Sample QuestionsJEE Main 24-Jan-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.
Let the position vectors of three vertices of a triangle be $4 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}-3 \overrightarrow{\mathrm{r}},-5 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}$ and $2 \vec{p}-\vec{q}+2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{r}}}{4}$ and $\alpha \overrightarrow{\mathrm{p}}+\beta \overrightarrow{\mathrm{q}}+\gamma \overrightarrow{\mathrm{r}}$ respectively, then $\alpha+2 \beta+5 \gamma$ is equal to:
Answer: A.
View full solution →Let $\mathrm{f}:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f^{\prime}(x)=2 x f(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to:
Answer: C.
View full solution →If $\alpha>\beta>\gamma>0$, then the expression
$\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^{2}\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^{2}\right)}{(\beta-\gamma)}\right\}$
$+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^{2}\right)}{(\gamma-\alpha)}\right\}$ is equal to:
Answer: D.
View full solution →The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :
Answer: A.
View full solution →The equation of the chord, of the ellipse $\frac{\mathrm{x}^{2}}{25}+\frac{\mathrm{y}^{2}}{16}=1$, whose mid-point is $(3,1)$ is :
- ✓
$48 x+25 y=169$
- B
$4 x+122 y=134$
- C
$25 x+101 y=176$
- D
$5 x+16 y=31$
Answer: A.
View full solution →If $\int \frac{2 x^{2}+5 x+9}{\sqrt{x^{2}+x+1}} d x=x \sqrt{x^{2}+x+1}+\alpha \sqrt{x^{2}+x+1}+$ $\beta \log _{e}\left|x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right|+C$, where $C$ is the constant of integration, then $\alpha+2 \beta$ is equal to ____________ .
View full solution →Let $\mathrm{H}_{1}: \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{y^{2}}{\mathrm{~b}^{2}}=1$ and $\mathrm{H}_{2}:-\frac{\mathrm{x}^{2}}{\mathrm{~A}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~B}^{2}}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_{1}=\sqrt{\frac{5}{2}}$ and $e_{2}$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 \mathrm{e}_{2}^{2}$ is equal to ____________ .
View full solution →Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$, then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to ____________ .
View full solution →Let $P$ be the image of the point $Q(7,-2,5)$ in the line $L: \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $R(5, p, q)$ be a point on L . Then the square of the area of $\triangle \mathrm{PQR}$ is ____________ .
View full solution →Number of functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ____________ .
View full solution →In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of $P_1$ and $P_2$ are orthogonal to each other. The polarizer $P_3$ covers both the slits with its transmission axis at $45^{\circ}$ to those of $P$ and $P_2$. An unpolarized light of wavelength $\lambda$ and intensity $I_0$ is incident on $P_1$ and $P_2$. The intensity at a point after $P_3$ where the path difference between the light waves from $s_1$ and $s_2$ is $\frac{\lambda}{3}$, is
- A
$\frac{I_6}{2}$
- B
$\frac{I_0}{4}$
- ✓
$I_0$
- D
$\frac{ I _0}{3}$
Answer: C.
View full solution →Young's double slit interference apparatus is immersed in a liquid of refractive index 1.44. It has slit separation of 1.5 mm . The slits are illuminated by a parallel beam of light whose wavelength in air is 690 nm . The fringe-width on a screen placed behind the plane of slits at a distance of 0.72 m , will be :
Answer: A.
View full solution →The magnitude of heat exchanged by a system for the given cyclic process ABCA (as shown in figure) is (in SI unit)
- A
$10 \pi$
- ✓
$5 \pi$
- C
- D
$40 \pi$
Answer: B.
View full solution →Given below are two statements. One is labelled as
Assertion (A) and the other is labelled as Reason(R).
Assertion (A) : A electron in a certain region of uniform magnetic field is moving with constant velocity in a straight line path.
Reason (A) : The magnetic field in that region is along the direction of velocity of the electron.
In the light of the above statements, choose the correct answer from the options given below :
- A
(A) is false but (R) is true
- ✓
Both (A) and (R) are true and (R) is the correct explanation of (A)
- C
Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
- D
(A) is true but (R) is false
Answer: B.
View full solution →The temperature of a body in air falls from $40^{\circ} C$ to $24^{\circ} C$ in 4 minutes. The temperature of the air is $16^{\circ} C$. The temperature of the body in the next 4 minutes will be :
Answer: C.
View full solution →The ratio of the power of a light source $S_1$ to that the light source $S _2$ is $2 . S _1$ is emitting $2 \times 10^{15}$ photons per second at 600 nm . If the wavelength of the source $S _2$ is 300 nm , then the number of photons per second emitted by $S _2$ is ___________ $\times 10^{14}$. (5)
View full solution →The increase in pressure required to decrease the volume of a water sample by $0.2 \%$ is $P \times 10^{\circ} Nm ^{-2}$. Bulk modulus of water is $2.15 \times 10^9 Nm ^{-2}$. The value of $P$ is _____________
View full solution →A tightly wound long solenoid carries a current of
1.5 A. An electron is executing uniform circular motion inside the solenoid with a time period of 75 ns . The number of turns per metre in the solenoid is __________ .
[Take mass of electron $m _e=9 \times 10^{31} kg$, charge of electron $\left| q _e\right|=1.6 \times 10^{-19} C$,$
\left.\mu_{o}=4 \pi \times 10^{-7} \frac{N}{A^2}, 1 ns=10^{-9} s\right]
$
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A string of length $L$ is fixed at one end and carries a mass of M at the other end. The mass makes $\left(\frac{3}{\pi}\right)$ rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is ___________ ML.Acceleration due to gravity on the surface of earth is ' g '. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is ____________ g.
$
\begin{array}{l}
S(g)+\frac{3}{2} O_2(g) \rightarrow SO_3(g)+2 x \text { kcal } \\
SO_2(g)+\frac{1}{2} O_2(g) \rightarrow SO_3(g)+y kcal
\end{array}
$
The heat of formation of $SO _2(g)$ is given by :
- A
$\frac{2 x }{ y } kcal$
- ✓
$y-2 x$ kcal
- C
$2 x+y$ kcal
- D
$x+y$ kcal
Answer: B.
View full solution →Which of the following mixing of 1 M base and 1 M acid leads to the largest increase in temperature?
Answer: A.
View full solution →For hydrogen atom, the orbital/s with lowest energy is/are :
(A) 4 s
(B) $3 p_x$
(C) $3 d_{x^2-y^2}$
(D) $3 d_{z^2}$
(E) $4 p _2$
Choose the correct answer from the options given below :
Answer: D.
View full solution →The elemental composition of a compound is $54.2 \%, C , 9.2 \% H$ and $36.6 \% O$. If the molar mass of the compound is $132 g mol ^{-1}$, the molecular formula of the compound is :
[Given : The relative atomic mass of $C : H : O =$ 12:1:16]
- A
$C _4 H _9 O _3$
- B
$C _6 H _{12} O _6$
- ✓
$C _6 H _{12} O _3$
- D
$C _4 H _8 O _2$
Answer: C.
View full solution →The structure of the major product formed in the following reaction is :
Answer: B.
View full solution →The hydrocarbon (X) with molar mass $80 g mol ^{-1}$ and $90 \%$ carbon has ___________ degree of unsaturation.
View full solution →The observed and normal masses of compound $MX _2$ are 65.6 and 164 respectively. The percent degree of ionisation of $MX _2$ is ____________%. (Nearest integer)
View full solution →In Carius method of estimation of halogen, 0.25 g of an organic compound gave 0.15 g of silver bromide $( AgBr )$. The percentage of Bromine in the organic compound is _________ $\times 10^{-1} \%$ (Nearest integer).
(Given : Molar mass of Ag is 108 and Br is $80 g mol ^{-1}$ )
View full solution →The possible number of stereoisomers for 5-phenylpent-4-en-2-ol is ___________
Consider a complex reaction taking place in three steps with rate constants $k _1, k _2$ and $k _3$ respectively. The overall rate constant $k$ is given by the expression $k=\sqrt{\frac{k_1 k_3}{k_2}}$. If the activation energies of the three steps are 60,30 and $10 kJ mol ^{-1}$ respectively, then the overall energy of activation in $kJ mol ^{-1}$ is ___________ . (Nearest integer)
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