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Day 521: JEE Mains Maths

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

  1. Question 1

    Three Dimensional Geometry. Asked once: 30 Jan 2023

    82
    If λ1<λ2\lambda_{1}<\lambda_{2} are two values of λ\lambda such that the angle between the planes P1:r→⋅(3ı^−5ȷ^+k^)=7P_{1}:\overrightarrow{r}\cdot (3\widehat{\imath}- 5\widehat{\jmath}+\widehat{k}) = 7 and P2:r→⋅(λı^+ȷ^−3k^)=9P_{2}:\overrightarrow{r}\cdot (\lambda\widehat{\imath}+\widehat{\jmath}- 3\widehat{k}) = 9 is sin⁡−1(265)\sin^{- 1}\left( \frac{2\sqrt{6}}{5} \right), then the square of the length of perpendicular from the point (38λ1,10λ2,2)\left( 38\lambda_{1},10\lambda_{2},2 \right) to the plane P1P_{1} is

  2. Question 2

    Trigonometric Identities. Asked once: 2021 Paper 19

    77
    A 10 inches long pencil ABAB with mid point CC and a small eraser PP are placed on the horizontal top of a table such that PC=5PC =\sqrt{5} inches and ∠PCB=tan⁡−1(2)\angle PCB =\tan^{- 1}(2). The acute angle through which the pencil must be rotated about CC so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is:

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  3. Question 3

    Vector Algebra. Asked once: 12 Apr 2023

    71
    Let λ∈Z,a=λi^+j^−k^\lambda \in Z,a = \lambda\widehat{i}+\widehat{j}-\widehat{k} and b=3i^−j^+2k^b = 3\widehat{i}-\widehat{j}+ 2\widehat{k}. Let c→\overrightarrow{c} be a vector such that (a→+b→+c→)×c→=0→,a→⋅c→=−17(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}) \times\overrightarrow{c}=\overrightarrow{0},\overrightarrow{a}\cdot\overrightarrow{c}= - 17 and  b→⋅c→=−20\ \overrightarrow{b}\cdot\overrightarrow{c}= - 20. Then ∣c→×(λi^+j^+k^)∣2|\overrightarrow{c}\times (\lambda\widehat{i}+\widehat{j}+\widehat{k})|^{2} is equal to

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  4. Question 4

    Application of Derivatives. Asked once: 27 Jan 2024

    82
    Let for a differentiable function f:(0,∞)→Rf:(0,\infty) \rightarrow R, f(x)−f(y)≥log⁡e(xy)+x−y,∀x,y∈(0,∞)f(x) - f(y) \geq\log_{e}\left( \frac{x}{y} \right)+ x - y,\forall x,y \in (0,\infty). Then ∑n=120f′(1n2)\sum_{n = 1}^{20} f^{'}\left( \frac{1}{n^{2}} \right) is equal to.

  5. Question 5

    Application of Integrals. Asked once: 6 Apr 2024

    79
    Let the area of the region enclosed by the curves y=3x,2y=27−3xy = 3x,2y = 27 - 3x and y=3x−xxy = 3x - x\sqrt{x} be A. Then 10 A10\text{ }A is equal to

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