WB JEE 2024 | WB JEE Year Wise Previous Years Questions

Chemistry

Mathematics

1. All values of a for which the inequality $$\frac{1}{\sqrt{a}} \int_\limits1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt 2. For any integer $$\mathrm{n}, \int_\limits0^\pi \mathrm{e}^{\cos ^2 x} \cdot \cos ^3(2 n+1) x \mathrm{~d} x$$ has the va 3. Let $$\mathrm{f}$$ be a differential function with $$\lim _\limits{x \rightarrow \infty} \mathrm{f}(x)=0$$. If $$\mathrm 4. If $$x y^{\prime}+y-e^x=0, y(a)=b$$, then $$\lim _\limits{x \rightarrow 1} y(x)$$ is 5. The area bounded by the curves $$x=4-y^2$$ and the Y-axis is 6. $$f(x)=\cos x-1+\frac{x^2}{2!}, x \in \mathbb{R}$$ Then $$\mathrm{f}(x)$$ is 7. Let $$\mathrm{y}=\mathrm{f}(x)$$ be any curve on the $$\mathrm{X}-\mathrm{Y}$$ plane & $$\mathrm{P}$$ be a point on the 8. If a particle moves in a straight line according to the law $$x=a \sin (\sqrt{\lambda} t+b)$$, then the particle will co 9. A unit vector in XY-plane making an angle $$45^{\circ}$$ with $$\hat{i}+\hat{j}$$ and an angle $$60^{\circ}$$ with $$3 \ 10. Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be given by $$\mathrm{f}(x)=\left|x^2-1\right|$$, then 11. Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $$a_n$ 12. If for the series $$a_1, a_2, a_3$$, ...... etc, $$\mathrm{a}_{\mathrm{r}}-\mathrm{a}_{\mathrm{r}+\mathrm{i}}$$ bears a 13. If $$z_1$$ and $$z_2$$ be two roots of the equation $$z^2+a z+b=0, a^2 14. If $$\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 15. If $$\left(x^2 \log _x 27\right) \cdot \log _9 x=x+4$$ then the value of $$x$$ is 16. If $$\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}$$ and $$\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}$ 17. Let $$\mathrm{N}$$ be the number of quadratic equations with coefficients from $$\{0,1,2, \ldots, 9\}$$ such that 0 is a 18. If $$a, b, c$$ are distinct odd natural numbers, then the number of rational roots of the equation $$a x^2+b x+c=0$$ 19. The numbers $$1,2,3, \ldots \ldots, \mathrm{m}$$ are arranged in random order. The number of ways this can be done, so t 20. If $$A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$$ and $$\theta=\ 21. If $$\left(1+x+x^2+x^3\right)^5=\sum_\limits{k=0}^{15} a_k x^k$$ then $$\sum_\limits{k=0}^7(-1)^{\mathbf{k}} \cdot a_{2 22. The coefficient of $$a^{10} b^7 c^3$$ in the expansion of $$(b c+c a+a b)^{10}$$ is 23. $$ \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} \e 24. If $$\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \cdot A \cdot\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{ar 25. $$ \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}\righ 26. 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 27. Let $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$\mathrm{f}(x)=\frac{\mathrm{e}^{|x|}-\m 28. Let A be the set of even natural numbers that are 29. Two smallest squares are chosen one by one on a chess board. The probability that they have a side in common is 30. Two integers $$\mathrm{r}$$ and $$\mathrm{s}$$ are drawn one at a time without replacement from the set $$\{1,2, \ldots, 31. A biased coin with probability $$\mathrm{p}(0 32. The expression $$\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)$$ is 33. If $$0 34. The equation $$\mathrm{r} \cos \theta=2 \mathrm{a} \sin ^2 \theta$$ represents the curve 35. 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 equat 36. In $$\triangle \mathrm{ABC}$$, co-ordinates of $$\mathrm{A}$$ are $$(1,2)$$ and the equation of the medians through $$\m 37. A line of fixed length $$\mathrm{a}+\mathrm{b} . \mathrm{a} \neq \mathrm{b}$$ moves so that its ends are always on two f 38. With origin as a focus and $$x=4$$ as corresponding directrix, a family of ellipse are drawn. Then the locus of an end o 39. Chords $$\mathrm{AB}$$ & $$\mathrm{CD}$$ of a circle intersect at right angle at the point $$\mathrm{P}$$. If the length 40. The plane $$2 x-y+3 z+5=0$$ is rotated through $$90^{\circ}$$ about its line of intersection with the plane $$x+y+z=1$$. 41. If the relation between the direction ratios of two lines in $$\mathbb{R}^3$$ are given by $$l+\mathrm{m}+\mathrm{n}=0,2 42. $$\triangle \mathrm{OAB}$$ is an equilateral triangle inscribed in the parabola $$\mathrm{y}^2=4 \mathrm{a} x, \mathrm{a 43. For every real number $$x \neq-1$$, let $$\mathrm{f}(x)=\frac{x}{x+1}$$. Write $$\mathrm{f}_1(x)=\mathrm{f}(x)$$ & for $ 44. If $$\mathrm{U}_{\mathrm{n}}(\mathrm{n}=1,2)$$ denotes the $$\mathrm{n}^{\text {th }}$$ derivative $$(\mathrm{n}=1,2)$$ 45. The equation $$2^x+5^x=3^x+4^x$$ has 46. Consider the function $$\mathrm{f}(x)=(x-2) \log _{\mathrm{e}} x$$. Then the equation $$x \log _{\mathrm{e}} x=2-x$$ 47. If $$\alpha, \beta$$ are the roots of the equation $$a x^2+b x+c=0$$ then $$\lim _\limits{x \rightarrow \beta} \frac{1-\ 48. If $$\mathrm{f}(x)=\frac{\mathrm{e}^x}{1+\mathrm{e}^x}, \mathrm{I}_1=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\ 49. Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function and $$f(1)=4$$. Then the value of $$\lim _\lim 50. $$ \text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\ma 51. Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. t 52. Consider the function $$\mathrm{f}(x)=x(x-1)(x-2) \ldots(x-100)$$. Which one of the following is correct? 53. In a plane $$\vec{a}$$ and $$\vec{b}$$ are the position vectors of two points A and B respectively. A point $P$ with pos 54. Five balls of different colours are to be placed in three boxes of different sizes. The number of ways in which we can p 55. Let $$A=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1\end{array}\right), B=\left(\begin{array}{l}2 \\ 1 56. If $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are in A.P. with common difference $$\theta$$, then the sum of the series $$ 57. For the real numbers $$x$$ & $$y$$, we write $$x$$ p y iff $$x-y+\sqrt{2}$$ is an irrational number. Then relation p is 58. Let $$A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$$, then 59. $$ \text { If } 1000!=3^n \times m \text { where } m \text { is an integer not divisible by } 3 \text {, then } n= $$ 60. If $$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)=$$ 61. If two circles which pass through the points $$(0, a)$$ and $$(0,-a)$$ and touch the line $$\mathrm{y}=\mathrm{m} x+\mat 62. The locus of the midpoint of the system of parallel chords parallel to the line $$y=2 x$$ to the hyperbola $$9 x^2-4 y^2 63. Angle between two diagonals of a cube will be 64. $$ \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}\ 65. $$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$ 66. The acceleration f $$\mathrm{ft} / \mathrm{sec}^2$$ of a particle after a time $$\mathrm{t}$$ sec starting from rest is 67. Let $$\Gamma$$ be the curve $$\mathrm{y}=\mathrm{be}^{-x / a}$$ & $$\mathrm{L}$$ be the straight line $$\frac{x}{\mathrm 68. If $$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 69. If the quadratic equation $$a x^2+b x+c=0(a>0)$$ has two roots $$\alpha$$ and $$\beta$$ such that $$\alpha2$$, then 70. If $$\mathrm{a}_{\mathrm{i}}, \mathrm{b}_{\mathrm{i}}, \mathrm{c}_{\mathrm{i}} \in \mathbb{R}(\mathrm{i}=1,2,3)$$ and $$ 71. The function $$\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$\mathrm{f}(x)=\mathrm{e}^x+\mathrm{e}^{-x}$$ 72. A square with each side equal to '$$a$$' above the $$x$$-axis and has one vertex at the origin. One of the sides passing 73. If $$\mathrm{ABC}$$ is an isosceles triangle and the coordinates of the base points are $$B(1,3)$$ and $$C(-2,7)$$. The 74. $$ \text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are } $$ 75. Choose the correct statement :

Physics

1. Let $$\theta$$ be the angle between two vectors $$\vec{A}$$ and $$\vec{B}$$. If $$\hat{a}_{\perp}$$ is the unit vector p 2. The Power $$(\mathrm{P})$$ radiated from an accelerated charged particle is given by $$\mathrm{P} \propto \frac{(q \math 3. Two convex lens $$(\mathrm{L}_1$$ and $$\mathrm{L}_2)$$ of equal focal length $$\mathrm{f}$$ are placed at a distance $$ 4. Which of the following quantity has the dimension of length ? (h is Planck's constant, m is the mass of electron and c i 5. The speed distribution for a sample of $$\mathrm{N}$$ gas particles is shown below. $$\mathrm{P}(\mathrm{v})=0$$ for $$\ 6. The internal energy of a thermodynamic system is given by $$U=a s^{4 / 3} V^\alpha$$ where $$\mathrm{s}$$ is entropy, $$ 7. A particle of mass '$$m$$' moves in one dimension under the action of a conservative force whose potential energy has th 8. Longitudinal waves cannot 9. A $$2 \mathrm{~V}$$ cell is connected across the points $$\mathrm{A}$$ and $$\mathrm{B}$$ as shown in the figure. Assume 10. 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 th 11. The elastic potential energy of a strained body is 12. Which of the following statement(s) is/are truc in respect of nuclear binding energy ? (i) The mass energy of a nucleus 13. A satellite of mass $$\mathrm{m}$$ rotates round the earth in a circular orbit of radius R. If the angular momentum of t 14. What force $$\mathrm{F}$$ is required to start moving this $$10 \mathrm{~kg}$$ block shown in the figure if it acts at a 15. 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 ref 16. Consider a circuit where a cell of emf $$E_0$$ and internal resistance $$\mathrm{r}$$ is connected across the terminal 17. The equivalent capacitance of a combination of connected capacitors shown in the figure between the points $$\mathrm{P}$ 18. In a single-slit diffraction experiment, the slit is illuminated by light of two wavelengths $$\lambda_1$$ and $$\lambda 19. The acceleration-time graph of a particle moving in a straight line is shown in the figure. If the initial velocity of t 20. The position vector of a particle of mass $$\mathrm{m}$$ moving with a constant velocity $$\vec{v}$$ is given by $$\vec{ 21. The position of the centre of mass of the uniform plate as shown in the figure is 22. In a series LCR circuit, the rms voltage across the resistor and the capacitor are $$30 \mathrm{~V}$$ and $$90 \mathrm{ 23. A small ball of mass m is suspended from the ceiling of a floor by a string of length $$\mathrm{L}$$. The ball moves al 24. If $$\hat{n}_1, \hat{n}_2$$ and $$\hat{\mathrm{t}}$$ represent, unit vectors along the incident ray, reflected ray and n 25. A beam of light of wavelength $$\lambda$$ falls on a metal having work function $$\phi$$ placed in a magnetic field B. T 26. A charged particle moving with a velocity $$\vec{v}=v_1 \hat{i}+v_2 \hat{j}$$ in a magnetic field $$\vec{B}$$ experience 27. Two straight conducting plates form an angle $$\theta$$ where their ends are joined. A conducting bar in contact with th 28. Three point charges $$\mathrm{q},-2 \mathrm{q}$$ and $$\mathrm{q}$$ are placed along $$x$$ axis at $$x=-{a}, 0$$ and $a$ 29. A body floats with $$\frac{1}{n}$$ of its volume keeping outside of water. If the body has been taken to height $$\mathr 30. A small sphere of mass m and radius r slides down the smooth surface of a large hemispherical bowl of radius R. If the s 31. When a convex lens is placed above an empty tank, the image of a mark at the bottom of the tank, which is 45 cm from the 32. In the given network of AND and OR gates, output Q can be written as (assuming n even) 33. Water is filled in a cylindrical vessel of height $$\mathrm{H}$$. A hole is made at height $$\mathrm{z}$$ from the botto 34. A metal plate of area $$10^{-2} \mathrm{~m}^2$$ rests on a layer of castor oil, $$2 \times 10^{-3} \mathrm{~m}$$ thick, 35. The following figure shows the variation of potential energy $$V(x)$$ of a particle with distance $$x$$. The particle ha 36. Monochromatic light of wavelength $$\lambda=4770 \mathop A\limits^o $$ is incident separately on the surfaces of four di 37. Consider the integral form of the Gauss' law in electrostatics $$\oint {\overrightarrow E .d\overrightarrow S } = {Q \o 38. A uniform rod $$\mathrm{AB}$$ of length $$1 \mathrm{~m}$$ and mass $$4 \mathrm{~kg}$$ is sliding along two mutually per 39. The variation of impedance $$\mathrm{Z}$$ of a series $$\mathrm{L C R}$$ circuit with frequency of the source is shown 40. The electric field of a plane electromagnetic wave in a medium is given by $$ \overrightarrow{\mathrm{E}}(x, y, z, t)=\m

WB JEE 2024

MCQ (Single Correct Answer)

-0.25

$$ \text { If } \int \frac{\log _e\left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \mathrm{~d} x=\mathrm{f}(\mathrm{g}(x))+\mathrm{c} \text { then } $$

$$\mathrm{f}(x)=\frac{x^2}{2}, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)$$

$$\mathrm{f}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right), \mathrm{g}(x)=\frac{x^2}{2}$$

$$\mathrm{f}(x)=x^2, \mathrm{~g}(x)=\log _{\mathrm{e}}\left(x+\sqrt{1+x^2}\right)$$

$$\mathrm{f}(x)=\log _{\mathrm{e}}\left(x-\sqrt{1+x^2}\right), \mathrm{g}(x)=x^2$$

WB JEE 2024

MCQ (Single Correct Answer)

-0.5

Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. then,

$$I(R)>\frac{\pi}{2 R}\left(1-e^{-R}\right)$$

$$I(R)<\frac{\pi}{2 R}\left(1-e^{-R}\right)$$

$$I(R)=\frac{\pi}{2 R}\left(1-e^{-R}\right)$$

$$I(R) \text { and } \frac{\pi}{2 R}(1-e^{-R}) \text { are not comparable }$$

WB JEE 2024

MCQ (Single Correct Answer)

-0.5

Consider the function $$\mathrm{f}(x)=x(x-1)(x-2) \ldots(x-100)$$. Which one of the following is correct?

This function has 100 local maxima

This function has 50 local maxima

This function has 51 local maxima

Local minima do not exist for this function

WB JEE 2024

MCQ (Single Correct Answer)

-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

$$\frac{|\vec{a}-\vec{b}|}{c}$$

$$\frac{|\vec{a}+\vec{b}|}{c}$$

$$\frac{|\vec{a}-\vec{b}|}{2 c}$$

$$\frac{|\vec{a}+\vec{b}|}{2 c}$$

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