Sample QuestionsJEE Main 4-April-2025 Paper - Shift 1 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $10 \sin ^{4} \theta+15 \cos ^{4} \theta=6$, then the value of $\frac{27 \operatorname{cosec}^{6} \theta+8 \sec ^{6} \theta}{16 \sec ^{8} \theta}$ is :
- ✓
$\frac{2}{5}$
- B
$\frac{3}{4}$
- C
$\frac{3}{5}$
- D
$\frac{1}{5}$
Answer: A.
View full solution →A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $\text X$ denote the number of defective pens. Then the variance of $\text X$ is
- A
$\frac{11}{15}$
- ✓
$\frac{28}{75}$
- C
$\frac{2}{15}$
- D
$\frac{3}{5}$
Answer: B.
View full solution →Consider the equation $\mathrm{x}^{2}+4 \mathrm{x}-\mathrm{n}=0$, where $n \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to
Answer: C.
View full solution →The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is
- A
$\frac{6}{5}$
- B
$\frac{50}{3}$
- C
$\frac{10}{3}$
- ✓
$\frac{18}{5}$
Answer: D.
View full solution →The value of $\int_{-1}^{1} \frac{(1+\sqrt{|x|-x}) e^{x}+(\sqrt{|x|-x}) e^{-x}}{e^{x}+e^{-x}} d x$ is equal to
- A
$3-\frac{2 \sqrt{2}}{3}$
- B
$2+\frac{2 \sqrt{2}}{3}$
- C
$1-\frac{2 \sqrt{2}}{3}$
- ✓
$1+\frac{2 \sqrt{2}}{3}$
Answer: D.
View full solution →Let $m$ and $n$ be the number of points at which the function $f(\mathrm{x})=\max \left\{\mathrm{x}, \mathrm{x}^{3}, \mathrm{x}^{5}, \ldots \ldots, \mathrm{x}^{21}\right\}, \mathrm{x} \in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $\mathrm{m}+\mathrm{n}$ is equal to _____________.
View full solution →Let $C$ be the circle $x^{2}+(y-1)^{2}=2, E_{1}$ and $E_{2}$ be two ellipses whose centres lie at the origin and major axes lie on $x$-axis and $y$-axis respectively. Let the straight line $\mathrm{x}+\mathrm{y}=3$ touch the curves $C,$ $E_{1}$ and $E_{2}$ at $P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$ and $R\left(x_{3}, y_{3}\right)$ respectively. Given that P is the mid-point of the line segment $\text{QR}$ and $\mathrm{PQ}=\frac{2 \sqrt{2}}{3}$, the value of $9\left(x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}\right)$ is equal to _____________.
View full solution →Let $\mathrm{A}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2-\mathrm{i}|=3\}$, $B=\{z \in C: \operatorname{Re}(z-i z)=2\}$ and $S=A \cap B$. Then $\sum_{z \in S}|z|^{2}$ is equal to _____________.
View full solution →Let $A=\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta\end{array}\right].$ If for some $\theta \in(0, \pi),$ $A^{2}=A^{T},$ then the sum of the diagonal elements of the matrix $(\mathrm{A}+\mathrm{I})^{3}+(\mathrm{A}-\mathrm{I})^{3}-6 \mathrm{~A}$ is equal to _____________.
View full solution →If the area of the region $\{(x, y):|x-5| \leq y \leq 4 \sqrt{x}\}$ is A , then 3A is equal to _____________.
View full solution →Considering the Bohr model of hydrogen like atoms, the ratio of the ratio of the radius $5^{\text {th }}$ orbit of the electron in $\mathrm{Li}^{2+}$ and $\mathrm{He}^{+}$is
- A
$\frac{3}{2}$
- B
$\frac{4}{9}$
- C
$\frac{9}{4}$
- ✓
$\frac{2}{3}$
Answer: D.
View full solution →Two small spherical balls of mass 10g each with charges $-2 \mu \mathrm{C}$ and $2 \mu \mathrm{C}$, are attached to two ends of very light rigid rod of length 20 cm. The arrangement is now placed near an infinite nonconducting charge sheet with uniform charge density of $100 \mu \mathrm{C} / \mathrm{m}^{2}$ such that length of rod makes an angle of $30^{\circ}$ with electric field generated by charge sheet. Net torque acting on the rod is:
(Take $\varepsilon_{0}: 8.85 \times 10^{-12} \mathrm{C}^{2} / \mathrm{Nm}^{2}$)
Answer: B.
View full solution →Two infinite identical charged sheets and a charged spherical body of charge density '$\rho$' are arranged as shown in figure. Then the correct relation between the electrical fields at $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D points is
- A
$\vec{\mathrm{E}}_{\mathrm{A}}=\vec{\mathrm{E}}_{\mathrm{B}} ; \vec{\mathrm{E}}_{\mathrm{C}}=\vec{\mathrm{E}}_{\mathrm{D}}$
- B
$\vec{\mathrm{E}}_{\mathrm{A}}>\vec{\mathrm{E}}_{\mathrm{B}} ; \vec{\mathrm{E}}_{\mathrm{C}}=\vec{\mathrm{E}}_{\mathrm{D}}$
- ✓
$\vec{\mathrm{E}}_{\mathrm{C}} \neq \vec{\mathrm{E}}_{\mathrm{D}} ; \vec{\mathrm{E}}_{\mathrm{A}}>\vec{\mathrm{E}}_{\mathrm{B}}$
- D
$\left|\vec{\mathrm{E}}_{\mathrm{A}}\right|=\left|\vec{\mathrm{E}}_{\mathrm{B}}\right| ; \vec{\mathrm{E}}_{\mathrm{C}}>\vec{\mathrm{E}}_{\mathrm{D}}$
Answer: C.
View full solution →Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $\theta_{1}$ and $\theta_{2}$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?
- A
$\theta_{1} l_{2}^{2}=\theta_{2} l_{1}^{2}$
- B
$\theta_{1} l_{1}=\theta_{2} l_{2}$
- C
$\theta_{1} l_{1}^{2}=\theta_{2} l_{2}^{2}$
- ✓
$\theta_{1} l_{2}=\theta_{2} l_{1}$
Answer: D.
View full solution →In an experiment with a closed organ pipe, it is filled with water by $\left(\frac{1}{5}\right)$ th of its volume. The frequency of the fundamental note will change by
- ✓
$25 \%$
- B
$20 \%$
- C
$-20 \%$
- D
$-25 \%$
Answer: A.
View full solution →Conductor wire ABCDE with each arm 10 cm in length is placed in magnetic field of $\frac{1}{\sqrt{2}}$ Tesla, perpendicular to its plane. When conductor is pulled towards right with constant velocity of $10 \mathrm{~cm} / \mathrm{s}$, induced emf between points A and E is _____________ mV.
View full solution →Four capacitor each of capacitance $16 \mu \mathrm{F}$ are connected as shown in the figure. The capacitance between points A and B is : _____________ (in $\mu \mathrm{F}$ ).
View full solution →Distance between object and its image (magnified by $-\frac{1}{3}$) is 30 cm. The focal length of the mirror used is $\left(\frac{\mathrm{x}}{4}\right) \mathrm{cm}$,
where magnitude of value of $x$ is _____________.
View full solution →Two slabs with square cross section of different materials $(1,2)$ with equal sides $(l)$ and thickness $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$ such that $\mathrm{d}_{2}=2 \mathrm{d}_{1}$ and $l>\mathrm{d}_{2}$. Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $\theta_{2}=2 \theta_{1}$. If the shear moduli of material 1 is $4 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}$, then shear moduli of material 2 is $x \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}$, where value of $x$ is _____________.
View full solution →A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{x}{5}}$ where $\mathrm{x}=$ _____________.
View full solution →Which one of the following about an electron occupying the 1s orbital in a hydrogen atom is incorrect? (Bohr's radius is represented by $\mathrm{a}_{0}$)
View full solution →Pair of transition metal ions having the same number of unpaired electrons is :
- ✓
$V ^{2+}, Co ^{2+}$
- B
$Ti ^{2+}, Co ^{2+}$
- C
$Fe ^{3+}, Cr ^{2+}$
- D
$Ti ^{3+}, Mn ^{2+}$
Answer: A.
View full solution →Given below are two statements.
In the light of the above statements, choose the correct answer from the options given below :
- A
Statement I is false but Statement II is true
- B
Both Statement I and Statement II are false
- ✓
Statement I is true but Statement II is false
- D
Both Statement I and Statement II are true
Answer: C.
View full solution →Predict the major product of the following reaction sequence :-
An organic compound (X) with molecular formula $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}$ is not readily oxidised. On reduction it gives $\left(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}(\mathrm{Y})\right.$ which reacts with HBr to give a bromide ( Z ) which is converted to Grignard reagent. This Grinard reagent on reaction with (X) followed by hydrolysis give 2,3-dimethylbutan-2-ol. Compounds (X), (Y) and (Z) respectively are :
- A
$\mathrm{CH}_{3} \mathrm{COCH}_{3}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{CH}_{3} \mathrm{CH}(\mathrm{Br}) \mathrm{CH}_{3}$
- ✓
$\mathrm{CH}_{3} \mathrm{COCH}_{3}, \mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{3}, \mathrm{CH}_{3} \mathrm{CH}(\mathrm{Br}) \mathrm{CH}_{3}$
- C
$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHO}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{Br}$
- D
$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHO}, \mathrm{CH}_{3} \mathrm{CH}=\mathrm{CH}_{2}, \mathrm{CH}_{3} \mathrm{CH}(\mathrm{Br}) \mathrm{CH}_{3}$
Answer: B.
View full solution →The total number of hydrogen bonds of a DNA-double Helix strand whose one strand has the following sequence of bases is _____________.
$5^{'}$ - G - G-C-A-A-A-T-C-G-G-C-T-A-3'
View full solution →The pH of a 0.01 M weak acid $\mathrm{HX}\left(\mathrm{K}_{\mathrm{a}}=4 \times 10^{-10}\right)$ is found to be 5. Now the acid solution is diluted with excess of water so that the pH of the solution changes to 6. The new concentration of the diluted weak acid is given as $x \times 10^{-4} \mathrm{M}$. The value of x is _____________(nearest integer)
View full solution →Fortification of food with iron is done using $\mathrm{FeSO}_{4} .7 \mathrm{H}_{2} \mathrm{O}$. The mass in grams of the $\mathrm{FeSO}_{4}.7 \mathrm{H}_{2} \mathrm{O}$ required to achieve 12 ppm of iron in 150 kg of wheat is _____________ (Nearest integer)
[Given : Molar mass of $\mathrm{Fe}, \mathrm{S}$ and O respectively are $56,32$ and $16 \mathrm{~g} \mathrm{~mol}^{-1}$]
View full solution →$\mathrm{KMnO}_{4}$ acts as an oxidising agent in acidic medium. '$X$' is the difference between the oxidation states of Mn in reactant and product. 'Y' is the number of '$d$' electrons present in the brown red precipitate formed at the end of the acetate ion test with neutral ferric chloride. The value of $\mathrm{X}+\mathrm{Y}$ is _____________.
View full solution →In Dumas' method for estimation of nitrogen 1g of an organic compound gave 150 mL of nitrogen collected at 300 K temperature and 900 mm Hg pressure. The percentage composition of nitrogen in the compound is _____________ $\%$ (nearest integer).
(Aqueous tension at $300 \mathrm{~K}=15 \mathrm{~mm} \mathrm{Hg}$)
View full solution →