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Day 443: JEE Mains Maths

5 questions. Try each one first, then open its answer.

  1. Question 1

    Determinants. Asked once: 23 Jan 2026 S1

    61
    Among the statements : Statements (I) : If ∣1cos⁡αcos⁡βcos⁡α1cos⁡γcos⁡βcos⁡γ1∣=∣0cos⁡αcos⁡βcos⁡α0cos⁡γcos⁡βcos⁡γ0∣\left| \begin{matrix} 1 & \cos\alpha & \cos\beta \\ \cos\alpha & 1 & \cos\gamma \\ \cos\beta & \cos\gamma & 1 \end{matrix} \right| = \left| \begin{matrix} 0 & \cos\alpha & \cos\beta \\ \cos\alpha & 0 & \cos\gamma \\ \cos\beta & \cos\gamma & 0 \end{matrix} \right| , then cos⁡2α+cos⁡2β+cos⁡2γ=32\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = \frac{3}{2}, and Statements (II) : If ∣x2+xx+1x−22x2+3x−13x3x−3x2+2x+32x−12x−1∣\left| \begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2x^{2} + 3x - 1 & 3x & 3x - 3 \\ x^{2} + 2x + 3 & 2x - 1 & 2x - 1 \end{matrix} \right| =px+q= px + q, then p2=196q2p^{2} = 196q^{2},

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  2. Question 2

    Differential Equations. Asked once: 2 Apr 2025

    72
    Let f:R→Rf:R \rightarrow R be a thrice differentiable odd function satisfying f′(x)≥0,f′(x)=f(x),f(0)=0,f′(0)=3f'(x) \geq 0,f'(x) = f(x),f(0) = 0,f'(0) = 3. Then 9f(log⁡e3)9f\left( \log_{e}3 \right) is equal to .

  3. Question 3

    Differentiation. Asked once: 4 Apr 2026 S1

    67
    Let f be a real polynomial of degree n such that f(x)=f′(x)f′′(x)f(x) =f^{'}(x)f^{''}(x), for all x∈Rx \in\mathbb{R}. If f(0)=0f(0) = 0, then 36 (f′(2)+f′′(2)+∫02f(x)dx)\left( f^{'}(2) +f^{''}(2) +\int_{0}^{2} f(x)dx \right) is equal to :

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  4. Question 4

    Indefinite Integration. Asked once: 24 Jan 2026 S1

    61
    Let f(t)=∫(1−sin⁡(log⁡et)1−cos⁡(log⁡et))dt,t>1f(t) = \int\left( \frac{1 - \sin\left( \log_{e}t \right)}{1 - \cos\left( \log_{e}t \right)} \right)dt,t > 1. If f(eπ/2)=−eπ/2f\left( e^{\pi/2} \right)= -e^{\pi/2} and f(eπ/4)=αeπ/4f\left( e^{\pi/4} \right)= \alpha e^{\pi/4}, then α\alpha equals

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  5. Question 5

    Inverse Trigonometric Functions. Asked once: 2 Apr 2025

    148
    If y=cos⁡(π3+cos⁡−1x2)y = \cos\left( \frac{\pi}{3} + \cos^{- 1}\frac{x}{2} \right), then (x−y)2+3y2(x - y)^{2} + 3y^{2} is equal to