The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to
- A$6 e$
- B$4 e$
- C$3 e$
- D$2 e$
Question types
JEE Main 3-April-2025 Paper - Shift 2 question types
75 questions across 6 question groups — pick any mix to generate a JEE paper with step-by-step answer keys.
75
Questions
6
Question groups
5
Question types
01
SECTION - A [MATHS - MCQ]
20 Q→02SECTION - B [MATHS - NUMERIC]
5 Q→03SECTION - A [PHYSICS MCQ]
20 Q→04SECTION - B [PHYSICS - NUMERIC]
5 Q→05SECTION - A [CHEMISTY - MCQ]
20 Q→06SECTION - B [CHEMISTY - NUMERIC]
5 Q→Sample Questions
JEE Main 3-April-2025 Paper - Shift 2 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The distance of the point $(7,10,11)$ from the line $\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6}$ is
- A18
- B14
- C12
- D16
The shortest distance between the curves $y^{2}=8 x$ and $x^{2}+y^{2}+12 y+35=0$ is :
- A$2 \sqrt{3}-1$
- B$\sqrt{2}$
- C$3 \sqrt{2}-1$
- D$2 \sqrt{2}-1$
Let C be the circle of minimum area enclosing the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with eccentricity $\frac{1}{2}$ and foci $( \pm 2,0)$. Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 29 is parallel to the major axis of E and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle PQR is :
- A$6(3+\sqrt{2})$
- B$8(3+\sqrt{2})$
- C$62+\sqrt{3}$
- D$82+\sqrt{3}$
The number of solutions of equation $4-\sqrt{3} \sin x$ $-2 \sqrt{3} \cos ^{2} x=-\frac{4}{1+\sqrt{3}}, x \in\left[-2 \pi, \frac{5 \pi}{2}\right]$ is
- A4
- B3
- C6
- D5
If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^{2}-y^{2}-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to __________ .
View full solution →Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}}$ be a vector such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=4$. Then $|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{d}})|^{2}$ is equal to __________ .
View full solution →If $\operatorname{Lim}_{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^{2}}}=p$, then $96 \log _{e} p$ is equal to __________.
View full solution →Let $\left(1+x+x^{2}\right)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots .+a_{20} x^{20}$. If
$\left(a_{1}+a_{3}+a_{5}+\ldots .+a_{19}\right)-11 \mathrm{a}_{2}=121 \mathrm{k}$, then k is equal to __________ .
View full solution →$\left(a_{1}+a_{3}+a_{5}+\ldots .+a_{19}\right)-11 \mathrm{a}_{2}=121 \mathrm{k}$, then k is equal to __________ .
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1$. Let B be the inverse of the matrix $\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^{2}\right)\right)$. Then $|(\lambda B+1)|$ is equal to __________.
View full solution →Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : If oxygen ion $\left(\mathrm{O}^{-2}\right)$ and Hydrogen ion $\left(\mathrm{H}^{+}\right)$enter normal to the magnetic field with equal momentum, then the path of $\mathrm{O}^{-2}$ ion has a smaller curvature than that of $\mathrm{H}^{+}$.
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statement, choose the correct answer from the options given below
Assertion A : If oxygen ion $\left(\mathrm{O}^{-2}\right)$ and Hydrogen ion $\left(\mathrm{H}^{+}\right)$enter normal to the magnetic field with equal momentum, then the path of $\mathrm{O}^{-2}$ ion has a smaller curvature than that of $\mathrm{H}^{+}$.
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statement, choose the correct answer from the options given below
- A$\mathbf{A}$ is true but $\mathbf{R}$ is false
- BBoth $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$
- C$\mathbf{A}$ is false but $\mathbf{R}$ is true
- DBoth $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$
A block of mass 1 kg , moving along x with speed $\mathrm{v}_{\mathrm{i}}=10 \mathrm{~m} / \mathrm{s}$ enters a rough region ranging from $\mathrm{x}=0.1 \mathrm{~m}$ to $\mathrm{x}=1.9 \mathrm{~m}$. The retarding force acting on the block in this range is $\mathrm{F}_{\mathrm{r}}=-\mathrm{kx} \mathrm{N}$, with $\mathrm{k}=10 \mathrm{~N} / \mathrm{m}$. Then the final speed of the block as it crosses rough region is
- A$10 \mathrm{~m} / \mathrm{s}$
- B$4 \mathrm{~m} / \mathrm{s}$
- C$6 \mathrm{~m} / \mathrm{s}$
- D$8 \mathrm{~m} / \mathrm{s}$
A motor operating on 100 V draws a current of 1 A . If the efficiency of the motor is $91.6 \%$, then the loss of power in units of $\mathrm{cal} / \mathrm{s}$ is
- A4
- B8.4
- C2
- D6.2
Pressure of an ideal gas, contained in a closed vessel, is increased by $0.4 \%$ when heated by $1^{\circ} \mathrm{C}$. Its initial temperature must be :
- A$25^{\circ} \mathrm{C}$
- B2500 K
- C250 K
- D$250^{\circ} \mathrm{C}$
Match the LIST-I with LIST-II
Choose the correct answer from the options given below :
| LIST-I | LIST-II | ||
| A. | Boltzmann constant | I. | $\mathrm{ML}^{2} \mathrm{~T}^{-1}$ |
| B. | Coefficient of viscosity | II. | MLT $^{-3} \mathrm{~K}^{-1}$ |
| C. | Planck's constant | III. | $\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}$ |
| D. | Thermal conductivity | IV. | $\mathrm{ML}^{-1} \mathrm{~T}^{-1}$ |
- AA-III, B-IV, C-I, D-II
- BA-II, B-III, C-IV, D-I
- CA-III, B-II, C-I, D-IV
- DA-III, B-IV, C-II, D-I
A physical quantity C is related to four other quantities $\mathrm{p}, \mathrm{q}, \mathrm{r}$ and s as follows
$\mathrm{C}=\frac{\mathrm{pq}^{2}}{\mathrm{r}^{3} \sqrt{\mathrm{~s}}}$
The percentage errors in the measurement of $\mathrm{p}, \mathrm{q}, \mathrm{r}$ and s are $1 \%, 2 \% 3 \%$ and $2 \%$ respectively.
The percentage error in the measurement of C will be __________ %.
View full solution →$\mathrm{C}=\frac{\mathrm{pq}^{2}}{\mathrm{r}^{3} \sqrt{\mathrm{~s}}}$
The percentage errors in the measurement of $\mathrm{p}, \mathrm{q}, \mathrm{r}$ and s are $1 \%, 2 \% 3 \%$ and $2 \%$ respectively.
The percentage error in the measurement of C will be __________ %.
An electron in the hydrogen atom initially in the fourth excited state makes a transition to $\mathrm{n}^{\text {th }}$ energy state by emitting a photon of energy 2.86 eV . The integer value of $n$ will be __________ .
View full solution →Two cells of emf 1 V and 2 V and internal resistance $2 \Omega$ and $1 \Omega$, respectively, are connected in series with an external resistance of $6 \Omega$. The total current in the circuit is $I_{1}$. Now the same two cells in parallel configuration are connected to same external resistance. In this case, the total current drawn is $I_{2}$. The value of $\left(\frac{I_{1}}{I_{2}}\right)$ is $\frac{x}{3}$. The value of $x$ is __________ .
View full solution →The excess pressure inside a soap bubble $A$ in air is half the excess pressure inside another soap bubble B in air. If the volume of the bubble A is $n$ times the volume of the bubble B , then, the value of $n$ is __________ .
View full solution →Light from a point source in air falls on a spherical glass surface (refractive index, $\mu=1.5$ and radius of curvature $=50 \mathrm{~cm}$ ). The image is formed at a distance of 200 cm from the glass surface inside the glass. The magnitude of distance of the light source from the glass surface is __________ m.
View full solution →Consider the following statements related to temperature dependence of rate constants.
Identify the correct statements,
A. The Arrhenius equation holds true only for an elementary homogenous reaction.
B. The unit of A is same as that of k in Arrhenius equation.
C. At a given temperature, a low activation energy means a fast reaction.
D. A and Ea as used in Arrhenius equation depend on temperature.
E. When Ea >> RT. A and Ea become interdependent.
Choose the correct answer from the options given below :
Identify the correct statements,
A. The Arrhenius equation holds true only for an elementary homogenous reaction.
B. The unit of A is same as that of k in Arrhenius equation.
C. At a given temperature, a low activation energy means a fast reaction.
D. A and Ea as used in Arrhenius equation depend on temperature.
E. When Ea >> RT. A and Ea become interdependent.
Choose the correct answer from the options given below :
- AA, C and D Only
- BB, D and E Only
- CB and C Only
- DA and B Only
What is the correct IUPAC name of
- A3-Bromo-2-hydroxy-5-nitrobenzoic acid
- B3-Bromo-4-hydroxy-1-nitrobenzoic acid
- C2-Hydroxy-3-bromo-5-nitrobenzoic acid
- D5-Nitro-3-bromo-2-hydroxybenzoic acid
The correct orders among the following are
Atomic radius : $\mathrm{B}<\mathrm{Al}<\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}$
Electronegativity : $\mathrm{Al}<\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}<\mathrm{B}$
Density: $\mathrm{Tl}<\mathrm{In}<\mathrm{Ga}<\mathrm{Al}<\mathrm{B}$
1st Ionisation Energy : $\mathrm{In}<\mathrm{Al}<\mathrm{Ga}<\mathrm{Tl}<\mathrm{B}$
Choose the correct answer from the options given below :
Atomic radius : $\mathrm{B}<\mathrm{Al}<\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}$
Electronegativity : $\mathrm{Al}<\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}<\mathrm{B}$
Density: $\mathrm{Tl}<\mathrm{In}<\mathrm{Ga}<\mathrm{Al}<\mathrm{B}$
1st Ionisation Energy : $\mathrm{In}<\mathrm{Al}<\mathrm{Ga}<\mathrm{Tl}<\mathrm{B}$
Choose the correct answer from the options given below :
- AB and D Only
- BA and C Only
- CC and D Only
- DA and B Only
The sequence from the following that would result in giving predominantly $3,4,5$-Tribromoaniline is :
- A
- B
- C
- D
Given below are two statements :
Statement I : When a system containing ice in equilibrium with water (liquid) is heated, heat is absorbed by the system and there is no change in the temperature of the system until whole ice gets melted.
Statement II : At melting point of ice, there is absorption of heat in order to overcome intermolecular forces of attraction within the molecules of water in ice and kinetic energy of molecules is not increased at melting point.
In the light of the above statements, choose the correct answer from the options given below:
Statement I : When a system containing ice in equilibrium with water (liquid) is heated, heat is absorbed by the system and there is no change in the temperature of the system until whole ice gets melted.
Statement II : At melting point of ice, there is absorption of heat in order to overcome intermolecular forces of attraction within the molecules of water in ice and kinetic energy of molecules is not increased at melting point.
In the light of the above statements, choose the correct answer from the options given below:
- AStatement I is true but Statement II is false
- BBoth Statement I and Statement II are false
- CBoth Statement I and Statement II are true
- DStatement I is false but Statement II is true
The total number of structural isomers possible for the substituted benzene derivatives with the molecular formula $\mathrm{C}_{9} \mathrm{H}_{12}$ is __________ .
View full solution →Among, $\mathrm{Sc}, \mathrm{Mn}, \mathrm{Co}$ and Cu, identify the element with highest enthalpy of atomisation. The spin only magnetic moment value of that element in its +2 oxidation state is __________ BM (in nearest integer).
View full solution →A sample of n-octane (1.14 g) was completely burnt in excess of oxygen in a bomb calorimeter, whose heat capacity is $5 \mathrm{~kJ} \mathrm{~K}^{-1}$. As a result of combustion reaction, the temperature of the calorimeter is increased by 5 K . The magnitude of the heat of combustion of octane at constant volume is __________ $\mathrm{kJ} \mathrm{mol}^{-1}$ (nearest integer).
View full solution →
A perfect gas ( 0.1 mol ) having $\overline{\mathrm{C}}_{\mathrm{v}}=1.50 \mathrm{R}$ (independent of temperature) undergoes the above transformation from point 1 to point 4 . If each step is reversible, the total work done (w) while going from point 1 to point 4 is $(-)$ __________ J (nearest integer)
[Given : $\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ]
X g of nitrobenzene on nitration gave 4.2 g of m - dinitrobenzene.
X = __________ g. (nearest integer)
[Given : molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ) $\mathrm{C}: 12, \mathrm{H}: 1$, $\mathrm{O}: 16, \mathrm{~N}: 14]$
View full solution →X = __________ g. (nearest integer)
[Given : molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ) $\mathrm{C}: 12, \mathrm{H}: 1$, $\mathrm{O}: 16, \mathrm{~N}: 14]$
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