JEE Main 2025 (23 Jan Shift 1)

67 Questions • 75 Minutes • 2025

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If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

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One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

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Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ be $\hat i+2\hat j+\hat k$, $\hat i+3\hat j-2\hat k$ and $2\hat i+\hat j-\hat k$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{110}}3$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt2}$, then the position vector of $E$ is

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If $A$, $B$ and $(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1}))$ are non-singular matrices of same order, then the inverse of $A\left(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\right)^{-1}B$, is equal to

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Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is

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Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e2,k)$. If the curve satisfies the differential equation $2(3+y)e^{2x}dx-(7+e^{2x})dy=0$, then $k$ is equal to

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If the function $f(x)=\begin{cases}\frac2x\{\sin(k_1+1)x+\sin(k_2-1)x\},&x<0\\4,&x=0\\\frac2x\log_e\left(\frac{2+k_1x}{2+k_2x}\right),&x>0\end{cases}$ is continuous at $x=0$, then $k_1^2+k_2^2$ is equal to

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If the line $3x-2y+12=0$ intersects the parabola $4y=3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to

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Let $P$ be the foot of the perpendicular from the point $Q(10,-3,-1)$ on the line $\frac{x-3}7=\frac{y-2}{-1}=\frac{z+1}{-2}$. Then the area of the right angled triangle $PQR$, where $R$ is the point $(3,-2,1)$, is

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Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$, divides the arc $AC$ such that $\frac{\text{length of arc }AB}{\text{length of arc }BC}=\frac15$, and $\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$, then $\alpha+\sqrt2(\sqrt3-1)\beta$ is equal to

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Let $f(x)=\log_ex$ and $g(x)=\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}$. Then the domain of $f\circ g$ is

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If the system of equations $(\lambda-1)x+(\lambda-4)y+\lambda z=5$, $\lambda x+(\lambda-1)y+(\lambda-4)z=7$, $(\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

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The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

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Let $R=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:

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Let the area of a $\triangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be 35 square units. If its orthocenter and centroid are $O\left(2,\frac{14}5\right)$ and $C(c,d)$ respectively, then $c+2d$ is equal to

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The value of $\int_{e^2}^{e^4}\frac1x\left(\frac{e^{\left((\log_ex)^2+1\right)^{-1}}}{e^{\left((\log_ex)^2+1\right)^{-1}}+e^{\left((6-\log_ex)^2+1\right)^{-1}}}\right)dx$ is

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Let $\left|\frac{\bar z-i}{2\bar z+i}\right|=\frac13$, $z\in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0)$, $C$ and $(\alpha,0)$ is 11 square units, then $\alpha^2$ equals:

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The value of $(\sin70^\circ)\left(\cot10^\circ\cot70^\circ-1\right)$ is

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Let $I(x)=\int\frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37)-I(24)=\frac14\left(\frac1{b^{\frac1{13}}}-\frac1{c^{\frac1{13}}}\right)$, $b,c\in\mathbf N$, then $3(b+c)$ is equal to

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If $\frac\pi2\le x\le\frac{3\pi}4$, then $\cos^{-1}\left(\frac{12}{13}\cos x+\frac5{13}\sin x\right)$ is equal to

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Paper Info

Exam: JEE
Year: 2025
Questions: 67
Duration: 75 mins

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