Direction Cosines of Axes

Q. If a line makes angles 90°, 135°, 45° with x , y and z -axes respectively, find its direction cosines.

Q. The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are

1. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
2. $2,3,6$
3. $\frac{2}{11},\frac{3}{11},\frac{6}{11}$
4. None of these

Q.

Determine the direction cosines of the normal to plane and the distance from the origin:

x + y + z = 1

Q. If a line makes angles ${90}^{\circ },{135}^{\circ },{45}^{\circ }$ with the $x,y$ and $Z-\text{axes}$ respectively, find its direction cosines.

Q. A line makes acute angles $\alpha ,\beta ,\gamma$ with the coordinate axes such that $\cos \alpha \cos \beta =\cos \beta \cos \gamma =\frac{2}{9}$ and $\cos \gamma \cos \alpha =\frac{4}{9},$ then the value of $\cos \alpha +\cos \beta +\cos \gamma$ is:

1. $\frac{25}{9}$
2. $\frac{5}{9}$
3. $\frac{5}{3}$
4. $\frac{2}{3}$

Q. Choose the correct options for $\sin {630}^{\circ }\&\cos {810}^{\circ }.$

1. $\cos {810}^{\circ }=0$
2. $\sin {630}^{\circ }=1$
3. $\sin {630}^{\circ }=-1$
4. $\cos {810}^{\circ }=1$

Q. The direction cosines of the normal to the plane 2x + 3y - 6z = 5 are

1. $(2,3,-6)$
2. $\frac{2}{7},\frac{3}{7},-\frac{6}{7}$
3. $\frac{2}{5},\frac{3}{5},-\frac{6}{5}$
4. None of these

Q. If the ratio of the roots of $l{x}^2+nx+n=0$ is p:q, then

1. 
2. 
3. 
4. 

Q. The projection of the line segment joining the points A(-1, 0, 3) and B(2, 5, 1) on the line whose direction ratios are proportional to 6, 2, 3, is

1. $\frac{10}{7}$
2. $\frac{22}{7}$
3. $\frac{18}{7}$
4. none of these

Q. Assertion :The equation sin x = $e^x$ has infinite solutions. Reason: $y=e^x$ is unbounded functions.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. The area of the region bounded by the curves $f\left(x\right)=\sin \pi x$ and x-axis for $x\in [-1,2]$ is

1. $\frac{8}{\pi }$
2. $\frac{6}{\pi }$
3. $8\pi$
4. $6\pi$

Q. Let $O=(0,0)$; let $A$ and $B$ be points respectively on $x$-axis and $y$-axis such that $\angle OBA={60}^{\circ }$. Let $D$ be a point in the first quadrant such that $OAD$ is an equilateral triangle. Then the slope of $DB$ is

1. $\sqrt{3}$
2. $\sqrt{2}$
3. $\frac{1}{\sqrt{2}}$
4. $\frac{1}{\sqrt{3}}$

Q.

Which of the following is parallel to plane 3x - 3y +4z = 7 ?

1. 

2. 

3. 

4. .

Q. A mapping which is not injective from A to B.

1. $\{(2,5),(3,5),(3,7)\}.$
2. $\{(2,5),(3,5),(4,7)\}.$
3. $\{(2,5),(3,6),(4,7)\}.$
4. None of these

Q. A mapping from A to B which is not injective i.e. many-one.

Q. A line makes angles $\alpha ,\beta ,\gamma$ with the co-ordinate axes. If $\alpha +\beta ={90}^{\circ }$, then $\gamma$ =

1. $0$
2. None of these
3. ${90}^{\circ }$
4. ${180}^{\circ }$

Q. The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are

1. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
2. $2,3,6$
3. $\frac{2}{11},\frac{3}{11},\frac{6}{11}$
4. None of these

Q. The projections of a directed line segment on the coordinate axes $12,4,3$. The direction cosines of the line are

1. $-\frac{12}{13},-\frac{4}{13},\frac{3}{13}$
2. $\frac{12}{13},-\frac{4}{13},\frac{3}{13}$
3. $\frac{12}{13},\frac{4}{13},\frac{3}{13}$
4. none of these

Q.

The direction cosines of the normal to the plane 2x + 3y - 6z = 5 are

1. 
   

2. 
   

3. None of these

4. 
   

Q. The direction cosines of the normal to the plane 2x + 3y - 6z = 5 are

1. $(2,3,-6)$
2. $\frac{2}{7},\frac{3}{7},-\frac{6}{7}$
3. None of these
4. $\frac{2}{5},\frac{3}{5},-\frac{6}{5}$

Q. In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin,
$5y+8=0$

Q. A line makes angles $\alpha ,\beta ,\gamma$ with the co-ordinate axes. If $\alpha +\beta ={90}^{\circ }$, then $\gamma$ =

1. $0$
2. ${90}^{\circ }$
3. ${180}^{\circ }$
4. None of these

Q. The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are

1. $\frac{2}{7},\frac{3}{7},\frac{6}{7}$
2. None of these
3. $2,3,6$
4. $\frac{2}{11},\frac{3}{11},\frac{6}{11}$