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Day 554: JEE Mains Maths

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

  1. Question 1

    Quadratic Equations. Asked once: 25 Jan 2023

    88
    Let S={α:log⁡2(92α−4+13)−log⁡2(52⋅32α−4+1)=2}S =\left\{ \alpha:\log_{2}\left( 9^{2\alpha - 4}+ 13 \right)-\log_{2}\left( \frac{5}{2}\cdot3^{2\alpha - 4}+ 1 \right)= 2 \right\}. Then the maximum value of β\beta for which the equation x2−2(∑α∈Sα)2x+∑α∈S(α+1)2β=0x^{2}- 2\left( \sum_{\alpha \in S} \alpha \right)^{2}x +\sum_{\alpha \in S} (\alpha + 1)^{2}\beta = 0 has real roots, is

  2. Question 2

    Relations and Functions. Asked once: 25 Jan 2023

    155
    Let f(x)=2xn+λ,λ∈R,n∈Nf(x) = 2x^{n}+ \lambda,\lambda \in R,n \in N, and f(4)=133,f(5)=255f(4) = 133,f(5) = 255. Then the sum of all the positive integer divisors of (f(3)−f(2))(f(3) - f(2)) is

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  3. Question 3

    Sequences and Series. Asked once: 6 Apr 2024

    87
    Let the first term of a series be T1=6T_{1}= 6 and its rth r^{\text{th~}} term Tr=3Tr−1+6r,r=2,3,….,nT_{r}= 3T_{r- 1}+6^{r},r= 2,3,\ldots.,n. If the sum of the first nn terms of this series is 15(n2−12n+39)\frac{1}{5}\left( n^{2}- 12n+ 39 \right) (4.6n−5.3n+1)\left( 4.6^{n}-5.3^{n}+ 1 \right). Then nn is equal to

  4. Question 4

    Straight Lines. Asked once: 11 Apr 2023

    63
    Let RR be a rectangle given by the lines x=0,x=2x = 0,x = 2, y=0y = 0 and y=5y = 5. Let A(α,0)A(\alpha,0) and B(0,β),α∈[0,2]B(0,\beta),\alpha \in \lbrack 0,2\rbrack and β∈[0,5]\beta \in \lbrack 0,5\rbrack, be such that the line segment ABAB divides the area of the rectangle RR in the ratio 4:1. Then, the mid-point of ABAB lies on a

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  5. Question 5

    Three Dimensional Geometry. Asked once: 25 Jun 2022

    159
    Let PP be the plane passing through the intersection of the planes r→⋅(i^+3j^−k^)=5\overrightarrow{r}\cdot (\widehat{i}+ 3\widehat{j}-\widehat{k}) = 5 and r→⋅(2i^−j^+k^)=3\overrightarrow{r}\cdot (2\widehat{i}-\widehat{j}+\widehat{k}) = 3, and the point (2,1,−2)(2,1, - 2). Let the position vectors of the points XX and YY be i^−2j^+4k^\widehat{i}- 2\widehat{j}+ 4\widehat{k} and 5i^−j^+2k^5\widehat{i}-\widehat{j}+ 2\widehat{k} respectively. Then the points

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