Direction ratios and direction cosines — practice questions

Question 1 · difficulty L2 · understanding

Let $$\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$$ and $$\mathrm{R}(7,3,2)$$ be the vertices of $$\triangle \mathrm{PQR}$$. Then, the angle $$\angle \mathrm{QPR}$$ is

• A. $$\cos ^{-1}\left(\frac{7}{18}\right)$$
• B. $$\frac{\pi}{6}$$
• C. $$\cos ^{-1}\left(\frac{1}{18}\right)$$
• D. $$\frac{\pi}{3}$$

Question 2 · difficulty L3 · understanding

Let $$\mathrm{P}(-2,-1,1)$$ and $$\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $$\alpha,-1, \beta$$, where both $$\alpha$$ and $$\beta$$ are integers of minimum absolute values, then $$\alpha^{2}+\beta^{2}$$ is equal to ____________.

Question 3 · difficulty L3 · understanding

Let the direction cosines of two lines satisfy the equations : $4 l+m-n=0$ and $2 m n+10 n l+3 l m=0$. Then the cosine of the acute angle between these lines is :

• A. $\frac{10}{7 \sqrt{38}}$
• B. $\frac{10}{\sqrt{38}}$
• C. $\frac{10}{3 \sqrt{38}}$
• D. $\frac{20}{3 \sqrt{38}}$

Question 4 · difficulty L3 · understanding

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$-axes, respectively, is half of the angle that this line makes with the positive $x$-axes. Then the sum of all possible values of the angle $\beta$ is

• A. $\frac{\pi}{2}$
• B. $\pi$
• C. $\frac{3 \pi}{4}$
• D. $\frac{3 \pi}{2}$

Question 5 · difficulty L3 · understanding

Let $$P(x, y, z)$$ be a point in the first octant, whose projection in the $$x y$$-plane is the point $$Q$$. Let $$O P=\gamma$$; the angle between $$O Q$$ and the positive $$x$$-axis be $$\theta$$; and the angle between $$O P$$ and the positive $$z$$-axis be $$\phi$$, where $$O$$ is the origin. Then the distance of $$P$$ from the $$x$$-axis is

• A. $$\gamma \sqrt{1-\sin ^2 \phi \cos ^2 \theta}$$
• B. $$\gamma \sqrt{1+\cos ^2 \theta \sin ^2 \phi}$$
• C. $$\gamma \sqrt{1+\cos ^2 \phi \sin ^2 \theta}$$
• D. $$\gamma \sqrt{1-\sin ^2 \theta \cos ^2 \phi}$$

Question 6 · difficulty L3 · understanding

If the line $$\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$$ makes a right angle with the line $$\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$$, then $$4 \lambda+9 \mu$$ is equal to :

• A. 4
• B. 13
• C. 5
• D. 6

Question 7 · difficulty L3 · understanding

Consider a $\triangle A B C$ where $A(1,3,2), B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle B A C$ meets the line $B C$ at $D$, then the length of the projection of the vector $\overrightarrow{A D}$ on the vector $\overrightarrow{A C}$ is :

• A. $\frac{37}{2 \sqrt{38}}$
• B. $\sqrt{19}$
• C. $\frac{39}{2 \sqrt{38}}$
• D. $\frac{\sqrt{38}}{2}$

Question 8 · difficulty L3 · understanding

Let $$O$$ be the origin and the position vectors of $$A$$ and $$B$$ be $$2 \hat{i}+2 \hat{j}+\hat{k}$$ and $$2 \hat{i}+4 \hat{j}+4 \hat{k}$$ respectively. If the internal bisector of $$\angle \mathrm{AOB}$$ meets the line $$\mathrm{AB}$$ at $$\mathrm{C}$$, then the length of $$O C$$ is

• A. $$\frac{3}{2} \sqrt{34}$$
• B. $$\frac{2}{3} \sqrt{31}$$
• C. $$\frac{2}{3} \sqrt{34}$$
• D. $$\frac{3}{2} \sqrt{31}$$

Question 9 · difficulty L3 · understanding

If two straight lines whose direction cosines are given by the relations $$l + m - n = 0$$, $$3{l^2} + {m^2} + cnl = 0$$ are parallel, then the positive value of c is :

• A. 6
• B. 4
• C. 3
• D. 2

Question 10 · difficulty L3 · understanding

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :

• A. $${\pi \over 2}$$
• B. $$\pi - {\cos ^{ - 1}}\left( {{4 \over 9}} \right)$$
• C. $${\cos ^{ - 1}}\left( {{8 \over 9}} \right)$$
• D. $${\pi \over 3}$$