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Day 214: JEE Mains Maths

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

  1. Question 1

    Determinants. Asked once: 2021 Paper 5 (same type asked 2 times) See all 2 of this type

    86
    Let
    P=[−302056901401121206014] and A=[27ω2−1−ω10−ω−ω+1]P=\begin{bmatrix} - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}\text{~and~}A=\begin{bmatrix} 2 & 7 & \omega^{2} \\ - 1 & -\omega & 1 \\ 0 & -\omega & -\omega+ 1 \end{bmatrix}
    where ω=−1+i32\omega=\frac{- 1 +i\sqrt{3}}{2}, and I3I_{3} be the identity matrix of order 3. If the determinant of the matrix (P−1AP−I3)2\left( P^{- 1}AP -I_{3} \right)^{2} is αω2\alpha\omega^{2}, then the value of α\alpha is equal to

  2. Question 2

    Differentiation. Asked once: 25 Jan 2023 (same type asked 2 times) See all 2 of this type

    76
    Let y(x)=(1+x)(1+x2)(1+x4)(1+x8)(1+x16)y(x) = (1 + x)\left( 1 +x^{2} \right)\left( 1 +x^{4} \right)\left( 1 +x^{8} \right)\left( 1 +x^{16} \right). Then y′−y′′y^{'}-y^{''} at x=−1x = - 1 is equal to :

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  3. Question 3

    Sequences and Series. Asked once: 31 Jan 2023 (same type asked 2 times) See all 2 of this type

    162
    Let a1,a2,a3,…a_{1},a_{2},a_{3},\ldots be an A.P. If a7=3a_{7}= 3, the product a1a4a_{1}a_{4} is minimum and the sum of its first nn terms is zero, then n!−4an(n+2)n! - 4a_{n(n + 2)} is equal to:

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  4. Question 4

    Straight Lines. Asked once: 2021 Paper 18 (same type asked 2 times) See all 2 of this type

    79
    Let ABCABC be a triangle with A(−3,1)A( - 3,1) and ∠ACB=θ,0<θ<π2\angle ACB =\theta,0 <\theta<\frac{\pi}{2}. If the equation of the median through BB is 2x+y−3=02x + y - 3 = 0 and the equation of angle bisector of CC is 7x−4y−1=07x- 4y- 1 = 0, then tan⁡θ\tan\theta is equal to:

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  5. Question 5

    Indefinite Integration. Asked once: 2021 Paper 6 (same type asked 2 times) See all 2 of this type

    84
    For real numbers α,β,γ\alpha,\beta,\gamma and δ\delta, if
    ∫(x2−1)+tan⁡−1(x2+1x)(x4+3x2+1)tan⁡−1(x2+1x)dx=αlog⁡e(tan⁡−1(x2+1x))\begin{gathered}\int\frac{\left( x^{2}- 1 \right)+\tan^{- 1}\left( \frac{x^{2}+ 1}{x} \right)}{\left( x^{4}+ 3x^{2}+ 1 \right)\tan^{- 1}\left( \frac{x^{2}+ 1}{x} \right)}dx \\ =\alpha\log_{e}\left( \tan^{- 1}\left( \frac{x^{2}+ 1}{x} \right) \right)\end{gathered}
    +βtan⁡−1(γ(x2−1)x)+δtan⁡−1(x2+1x)+C+\beta\tan^{- 1}\left( \frac{\gamma\left( x^{2}- 1 \right)}{x} \right)+\delta\tan^{- 1}\left( \frac{x^{2}+ 1}{x} \right)+ C
    where CC is an arbitrary constant, then the value of 10(α+βγ+δ)10(\alpha+\beta\gamma+\delta) is equal to