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Day 487: JEE Mains Maths

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

  1. Question 1

    Inverse Trigonometric Functions. Asked once: 31 Jan 2023

    158
    Let (a,b)⊂(0,2π)(a,b) \subset (0,2\pi) be the largest interval for which sin⁡−1(sin⁡θ)−cos⁡−1(sin⁡θ)>0,θ∈(0,2π)\sin^{- 1}(\sin\theta) -\cos^{- 1}(\sin\theta) > 0,\theta \in (0,2\pi), holds. If αx2+βx+sin⁡−1(x2−6x+10)+cos⁡−1(x2−6x+10)=0\alpha x^{2}+ \beta x +\sin^{- 1}\left( x^{2}- 6x + 10 \right)+\cos^{- 1}\left( x^{2}- 6x + 10 \right)= 0 and α−β=b−a\alpha - \beta = b - a, then α\alpha is equal to :

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  2. Question 2

    Limits and Continuity. Asked once: 6 Apr 2024

    156
     lim⁡n→∞(12−1)(n−1)+(22−2)(n−2)+…..+((n−1)2−(n−1))⋅1(13+23+…..+n3)−(12+22+…..+n2)\ \lim_{n\rightarrow \infty} \frac{\left( 1^{2}- 1 \right)(n- 1) +\left( 2^{2}- 2 \right)(n- 2) + \ldots.. +\left( (n- 1)^{2}- (n- 1) \right)\cdot 1}{\left( 1^{3}+2^{3}+ \ldots.. +n^{3} \right)-\left( 1^{2}+2^{2}+ \ldots.. +n^{2} \right)} is equal to:

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  3. Question 3

    Matrices. Asked once: 23 Jan 2025

    68
    If A,BA,B and (adj(A−1)+adj(B−1))\left( adj\left( A^{- 1} \right)+ adj\left( B^{- 1} \right) \right) are non-singular matrices of same order, then the inverse of A(adj(A−1)+adj(B−1))−1 BA\left( adj\left( A^{- 1} \right)+ adj\left( B^{- 1} \right) \right)^{- 1}\text{ }B, is equal to

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  4. Question 4

    Permutations and Combinations. Asked once: 22 Jan 2025

    61
    From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

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  5. Question 5

    Probability. Asked once: 22 Jan 2025

    69
    Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is mn\frac{m}{n}, where gcd(m,n)=1gcd(m,n) = 1, then m+nm + n is equal to :

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