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Day 108: JEE Mains Maths

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

  1. Question 1

    Matrices. This type asked 2 times: 13 Apr 2023, 28 Jan 2025. This one: 28 Jan 2025 See all 2 of this type

    71
    Let M denote the set of all real matrices of order 3×33 \times 3 and let S={−3,−2,−1,1,2}\mathbf{S = \{ - 3, - 2, - 1,1,2\}}. Let S1={A=[aij]∈M:A=ATS_{1}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:A =A^{T} \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} S2={A=[aij]∈M:A=−ATS_{2}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:A = -A^{T} \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} S3={A=[aij]∈M:a11+a22+a33=0S_{3}=\left\{ A =\left\lbrack a_{ij} \right\rbrack\in M:a_{11}+a_{22}+a_{33}= 0 \right. and  aij∈S,∀i,j}\left. \ a_{ij}\in S,\forall i,j \right\} If n(S1∪S2∪S3)=125αn\left( S_{1}\cup S_{2}\cup S_{3} \right)= 125\alpha, then α\alpha equals.

  2. Question 2

    Probability. This type asked 2 times: 2021 Paper 11, 28 Jan 2025. This one: 28 Jan 2025 See all 2 of this type

    131
    Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S , one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :

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  3. Question 3

    Limits and Continuity. This type asked 2 times: 25 Jun 2022, 24 Jan 2025. This one: 24 Jan 2025 See all 2 of this type

    57
    lim⁡x→0csc⁡x(2cos⁡2x+3cos⁡x−cos⁡2x+sin⁡x+4)\lim_{x \rightarrow 0} \csc x\left( \sqrt{2\cos^{2}x + 3\cos x}-\sqrt{\cos^{2}x + \sin x + 4} \right) is

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  4. Question 4

    Straight Lines. This type asked 2 times: 27 Jan 2024, 24 Jan 2025. This one: 24 Jan 2025 See all 2 of this type

    133
    Let the points (112,α)\left( \frac{11}{2},\alpha \right) lie on or inside the triangle with sides x+y=11,x+2y=16x + y = 11,x + 2y = 16 and 2x+3y=292x + 3y = 29. Then the product of the smallest and the largest values of α\alpha is equal to :

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  5. Question 5

    Vector Algebra. This type asked 2 times: 10 Apr 2023, 24 Jan 2025. This one: 24 Jan 2025 See all 2 of this type

    131
    Let the position vectors of three vertices of a triangle be 4p→+q→−3r→,−5p→+q→+2r→4\overrightarrow{p}+\overrightarrow{q}- 3\overrightarrow{r}, - 5\overrightarrow{p}+\overrightarrow{q}+ 2\overrightarrow{r} and 2p→−q→+2r→2\overrightarrow{p}-\overrightarrow{q}+ 2\overrightarrow{r}. If the position vectors of the orthocenter and the circumcenter of the triangle are p→+q→+r→4\frac{\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}}{4} and αp→+βq→+γr→\alpha\overrightarrow{p}+\beta\overrightarrow{q}+\gamma\overrightarrow{r} respectively, then α+2β+5γ\alpha+ 2\beta+ 5\gamma is equal to:

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