MyGoalPrep LogoMyGoalPrep.com

All Questions / JEE Main 2025

JEE Main 2025 Previous Year Questions

Practice JEE Main 2025 exam questions from January shifts, reworded as MCQs with detailed solutions. Covering all three subjects — Physics, Chemistry, and Mathematics — these questions help you understand the latest JEE Main pattern and difficulty level.

184 questions • 184 math

Medium•Jee Main 2025
The value of (sin⁡70∘)(cot⁡10∘cot⁡70∘−1)(\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) is
Medium•Jee Main 2025
Let a1,a2,a3,…a_1, a_2, a_3, \ldots be a G.P. of increasing terms. If a1a5=28a_1 a_5 = 28 and a2+a4=29a_2 + a_4 = 29, then a6a_6 is equal to
Hard•Jee Main 2025
Let x=x(y)x = x(y) be the solution of the differential equation y2 dx+(x−1y) dy=0y^2 \, dx + (x - \frac{1}{y}) \, dy = 0. If $x
Medium•Jee Main 2025
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is mn\frac{m}{n}, where gcd⁡(m,n)=1\gcd(m, n) = 1, then m+nm + n is equal to
Medium•Jee Main 2025
The product of all solutions of the equation e5log⁡x2+3=x8,x>0e^{5 \log x^2 + 3} = x^8, x > 0, is
Medium•Jee Main 2025
Let the triangle PQR be the image of the triangle with vertices (1,3),(3,1)(1, 3), (3, 1) and (2,4)(2, 4) in the line x+2y=2x + 2y = 2. If the centroid of △PQR\triangle PQR is the point (α,β)(\alpha, \beta), then 15(α−β)15(\alpha - \beta) is equal to
Medium•Jee Main 2025
Let for f(x)=7tan⁡8x+7tan⁡6x−3tan⁡4x−3tan⁡2xf(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x, I1=∫0π/4f(x) dxI_1 = \int_{0}^{\pi/4} f(x) \, dx and I2=∫0π/4xf(x) dxI_2 = \int_{0}^{\pi/4} x f(x) \, dx. Then 7I1+12I27I_1 + 12I_2 is equal to
Medium•Jee Main 2025
Let the parabola y=x2+px−3y = x^2 + px - 3, meet the coordinate axes at the points P, Q and R. If the circle C with centre at (α,β)(\alpha, \beta) passes through the points P, Q and R, then the area of △PQR\triangle PQR is
Medium•Jee Main 2025
Let L1:x−12=y−23=z−34L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} and L2:x−32=y−43=z−54L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4} be two lines. Then which of the following points lies on the line of the shortest distance between L1L_1 and L2L_2?
Medium•Jee Main 2025
Let f(x)f(x) be a real differentiable function such that f(0)=1f(0) = 1 and f(x+y)=f(x)f(y)+f′(x)f(y)f(x + y) = f(x)f(y) + f'(x)f(y) for all x,y∈Rx, y \in \mathbb{R}. Then ∑n=1100log⁡2f(n)\sum_{n=1}^{100} \log_2 f(n) is equal to
Medium•Jee Main 2025
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is
Medium•Jee Main 2025
Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of 16(sec⁡−1x)2+(cosec⁡−1x)216 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 is
Medium•Jee Main 2025
For positive integers nn, if 4an=(n2+5n+6)4a_n = (n^2 + 5n + 6) and Sn=∑k=1n(1ak)S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right), then the value of 507S2025507S_{2025} is
Medium•Jee Main 2025
Let f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} be a twice differentiable function such that f(x+y)=f(x)f(y)f(x + y) = f(x)f(y) for all x,y∈Rx, y \in \mathbb{R}. If f′(0)=4af'(0) = 4a and ff satisfies f′′(x)−3af′(x)−f(x)=0,a>0f''(x) - 3af'(x) - f(x) = 0, a > 0, then the area of the region R={(x,y)∣0≤y≤f(ax),0≤x≤2}R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\} is
Medium•Jee Main 2025
The area of the region, inside the circle (x−23)2+y2=12(x - 2\sqrt{3})^2 + y^2 = 12 and outside the parabola y2=23xy^2 = 2\sqrt{3}x is
Medium•Jee Main 2025
Let the foci of a hyperbola be (1,14)(1, 14) and (1,−12)(1, -12). If it passes through the point (1,6)(1, 6), then the length of its latus-rectum is
Medium•Jee Main 2025
If ∑r=1nTr=(2n−1)(2n+1)(2n+3)(2n+5)64\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}, then lim⁡n→∞∑r=1n(1Tr)\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right) is equal to
Medium•Jee Main 2025
A coin is tossed three times. Let XX denote the number of times a tail follows a head. If μ\mu and σ2\sigma^2 denote the mean and variance of XX, then the value of 64(μ+σ2)64(\mu + \sigma^2) is
Medium•Jee Main 2025
The number of non-empty equivalence relations on the set {1,2,3}\{1, 2, 3\} is
Medium•Jee Main 2025
A circle CC of radius 2 lies in the second quadrant and touches both the coordinate axes. Let rr be the radius of a circle that has centre at the point (2,5)(2, 5) and intersects the circle CC at exactly two points. If the set of all possible values of rr is the interval (α,β)(\alpha, \beta), then 3β−2α3\beta - 2\alpha is equal to
Medium•Jee Main 2025
Let A={1,2,3,…,10}A = \{1, 2, 3, \ldots, 10\} and B={mn:m,n∈A,m<n and gcd⁡(m,n)=1}B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}. Then n(B)n(B) is equal to
Medium•Jee Main 2025
Let z1,z2z_1, z_2 and z3z_3 be three complex numbers on the circle ∣z∣=1|z| = 1 with arg⁡(z1)=π4,arg⁡(z2)=0\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0 and arg⁡(z3)=π4\arg(z_3) = \frac{\pi}{4}. If ∣z1zˉ2+z2zˉ3+z3zˉ1∣2=α+β3,α,β∈Z|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}, then the value of α2+β2\alpha^2 + \beta^2 is
Medium•Jee Main 2025
Let P(4,43)P(4, 4\sqrt{3}) be a point on the parabola y2=4axy^2 = 4ax and PQPQ be a focal chord of the parabola. If MM and NN are the foot of perpendiculars drawn from PP and QQ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to
Medium•Jee Main 2025
The sum of all values of θ∈[0,2π] \theta \in [0, 2\pi] satisfying 2sin⁡2θ=cos⁡2θ 2\sin^2 \theta = \cos 2\theta and 2cos⁡2θ=3sin⁡θ 2\cos^2 \theta = 3\sin \theta is
Medium•Jee Main 2025
Let the curve z(1+i)+zˉ(1−i)=4 z(1 + i) + \bar{z}(1 - i) = 4 , z∈C z \in \mathbb{C} , divide the region ∣z−3∣≤1 |z - 3| \leq 1 into two parts of areas α \alpha and β \beta . Then ∣α−β∣ |\alpha - \beta| equals
Medium•Jee Main 2025
Let E:x2a2+y2b2=1,a>b E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b and H:x2A2−y2B2=1 H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 . Let the distance between the foci of E E and the foci of H H be 23 2\sqrt{3} . If a−A=2 a - A = 2 , and the ratio of the eccentricities of E E and H H is 13 \frac{1}{3} , then the sum of the lengths of their latus rectums is equal to
Medium•Jee Main 2025
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
Medium•Jee Main 2025
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
Medium•Jee Main 2025
Let x216+y225=1 \frac{x^2}{16} + \frac{y^2}{25} = 1 , z∈C z \in C , be the equation of a circle with center at C C . If the area of the triangle, whose vertices are at the points (0,0) (0, 0) , C C and (α,0) (\alpha, 0) is 11 square units, then α2 \alpha^2 equals:
Medium•Jee Main 2025
Let the position vectors of the vertices A,BA, B and CC of a tetrahedron ABCDABCD be i+2j+k,i+3j=2k^\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k} and 2i+j−k2\mathbf{i} + \mathbf{j} - \mathbf{k} respectively. The altitude from the vertex DD to the opposite face ABCABC meets the median line segment through AA of the triangle ABCABC at the point EE. If the length of ADAD is 113\frac{\sqrt{11}}{3} and the volume of the tetrahedron is 8056\frac{\sqrt{805}}{6}, then the position vector of EE is
Medium•Jee Main 2025
If A,B,A, B, and (adj(A−1)+adj(B−1))(\text{adj} (A^{-1}) + \text{adj} (B^{-1})) are non-singular matrices of same order, then the inverse of A(adj(A−1)+adj(B−1))−1BA (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B, is equal to
Medium•Jee Main 2025
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
Hard•Jee Main 2025
Let a curve y=f(x)y = f(x) pass through the points (0,5)(0, 5) and (log⁡e2,k)(\log_e 2, k). If the curve satisfies the differential equation 2(3+y)e2xdx−(7+e2x)dy=02(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0, then kk is equal to
Medium•Jee Main 2025
If the function f(x)={2xsin⁡(k1x+k2−1)x,x<04,x=02xlog⁡e(2+k2x2+k2x),x>0f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases} is continuous at x=0x = 0, then k12+k22k_1^2 + k_2^2 is equal to
Medium•Jee Main 2025
If the line 3x−2y+12=03x - 2y + 12 = 0 intersects the parabola 4y=3x24y = 3x^2 at the points AA and BB, then at the vertex of the parabola, the line segment ABAB subtends an angle equal to
Medium•Jee Main 2025
Let P P be the foot of the perpendicular from the point Q(10,−3,−1) Q(10, -3, -1) on the line x−37=y−21=z+12 \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} . Then the area of the right angled triangle PQR PQR , where R R is the point (3,−2,1)(3, -2, 1), is \begin{align*}
Medium•Jee Main 2025
Let the arc AC AC of a circle subtend a right angle at the centre O O . If the point B B on the arc AC AC , divides the arc AC AC such that length of arc ABlength of arc BC=15 \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} , and OC→=αOA→+βOB→ \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} , then α+2(3−1)β \alpha + \sqrt{2(\sqrt{3} - 1)}\beta is equal to \begin{align*}
Medium•Jee Main 2025
Let f(x)=log⁡2x f(x) = \log_2 x and g(x)=x4−2x3+3x2−2x+22x2−2x+1 g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} . Then the domain of f∘g f \circ g is \begin{align*}
Medium•Jee Main 2025
(λ−1)x+(λ−4)y+λz=5(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 If the system of equations λx+(λ−1)y+(λ−4)z=7 \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 has infinitely many solutions, then λ2+λ \lambda^2 + \lambda is equal to \begin{align*}
Medium•Jee Main 2025
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 \begin{align*}
Medium•Jee Main 2025
Let R={(1,2),(2,3),(3,3)} R = \{(1, 2), (2, 3), (3, 3)\} be a relation defined on the set {1,2,3,4} \{1, 2, 3, 4\} . Then the minimum number of elements, needed to be added in R R so that R R becomes an equivalence relation, is \begin{align*}
Medium•Jee Main 2025
Let the area of a △PQR \triangle PQR with vertices P(5,4),Q(−2,4) P(5, 4), Q(-2, 4) and R(a,b) R(a, b) be 35 square units. If its orthocenter and centroid are O(2,127) O \left(2, \frac{12}{7}\right) and C(c,d) C(c, d) respectively, then c+2d c + 2d is equal to \begin{align*}
Medium•Jee Main 2025
The value of ∫R1x(e(log⁡2x)2+1−e(log⁡2x)2−1)dx \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx is \begin{align*}
Medium•Jee Main 2025
Let I(x)=∫dx(x−11)(x+15) I(x) = \int \frac{dx}{(x-11)(x+15)} . If I(37)−I(24)=14(1βx−1cx) I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) , b,c∈N b, c \in \mathbb{N} , then 3(b+c) 3(b + c) is equal to
Medium•Jee Main 2025
If π6≤x≤3π4 \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} , then cos⁡−1(1213cos⁡x+513sin⁡x) \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) is equal to
Medium•Jee Main 2025
The distance of the line x22=y63=z34 \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} from the point (1,4,0)(1, 4, 0) along the line x4=y22=z33 \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} is
Medium•Jee Main 2025
Let A={(x,y)∈R×R:∣x+y∣≥3} A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} and B={(x,y)∈R×R:∣x∣+∣y∣≤3} B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} . If C={(x,y)∈A∩B:x=0 or y=0} C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} , then ∑(x,y)∈C∣x+y∣ \sum_{(x, y) \in C} |x + y| is
Medium•Jee Main 2025
Let X=R×R X = \mathbb{R} \times \mathbb{R} . Define a relation R R on X X as: (a1,b1)R(a2,b2)  ⟺  b1=b2(a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 Statement I : R R is an equivalence relation. Statement II : For some (a,b)∈X(a, b) \in X, the set S={(x,y)∈X:(x,y)R(a,b)} S = \{(x, y) \in X : (x, y)R(a, b)\} represents a line parallel to y=x y = x . In the light of the above statements, choose the correct answer from the options given below
Medium•Jee Main 2025
Let ∫x3sin⁡x dx=g(x)+C \int x^3 \sin x \, dx = g(x) + C , where C C is the constant of integration. If 8(g(π2)+g′(π2))=απ3+βπ2+γ,α,β,γ∈Z 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} , then α+β−γ \alpha + \beta - \gamma equals
Medium•Jee Main 2025
A rod of length eight units moves such that its ends A A and B B always lie on the lines x−y+2=0 x - y + 2 = 0 and y+2=0 y + 2 = 0 , respectively. If the locus of the point P P , that divides the rod AB AB internally in the ratio 2:1 2 : 1 is 9(x2+αy2+βxy+γx+28y)−76=0 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 , then α−β−γ \alpha - \beta - \gamma is equal to
Medium•Jee Main 2025
If the square of the shortest distance between the lines x2m=y6n=z33 \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} and x3m=y3n=z33 \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} is mn \frac{m}{n} , where m,n m, n are coprime numbers, then m+n m + n is equal to
Medium•Jee Main 2025
lim⁡x→∞(2x2−3x+5)(3x−1)23(3x2+5x+4)(3x+2)3 \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} is equal to
Medium•Jee Main 2025
Let the point A A divide the line segment joining the points P(−1,−1,2) P(-1,-1,2) and Q(5,5,10) Q(5,5,10) internally in the ratio r:1(r>0) r : 1(r > 0) . If O O is the origin and OQ→⋅OA→=15∣OP→×OA→∣2 \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 = 10, then the value of r r is
Medium•Jee Main 2025
The length of the chord of the ellipse x24+y23=1 \frac{x^2}{4} + \frac{y^2}{3} = 1 , whose mid-point is (1,12)(1, \frac{1}{2}), is
Medium•Jee Main 2025
The system of equations x+2y+5z=9 x + 2y + 5z = 9 , has no solution if: (x) x+5y+λz=μ x + 5y + \lambda z = \mu ,
Medium•Jee Main 2025
Let the range of the function f(x)=6+16cos⁡x⋅cos⁡(x3−x)⋅cos⁡(x3+x)⋅sin⁡3x⋅cos⁡6x,x∈R f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} be [α,β][\alpha, \beta] . Then the distance of the point (α,β)(\alpha, \beta) from the line 3x+4y+12=03x + 4y + 12 = 0 is
Hard•Jee Main 2025
Let x=x(y) x = x(y) be the solution of the differential equation \[ y = (x−ydydx)sin⁡(xy),y>0 and x\left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x
Medium•Jee Main 2025
The equation of the chord, of the ellipse x225+y216=1\frac{x^2}{25} + \frac{y^2}{16} = 1, whose mid-point is (3,1)(3, 1), is: 25x+101y=17625x + 101y = 176
Medium•Jee Main 2025
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of 14 \frac{1}{4} cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is
Medium•Jee Main 2025
The number of complex numbers z z , satisfying ∣z∣=1|z| = 1 and ∣z2+zˉ2∣=1 |\frac{z}{2} + \frac{\bar{z}}{2}| = 1, is
Hard•Jee Main 2025
Let A=[aij] A = [a_{ij}] be a 3×3 3 \times 3 matrix such that A[011]=[001],A[101]=[100] and A[210]=[100] A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , then a23 a_{23} equals
Medium•Jee Main 2025
If I=∫0πsin⁡x2sin⁡x2cos⁡x2 dx I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx , then I=∫021xsin⁡xcos⁡xsin⁡4x+cos⁡4x dx I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx equals
Medium•Jee Main 2025
A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is
Medium•Jee Main 2025
Let the shortest distance from (a,0),a>0(a, 0), a > 0 to the parabola y2=4xy^2 = 4x be 4. Then the equation of the circle passing through the point (a,0)(a, 0) and the focus of the parabola, and having its centre on the axis of the parabola is
Medium•Jee Main 2025
If in the expansion of (1+x)p(1−x)q(1 + x)^p(1 - x)^q, the coefficients of xx and x2x^2 are 1 and -2, respectively, then p2+q2p^2 + q^2 is equal to
Medium•Jee Main 2025
If the area of the region {(x,y):−1≤x≤1,0≤y≤a+ex+1−e−x,a>0}\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\} is ex+1ex+1e\frac{e^{x+1} e^{x+1}}{e}, then the value of aa is
Medium•Jee Main 2025
Let circle CC be the image of x2+y2−2x+4y−4=0x^2 + y^2 - 2x + 4y - 4 = 0 in the line 2x−3y+5=02x - 3y + 5 = 0 and AA be the point on CC such that OAOA is parallel to xx-axis and AA lies on the right hand side of the centre OO of CC. If B(α,β)B(\alpha, \beta), with β<4\beta < 4, lies on CC such that the length of the arc ABAB is (1/6)th(1/6)^{th} of the perimeter of CC, then β−3α\beta - \sqrt{3}\alpha is equal to
Medium•Jee Main 2025
Let in a △ABC\triangle ABC, the length of the side ACAC be 66, the vertex BB be (1,2,3)(1, 2, 3) and the vertices A,CA, C lie on the line x−32=y−72=z−72\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}. Then the area (in sq. units) of △ABC\triangle ABC is
Medium•Jee Main 2025
Let the product of the focal distances of the point (3,13)\left(\sqrt{3}, \frac{1}{3}\right) on the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a>b)(a > b), be 74\frac{7}{4}. Then the absolute difference of the eccentricities of two such ellipses is
Medium•Jee Main 2025
If the system of equations 5x+λy+3z=125x + \lambda y + 3z = 12 and 100x−47y+μz=212100x - 47y + \mu z = 212 has infinitely many solutions, then μ−2λ\mu - 2\lambda is equal to
Medium•Jee Main 2025
For some n≠10n \neq 10, let the coefficients of the 55th, 66th and 77th terms in the binomial expansion of (1+x)n+4(1 + x)^{n+4} be in A.P. Then the largest coefficient in the expansion of (1+x)n+4(1 + x)^{n+4} is
Medium•Jee Main 2025
The product of all the rational roots of the equation (x2−9x+11)2−(x−4)(x−5)=3\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3, is equal to
Medium•Jee Main 2025
Let the line passing through the points (−1,2,1)(-1, 2, 1) and parallel to the line x+12=y+13=z−13\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3} intersect the line x+23=y−32=z+41\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1} at the point PP. Then the distance of PP from the point Q(4,−5,1)Q(4, -5, 1) is
Medium•Jee Main 2025
Let the lines 3x−4y−α=03x - 4y - \alpha = 0, 8x−11y−33=08x - 11y - 33 = 0, and 2x−3y+λ=02x - 3y + \lambda = 0 be concurrent. If the image of the point (1,2)(1, 2) in the line 2x−3y+λ=02x - 3y + \lambda = 0 is (5713,−4013)\left(\frac{57}{13}, \frac{-40}{13}\right), then ∣αλ∣|\alpha\lambda| is equal to
Medium•Jee Main 2025
If α\alpha and β\beta are the roots of the equation 2x2−3x−2i=02x^2 - 3x - 2i = 0, where i=−1i = \sqrt{-1}, then 16⋅Re(α19+β19+α11+β11α5+β5)⋅Im(α19+β19+α11+β11α5+β5)16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) is equal to
Medium•Jee Main 2025
For a statistical data x1,x2,…,x10x_1, x_2, \ldots, x_{10} of 10 values, a student obtained the mean as 5.5 and ∑i=110xi2=371\sum_{i=1}^{10} x_i^2 = 371. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
Medium•Jee Main 2025
The area of the region {(x,y):x2+4x+2≤y≤∣x+2∣}\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\} is equal to
Medium•Jee Main 2025
Let Sn=12+19+112+121+…S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots upto nn terms. If the sum of the first six terms of an A.P. with first term −p-p and common difference pp is 2025S2025\sqrt{2025} S_{2025}, then the absolute difference between 20th and 15th terms of the A.P. is
Medium•Jee Main 2025
Let f:R→{0}→Rf : R \to \{0\} \to R be a function such that f(x)=6f(1x)=353x−52f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}. If the limit as x→0x \to 0 (x1x+f(x))=β;α,β∈R\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R, then α+2β\alpha + 2\beta is equal to
Medium•Jee Main 2025
If I(m,n)=∫01xm−1(1−x)n−1 dx,m,n>0I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0, then I(9,14)+I(10,13)I(9, 14) + I(10, 13) is
Medium•Jee Main 2025
AA and BB alternately throw a pair of dice. AA wins if he throws a sum of 5 before BB throws a sum of 8, and BB wins if he throws a sum of 8 before AA throws a sum of 5. The probability, that AA wins if AA makes the first throw, is
Medium•Jee Main 2025
Let f(x)=2x2+162x3+2x2+4+32f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}. Then the value of 8(f(115)+f(215)+…+f(5915))8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right) is equal to
Hard•Jee Main 2025
Let y=y(x)y = y(x) be the solution of the differential equation (xy−5x21+x2) dx+(1+x2) dy=0,y(0)=0(xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0. Then y(3)y(\sqrt{3}) is equal to
Medium•Jee Main 2025
lim⁡x→0csc⁡x(2cos⁡2x+3cos⁡x−cos⁡2x+sin⁡x+4)\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right) is
Medium•Jee Main 2025
Consider the region R={(x,y):x≤y≤9−111x2,x≥0} R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} . The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R R , is
Medium•Jee Main 2025
Let a⃗=i^+2j^+3k^,b⃗=3i^+j^−k^ \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} and c⃗ \vec{c} be three vectors such that c⃗ \vec{c} is coplanar with a⃗ \vec{a} and b⃗ \vec{b} . If the vector C⃗ \vec{C} is perpendicular to b⃗ \vec{b} and a⃗⋅c⃗=5 \vec{a} \cdot \vec{c} = 5 , then ∣c⃗∣ |\vec{c}| is equal to
Medium•Jee Main 2025
Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to
Medium•Jee Main 2025
If the system of equations 2x+λy+5z=52x + \lambda y + 5z = 5 has infinitely many solutions, then λ+μ\lambda + \mu is equal to: 14x+3y+μz=3314x + 3y + \mu z = 33
Medium•Jee Main 2025
Let A={x∈(0,π)−{π2}:log⁡2/π∣sin⁡x∣+log⁡2/π∣cos⁡x∣=2}A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\} and B={x≥0:x−4−3x−2+6=0}B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}. Then n(A∪B)n(A \cup B) is equal to
Medium•Jee Main 2025
The area of the region enclosed by the curves y=exy = e^x, y=∣ex−1∣y = |e^x - 1| and y-axis is
Medium•Jee Main 2025
Let the points (112,α)(\frac{11}{2}, \alpha) lie on or inside the triangle with sides x+y=11x + y = 11, x+2y=16x + 2y = 16 and 2x+3y=292x + 3y = 29. Then the product of the smallest and the largest values of α\alpha is equal to
Medium•Jee Main 2025
Let f:(0,∞)→Rf : (0, \infty) \rightarrow \mathbb{R} be a function which is differentiable at all points of its domain and satisfies the condition x2f′(x)=2xf(x)+3x^2 f'(x) = 2x f(x) + 3, with $f
Medium•Jee Main 2025
If 7=5×1+17(5+α)+172(5+2α)+173(5+3α)+⋯∞7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty, then the value of α\alpha is
Medium•Jee Main 2025
Let [x][x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x)=[x]+[x−2]f(x) = [x] + [x - 2], −2<x<3-2 < x < 3, is not continuous and not differentiable. Then m+nm + n is equal to
Hard•Jee Main 2025
Let A=[aij]A = [a_{ij}] be a square matrix of order 2 with entries either 0 or 1. Let EE be the event that AA is an invertible matrix. Then the probability P(E)P(E) is
Medium•Jee Main 2025
Let the position vectors of three vertices of a triangle be 4p+q−3r,−5p+q+2r4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r} and 2p−q+2r2\mathbf{p} - \mathbf{q} + 2\mathbf{r}. If the position vectors of the orthocenter and the circumcenter of the triangle are 5p+2q+3r14\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14} and αp+βq+γr\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r} respectively, then α+2β+5γ\alpha + 2\beta + 5\gamma is equal to
Medium•Jee Main 2025
Let a=3i−j+2k,b=a×(i−2k)\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k}) and c=b×k\mathbf{c} = \mathbf{b} \times \mathbf{k}. Then the projection of c−2j\mathbf{c} - 2\mathbf{j} on a\mathbf{a} is
Medium•Jee Main 2025
The number of real solution(s) of the equation x2+3x+2=min⁡{∣x−3∣,∣x+2∣}x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\} is
Medium•Jee Main 2025
The function f:(−∞,∞)→(−∞,1), defined by f(x)=x2−2xx2+2f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2} is
Medium•Jee Main 2025
In an arithmetic progression, if S10=1030S_{10} = 1030 and S12=57S_{12} = 57, then S30−S10S_{30} - S_{10} is equal to
Medium•Jee Main 2025
Suppose AA and BB are the coefficients of 30th and 12th terms respectively in the binomial expansion of (1+x)2n−1(1 + x)^{2n-1}. If 2A=5B2A = 5B, then nn is equal to
Medium•Jee Main 2025
Let (2,3)(2, 3) be the largest open interval in which the function f(x)=2log⁡e(x−2)−x2+ax+1f(x) = 2\log_e(x - 2) - x^2 + ax + 1 is strictly increasing and (b,c)(b, c) be the largest open interval, in which the function g(x)=(x−1)3(x+2−a)2g(x) = (x - 1)^3(x + 2 - a)^2 is strictly decreasing. Then 100(a+b−c)100(a + b - c) is equal to
Hard•Jee Main 2025
For some a,ba, b, let f(x)=a+sin⁡xx∣11ba1+sin⁡xba1b+sin⁡x∣,x≠0,lim⁡x→0f(x)=λ+μa+νbf(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b. Then (λ+μ+ν)2(\lambda + \mu + \nu)^2 is equal to
Medium•Jee Main 2025
If the equation of the parabola with vertex V(32,3)V\left(\frac{3}{2}, 3\right) and the directrix x+2y=0x + 2y = 0 is αx2+βy2−γxy−30x−60y+225=0\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0, then α+β+γ\alpha + \beta + \gamma is equal to
Medium•Jee Main 2025
If α>β>γ>0\alpha > \beta > \gamma > 0, then the expression cot⁡−1{β+(1+β2)(α−β)}+cot⁡−1{γ+(1+γ2)(β−γ)}+cot⁡−1{α+(1+α2)(γ−α)}\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\} is equal to
Medium•Jee Main 2025
Let O O be the origin, the point A A be z1=3+22i z_1 = \sqrt{3} + 2\sqrt{2}i , the point B(z2) B(z_2) be such that 3∣z2∣=∣z1∣ \sqrt{3} |z_2| = |z_1| and arg⁡(z2)=arg⁡(z1)+π6 \arg(z_2) = \arg(z_1) + \frac{\pi}{6} . Then
Medium•Jee Main 2025
Let f:R→R f : \mathbb{R} \to \mathbb{R} be a function defined by f(x)=(2+3a)x2+(2a+72)x+b,a≠1. f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. If f(x+y)=f(x)+f(y)+1−12xy f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy , then the value of 28∑i=15∣f(i)∣ 28 \sum_{i=1}^{5} |f(i)| is
Medium•Jee Main 2025
Let ABCD ABCD be a trapezium whose vertices lie on the parabola y2=4x y^2 = 4x . Let the sides AD AD and BC BC of the trapezium be parallel to y y -axis. If the diagonal AC AC is of length 254 \frac{25}{4} and it passes through the point (1,0) (1, 0) , then the area of ABCD ABCD is
Medium•Jee Main 2025
The sum of all local minimum values of the function f(x)={1−2x,x<−113(7+2∣x∣),−1≤x≤2112(x−4)(x−5),x>2 f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} is
Medium•Jee Main 2025
Let nCr−1=28,nCr=56 ^nC_{r-1} = 28, ^nC_r = 56 and nCr+1=70 ^nC_{r+1} = 70 . Let A(4cos⁡t,4sin⁡t),B(2sin⁡t,−2cos⁡t) A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) and C(3r−n,r2−n−1) C(3r - n, r^2 - n - 1) be the vertices of a triangle ABC ABC , where t t is a parameter. If (3x−1)2+(3y)2=α (3x - 1)^2 + (3y)^2 = \alpha , is the locus of the centroid of triangle ABC ABC , then α \alpha equals
Medium•Jee Main 2025
Let the equation of the circle, which touches x x -axis at the point (a,0) (a, 0) , a>0 a > 0 and cuts off an intercept of length b b on y y -axis be x2+y2−αx+βy+γ=0 x^2 + y^2 - \alpha x + \beta y + \gamma = 0 . If the circle lies below x x -axis, then the ordered pair (2a,b2)(2a, b^2) is equal to
Medium•Jee Main 2025
If f(x)=x22x2+2,x∈R, f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, then ∑k=181f(k82) \sum_{k=1}^{81} f \left( \frac{k}{82} \right) is equal to
Medium•Jee Main 2025
Two number k1 k_1 and k2 k_2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1+jk2,(i=−1) i^{k_1} + j^{k_2}, (i = \sqrt{-1}) is non-zero, equals
Medium•Jee Main 2025
If the image of the point (4,4,3)(4, 4, 3) in the line x−12=y−23=z−13\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3} is (α,β,γ)(\alpha, \beta, \gamma), then α+β+γ\alpha + \beta + \gamma is equal to \[
Medium•Jee Main 2025
cos⁡(sin⁡−135+sin⁡−1513+sin⁡−13365)\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right) is equal to: \[
Medium•Jee Main 2025
Let A(x,y,z)A(x, y, z) be a point in xyxy-plane, which is equidistant from three points (0,3,2),(2,0,3)(0, 3, 2), (2, 0, 3) and (0,0,1)(0, 0, 1). Let B=(1,4,−1)B = (1, 4, -1) and C=(2,0,−2)C = (2, 0, -2). Then among the statements (S1) : △ABC\triangle ABC is an isosceles right angled triangle, and (S2) : the area of △ABC\triangle ABC is 922\frac{9\sqrt{2}}{2}, \[
Medium•Jee Main 2025
The area (in sq. units) of the region {(x,y):0≤y≤2∣x∣+1,0≤y≤x2+1,∣x∣≤3}\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \} is \[
Medium•Jee Main 2025
The sum of the squares of all the roots of the equation x2+∣2x−3∣−4=0x^2 + |2x - 3| - 4 = 0, is \[
Medium•Jee Main 2025
Let TrT_r be the rthr^{th} term of an A.P. If for some m,Tm=125,T25=125m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}, and 20∑r=125Tr=1320 \sum_{r=1}^{25} T_r = 13, then 5m∑r=mm+2Tr5m \sum_{r=m}^{m+2} T_r is equal to \[
Medium•Jee Main 2025
Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If xx denote the number of defective oranges, then the variance of xx is \[
Medium•Jee Main 2025
Let for some function y=f(x),∫0xtf(t)dt=x2f(x),x>0y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0 and f(2)=3f(2) = 3. Then f(6)f(6) is equal to \[
Medium•Jee Main 2025
If ∫9x2cos⁡πx(1+x2)2dx=π(αx2+β),α,β∈Z\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}, then (α+β)2(\alpha + \beta)^2 equals \[
Medium•Jee Main 2025
Let {an}\{a_n\} be a sequence such that a0=0,a1=12a_0 = 0, a_1 = \frac{1}{2} and 2an+2=5an+1−3an,n=0,1,2,3,…2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots. Then ∑k=1100ak\sum_{k=1}^{100} a_k is equal to
Medium•Jee Main 2025
The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is
Medium•Jee Main 2025
The relation R={(x,y):x,y∈Z and x+y is even} R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} is
Hard•Jee Main 2025
Let A=[12−201] A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} and P=[cos⁡θ−sin⁡θsin⁡θcos⁡θ],θ≥0. P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. If B=PAPT,C=PTBTP B = PAP^T, C = P^TB^TP and the sum of the diagonal elements of C C is mn \frac{m}{n} , where gcd⁡(m,n)=1 \gcd(m, n) = 1 , then m+n m + n is
Medium•Jee Main 2025
If the components of a⃗=αi^+βj^+γk^ \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} along and perpendicular to b⃗=3i^+j^−k^ \vec{b} = 3\hat{i} + \hat{j} - \hat{k} respectively, are 1610(3i^+j^−k^) \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) and 110(−4i^−5j^−17k^) \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) , then α2+β2+γ2 \alpha^2 + \beta^2 + \gamma^2 is equal to
Medium•Jee Main 2025
Let A,B,C A, B, C be three points in xy xy -plane, whose position vectors are given by 3i^+j^+3j^ \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} and i^+(1−a)j^ \hat{i} + (1 - a)\hat{j} respectively with respect to the origin O O . If the distance of the point C C from the line bisecting the angle between the vectors OA→ \overrightarrow{OA} and OB→ \overrightarrow{OB} is a2 \frac{a}{\sqrt{2}} , then the sum of all the possible values of a a is
Medium•Jee Main 2025
Let the coefficients of three consecutive terms Tr,Tr+1, T_r, T_{r+1}, and Tr+2 T_{r+2} in the binomial expansion of (a+b)12 (a + b)^{\frac{1}{2}} be in a G.P. and let p p be the number of all possible values of r r . Let q q be the sum of all rational terms in the binomial expansion of (3+4)12 (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} . Then p+q p + q is equal to
Medium•Jee Main 2025
Let [x] [x] denote the greatest integer less than or equal to x x . Then the domain of f(x)=sec⁡−1(2[x]+1) f(x) = \sec^{-1}(2[x] + 1) is
Medium•Jee Main 2025
Let S S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S S , one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is
Medium•Jee Main 2025
If ∑r=1151sin⁡(r2+12)sin⁡(r2+32) \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} = a3+b a\sqrt{3} + b , a,b∈Z a, b \in \mathbb{Z} , then a2+b2 a^2 + b^2 is equal to
Medium•Jee Main 2025
Let f f be a real valued continuous function defined on the positive real axis such that g(x)=∫0xtf(t) dt g(x) = \int_0^x t f(t) \, dt . If g(x2)=x6+x7 g(x^2) = x^6 + x^7 , then value of ∑r=115f(x3) \sum_{r=1}^{15} f(x^3) is
Medium•Jee Main 2025
Let f:[0,3]→A f : [0, 3] \rightarrow A be defined by f(x)=2x3−15x2+36x+7 f(x) = 2x^3 - 15x^2 + 36x + 7 and g:[0,∞)→B g : [0, \infty) \rightarrow B be defined by g(x)=x2025x2025+1 g(x) = \frac{x^{2025}}{x^{2025} + 1} . If both the functions are onto and S={x∈Z:x∈A or x∈B} S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} , then n(S) n(S) is equal to
Medium•Jee Main 2025
Bag B1B_1 contains 6 white and 4 blue balls, Bag B2B_2 contains 4 white and 6 blue balls, and Bag B3B_3 contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag B2B_2, is
Medium•Jee Main 2025
Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a twice differentiable function such that f(2)=1f(2) = 1. If F(x)=xf(x)F(x) = xf(x) for all x∈Rx \in \mathbb{R}, ∫x2xF′(x) dx=6\int_{x}^{2} x F'(x)\,dx = 6 and ∫x2x2F′′(x) dx=40\int_{x}^{2} x^2 F''(x)\,dx = 40, then F′(2)+∫x2F(x) dxF'(2) + \int_{x}^{2} F(x)\,dx is equal to
Medium•Jee Main 2025
Let f:R∖{0}→(−∞,1)f : \mathbb{R} \setminus \{0\} \to (-\infty, 1) be a polynomial of degree 2, satisfying f(x)f(1x)=f(x)+f(1x)f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right). If f(K)=−2Kf(K) = -2K, then the sum of squares of all possible values of KK is
Medium•Jee Main 2025
If AA and BB are the points of intersection of the circle x2+y2−8x=0x^2 + y^2 - 8x = 0 and the hyperbola x2y2−y2x2=1\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1 and a point PP moves on the line 2x−3y+4=02x - 3y + 4 = 0, then the centroid of △PAB\triangle PAB lies on the line
Medium•Jee Main 2025
If f(x)=∫1xx1/4(1+x1/4)1dxf(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}, f(0)=−6f(0) = -6, then $f
Medium•Jee Main 2025
The area of the region bounded by the curves x(1+y2)=1x \left( 1 + y^2 \right) = 1 and y2=2xy^2 = 2x is
Medium•Jee Main 2025
The square of the distance of the point (157,227,7)\left( \frac{15}{7}, \frac{22}{7}, 7 \right) from the line x+13=y+35=z+57\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7} in the direction of the vector i^+4j^+7k^\hat{i} + 4\hat{j} + 7\hat{k} is
Medium•Jee Main 2025
If the midpoint of a chord of the ellipse x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 1 is (2,4/3)(\sqrt{2}, 4/3), and the length of the chord is 253\frac{2\sqrt{5}}{3}, then α\alpha is
Medium•Jee Main 2025
If α+iβ\alpha + i\beta and γ+iδ\gamma + i\delta are the roots of x2−(3−2i)x−(2i−2)=0x^2 - (3 - 2i)x - (2i - 2) = 0, i=−1i = \sqrt{-1}, then αγ+βδ\alpha\gamma + \beta\delta is equal to
Medium•Jee Main 2025
Two equal sides of an isosceles triangle are along −x+2y=4-x + 2y = 4 and x+y=4x + y = 4. If mm is the slope of its third side, then the sum, of all possible distinct values of mm, is
Medium•Jee Main 2025
Let x1,x2,…,x10 x_1, x_2, \ldots, x_{10} be ten observations such that ∑i=110(xi−2)=30, \sum_{i=1}^{10} (x_i - 2) = 30, ∑i=110(xi−β)2=98,β>2, \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, and their variance is 45. \frac{4}{5}. If μ \mu and σ2 \sigma^2 are respectively the mean and the variance of 2(x1−1)+4β, 2(x_1 - 1) + 4\beta, 2(x2−1)+4β,…,2(x10−1)+4β, 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, then ∂μ∂β \frac{\partial \mu}{\partial \beta} is equal to
Medium•Jee Main 2025
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is
Medium•Jee Main 2025
The number of solutions of the equation (9x−9x+2)(2x−7x+3)=0 \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 is
Medium•Jee Main 2025
Define a relation R R on the interval [0,π4] [0, \frac{\pi}{4}] by xRy xRy if and only if sec⁡2x−tan⁡2y=1. \sec^2 x - \tan^2 y = 1. Then R R is
Medium•Jee Main 2025
Two parabolas have the same focus (4,3) (4, 3) and their directrices are the x x -axis and the y y -axis, respectively. If these parabolas intersect at the points A A and B, B, then (AB)2 (AB)^2 is equal to
Medium•Jee Main 2025
Let P P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set P P is
Medium•Jee Main 2025
Let a⃗=i^+2j^+k^ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} and b⃗=2i^+7j^+3k^. \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. Let L1:r⃗=(−i^+2j^+k^)+λa⃗,λ∈R L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} and L2:r⃗=(i^+k^)+μb⃗,μ∈R L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} be two lines. If the line L3 L_3 passes through the point of intersection of L1 L_1 and L2, L_2, and is parallel to a⃗+b⃗, \vec{a} + \vec{b}, then L3 L_3 passes through the point
Medium•Jee Main 2025
Let a⃗=2i^−j^+3k^,b⃗=3i^−5j^+k^ \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} and c⃗ \vec{c} be a vector such that a⃗×c⃗=c⃗×b⃗ \vec{a} \times \vec{c} = \vec{c} \times \vec{b} and (a⃗+c⃗)⋅(b⃗+c⃗)=168. (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. Then the maximum value of ∣c⃗∣2 |\vec{c}|^2 is
Medium•Jee Main 2025
The integral 80∫0π(sin⁡θ+cos⁡θ9+16sin⁡2θ)dθ 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta is equal to
Medium•Jee Main 2025
Let the ellipse E1:x2a2+y2b2=1,a>b E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b and E2:x2A2+y2B2=1,A<B E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B have same eccentricity 13 \frac{1}{\sqrt{3}} . Let the product of their lengths of latus rectums be 323 \frac{32}{\sqrt{3}} , and the distance between the foci of E1 E_1 be 4. If E1 E_1 and E2 E_2 meet at A,B,C A, B, C and D D , then the area of the quadrilateral ABCD ABCD equals:
Hard•Jee Main 2025
Let A=[aij]=[log⁡5128log⁡45log⁡58log⁡425] A = [a_{ij}] = \begin{bmatrix} \log_5 128 \log_4 5 \\ \log_5 8 \log_4 25 \end{bmatrix} . If Aij A_{ij} is the cofactor of aij a_{ij} , Cij=∑k=12aikAjk,1≤i,j≤2 C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 , and C=[Cij] C = [C_{ij}] , then 8∣C∣ 8|C| is equal to:
Medium•Jee Main 2025
Let ∣z1−8−2i∣≤1 |z_1 - 8 - 2i| \leq 1 and ∣z2−2+6i∣≤2,z1,z2∈C |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} . Then the minimum value of ∣z1−z2∣ |z_1 - z_2| is:
Medium•Jee Main 2025
Let L1:x−12=y−13=z−14 L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} and L2:x+12=y−23=z1 L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} be two lines. Let L3 L_3 be a line passing through the point (α,β,γ)(\alpha, \beta, \gamma) and be perpendicular to both L1 L_1 and L2 L_2 . If L3 L_3 intersects L1 L_1 , then ∣5α−11β−8γ∣ |5\alpha - 11\beta - 8\gamma| equals:
Hard•Jee Main 2025
Let M M and m m respectively be the maximum and the minimum values of f(x)=[1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x],x∈R f(x) = \begin{bmatrix} 1 + \sin^2 x \cos^2 x 4\sin 4x \\ \sin^2 x 1 + \cos^2 x 4\sin 4x \\ \sin^2 x \cos^2 x 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} Then M4−m4 M^4 - m^4 is equal to:
Medium•Jee Main 2025
Let ABC ABC be a triangle formed by the lines 7x−6y+3=0,x+2y−31=0 7x - 6y + 3 = 0, x + 2y - 31 = 0 and 9x−2y−19=0 9x - 2y - 19 = 0 . Let the point (h,k)(h, k) be the image of the centroid of ΔABC \Delta ABC in the line 3x+6y−53=0 3x + 6y - 53 = 0 . Then h2+k2+hk h^2 + k^2 + hk is equal to:
Medium•Jee Main 2025
The value of lim⁡n→∞(∑k=1nk4+4k2+11k+5(k+3)!) \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) is:
Medium•Jee Main 2025
The least value of n n for which the number of integral terms in the Binomial expansion of (7+11)n (\sqrt{7} + \sqrt{11})^n is 183, is:
Hard•Jee Main 2025
Let y=y(x) y = y(x) be the solution of the differential equation cos⁡x(log⁡e(cos⁡x))2dy+(sin⁡x−3ysin⁡xlog⁡e(cos⁡x))dx=0,x∈(0,π2). \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right). If y(π6)=−1log⁡22 y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} , then y(π8) y \left( \frac{\pi}{8} \right) is equal to:
Medium•Jee Main 2025
Let the line x+y=1 x + y = 1 meet the circle x2+y2=4 x^2 + y^2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:
Medium•Jee Main 2025
Let the area of the region {(x,y):2y≤x2+3,y≥∣x∣,∣y∣≤∣x−1∣} \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} be A. Then 6A 6A is equal to:
Medium•Jee Main 2025
Let f(x)=∫01(t2−9t+20) dt,1≤x≤5. f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. If the range of f f is [α,β][\alpha, \beta], then 4(α+β) 4(\alpha + \beta) equals
Medium•Jee Main 2025
Let a⃗ \vec{a} be a unit vector perpendicular to the vectors b⃗=i^−2j^+3k^ \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} and c⃗=2i^+3j^−k^ \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} , and makes an angle of cos⁡−1(−12) \cos^{-1}\left(-\frac{1}{2}\right) with the vector i^+j^+k^ \hat{i} + \hat{j} + \hat{k} . If a⃗ \vec{a} makes an angle of π3 \frac{\pi}{3} with the vector i^+αj^+k^ \hat{i} + \alpha\hat{j} + \hat{k} , then the value of α \alpha is
Medium•Jee Main 2025
Let α,β(α≠β)\alpha, \beta (\alpha \neq \beta) be the values of mm, for which the equations x+y+z=1,x+2y+4z=mx + y + z = 1, x + 2y + 4z = m and x+4y+10z=m2x + 4y + 10z = m^2 have infinitely many solutions. Then the value of ∑n=110(nα+nβ)\sum_{n=1}^{10} (n^\alpha + n^\beta) is equal to
Hard•Jee Main 2025
If for the solution curve y=f(x) y = f(x) of the differential equation dydx+(tan⁡x)y=2+sec⁡x(1+2sec⁡x)2 \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} , x∈(−π2,π2) x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) , then f(π4) f\left(\frac{\pi}{4}\right) is equal to
Medium•Jee Main 2025
Let P P be the foot of the perpendicular from the point (1,2,2) (1, 2, 2) on the line L:x−11=y+12=z−22 L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} . Let the line r⃗=(−i^+j^−2k^)+λ(i^−j^+k^),λ∈R \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} , intersect the line L L at Q Q . Then 2(PQ)2 2(PQ)^2 is equal to
Hard•Jee Main 2025
Let A=[aij] A = [a_{ij}] be a matrix of order 3×3 3 \times 3 , with aij=(2)i+j a_{ij} = (\sqrt{2})^{i+j} . If the sum of all the elements in the third row of A2 A^2 is α+β2 \alpha + \beta\sqrt{2} , α,β∈Z \alpha, \beta \in \mathbb{Z} , then α+β \alpha + \beta is equal to
Medium•Jee Main 2025
Let the line x+y=1 x + y = 1 meet the axes of x x and y y at A A and B B , respectively. A right angled triangle AMN AMN is inscribed in the triangle OAB OAB , where O O is the origin and the points M M and N N lie on the lines OB OB and AB AB , respectively. If the area of the triangle AMN AMN is 45 \frac{4}{5} of the area of the triangle OAB OAB and AN:NB=λ:1 AN : NB = \lambda : 1 , then the sum of all possible value(s) of λ \lambda is
Medium•Jee Main 2025
If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged in a dictionary, then the word at 440th position in this arrangement, is
Medium•Jee Main 2025
If the set of all a∈R a \in \mathbb{R} , for which the equation 2x2+(a−5)x+15=3a 2x^2 + (a - 5)x + 15 = 3a has no real root, is the interval (α,β)(\alpha, \beta), and X={x∈Z:α<x<β} X = \{x \in \mathbb{Z} : \alpha < x < \beta\} , then ∑x∈Xx2 \sum_{x \in X} x^2 is equal to
Hard•Jee Main 2025
Let A=[aij] A = [a_{ij}] be a 2×2 2 \times 2 matrix such that aij∈{0,1} a_{ij} \in \{0, 1\} for all i i and j j . Let the random variable X X denote the possible values of the determinant of the matrix A A . Then, the variance of X X is
Medium•Jee Main 2025
Let the function f(x)=(x2+1)∣x2−ax+2∣+cos⁡∣x∣ f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| be not differentiable at the two points x=α=2 x = \alpha = 2 and x=β x = \beta . Then the distance of the point (α,β)(\alpha, \beta) from the line 12x+5y+10=012x + 5y + 10 = 0 is equal to
Medium•Jee Main 2025
Let the area enclosed between the curves ∣y∣=1−x2 |y| = 1 - x^2 and x2+y2=1 x^2 + y^2 = 1 be α \alpha . If 9α=βπ+γ 9\alpha = \beta \pi + \gamma , β,γ \beta, \gamma are integers, then the value of ∣β−γ∣|\beta - \gamma| equals.
Medium•Jee Main 2025
The remainder, when 7103 7^{10^3} is divided by 23, is equal to
Medium•Jee Main 2025
If αx+βy=109 \alpha x + \beta y = 109 is the equation of the chord of the ellipse x2α+y2β=1 \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 , whose mid point is (12,14) \left( \frac{1}{2}, \frac{1}{4} \right) , then α+β \alpha + \beta is equal to
Medium•Jee Main 2025
If the domain of the function log⁡5(18x−x2−77) \log_5 (18x - x^2 - 77) is (α,β) (\alpha, \beta) and the domain of the function log⁡(x−1)(2x2+3x−2x2−3x−4) \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) is (γ,δ) (\gamma, \delta) , then α2+β2+γ2 \alpha^2 + \beta^2 + \gamma^2 is equal to
Medium•Jee Main 2025
Let a circle C C pass through the points (4,2) (4, 2) and (0,2) (0, 2) , and its centre lie on 3x+2y+2=0 3x + 2y + 2 = 0 . Then the length of the chord, of the circle C C , whose mid-point is (1,2) (1, 2) , is
Medium•Jee Main 2025
Let a straight line L L pass through the point P(2,−1,3) P(2, -1, 3) and be perpendicular to the lines x−12=y+11=z−3−2 \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} and x−31=y−2−1=z+24 \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} . If the line L L intersects the yz yz -plane at the point Q Q , then the distance between the points P P and Q Q is
Medium•Jee Main 2025
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is 2945 \frac{29}{45} , then n n is equal to
Medium•Jee Main 2025
Let S=N∪{0}S = N \cup \{0\}. Define a relation RR from SS to RR by R={(x,y):log⁡ey=xlog⁡e(23),x∈S,y∈R}R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \} Then, the sum of all the elements in the range of RR is equal to
Medium•Jee Main 2025
If sin⁡x+sin⁡2x=1,x∈(0,π2)\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right), then (cos⁡12x+xtan⁡12x)+3(cos⁡10x+tan⁡10x+cos⁡8x+tan⁡8x)+(cos⁡6x+tan⁡6x)(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x) is equal to