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[5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer…
[5×1] In sub-parts (i) and (ii) choose the correct options and in sub-parts (iii) to (v), answer the questions as instructed.
(i)
If the vectors $a\hat{i} + 3\hat{j} - 2\hat{k}$ and $3\hat{i} - 4\hat{j} + b\hat{k}$ are collinear, then $(a, b) =$
- (a)$\left(\frac{9}{4}, \frac{8}{3}\right)$
- (b)$\left(-\frac{9}{4}, \frac{8}{3}\right)$
- (c)$\left(\frac{9}{4}, -\frac{8}{3}\right)$
- (d)$\left(-\frac{9}{4}, -\frac{8}{3}\right)$
(ii)
The intercepts made by the plane $3x - 2y + 4z = 12$ on the coordinate axes are:
- (a)$6, -4, 3$
- (b)$4, -6, 3$
- (c)$2, -3, 4$
- (d)$\frac{1}{4}, -\frac{1}{6}, \frac{1}{3}$
(iii)
Find the angle between the vectors $\vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} - 9\hat{j} + 6\hat{k}$.
(iv)
Find the unit vector parallel to the vector: $3\hat{i} + 6\hat{j} - 2\hat{k}$
(v)
Find the equation of the plane passing through $(-2, 1, 3)$ and perpendicular to the line having direction ratios $\langle 3, 1, 5 \rangle$.
Answer
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From ISC 2021 Specimen Mathematics Paper 1, question 15.