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Mathematics · 2025 · 23 Jan · Shift 2

25 questions from this JEE Main paper

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2025 · 23 Jan · Shift 2
Mathematics

Mathematics · 2025 · 23 Jan · Shift 2

25 questions from this JEE Main paper

Definite Integration1 questions Complex Numbers2 questions Ellipse1 questions Vector Algebra1 questions Probability1 questions 3D Geometry2 questions Sets and Relations2 questions Area Under the Curves1 questions Trigonometric Ratio and Identites1 questions Matrices and Determinants2 questions Limits Continuity and Differentiability1 questions Binomial Theorem1 questions Straight Lines and Pair of Straight Lines1 questions Indefinite Integrals1 questions Application of Derivatives1 questions Parabola2 questions Differential Equations1 questions Permutations and Combinations1 questions Sequences and Series1 questions Statistics1 questions

Q26.If $I = \int_{0}^{\frac{π}{2}} \frac{s i n ^{\frac{3}{2}} x}{s i n ^{\frac{3}{2}} x + c o s ^{\frac{3}{2}} x} d x$ , then $\int_{0}^{2 I} \frac{x s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ equals :
Mathematics · Definite Integration · MCQ
Q27.The number of complex numbers $z$ , satisfying $∣ z ∣ = 1$ and $\frac{z}{z ˉ} + \frac{z ˉ}{z} = 1$ , is :
Mathematics · Complex Numbers · MCQ
Q28.The length of the chord of the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ , whose mid-point is $(1, \frac{1}{2})$ , is :
Mathematics · Ellipse · MCQ
Q29.Let the point A divide the line segment joining the points $P (- 1, - 1, 2)$ and $Q (5, 5, 10)$ internally in the ratio $r : 1 (r > 0)$ . If O is the origin and $(OQ \cdot OA) - \frac{1}{5} ∣ OP \times OA ∣^{2} = 10$ …
Mathematics · Vector Algebra · MCQ
Q30.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 : Includes diagram
Mathematics · Probability · MCQ
Q31.If the square of the shortest distance between the lines $\frac{x - 2}{1} = \frac{y - 1}{2} = \frac{z + 3}{- 3}$ and $\frac{x + 1}{2} = \frac{y + 3}{4} = \frac{z + 5}{- 5}$ is $\frac{m}{n}$ , where $m$ , $n$ are coprime numbers, then $m + n$ is equal to :
Mathematics · 3D Geometry · MCQ
Q32.Let $X = R \times R$ . Define a relation R on X as : $(a_{1}, b_{1}) R (a_{2}, b_{2}) \Leftrightarrow b_{1} = b_{2}$ Statement I: $R$ is an equivalence relation. Statement II : For some $(a, b) \in X$ …
Mathematics · Sets and Relations · MCQ
Q33.If the area of the region ${(x, y) : - 1 \leq x \leq 1, 0 \leq y \leq a + e^{∣ x ∣} - e^{- x}, a > 0}$ is $\frac{e ^{2} + 8 e + 1}{e}$ , then the value of $a$ is :
Mathematics · Area Under the Curves · MCQ
Q34.Let the range of the function $f (x) = 6 + 16 cos x \cdot cos (\frac{π}{3} - x) \cdot cos (\frac{π}{3} + x) \cdot sin 3 x \cdot cos 6 x, x \in R$ be $[α, β]$ . Then the distance of the point $(α, β)$ …
Mathematics · Trigonometric Ratio and Identites · MCQ
Q35.The system of equations $x + y + z = 6, x + 2 y + 5 z = 9, x + 5 y + λ z = μ,$ has no solution if
Mathematics · Matrices and Determinants · MCQ
Q36.The distance of the line $\frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ from the point $(1, 4, 0)$ along the line $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}$ is :
Mathematics · 3D Geometry · MCQ
Q37. $x \to \infty lim \frac{( 2 x ^{2} - 3 x + 5 ) ( 3 x - 1 ) ^{\frac{x}{2}}}{( 3 x ^{2} + 5 x + 4 ) ( 3 x + 2 ) ^{x}}$ is equal to :
Mathematics · Limits Continuity and Differentiability · MCQ
Q38.Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A 010 = 001, A 413 = 010$ …
Mathematics · Matrices and Determinants · MCQ
Q39.Let $A = {(x, y) \in R \times R : ∣ x + y ∣ ⩾ 3}$ and $B = {(x, y) \in R \times R : ∣ x ∣ + ∣ y ∣ \leq 3}$ . If $C = {(x, y) \in A \cap B : x = 0$ or $y = 0}$ , then $\sum_{(x, y) \in C} ∣ x + y ∣$ …
Mathematics · Sets and Relations · MCQ
Q40.If in the expansion of $(1 + x)^{p} (1 - x)^{q}$ , the coefficients of $x$ and $x^{2}$ are 1 and -2 , respectively, then $p^{2} + q^{2}$ is equal to :
Mathematics · Binomial Theorem · MCQ
Q41.A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ , respectively. If the locus of the point $P$ , that divides the rod $A B$ internally in the ratio $2 : 1$ is $9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76 = 0$ …
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q42.Let $\int x^{3} sin x d x = g (x) + C$ , where $C$ is the constant of integration. If $8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ, α, β, γ \in Z$ , then $α + β - γ$ …
Mathematics · Indefinite Integrals · MCQ
Q43.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 $81 cm^{3} / min$ and the thickness of the…
Mathematics · Application of Derivatives · MCQ
Q44.Let the shortest distance from $(a, 0), a > 0$ , to the parabola $y^{2} = 4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :
Mathematics · Parabola · MCQ
Q45.Let $x = x (y)$ be the solution of the differential equation $y = (x - y \frac{d x}{d y}) sin (\frac{x}{y}), y > 0$ and $x (1) = \frac{π}{2}$ . Then $cos (x (2))$ is equal to :
Mathematics · Differential Equations · MCQ
Q46.Let $x^{2} - ax - b = 0$ be the roots of the equation $x^{2} - ax - b = 0$ with $Im (α) < Im (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 57 i, P_{4} = - 37 i, P_{5} = 117 i$ …
Mathematics · Complex Numbers · Numerical
Q47.The focus of the parabola $y^{2} = 4 x + 16$ is the centre of the circle $C$ of radius 5 . If the values of $λ$ , for which C passes through the point of intersection of the lines $3 x - y = 0$ and $x + λ y = 4$ , are $λ_{1}$ and $λ_{2}, λ_{1} < λ_{2}$ …
Mathematics · Parabola · Numerical
Q48.The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is .
Mathematics · Permutations and Combinations · Numerical
Q49.The roots of the quadratic equation $3 x^{2} - p x + q = 0$ are $1 0^{th}$ and $1 1^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$ . If the sum of the first 11 terms of this arithmetic progression is…
Mathematics · Sequences and Series · Numerical
Q50.The variance of the numbers $8, 21, 34, 47, \dots, 320$ is .
Mathematics · Statistics · Numerical