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Mathematics · 2025 · 7 Apr · Shift 2

25 questions from this JEE Main paper

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2025 · 7 Apr · Shift 2
Mathematics

Mathematics · 2025 · 7 Apr · Shift 2

25 questions from this JEE Main paper

Application of Derivatives1 questions Differential Equations1 questions Complex Numbers1 questions Sequences and Series2 questions Probability2 questions Functions1 questions Quadratic Equation and Inequalities1 questions Ellipse2 questions Hyperbola2 questions 3D Geometry2 questions Area Under the Curves1 questions Trigonometric Functions and Equations1 questions Sets and Relations1 questions Vector Algebra1 questions Matrices and Determinants1 questions Straight Lines and Pair of Straight Lines1 questions Binomial Theorem1 questions Limits Continuity and Differentiability2 questions Indefinite Integrals1 questions

Q26.Let f : ℝ $\to$ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $x \to 0 lim \frac{f ( x )}{x ^{2}} = 5$ , then f(2) is equal to :
Mathematics · Application of Derivatives · MCQ
Q27.Let y = y(x) be the solution of the differential equation $(x^{2} + 1) y^{'} - 2 x y = (x^{4} + 2 x^{2} + 1) cos x$ , $y (0) = 1$ . Then $- 3 \int 3 y (x) d x$ is :
Mathematics · Differential Equations · MCQ
Q28.If the locus of z ∈ ℂ, such that Re $(\frac{z - 1}{2 z + i}) + Re (\frac{z - 1}{2 z - i}) = 2$ , is a circle of radius r and center $(a, b)$ , then $\frac{15 ab}{r ^{2}}$ is equal to :
Mathematics · Complex Numbers · MCQ
Q29.Let $a_{n}$ be the $n^{t h}$ term of an A.P. If $S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = 700$ , $a_{6} = 7$ and $S_{7} = 7$ , then $a_{n}$ is equal to :
Mathematics · Sequences and Series · MCQ
Q30.A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $n^{2} - m^{2}$ is equal…
Mathematics · Probability · MCQ
Q31.If the range of the function $f (x) = \frac{5 - x}{x ^{2} - 3 x + 2}, x e q 1, 2,$ is $(- \infty, α] \cup [β, \infty)$ , then $α^{2} + β^{2}$ is equal to :
Mathematics · Functions · MCQ
Q32.The number of real roots of the equation $x ∣ x - 2∣ + 3∣ x - 3∣ + 1 = 0$ is :
Mathematics · Quadratic Equation and Inequalities · MCQ
Q33.Let the length of a latus rectum of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ be 10. If its eccentricity is the minimum value of the function $f (t) = t^{2} + t + \frac{11}{12}$ , $t \in R$ , then $a^{2} + b^{2}$ is equal to :
Mathematics · Ellipse · MCQ
Q34.Let e1 and e2 be the eccentricities of the ellipse $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{25} = 1$ and the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1$ , respectively. If b < 5 and e1e2 = 1, then the eccentricity of the ellipse having its…
Mathematics · Hyperbola · MCQ
Q35.If the equation of the line passing through the point $(0, - \frac{1}{2}, 0)$ and perpendicular to the lines $r = λ (\hat{i} + a \hat{j} + b \hat{k})$ and $r = (\hat{i} - \hat{j} - 6 \hat{k}) + μ (- b \hat{i} + a \hat{j} + 5 \hat{k})$ …
Mathematics · 3D Geometry · MCQ
Q36.If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :
Mathematics · Sequences and Series · MCQ
Q37.If the area of the region ${(x, y) : 1 + x^{2} \leq y \leq min {x + 7, 11 - 3 x}}$ is $A$ , then $3 A$ is equal to :
Mathematics · Area Under the Curves · MCQ
Q38.The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :
Mathematics · Trigonometric Functions and Equations · MCQ
Q39.Consider the lines L1: x - 1 = y - 2 = z and L2: x - 2 = y = z - 1. Let the feet of the perpendiculars from the point P(5, 1, -3) on the lines L1 and L2 be Q and R respectively. If the area of the triangle PQR is A, then 4A2 is equal to :
Mathematics · 3D Geometry · MCQ
Q40.Let A = { ( $α, β$ ) $\in R \times R$ : | $α$ - 1| $\leq 4$ and | $β$ - 5| $\leq 6$ } and B = { ( $α, β$ ) $\in R \times R$ : 16( $α$ - $2)^{2}$ + 9( $β$ - $6)^{2} \leq 144$ }. Then
Mathematics · Sets and Relations · MCQ
Q41.Let $a$ and $b$ be the vectors of the same magnitude such that $\frac{∣ a + b ∣ + ∣ a - b ∣}{∣ a + b ∣ - ∣ a - b ∣} = 2 + 1.$ Then $\frac{∣ a + b ∣ ^{2}}{∣ a ∣ ^{2}}$ is :
Mathematics · Vector Algebra · MCQ
Q42.Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the…
Mathematics · Ellipse · MCQ
Q43.Let a random variable X take values 0, 1, 2, 3 with P(X=0)=P(X=1)=p, P(X=2)=P(X=3) and E(X2)=2E(X). Then the value of 8p−1 is :
Mathematics · Probability · MCQ
Q44.Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + λz = μ λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :
Mathematics · Matrices and Determinants · MCQ
Q45.If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :
Mathematics · Straight Lines and Pair of Straight Lines · MCQ
Q46.The sum of the series $2 \times 1 \times^{20} C_{4} - 3 \times 2 \times^{20} C_{5} + 4 \times 3 \times^{20} C_{6} - 5 \times 4 \times^{20} C_{7} + \dots\dots + 18 \times 17 \times^{20} C_{20}$ ,…
Mathematics · Binomial Theorem · Numerical
Q47.If the function $f (x) = \frac{t a n ( t a n x ) - s i n ( s i n x )}{t a n x - s i n x}$ is continuous at $x = 0$ , then $f (0)$ is equal to .
Mathematics · Limits Continuity and Differentiability · Numerical
Q48.If $\int (\frac{1}{x} + \frac{1}{x ^{3}}) (23 3 x^{- 24} + x^{- 26}) d x = - \frac{α}{3 ( α + 1 )} (3 x^{β} + x^{γ})^{\frac{α + 1}{α}} + C, x > 0, (α, β, γ \in Z)$ ,…
Mathematics · Indefinite Integrals · Numerical
Q49.Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2 a$ and $2 b$ , respectively, and one focus and the corresponding directrix of this hyperbola be $(- 5, 0)$ and $5 x + 9 = 0$ , respectively. If the product…
Mathematics · Hyperbola · Numerical
Q50.For $t > - 1$ , let $α_{t}$ and $β_{t}$ be the roots of the equation $((t + 2)^{1/7} - 1) x^{2} + ((t + 2)^{1/6} - 1) x + ((t + 2)^{1/21} - 1) = 0 . If t \to - 1^{+} lim α_{t} = a and t \to - 1^{+} lim β_{t} = b,$ …
Mathematics · Limits Continuity and Differentiability · Numerical