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Day 20: JEE Mains Maths

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

  1. Question 1

    Differential Equations. This type asked 6 times: 2021 Paper 18, 26 Jul 2022, 29 Jul 2022, 25 Jan 2023, 1 Feb 2024, 9 Apr 2024. This one: 9 Apr 2024 See all 6 of this type

    173
    For a differentiable function f:IR→IRf:IR \rightarrow IR, suppose f′(x)=3f(x)+αf^{'}(x) = 3f(x) + \alpha, where α∈IR,f(0)=1\alpha \in IR,f(0) = 1 and lim⁡x→−∞f(x)=7\lim_{x \rightarrow - \infty} f(x) = 7. Then 9f(−log⁡e3)9f\left( -\log_{e}3 \right) is equal to

  2. Question 2

    Vector Algebra. This type asked 6 times: 1 Feb 2023, 8 Apr 2023, 30 Jan 2024, 5 Apr 2024, 6 Apr 2024, 9 Apr 2024. This one: 9 Apr 2024 See all 6 of this type

    170
    Let a→=2i^+αj^+k^, b→=−i^+k^, c→=βj^−k^\overrightarrow{a}= 2\widehat{i}+ \alpha\widehat{j}+\widehat{k},\ \overrightarrow{b}= -\widehat{i}+\widehat{k},\ \overrightarrow{c}= \beta\widehat{j}-\widehat{k}, where α\alpha and β\beta are integers and αβ=−6\alpha\beta = - 6. Let the values of the ordered pair (α,β)(\alpha,\beta) for which the area of the parallelogram of diagonals a→+b→\overrightarrow{a}+\overrightarrow{b} and b→+c→\overrightarrow{b}+\overrightarrow{c} is 212\frac{\sqrt{21}}{2}, be (α1,β1)\left( \alpha_{1},\beta_{1} \right) and (α2,β2)\left( \alpha_{2},\beta_{2} \right). Then α12+β12−α2β2\alpha_{1}^{2}+\beta_{1}^{2}-\alpha_{2}\beta_{2} is equal to

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  3. Question 3

    Limits and Continuity. This type asked 6 times: 2021 Paper 1, 2021 Paper 24, 30 Jan 2023, 29 Jan 2024, 30 Jan 2024, 4 Apr 2024. This one: 4 Apr 2024 See all 6 of this type

    158
    Let f(x)=∫0x(t+sin⁡(1−et))dt,x∈Rf(x) =\int_{0}^{x} \left( t + \sin\left( 1 -e^{t} \right) \right)dt,x \in R. Then lim⁡x→0f(x)x3\lim_{x \rightarrow 0} \frac{f(x)}{x^{3}} is equal to

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  4. Question 4

    Application of Derivatives. This type asked 6 times: 2021 Paper 1, 2021 Paper 2, 2021 Paper 6, 28 Jul 2022, 29 Jan 2024, 1 Feb 2024. This one: 1 Feb 2024 See all 6 of this type

    79
    If 5f(x)+4f(1x)=x2−2,∀x≠05f(x) + 4f\left( \frac{1}{x} \right)=x^{2}- 2,\forall x \neq 0 and y=9x2f(x)y = 9x^{2}f(x), then yy is strictly increasing in:

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  5. Question 5

    Vector Algebra. This type asked 6 times: 2021 Paper 8, 28 Jul 2022, 29 Jan 2023, 8 Apr 2023, 12 Apr 2023, 15 Apr 2023. This one: 15 Apr 2023 See all 6 of this type

    79
    Let SS be the set of all (λ,μ)(\lambda,\mu) for which the vectors λi^−j^+k^,i^+2j^+μk^\lambda\widehat{i}-\widehat{j}+\widehat{k},\widehat{i}+ 2\widehat{j}+ \mu\widehat{k} and 3i^−4j^+5k^3\widehat{i}- 4\widehat{j}+ 5\widehat{k}, where λ−μ=5\lambda - \mu = 5, are coplanar, then ∑(λ,μ)∈S80(λ2+μ2)\sum_{(\lambda,\mu) \in S} 80\left( \lambda^{2}+\mu^{2} \right) is equal to :

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