(\sqrt{3} cm\) and refractive index \(\sqrt{2}\) The angle of incidence of a ray is equal to the critical angle for glass-air interface The lateral displacement of the ray when it passes through the plate is ___\(\times 10^{-2} cm\) (given \(\sin 15^{\circ}=026\) )

The wavelength of the radiation emitted is \(\lambda_0\) when an electron jumps from the second excited state to the first excited state of hydrogen atom If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be \(\frac{20}{x} \lambda_0\) The value of \(x\) is

The distance between two consecutive points with phase difference of \(60^{\circ}\) in a wave of frequency \(500 \,Hz\) is \(60\, m\) The velocity with which wave is traveling is ___\(km / s\)

As shown in the figure, In an experiment to determine Young's modulus of a wire the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of \(45^{\circ}\) with the load axis The length of wire is \(628 \,cm\) and its diameter is \(4 \,mm\) The Young's modulus is found to be \(x \times 10^4 Nm ^{-2}\)The value of \(x\) is

An LCR series circuit of capacitance \(625\ nF\) and resistance of \(50 \ \Omega\) is connected to an AC source of frequency \(20\ kHz\) For maximum value of amplitude of current in circuit, the value of inductance is ___ \(mH\). (Take \(\pi^2 =10\))

In the given circuit, the equivalent resistance between the terminal \(A\) and \(B\) is __\(\Omega\)

A uniform electric field of \(10 N / C\) is created between two parallel charged plates (as shown in figure) An electron enters the field symmetrically between the plates with a kinetic energy \(05 eV\) The length of each plate is \(10 cm\) The angle \((\theta)\) of deviation of the path of electron as it comes out of the field is ___ (in degree)

Inert gases have positive electron gain enthalpy Its correct order is

• (1) \(He \lt Ne \lt Kr _{ r }\lt Xe _{ e }\)
• (2) \(He \lt Xe \lt Kr \lt Ne\)
• (3) \(He \lt Kr \lt Xe _{ e }\lt Ne\)
• (4) \(Xe \lt Kr \lt Ne \lt He\)

The correct sequence of reagents for the preparation of Q and R is:

• (1)

  (i) Cr₂O₃, 770K,20atm (ii) CrO₂Cl₂,H₃O⁺; (iii) NaOH; (iv) H₃O⁺

  
   

• (2) (i) Mo₂O₃,Δ; (ii) CrO₂Cl₂,H₃O⁺; (iii) NaOH; (iv) H₃O⁺
   
• (3)

  (i) CrO₂Cl₂,H₃O⁺; (ii) Cr₂O₃,770K,20atm;(iii) NaOH; (iv) H₃O⁺

  
   

• (4) (i) KMnO_4,OH⁻; (ii) Mo₂O₃,Δ; (iii) NaOH; (iv) H₃O⁺
   

In the cumene to phenol preparation in presence of air, the intermediate is

• (1)
• (2)
• (3)
• (4)

Match items of Row I with those of Row II:

[(i)] \(\alpha\)-D-(-)-Fructofuranose 
[(ii)] \(\beta\)-D-(-)-Fructofuranose 
[(iii)] \(\alpha\)-D-(+)-Glucopyranose 
[(iv)] \(\beta\)-D-(+)-Glucopyranose 
Correct Match:

• (1) \(P \rightarrow\) iv, \(Q \rightarrow\) iii, \(R \rightarrow\) i., \(S \rightarrow ii\)
• (2) \(P \rightarrow\) iii, \(Q \rightarrow\) iv, \(R \rightarrow\) i., \(S \rightarrow ii\)
• (3) \(P \rightarrow\) iii, \(Q \rightarrow\) iv, \(R \rightarrow\) ii., \(S \rightarrow i\)
• (4) \(P \rightarrow\) i, \(Q \rightarrow\) ii, \(R \rightarrow\) iii., \(S \rightarrow iv\)

The compound which will have the lowest rate towards nucleophilic aromatic substitution on treatment with \(\mathrm{OH^-}\) is

• (1)

• (2)

• (3)

• (4)

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

Identify the product formed (A and E)

• (1)
• (2)
• (3)
• (4)

' \(25\) volume' hydrogen peroxide means

• (1) \(100 \,mL\) marketed solution contains \(25\, g\) of \(H _2 O _2\).
• (2) \(1 L\) marketed solution contains \(75 \,g\) of \(H _2 O _2\).
• (3) \(1 L\) marketed solution contains \(250\, g\) of \(H _2 O _2\).
• (4) \(1 L\) marketed solution contains \(25\, g\) of \(H _2 O _2\).

The radius of the \(2^{\text {nd }}\) orbit of \(Li ^{2+}\) is \(x\). The expected radius of the \(3^{\text {rd }}\) orbit of \(Be ^{3+}\) is

• (1) \(\frac{16}{27} \pi\)
• (2) \(\frac{4}{9} x\)
• (3) \(\frac{9}{4} x\)
• (4) \(\frac{27}{16} x\)

Which of the following conformations will be the most stable?

• (1)
• (2)
• (3)
• (4)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason \(R\):
Assertion A: Acetal / Ketal is stable in basic medium
Reason R: The high leaving tendency of alkoxide ion gives the stability to acetal ketal in basic medium
In the light of the above statements, choose the correct answer from the options given below :

• (1) Both \(A\) and \(R\) are true but \(R\) is NOT the correct explanation of \(A\)
• (2) A is false but \(R\) is true
• (3) A is true but \(R\) is false
• (4) Both \(A\) and \(R\) are true and \(R\) is the correct explanation of \(A\)

The correct order in aqueous medium of basic strength in case of methyl substituted amines is :

• (1) \(NH _3\gt Me _3 N \gt MeNH _2\gt Me _2 NH\)
• (2) \(Me _2 NH \gt Me _3 N \gt MeNH _2\gt NH _3\)
• (3) \(Me _2 NH \gt MeNH _2\gt Me _3 N \gt NH _3\)
• (4) \(Me _3 N \gt Me _2 NH \gt MeNH _2\gt NH _3\)

Match the List-I with List-II :

• (1) \(P \rightarrow\) iv, \(Q \rightarrow\) ii, \(R \rightarrow\) ii., \(S \rightarrow i\)
• (2) \(P \rightarrow i , Q \rightarrow iii , R \rightarrow\) ii, \(S \rightarrow\) iv
• (3) \(P \rightarrow\) iii, \(Q \rightarrow\) i, \(R \rightarrow\) iv, \(S \rightarrow\) ii
• (4) \(P \rightarrow\) i. \(Q \rightarrow\) iii, \(R \rightarrow\) iv, \(S \rightarrow\) ii

Reaction of thionyl chloride with white phosphorus forms a compound [A], which on hydrolysis gives [B], a dibasic acid \([ A ]\) and \([ B ]\) are respectively

• (1) \(P _4 O _6\) and \(H _3 PO _3\)
• (2) \(PCl _5\) and \(H _3 PO _4\)
• (3) \(PCl _3\) and \(H _3 PO _3\)
• (4) \(POCl _3\) and \(H _3 PO _4\)

Which one of the following reactions does not occur during extraction of copper?

• (1) \(CaO + SiO _2 \rightarrow CaSiO _3\)
• (2) \(2 Cu _2 S +3 O _2 \rightarrow 2 Cu _2 O +2 SO _2\)
• (3) \(FeO + SiO _2 \rightarrow FeSiO _3\)
• (4) \(2 FeS +3 O _2 \rightarrow 2 FeO +2 SO _2\)

Which of the following statements is incorrect for antibiotics?

• (1) An antibiotic must be a product of metabolism.
• (2) An antibiotic should be effective in low concentrations.
• (3) An antibiotic is a synthetic substance produced as a structural analogue of naturally occurring antibiotic.
• (4) An antibiotic should promote the growth or survival of microorganisms.

• (1) \(X =[ O ], Y = NO , A = O _2, B = O _3\)
• (2) \(X = N _2 O , Y =[ O ], A = O _3, B = NO\)
• (3) \(X =\frac{1}{2} O _2, Y = NO _2, A = O _3, B = O _2\)
• (4) \(X = NO , Y =[ O ], A = O _2, B = N _2 O _3\)

The variation of the rate of an enzyme catalyzed reaction with substrate concentration is correctly represented by graph

• (1)
• (2)
• (3)
• (4)

Compound A reacts with \(NH _4 Cl\) and forms a compound B.

Compound B reacts with \(H _2 O\) and excess of \(CO _2\) to form compound \(C\) which on passing through or reaction with saturated \(NaCl\) solution forms sodium hydrogen carbonate Compound \(A , B\) and \(C\), are respectively.

• (1) \(CaCl _2, NH _4^{+},\left( NH _4\right)_2 CO _3\)
• (2) \(CaCl _2, NH _3, NH _4 HCO _3\)
• (3) \(Ca ( OH )_2, NH _3, NH _4 HCO _3\)
• (4) \(Ca ( OH )_2, NH _4^{+},\left( NH _4\right)_2 CO _3\)

A cubic solid is made up of two elements \(X\) and \(Y\) Atoms of \(X\) are present on every alternate corner and one at the center of cube \(Y\) is at \(\frac{1}{3} rd\) of the total faces. The empirical formula of the compound is

• (1) \(X _2 Y _{1.5}\)
• (2) \(X_{3} Y_2\) 
• (3) \(XY _{2, 5}\)
• (4) \(X _{1.5} Y _2\)

The density of a monobasic strong acid (Molar mass \(242 \,g\) mol) is \(121 \,kg \,L\) The volume of its solution required for the complete neutralization of \(25 \,mL\) of \(024\, M\, NaOH\) is ___\(\times 10^{-2} mL\) (Nearest integer)

A litre of buffer solution contains \(01\) mole of each of \(NH _3\) and \(NH _4 CL\) On the addition of \(002\) mole of \(HCl\) by dissolving gaseous \(HCl\), the \(pH\) of the solution is found to be ____ \(\times 10^{-3}\) (Nearest integer) [Given : \(pK _5\left( NH _3\right)=4745\) \( \log 2=0301\) \(\log 3=0477\) \(T =298 K ]\)

\(Pt(s) ∣ H2​(g)(1atm) ∣ H+(aq, [H+]=1)\, ∥\, Fe3+(aq), Fe2+(aq) ∣ Pt(s)\)
Given\( E^∘_{Fe^{3+}Fe^{2+}}\)\(=0.771V\) and \(E^∘_{H^{+1/2}H_2}​=0\,V,T=298K\)
If the potential of the cell is 0.712V, the ratio of concentration of \(Fe2+\) to \(Fe3+\) is

The osmotic pressure of solutions of PVC in cyclohexanone at \(300 K\) are plotted on the graph The molar mass of \(PVC\) is ____\(g \,mol ^{-1}\) (Nearest integer)

( Given : R = 0.083 L atm K^-1 mol^-1 )

In sulphur estimation, \(0471 \,g\) of an organic compound gave \(14439 \,g\) of barium sulphate The percentage of sulphur in the compound is ___(Nearest Integer) (Grven: Atomic mass Ba: \(137 \,u , S : 32\, u , O : 16 \,u\) )

An athlete is given \(100\, g\) of glucose \(\left( C _6 H _{12} O _6\right)\) for energy This is equivalent to \(1800\, kJ\) of energy The \(50 \%\) of this energy gained is utilized by the athlete for sports activities at the event In order to avoid storage of energy, the weight of extra water he would need to perspire is ___ \(g\) (Nearest integer) Assume that there is no other way of consuming stored energy Given: The enthalpy of evaporation of water is \(45 \,kJ \,m o l^{-1}\) Molar mass of \(C H\, \&\, O\) are 12,1 and \(16\, g \,mol ^{-1}\)

How many of the following metal ions have similar value of spin only magnetic moment in gaseous state? (Given: Atomic number: \(V, 23; Cr, 24; Fe, 26; Ni, 28\)) \(V ^{3+} \cdot Ci ^{3+} \cdot Fe ^{2+} \cdot Ni ^{3+}\)

The number of paramagnetic species from the following is ________
\( {\left[ Ni ( CN )_4\right]^{2-} \cdot\left[ Ni ( CO )_4\right],\left[ NiCl _4\right]^{2-}} \) \( {\left[ Fe ( CN )_6\right]^{4-},\left[ Cu \left( NH _3\right)_4\right]^{2+}}\) \( {\left[ Fe ( CN )_6\right]^{3-} \text { and }\left[ Fe \left( H _2 O \right)_6\right]^{2+}}\)

For the first order reaction \(A \rightarrow B\), the half life is \(30 min\) The time taken for \(75 \%\) completion of the reaction is min (Nearest integer) Given : \(\log _g 2=03010\) \( \log 3=04771 \) \( \log 5=06989\)

The total number of lone pairs of electrons on oxygen atoms of ozone is _______