JEE-MAIN 2025 Mathematics – Key Facts

FILL_IN_THE_BLANK flashcards covering formulas, results and key numeric answers from the JEE-MAIN 2025 Mathematics papers.

If y = y(x) satisfies (dy/dx)+2y sec²x = 2 sec²x + 3 tan x sec²x with y(0)=5/4, then y(π/4) equals .

21

The sum of the first ten terms of the series 4.1+4.2+4.3+… is (m⁄n) with gcd(m,n)=1; m + n equals .

441

For y = cos(π/3 – x/2), the expression (x – y)² + 3y² equals .

3

In ΔABC with A(4,–2), B(1,1), C(9,–3), the maximum area of the parallelogram AFDE (D,E,F on BC,CA,AB) is square units.

3

If the positive roots of (1–a)x²+2(a–3)x+9=0 exist for a∈(–∞,–α]∪[β,γ), then 2α + β + γ = .

7

The real number α satisfying ⌊∫₀¹ eˣ/(eˣ–1) dx⌋ = α has α³ = .

8

For the odd differentiable function f with f'(x) ≥ 0, f''(x) = f(x), f(0)=0, f'(0)=3, the value 9f(logₑ3) equals .

36

If the area of the region {(x,y):4≤x²+y²≤x,y≥0} is πβ, then α+β = .

22

The absolute difference between the squares of the radii of the two circles through (–9,4) tangent to x+y=3 and x–y=3 is .

768

For the 3×3 matrix A with |A| = –1, letting B = (adj A adj A²)⁻¹, the value |λB + I| (for λ=1) equals .

38

In the expansion of (1 + x + x²)¹⁰, (a₁+a₃+…+a₁₉) – 11a₂ = 121k; here k = .

239

If limₓ→0 (tan x)/(x)^{1/2} = p, then 96 logₑ p = .

32

With vectors a = i +2j –k, b = 3i –3j +3k and c = 2i –j +2k, the quantity |(a·d)×d| where d×b = c×a and a·c = 3 equals .

128

For the hyperbola with foci (4,2),(8,2), 3x² – y² – αx + βy + γ = 0, we have α + β + γ = .

141

Choosing a random 5-letter word from all permutations of A,B,C,D,E with probabilities doubling successively, P(CDBEA)=β/α; here α+β = .

183

For the hyperbola (x²/a²) – (y²/b²) = 1 whose focal-distance product at some point is 32, if its conjugate-axis length is p and latus-rectum length is q, then p² + q² = .

120

Unit vectors â , b̂ , ĉ satisfy (â × ĉ)·(b̂ × ĉ)=1 and (λâ+μb̂)⊥ĉ; if |λ|+|μ| minimal, then λ²+μ² = .

5

The count of 7-digit numbers with even digit-sum equals m·n·10ⁿ (m,n∈{1,…,9}); m + n = .

14

Area bounded by y = max{x, x³, x⁵,…,x²¹} is continuous except at m points and differentiable except at n points; m + n = .

3

For matrix A with A²(A–2I) – 4(A–I) = 0 and A⁵ = αA²+βA+γI, we have α + β + γ = .

12

Shortest distance between lines through (7,6,2) ∥(3,–2,4) and (5,3,4) ∥(2,1,3) equals .

(√38)/|23| (value from question: option 1 gave √38/|23|) – accept √38/|23|

Ellipse whose minor-axis length is one-fourth of focal distance has eccentricity .

4/√17

ODE (x²+1) y' – 2xy = (x⁴+2x²+1) cos x with y(0)=1 gives ∫₀³ y(x) dx = .

24

For ∫₀^π (x+3)sin x /(1+3cos x) dx, the value equals .

(6π/3)+? — actual answer option (3): (6π+π)/3? Provided answer 3 → (π+6)/3? We leave as (π+6)/3

Number of distinct pairs (m,n) of 2-digit numbers with gcd(m,n)=6 is .

64

Product of last two digits of 1919^1919 equals .

63

Circle touching y-axis at (0,α) and having PS as diameter for focal chord at 60° of y²=4x gives 5α² = .

15

If ΔABC has sides along 4x–7y+10=0, x+y=5, 7x+4y=15, then distance between its orthocentre and that of triangle formed by x=0,y=0,x+y=1 is .

√20

Domain of f(x)=log₇(1–log₄(x²–9x+18)) is (α,β)∪(γ,δ); α+β+γ+δ = .

18

Area of region bounded by |x–5| ≤ y ≤ √(4·x) equals A; 3A = .

368