WB JEE 2024
JEE 2024 Previous Year
3 hDuration
200Total Marks
153Questions
3Sections
Instructions
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Paper Structure
Chemistry
Chemistry
Q1.
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Toluene reacts with mixed acid at $$25^{\circ} \mathrm{C}$$ to produce
Q2.
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How many $$\mathrm{P}-\mathrm{O}-\mathrm{P}$$ linkages are there in $$\mathrm{P}_4 \mathrm{O}_{10}$$
Q3.
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The number(s) of $$-\mathrm{OH}$$ group(s) present in $$\mathrm{H}_3 \mathrm{PO}_3$$ and $$\mathrm{H}_3 \mathrm{PO}_4$$ is/are
Q4.
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Which of the following species exhibits both LMCT and paramagnetism?
Q5.
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Identify the ion having $$4 f^6$$ electronic configuration.
Q6.
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The equivalent weight of $$\mathrm{Na}_2 \mathrm{S}_2 \mathrm{O}_3(\mathrm{Gram}$$ molecular weight $$=\mathrm{M})$$ in the given reaction is
$$\mathrm{I}_2+2 \mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3=2 \mathrm{NaI}+\mathrm{Na}_2 \mathrm{~S}_4 \mathrm{O}_6$$
Q7.
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Which hydrogen like species will have the same radius as that of $$1^{\text {st }}$$ Bohr orbit of hydrogen atom?
Q8.
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Consider an electron moving in the first Bohr orbit of a $$\mathrm{He}^{+}$$ ion with a velocity $$v_1$$. If it is allowed to move in the third Bohr orbit with a velocity $$v_3$$, then indicate the correct $$v_3: v_1$$ ratio.
Q9.
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Correct solubility order of $$\mathrm{AgF}, \mathrm{AgCl}, \mathrm{AgBr}, \mathrm{AgI}$$ in water is
Q10.
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What will be the change in acidity if
(i) $$\mathrm{CuSO}_4$$ is added in saturated $$(\mathrm{NH}_4)_2 \mathrm{SO}_4$$ solution
(ii) $$\mathrm{SbF}_5$$ is added in anhydrous $$\mathrm{HF}$$
Q11.
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$$\mathrm{pH}$$ of $$10^{-8}(\mathrm{M}) \mathrm{~HCl}$$ solution is
Q12.
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The specific conductance $$(\mathrm{k})$$ of $$0.02(\mathrm{M})$$ aqueous acetic acid solution at $$298 \mathrm{~K}$$ is $$1.65 \times 10^{-4} \mathrm{~S} \mathrm{~cm}^{-1}$$. The degree of dissociation of acetic acid is
$$[\lambda_{\mathrm{O}^{+}}+=349.1 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1} \text { and } \lambda_{{ }^{\circ} \mathrm{CH}_3 \mathrm{COO}^{-}}=40.9 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}]$$
Q13.
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At 25$$^\circ$$C, the ionic product of water is 10$$^{-14}$$. The free energy change for the self-ionization of water in kCal mol$$^{-1}$$ is close to
Q14.
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Number of moles of ions produced by complete dissociation of one mole of Mohr's salt in water is
Q15.
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Equal volumes of aqueous solution of $$0.1(\mathrm{M}) \mathrm{HCl}$$ and $$0.2(\mathrm{M}) \mathrm{H}_2 \mathrm{SO}_4$$ are mixed. The concentration of $$\mathrm{H}^{+}$$ ions in the resulting solution is
Q16.
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The product 'P' in the above reaction is
The product 'P' in the above reaction is
Q17.
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Which of the following contains maximum number of lone pairs on the central atom?
Q18.
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For a first order reaction with rate constant $$\mathrm{k}$$, the slope of the plot of $$\log$$ (reactant concentration) against time is
Q19.
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The reactivity order of the following molecules towards $$\mathrm{S}_{\mathrm{N}} 1$$ reaction is
$$\begin{array}{ccc}
\text { Allyl chloride } & \text { Chlorobenzene } & \text { Ethyl chloride } \\
\text { (I) } & \text { (II) } & \text { (III) }
\end{array}$$
Q20.
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The major product of the following reaction is :
Q21.
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Identify the incorrect statement among the following :
Q22.
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Which of the following statements is true about equilibrium constant and rate constant of a single step chemical reaction?
Q23.
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Which of the following statements is correct for a spontaneous polymerization reaction ?
Q24.
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For a spontaneous process, the incorrect statement is
Q25.
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The compressibility factor for a van der Waal gas at high pressure is
Q26.
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The correct acidity order of phenol (I), 4-hydroxybenzaldehyde (II) and 3-hydroxybenzaldehyde (III) is
Q27.
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In the following sequence of reaction compound 'M' is
Q28.
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Ozonolysis of $$\underline{o}$$-xylene produces
Q29.
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Which of the following statement/statements is/are correct ?
Q30.
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The correct order of boiling point of the given aqueous solutions is
Q31.
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Which of the following ion/ions is/are diamagnetic ?
Q32.
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After the emission of a $$\beta$$-particle followed by an $$\alpha$$-particle from $${ }_{83}^{214} \mathrm{Bi}$$, the number of neutrons in the atom is -
Q33.
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Which of the following represent(s) the enantiomer of Y ?
Q34.
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The compound that does not give positive test for nitrogen in Lassaigne's test is
Q35.
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$$\mathrm{Q}$$ and $$\mathrm{R}$$ in the above reaction sequences are respectively
$$\mathrm{Q}$$ and $$\mathrm{R}$$ in the above reaction sequences are respectively
Q36.
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The decreasing order of reactivity of the following alkenes towards $$\mathrm{HBr}$$ addition is
Q37.
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Which of the following statements about the $$\mathrm{S}_{\mathrm{N}} 2$$ reaction mechanism is/are true?
Q38.
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The compounds A and B are respectively
Q39.
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Metallic conductors and semiconductors are heated separately. What are the changes with respect to conductivity?
Q40.
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Identify the correct statement(s) :
Mathematics
Mathematics
Q1.
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If $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are in A.P. with common difference $$\theta$$, then the sum of the series
$$
\sec \alpha_1 \sec \alpha_2+\sec \alpha_2 \sec \alpha_3+\ldots .+\sec \alpha_{n-1} \sec \alpha_n=k\left(\tan \alpha_n-\tan \alpha_1\right)$$
where $$\mathrm{k}=$$
Q2.
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Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $$a_n$$ and $$b_n$$ be the $$n^{\text {th }}$$ term of A.P. and G.P. respectively then
Q3.
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If for the series $$a_1, a_2, a_3$$, ...... etc, $$\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}$$ bears a constant ratio with $$\mathrm{a}_{\mathrm{r}} \cdot \mathrm{a}_{\mathrm{r}+1}$$; then $$\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3 \ldots .$$. are in
Q4.
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The area bounded by the curves $$x=4-y^2$$ and the Y-axis is
Q5.
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Consider the function $$\mathrm{f}(x)=(x-2) \log _{\mathrm{e}} x$$. Then the equation $$x \log _{\mathrm{e}} x=2-x$$
Q6.
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Two smallest squares are chosen one by one on a chess board. The probability that they have a side in common is
Q7.
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Two integers $$\mathrm{r}$$ and $$\mathrm{s}$$ are drawn one at a time without replacement from the set $$\{1,2, \ldots, \mathrm{n}\}$$. Then $$\mathrm{P}(\mathrm{r} \leq \mathrm{k} / \mathrm{s} \leq \mathrm{k})=$$
(k is an integer < n)
Q8.
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A biased coin with probability $$\mathrm{p}(0<\mathrm{p}<1)$$ of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $$\frac{2}{5}$$, then $$\mathrm{p}=$$
Q9.
mcq multi
+2 / 0
If the quadratic equation $$a x^2+b x+c=0(a>0)$$ has two roots $$\alpha$$ and $$\beta$$ such that $$\alpha2$$, then
Q10.
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Let $$\mathrm{N}$$ be the number of quadratic equations with coefficients from $$\{0,1,2, \ldots, 9\}$$ such that 0 is a solution of each equation. Then the value of $$\mathrm{N}$$ is
Q11.
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If $$\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}$$ and $$\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}$$ where $$\mathrm{ac} \neq 0$$, then $$\mathrm{P}(x) \cdot \mathrm{Q}(x)=0$$ has (a, b, c, d are real)
Q12.
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If $$a, b, c$$ are distinct odd natural numbers, then the number of rational roots of the equation $$a x^2+b x+c=0$$
Q13.
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$$
\text { Let } f(x)=\left|\begin{array}{ccc}
\cos x & x & 1 \\
2 \sin x & x^3 & 2 x \\
\tan x & x & 1
\end{array}\right| \text {, then } \lim _\limits{x \rightarrow 0} \frac{f(x)}{x^2}=
$$
Q14.
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If $$\alpha, \beta$$ are the roots of the equation $$a x^2+b x+c=0$$ then $$\lim _\limits{x \rightarrow \beta} \frac{1-\cos \left(a x^2+b x+c\right)}{(x-\beta)^2}$$ is
Q15.
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Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be given by $$\mathrm{f}(x)=\left|x^2-1\right|$$, then
Q16.
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$$f(x)=\cos x-1+\frac{x^2}{2!}, x \in \mathbb{R}$$ Then $$\mathrm{f}(x)$$ is
Q17.
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Consider the function $$\mathrm{f}(x)=x(x-1)(x-2) \ldots(x-100)$$. Which one of the following is correct?
Q18.
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Let $$\mathrm{y}=\mathrm{f}(x)$$ be any curve on the $$\mathrm{X}-\mathrm{Y}$$ plane & $$\mathrm{P}$$ be a point on the curve. Let $$\mathrm{C}$$ be a fixed point not on the curve. The length $$\mathrm{PC}$$ is either a maximum or a minimum, then
Q19.
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If a particle moves in a straight line according to the law $$x=a \sin (\sqrt{\lambda} t+b)$$, then the particle will come to rest at two points whose distance is [symbols have their usual meaning]
Q20.
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+2 / 0
The acceleration f $$\mathrm{ft} / \mathrm{sec}^2$$ of a particle after a time $$\mathrm{t}$$ sec starting from rest is given by $$\mathrm{f}=6-\sqrt{1.2 \mathrm{t}}$$. Then the maximum velocity $$\mathrm{v}$$ and time $$\mathrm{T}$$ to attend this velocity are
Q21.
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If $$\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \cdot A \cdot\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$$, then $$A=$$
Q22.
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Let $$A=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1\end{array}\right), B=\left(\begin{array}{l}2 \\ 1 \\ 7\end{array}\right)$$
Then for the validity of the result $$\mathrm{AX}=\mathrm{B}, \mathrm{X}$$ is
Q23.
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Let $$A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$$, then
Q24.
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If $$A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$$ and $$\theta=\frac{2 \pi}{7}$$, then $$A^{100}=A \times A \times \ldots .(100$$ times) is equal to
Q25.
mcq single
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$$
\text { If }\left|\begin{array}{lll}
x^k & x^{k+2} & x^{k+3} \\
y^k & y^{k+2} & y^{k+3} \\
z^k & z^{k+2} & z^{k+3}
\end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \text {, then }
$$
Q26.
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+2 / 0
If $$\mathrm{a}_{\mathrm{i}}, \mathrm{b}_{\mathrm{i}}, \mathrm{c}_{\mathrm{i}} \in \mathbb{R}(\mathrm{i}=1,2,3)$$ and $$x \in \mathbb{R}$$ and $$\left|\begin{array}{lll}\mathrm{a}_1+b_1 x & a_1 x+b_1 & c_1 \\ \mathrm{a}_2+b_2 x & a_2 x+b_2 & c_2 \\ \mathrm{a}_3+b_3 x & a_3 x+b_3 & c_3\end{array}\right|=0$$, then
Q27.
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If the relation between the direction ratios of two lines in $$\mathbb{R}^3$$ are given by
$$l+\mathrm{m}+\mathrm{n}=0,2 l \mathrm{~m}+2 \mathrm{mn}-l \mathrm{n}=0$$
then the angle between the lines is
($$l, \mathrm{~m}, \mathrm{n}$$ have their usual meaning)
Q28.
mcq single
+2 / 0.5
Angle between two diagonals of a cube will be
Q29.
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+1 / 0.25
The plane $$2 x-y+3 z+5=0$$ is rotated through $$90^{\circ}$$ about its line of intersection with the plane $$x+y+z=1$$. The equation of the plane in new position is
Q30.
mcq single
+2 / 0.5
If $$A$$ and $$B$$ are acute angles such that $$\sin A=\sin ^2 B$$ and $$2 \cos ^2 A=3 \cos ^2 B$$, then $$(A, B)=$$
Q31.
mcq single
+1 / 0.25
The expression $$\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)$$ is
Q32.
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+1 / 0.25
If $$0< \theta<\frac{\pi}{2}$$ and $$\tan 3 \theta \neq 0$$, then $$\tan \theta+\tan 2 \theta+\tan 3 \theta=0$$ if $$\tan \theta \cdot \tan 2 \theta=\mathrm{k}$$ where $$\mathrm{k}=$$
Q33.
mcq single
+1 / 0.25
$$
\text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then }
$$
Q34.
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$$\triangle \mathrm{OAB}$$ is an equilateral triangle inscribed in the parabola $$\mathrm{y}^2=4 \mathrm{a} x, \mathrm{a}>0$$ with O as the vertex, then the length of the side of $$\triangle \mathrm{O A B}$$ is
Q35.
mcq multi
+2 / 0
If $$n$$ is a positive integer, the value of $$(2 n+1){ }^n C_0+(2 n-1){ }^n C_1+(2 n-3){ }^n C_2 +\ldots .+1 \cdot{ }^n C_n$$ is
Q36.
mcq single
+1 / 0.25
If $$\left(1+x+x^2+x^3\right)^5=\sum_\limits{k=0}^{15} a_k x^k$$ then $$\sum_\limits{k=0}^7(-1)^{\mathbf{k}} \cdot a_{2 k}$$ is equal to
Q37.
mcq single
+1 / 0.25
The coefficient of $$a^{10} b^7 c^3$$ in the expansion of $$(b c+c a+a b)^{10}$$ is
Q38.
mcq single
+1 / 0.25
If $$\left(x^2 \log _x 27\right) \cdot \log _9 x=x+4$$ then the value of $$x$$ is
Q39.
mcq multi
+2 / 0
If $$\mathrm{ABC}$$ is an isosceles triangle and the coordinates of the base points are $$B(1,3)$$ and $$C(-2,7)$$. The coordinates of $$A$$ can be
Q40.
mcq single
+1 / 0.25
If $$(1,5)$$ be the midpoint of the segment of a line between the line $$5 x-y-4=0$$ and $$3 x+4 y-4=0$$, then the equation of the line will be
Q41.
mcq multi
+2 / 0
A square with each side equal to '$$a$$' above the $$x$$-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle $$\alpha$$ $$\left(0<\alpha< \frac{\pi}{4}\right)$$ with the positive direction of the axis. Equation of the diagonals of the square
Q42.
mcq multi
+2 / 0
Let $$\Gamma$$ be the curve $$\mathrm{y}=\mathrm{be}^{-x / a}$$ & $$\mathrm{L}$$ be the straight line $$\frac{x}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$$ where $$\mathrm{a}, \mathrm{b} \in \mathbb{R}$$.
Then
Q43.
mcq single
+1 / 0.25
In $$\triangle \mathrm{ABC}$$, co-ordinates of $$\mathrm{A}$$ are $$(1,2)$$ and the equation of the medians through $$\mathrm{B}$$ and C are $$x+\mathrm{y}=5$$ and $$x=4$$ respectively. Then the midpoint of $$\mathrm{BC}$$ is
Q44.
mcq single
+1 / 0.25
In R, a relation p is defined as follows:
$$\forall a, b \in \mathbb{R}, a p$$ holds iff $$a^2-4 a b+3 b^2=0$$. Then
Q45.
mcq single
+1 / 0.25
Let A be the set of even natural numbers that are < 8 & B be the set of prime integers that are $$<7$$ The number of relations from A to B are
Q46.
mcq single
+2 / 0.5
For the real numbers $$x$$ & $$y$$, we write $$x$$ p y iff $$x-y+\sqrt{2}$$ is an irrational number. Then relation p is
Q47.
mcq single
+1 / 0.25
If $$\cos \theta+i \sin \theta, \theta \in \mathbb{R}$$, is a root of the equation
$$a_0 x^n+a_1 x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots . a_n \in \mathbb{R}, a_0 \neq 0,$$
then the value of $$a_1 \sin \theta+a_2 \sin 2 \theta+\ldots .+a_n \sin n \theta$$ is
Q48.
mcq single
+1 / 0.25
If $$z_1$$ and $$z_2$$ be two roots of the equation $$z^2+a z+b=0, a^2<4 b$$, then the origin, $$\mathrm{z}_1$$ and $$\mathrm{z}_2$$ form an equilateral triangle if
Q49.
mcq single
+2 / 0.5
In a plane $$\vec{a}$$ and $$\vec{b}$$ are the position vectors of two points A and B respectively. A point $P$ with position vector $$\overrightarrow{\mathrm{r}}$$ moves on that plane in such a way that $$|\overrightarrow{\vec{r}}-\vec{a}| \sim|\vec{r}-\vec{b}|=c$$ (real constant). The locus of P is a conic section whose eccentricity is
Q50.
mcq single
+2 / 0.5
The locus of the midpoint of the system of parallel chords parallel to the line $$y=2 x$$ to the hyperbola $$9 x^2-4 y^2=36$$ is
Q51.
mcq single
+2 / 0
The function $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$\mathrm{f}(x)=\mathrm{e}^x+\mathrm{e}^{-x}$$ is :
Q52.
mcq multi
+2 / 0
Choose the correct statement :
Q53.
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Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$\mathrm{f}(x)=\frac{\mathrm{e}^{|x|}-\mathrm{e}^{-x}}{\mathrm{e}^x+\mathrm{e}^{-x}}$$, then
Q54.
mcq single
+1 / 0.25
For every real number $$x \neq-1$$, let $$\mathrm{f}(x)=\frac{x}{x+1}$$.
Write $$\mathrm{f}_1(x)=\mathrm{f}(x)$$ & for $$\mathrm{n} \geq 2, \mathrm{f}_{\mathrm{n}}(x)=\mathrm{f}\left(\mathrm{f}_{\mathrm{n}-1}(x)\right)$$. Then $$\mathrm{f}_1(-2) \cdot \mathrm{f}_2(-2) \ldots . . \mathrm{f}_{\mathrm{n}}(-2)$$ must be
Q55.
mcq single
+1 / 0.25
The equation $$2^x+5^x=3^x+4^x$$ has
Q56.
mcq single
+1 / 0.25
If $$x y^{\prime}+y-e^x=0, y(a)=b$$, then $$\lim _\limits{x \rightarrow 1} y(x)$$ is
Q57.
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+1 / 0.25
Let $$\mathrm{f}$$ be a differential function with $$\lim _\limits{x \rightarrow \infty} \mathrm{f}(x)=0$$. If $$\mathrm{y}^{\prime}+\mathrm{yf}^{\prime}(x)-\mathrm{f}(x) \mathrm{f}^{\prime}(x)=0$$, $$\lim _\limits{x \rightarrow \infty} y(x)=0$$ then
Q58.
mcq single
+1 / 0.25
Chords $$\mathrm{AB}$$ & $$\mathrm{CD}$$ of a circle intersect at right angle at the point $$\mathrm{P}$$. If the length of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is
Q59.
mcq single
+2 / 0.5
If two circles which pass through the points $$(0, a)$$ and $$(0,-a)$$ and touch the line $$\mathrm{y}=\mathrm{m} x+\mathrm{c}$$, cut orthogonally then
Q60.
mcq single
+1 / 0.25
A line of fixed length $$\mathrm{a}+\mathrm{b} . \mathrm{a} \neq \mathrm{b}$$ moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length a and b is
Q61.
mcq single
+1 / 0.25
With origin as a focus and $$x=4$$ as corresponding directrix, a family of ellipse are drawn. Then the locus of an end of minor axis is
Q62.
mcq single
+1 / 0.25
The equation $$\mathrm{r} \cos \theta=2 \mathrm{a} \sin ^2 \theta$$ represents the curve
Q63.
mcq single
+2 / 0.5
$$
\text { If } y=\tan ^{-1}\left[\frac{\log _e\left(\frac{e}{x^2}\right)}{\log _e\left(e x^2\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log _e x}{1-6 \cdot \log _e x}\right] \text {, then } \frac{d^2 y}{d x^2}=
$$
Q64.
mcq single
+1 / 0.25
If $$\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)$$ denotes the $$\mathrm{n}^{\text {th }}$$ derivative $$(\mathrm{n}=1,2)$$ of $$\mathrm{U}(x)=\frac{\mathrm{L} x+\mathrm{M}}{x^2-2 \mathrm{~B} x+\mathrm{C}}$$ (L, M, B, C are constants), then $$\mathrm{PU}_2+\mathrm{QU}_1+\mathrm{RU}=0$$, holds for
Q65.
mcq single
+2 / 0.5
Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is
Q66.
mcq single
+2 / 0.5
$$
\text { If } 1000!=3^n \times m \text { where } m \text { is an integer not divisible by } 3 \text {, then } n=
$$
Q67.
mcq single
+1 / 0.25
The numbers $$1,2,3, \ldots \ldots, \mathrm{m}$$ are arranged in random order. The number of ways this can be done, so that the numbers $$1,2, \ldots \ldots ., \mathrm{r}(\mathrm{r}<\mathrm{m})$$ appears as neighbours is
Q68.
mcq single
+1 / 0.25
For any integer $$\mathrm{n}, \int_\limits0^\pi \mathrm{e}^{\cos ^2 x} \cdot \cos ^3(2 n+1) x \mathrm{~d} x$$ has the value :
Q69.
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+1 / 0.25
All values of a for which the inequality $$\frac{1}{\sqrt{a}} \int_\limits1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{d} x<4$$ is satisfied, lie in the interval
Q70.
mcq single
+1 / 0.25
If $$\mathrm{f}(x)=\frac{\mathrm{e}^x}{1+\mathrm{e}^x}, \mathrm{I}_1=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} x \mathrm{~g}(x(1-x)) \mathrm{d} x$$ and $$\mathrm{I}_2=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} \mathrm{g}(x(1-x)) \mathrm{d} x$$, then the value of $$\frac{I_2}{I_1}$$ is
Q71.
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+1 / 0.25
Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function and $$f(1)=4$$. Then the value of $$\lim _\limits{x \rightarrow 1} \int_\limits4^{f(x)} \frac{2 t}{x-1} d t$$, if $$f^{\prime}(1)=2$$ is
Q72.
mcq multi
+2 / 0
$$
\text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are }
$$
Q73.
mcq single
+2 / 0.5
Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. then,
Q74.
mcq single
+2 / 0.5
$$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$
Physics
Physics
Q1.
mcq single
+2 / 0.5
The following figure shows the variation of potential energy $$V(x)$$ of a particle with distance $$x$$. The particle has
Q2.
mcq single
+1 / 0.25
A $$2 \mathrm{~V}$$ cell is connected across the points $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in the figure. Assume that the resistance of each diode is zero in forward bias and infinity in reverse bias. The current supplied by the cell is
Q3.
mcq single
+2 / 0.5
In the given network of AND and OR gates, output Q can be written as (assuming n even)
Q4.
mcq single
+1 / 0.25
Let $$\theta$$ be the angle between two vectors $$\vec{A}$$ and $$\vec{B}$$. If $$\hat{a}_{\perp}$$ is the unit vector perpendicular to $$\vec{A}$$, then the direction of $$
\overrightarrow{\mathrm{B}}-\mathrm{B} \sin \theta \hat{\mathrm{a}}_{\perp} \text { is }$$
Q5.
mcq single
+1 / 0.25
In a single-slit diffraction experiment, the slit is illuminated by light of two wavelengths $$\lambda_1$$ and $$\lambda_2$$. It is observed that the $$2^{\text {nd }}$$ order diffraction minimum for $$\lambda_1$$ coincides with the $$3^{\text {rd }}$$ diffraction minimum for $$\lambda_2$$. Then
Q6.
mcq single
+1 / 0.25
Light of wavelength $$6000 \mathop A\limits^o$$ is incident on a thin glass plate of r.i. 1.5 such that the angle of refraction into the plate is $$60^{\circ}$$. Calculate the smallest thickness of the plate which will make dark fringe by reflected beam interference.
Q7.
mcq single
+1 / 0.25
A particle of mass '$$m$$' moves in one dimension under the action of a conservative force whose potential energy has the form of $$U(x)=-\frac{\alpha x}{x^2+\beta^2}$$ where $$\alpha$$ and $$\beta$$ are dimensional parameters. The angular frequency of the oscillation is proportional to
Q8.
mcq single
+1 / 0.25
The position vector of a particle of mass $$\mathrm{m}$$ moving with a constant velocity $$\vec{v}$$ is given by $$\vec{r}=x(t) \hat{i}+b \hat{j}$$, where $$\mathrm{b}$$ is a constant. At an instant, $$\vec{r}$$ makes an angle $$\theta$$ with the $$x$$-axis as shown in the figure. The variation of the angular momentum of the particle about the origin with $$\theta$$ will be
Q9.
mcq single
+1 / 0.25
A small sphere of mass m and radius r slides down the smooth surface of a large hemispherical bowl of radius R. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point $$\mathrm{A}$$ of the bowl will be [given, moment of inertia of sphere $$=\frac{2}{5} \mathrm{mr}^2$$]
Q10.
mcq single
+1 / 0.25
A small ball of mass m is suspended from the ceiling of a floor by a string of length $$\mathrm{L}$$. The ball moves along a horizontal circle with constant angular velocity $$\omega$$, as shown in the figure. The torque about the centre (O) of the horizontal circle is
A small ball of mass m is suspended from the ceiling of a floor by a string of length $$\mathrm{L}$$. The ball moves along a horizontal circle with constant angular velocity $$\omega$$, as shown in the figure. The torque about the centre (O) of the horizontal circle is
Q11.
mcq multi
+2 / 0
A uniform rod $$\mathrm{AB}$$ of length $$1 \mathrm{~m}$$ and mass $$4 \mathrm{~kg}$$ is sliding along two mutually perpendicular frictionless walls OX and OY. The velocity of the two ends of the $$\operatorname{rod} \mathrm{A}$$ and $$\mathrm{B}$$ are $$3 \mathrm{~m} / \mathrm{s}$$ and $$4 \mathrm{~m} / \mathrm{s}$$ respectively, as shown in the figure. Then which of the following statement(s) is/are correct?
A uniform rod $$\mathrm{AB}$$ of length $$1 \mathrm{~m}$$ and mass $$4 \mathrm{~kg}$$ is sliding along two mutually perpendicular frictionless walls OX and OY. The velocity of the two ends of the $$\operatorname{rod} \mathrm{A}$$ and $$\mathrm{B}$$ are $$3 \mathrm{~m} / \mathrm{s}$$ and $$4 \mathrm{~m} / \mathrm{s}$$ respectively, as shown in the figure. Then which of the following statement(s) is/are correct?
Q12.
mcq single
+1 / 0.25
The position of the centre of mass of the uniform plate as shown in the figure is
The position of the centre of mass of the uniform plate as shown in the figure is
Q13.
mcq single
+1 / 0.25
Two straight conducting plates form an angle $$\theta$$ where their ends are joined. A conducting bar in contact with the plates and forming an isosceles triangle with them starts at the vertex at time $$t=0$$ and moves with constant velocity $$\vec{v}$$ to the right as shown in figure. A magnetic field $$\vec{B}$$ points out of the page. The magnitude of emf induced at $$t=1$$ second will be
Q14.
mcq single
+1 / 0.25
The speed distribution for a sample of $$\mathrm{N}$$ gas particles is shown below. $$\mathrm{P}(\mathrm{v})=0$$ for $$\mathrm{v}>2 \mathrm{v}_0$$. How many particles have speeds between $$1.2 \mathrm{v}_0$$ and $$1.8 \mathrm{v}_0$$ ?
Q15.
mcq single
+1 / 0.25
The internal energy of a thermodynamic system is given by $$U=a s^{4 / 3} V^\alpha$$ where $$\mathrm{s}$$ is entropy, $$\mathrm{V}$$ is volume and '$$\mathrm{a}$$' and '$$\alpha$$' are constants. The value of $$\alpha$$ is
Q16.
mcq single
+1 / 0.25
A body floats with $$\frac{1}{n}$$ of its volume keeping outside of water. If the body has been taken to height $$\mathrm{h}$$ inside water and released, it will come to the surface after time t. Then
Q17.
mcq single
+2 / 0.5
A metal plate of area $$10^{-2} \mathrm{~m}^2$$ rests on a layer of castor oil, $$2 \times 10^{-3} \mathrm{~m}$$ thick, whose coefficient of viscosity is $$1.55 \mathrm{~Ns} \mathrm{~m}^{-2}$$. The approximate horizontal force required to move the plate with a uniform speed of $$3 \times 10^{-2} \mathrm{~ms}^{-1}$$ is
Q18.
mcq single
+2 / 0.5
Water is filled in a cylindrical vessel of height $$\mathrm{H}$$. A hole is made at height $$\mathrm{z}$$ from the bottom, as shown in the figure. The value of z for which the range (R) of the emerging water through the hole will be maximum for
Q19.
mcq single
+1 / 0.25
The elastic potential energy of a strained body is
Q20.
mcq single
+1 / 0.25
A beam of light of wavelength $$\lambda$$ falls on a metal having work function $$\phi$$ placed in a magnetic field B. The most energetic electrons, perpendicular to the field are bent in circular arcs of radius R. If the experiment is performed for different values of $$\lambda$$, then $$\mathrm{B}^2$$ vs. $$\frac{1}{\lambda}$$ graph will look like (keeping all other quantities constant)
Q21.
mcq single
+2 / 0
Monochromatic light of wavelength $$\lambda=4770 \mathop A\limits^o $$ is incident separately on the surfaces of four different metals A, B, C and D. The work functions of A, B, C and D are $$4.2 \mathrm{~eV}, 3.7 \mathrm{~eV}, 3.2 \mathrm{~eV}$$ and $$2.3 \mathrm{~eV}$$, respectively. The metal / metals from which electrons will be emitted is / are
Q22.
mcq multi
+2 / 0
Consider the integral form of the Gauss' law in electrostatics
$$\oint {\overrightarrow E .d\overrightarrow S } = {Q \over {{\varepsilon _0}}}$$
Which of the following statements are correct?
Q23.
mcq single
+1 / 0.25
A charge Q is placed at the centre of a cube of sides a. The total flux of electric field through the six surfaces of the cube is
Q24.
mcq single
+1 / 0.25
Three point charges $$\mathrm{q},-2 \mathrm{q}$$ and $$\mathrm{q}$$ are placed along $$x$$ axis at $$x=-{a}, 0$$ and $a$ respectively. As $$\mathrm{a} \rightarrow 0$$ and $$\mathrm{q} \rightarrow \infty$$ while $$\mathrm{q} \mathrm{a}^2=\mathrm{Q}$$ remains finite, the electric field at a point P, at a distance $$x(x \gg a)$$ from $$x=0$$ is $$\overrightarrow{\mathrm{E}}=\frac{\alpha \mathrm{Q}}{4 \pi \epsilon_0 x^\beta} \hat{i}$$. Then
Q25.
mcq single
+1 / 0.25
A satellite of mass $$\mathrm{m}$$ rotates round the earth in a circular orbit of radius R. If the angular momentum of the satellite is J, then its kinetic energy $$(\mathrm{K})$$ and the total energy (E) of the satellite are
Q26.
mcq single
+1 / 0.25
The equivalent capacitance of a combination of connected capacitors shown in the figure between the points $$\mathrm{P}$$ and $$\mathrm{N}$$ is
Q27.
mcq single
+1 / 0.25
The acceleration-time graph of a particle moving in a straight line is shown in the figure. If the initial velocity of the particle is zero then the velocity-time graph of the particle will be
Q28.
mcq single
+2 / 0.5
When a convex lens is placed above an empty tank, the image of a mark at the bottom of the tank, which is 45 cm from the lens is formed 36 cm above the lens. When a liquid is poured in the tank to a depth of 40 cm, the distance of the image of the mark above the lens is 48 cm. The refractive index of the liquid is
Q29.
mcq single
+1 / 0.25
If $$\hat{n}_1, \hat{n}_2$$ and $$\hat{\mathrm{t}}$$ represent, unit vectors along the incident ray, reflected ray and normal to the surface respectively, then
Q30.
mcq single
+1 / 0.25
Two convex lens $$(\mathrm{L}_1$$ and $$\mathrm{L}_2)$$ of equal focal length $$\mathrm{f}$$ are placed at a distance $$\frac{\mathrm{f}}{2}$$ apart. An object is placed at a distance $$4 \mathrm{f}$$ in the left of $$\mathrm{L_1}$$ as shown in figure. The final image is at
Q31.
mcq single
+1 / 0.25
Which of the following quantity has the dimension of length ?
(h is Planck's constant, m is the mass of electron and c is the velocity of light)
Q32.
mcq single
+1 / 0.25
The Power $$(\mathrm{P})$$ radiated from an accelerated charged particle is given by $$\mathrm{P} \propto \frac{(q \mathrm{a})^{\mathrm{m}}}{\mathrm{c}^{\mathrm{n}}}$$ where $$\mathrm{q}$$ is the charge, $$\mathrm{a}$$ is the acceleration of the particle and $$\mathrm{c}$$ is speed of light in vacuum. From dimensional analysis, the value of $$m$$ and $$n$$ respectively, are
Q33.
mcq single
+1 / 0.25
Which of the following statement(s) is/are truc in respect of nuclear binding energy ?
(i) The mass energy of a nucleus is larger than the total mass energy of its individual protons and neutrons.
(ii) If a nucleus could be separated into its nucleons, an energy equal to the binding energy would have to be transferred to the particles during the separating process.
(iii) The binding energy is a measure of how well the nucleons in a nucleus are held together.
(iv) The nuclear fission is somehow related to acquiring higher binding energy.
Q34.
mcq single
+1 / 0.25
Longitudinal waves cannot
Q35.
mcq single
+1 / 0.25
A charged particle moving with a velocity $$\vec{v}=v_1 \hat{i}+v_2 \hat{j}$$ in a magnetic field $$\vec{B}$$ experiences a force $$\vec{F}=F_1 \hat{i}+F_2 \hat{j}$$. Here $$v_1, v_2, F_1, F_2$$ all are constants. Then $$\overrightarrow{\mathrm{B}}$$ can be
Q36.
mcq single
+1 / 0.25
Consider a circuit where a cell of emf $$E_0$$ and internal resistance $$\mathrm{r}$$ is connected across the terminal $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in figure. The value of $$\mathrm{R}$$ for which the power generated in the circuit is maximum, is given by
Consider a circuit where a cell of emf $$E_0$$ and internal resistance $$\mathrm{r}$$ is connected across the terminal $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in figure. The value of $$\mathrm{R}$$ for which the power generated in the circuit is maximum, is given by
Q37.
mcq multi
+2 / 0
The electric field of a plane electromagnetic wave in a medium is given by
$$
\overrightarrow{\mathrm{E}}(x, y, z, t)=\mathrm{E}_0 \hat{\mathrm{n}} \mathrm{e}^{i k_o[(x+y+z)-c t]}
$$
where $$\mathrm{c}$$ is the speed of light in free space. $$\overrightarrow{\mathrm{E}}$$ field is polarized in the $$x-\mathrm{z}$$ plane. The speed of wave is $$v$$ in the medium. Then
Q38.
mcq single
+1 / 0.25
In a series LCR circuit, the rms voltage across the resistor and the capacitor are $$30 \mathrm{~V}$$ and $$90 \mathrm{~V}$$ respectively. If the applied voltage is $$50 \sqrt{2} \sin \omega t$$, then the peak voltage across the inductor is
In a series LCR circuit, the rms voltage across the resistor and the capacitor are $$30 \mathrm{~V}$$ and $$90 \mathrm{~V}$$ respectively. If the applied voltage is $$50 \sqrt{2} \sin \omega t$$, then the peak voltage across the inductor is
Q39.
mcq multi
+2 / 0
The variation of impedance $$\mathrm{Z}$$ of a series $$\mathrm{L C R}$$ circuit with frequency of the source is shown in the figure. Which of the following statement(s) is/are true ?
The variation of impedance $$\mathrm{Z}$$ of a series $$\mathrm{L C R}$$ circuit with frequency of the source is shown in the figure. Which of the following statement(s) is/are true ?