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Day 543: MHT-CET Maths

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

  1. Question 1

    Trigonometric Functions. Asked once: 10 May 2023 S1

    144
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=π\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=\pi and x2+y2+z2+kxyz=1x^2+y^2+z^2+kxyz=1, then k is

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  2. Question 2

    Vectors. Asked once: 9 May 2023 S2

    111
    The vector a⃗=αi^+2j^+βk^\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k} lies in the plane of the vectors b⃗=i^+j^\vec{b} = \hat{i}+\hat{j} and c⃗=j^+k^\vec{c} = \hat{j}+\hat{k} and bisects the angle between b⃗\vec{b} and c⃗\vec{c}. Then which one of the following gives possible values of α\alpha and β\beta?

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  3. Question 3

    Indefinite Integration. Asked once: 16 May 2023 S1

    147
    The integral ∫2x3−1x4+x dx\int\frac{2x^3-1}{x^4+x}\,dx is equal to

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  4. Question 4

    Trigonometric Functions. Asked once: 9 May 2023 S2

    138
    The smallest positive value of x in degrees satisfying the equation tan⁡(x+100∘)=tan⁡(x+50∘)tan⁡(x)tan⁡(x−50∘)\tan(x+100^\circ) = \tan(x+50^\circ)\tan(x)\tan(x-50^\circ) is

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  5. Question 5

    Vectors. Asked once: 9 May 2023 S1

    130
    Suppose that p⃗\vec{p}, q⃗\vec{q} and r⃗\vec{r} are three non-coplanar vectors in R3\mathbb{R}^3. Let the components of a vector s⃗\vec{s} along p⃗\vec{p}, q⃗\vec{q} and r⃗\vec{r} be 4, 3 and 5 respectively. If the components of this vector s⃗\vec{s} along (−p⃗+q⃗+r⃗)(-\vec{p}+\vec{q}+\vec{r}), (p⃗−q⃗+r⃗)(\vec{p}-\vec{q}+\vec{r}) and (−p⃗−q⃗+r⃗)(-\vec{p}-\vec{q}+\vec{r}) are xx, yy and zz respectively, then the value of 2x+y+z2x+y+z is

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