Exam Questions & Step-by-Step Markschemes (312 Problems)
Question 1 • 2022
2 Marks
In a parallelogram PQRS, \overrightarrow{PQ} = 3\hat{i} - 2\hat{j} + 2\hat{k} and \overrightarrow{PS} = -\hat{i} - 2\hat{k} . Find |\overrightarrow{PR}| and |\overrightarrow{QS}| .
Answer: |\overrightarrow{PR}| = 2\sqrt{2} , |\overrightarrow{QS}| = 6
Question 2 • 2022
2 Marks
Find the values of \lambda , for which the distance of point (2, 1, \lambda) from plane 3x + 5y + 4z = 11 is 2\sqrt{2} units.
Answer: \lambda = \pm 5
Question 3 • 2022
3 Marks
If \vec{a} , \vec{b} , \vec{c} and \vec{d} are four non-zero vectors such that \vec{a} \times \vec{b} = \vec{c} \times \vec{d} and \vec{a} \times \vec{c} = 4\vec{b} \times \vec{d} , then show that (\vec{a} - 2\vec{d}) is parallel to (2\vec{b} - \vec{c}) where \vec{a} \neq 2\vec{d} , \vec{c} \neq 2\vec{b} .
Answer: Proved.
Question 4 • 2022
3 Marks
The two adjacent sides of a parallelogram are represented by 2\hat{i} - 4\hat{j} - 5\hat{k} and 2\hat{i} + 2\hat{j} + 3\hat{k} . Find the unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram also.
Answer: Unit vectors \pm\frac{1}{\sqrt{6}}(2\hat{i} - \hat{j} - \hat{k}) and \pm\frac{1}{5}(3\hat{j} + 4\hat{k}) ; area = 2\sqrt{101} sq units
Question 5 • 2022
3 Marks
Find the vector equation of the plane passing through the intersection of the planes \vec{r} \cdot (2\hat{i} + 2\hat{j} - 3\hat{k}) = 7 and \vec{r} \cdot (2\hat{i} + 5\hat{j} + 3\hat{k}) = 9 and through the point (2, 1, 3).
Answer: \vec{r} \cdot (38\hat{i} + 68\hat{j} + 3\hat{k}) = 153
Question 6 • 2022
4 Marks
Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such wires lie along the following lines : l_1 : \frac{x + 1}{3} = \frac{y - 3}{-2} = \frac{z + 2}{-1} l_2 : \frac{x}{-1} = \frac{y - 7}{3} = \frac{z + 7}{-2} Based on the given information, answer the following questions : (i) Are the lines l_1 and l_2 coplanar ? Justify your answer. (2) (ii) Find the point of intersection of the lines l_1 and l_2 . (2)
Answer: (i) Yes, the lines are coplanar. (ii) (2, 1, -3)