Question Set C

CBSE 12th Standard Maths Three - Dimensional Geometry HOT Questions Study Material - QB365

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Questions:

Multiple Choice Question

40 x 1 = 40

The lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{4-z}{k}\) and \(\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{-2}\) are mutually perpendicular, if the value of k is

(a)

\(-\frac{2}{3}\)

(b)

\(\frac{2}{3}\)

(c)

-2

(d)

The direction cosines of vector \(\overrightarrow{B A}\), where coordinates of A and B are (1, 2, -1) and (3, 4, 0) respectively, are

(a)

-2, -2, -1

(b)

\(-\frac{2}{3},-\frac{2}{3},-\frac{1}{3}\)

(c)

2, 2, 1

(d)

\(\frac{2}{3}, \frac{2}{3}, \frac{1}{3}\)

If a line makes angles of 90°, 135° and 45° with the X, Y and Z-axes respectively, then its direction cosines are

(a)

\(0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

(b)

\(-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\)

(c)

\(\frac{1}{\sqrt{2}}, 0,-\frac{1}{\sqrt{2}}\)

(d)

\(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

The point (x, y, 0) on the XY-plane divides the line segment joining the points (1, 2, 3) and (3, 2, 1l) in the ratio

(a)

1 : 2 internally

(b)

2 : 1 internally

(c)

3 : 1 internally

(d)

3 : 1 externally

The Cartesian equation of the line passing through the point (1, - 3, 2) and parallel to the line \(\vec{r}=(2+\lambda) \hat{i}+\lambda \hat{j}+(2 \lambda-1) \hat{k}\) is

(a)

\(\frac{x-1}{2}=\frac{y+3}{0}=\frac{z-2}{-1}\)

(b)

\(\frac{x+1}{1}=\frac{y-3}{1}=\frac{z+2}{2}\)

(c)

\(\frac{x+1}{2}=\frac{y-3}{0}=\frac{z+2}{-1}\)

(d)

\(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)

If line \(\frac{x-1}{2}=\frac{y+3}{1}=\frac{z-5}{-1}\) is parallel to the plane px + 3y - z + 5 = 0, then the value of 'p' is

(a)

(b)

-2

(c)

\(\frac{1}{2}\)

(d)

none of these

Direction ratios of the line \(\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3} \text { are }\)

(a)

2,6,3

(b)

-2,6,3

(c)

2, - 6, 3

(d)

none of these

A line makes equal angles with axes, direction cosines of line are

(a)

1, 1, 1

(b)

\(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\)

(c)

\(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)

(d)

\(\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)

Direction cosines of a unit vector perpendicular to the plane \(\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})-1=0 \text { are }\)

(a)

\(6,-3,-2\)

(b)

\(\frac{6}{7},-\frac{3}{7},-\frac{2}{7}\)

(c)

\(-\frac{6}{7}, \frac{3}{7},-\frac{2}{7}\)

(d)

none of these

10.

The ratio in which the line segment joining the points (2, -4, 5) and (3, 5, 4) is divided by YZ-plane is

(a)

5 : 4 internally

(b)

4 : 5 internally

(c)

2 : 3 externally

(d)

none of these

11.

Lquation of the line passing through the point (2, 1, 3) and perpendicular to the lines \(\frac{x-1}{1}=\frac{y-2}{3}=\frac{z-3}{3}\) \(\text { and } \frac{x}{-3}=\frac{y}{2}=\frac{z}{5} \text { is }\)

(a)

\(\frac{x-1}{2}=\frac{y-2}{-7^{\circ}}=\frac{z-3}{4}\)

(b)

\(\frac{x}{-2}=\frac{y}{7}=\frac{z}{-4}\)

(c)

\(\frac{x-2}{-2}=\frac{y-1}{7}=\frac{z-3}{-4}\)

(d)

none of these

12.

Angle between the lines with direction ratios 2, 1,2 and 3,2, -6 is

(a)

\(\cos ^{-1}(-4)\)

(b)

\(\cos ^{-1}\left(-\frac{4}{21}\right)\)

(c)

\(-\frac{4}{21}\)

(d)

none of these

13.

Direction ratios of a line passing through the points (2, 1, 0) and (3, 2, 1) are

(a)

(1, 1, -1)

(b)

1 ,1, -1

(c)

< 5, 3, -1 >

(d)

none of these

14.

Intercept cut by the plane 2x - y + 2z + 7 = 0 on the x-axis is

(a)

(b)

\(\frac{7}{2}\)

(c)

\(-\frac{7}{2}\)

(d)

-2

15.

The planes \(2 x-y+4 z=5\) and \(5 x-2.5 y+10 z=6\) are

(a)

perpendicular

(b)

parallel

(c)

intersect along Y-axis

(d)

passes through \(\left(0,0, \frac{5}{4}\right)\)

16.

The coordinates of a point on the line \(\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}\) at a distance of \(\frac{6}{\sqrt{2}}\) from the point (1, 2, 3) is

(a)

(56, 43, 111)

(b)

\(\left(\frac{56}{17}, \frac{43}{17} \cdot \frac{111}{17}\right)\)

(c)

(2, 1, 3)

(d)

(-2, -1, -3)

17.

The equation of X -axis in space is

(a)

x = 0, y = 0

(b)

x = 0, z = 0

(c)

x = 0

(d)

y = 0, z = 0

18.

The reflection of the point \((\alpha, \beta, \gamma)\) in the XY -plane is

(a)

\((\alpha, \beta, 0)\)

(b)

\((0,0, \gamma)\)

(c)

\((-\alpha,-\beta, \gamma)\)

(d)

\((\alpha, \beta,-\gamma)\)

19.

If the plane \(2 x-3 y+6 z-11=0\) makes an angle \(\sin ^{-1} \alpha\) with X-axis, then the value of \(\alpha\) is

(a)

\(\frac{\sqrt{3}}{2}\)

(b)

\(\frac{\sqrt{2}}{3}\)

(c)

\(\frac{2}{7}\)

(d)

\(\frac{3}{7}\)

20.

Distance between the two planes \(2 x+3 y+4 z=4\) and 4x + 6Y + 8z = 12 and \(4 x+6 y+8 z=12\) is

(a)

2 units

(b)

4units

(c)

8 units

(d)

\(\frac{2}{\sqrt{29}} \text { units }\)

21.

The locus represented by xy + yz = 0 is

(a)

a pair of perpendicular lines

(b)

a pair of parallel lines

(c)

a pair of parallel planes

(d)

a pair of perpendicular planes

22.

Two lines \(L_{1}: x=5, \frac{y}{3-\alpha}=\frac{z}{-2} \text { and } L_{2}: x=\alpha\) \(\frac{y}{-1}=\frac{z}{2-\alpha}\) are coplanar. Then \(\alpha\) can take values

(a)

1, 4, 5

(b)

1, 2, 5

(c)

3, 4, 5

(d)

2, 4, 5

23.

The distance of the plane \(\vec{r}\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1\) from the origin is

(a)

(b)

(c)

\(\frac{1}{7}\)

(d)

None of these

24.

The point of intersection of the lines \(\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{1} \text { and } \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) is

(a)

(-1,-1,-1

(b)

(-1,-1,1)

(c)

(1,-1,-1

(d)

(-1,1,-1)

25.

If the lines \(x=a y+b, z=c y+d \text { and } x=a^{\prime} y+b^{\prime}\) \(z=c^{\prime} y+d^{\prime}\) are perpendicular, then

(a)

\(a a^{\prime}+c c^{\prime}=1\)

(b)

\(a a^{\prime}+c c^{\prime}=-1\)

(c)

\(a b+c d=a^{\prime} b^{\prime}+c^{\prime} d^{\prime}\)

(d)

\(a a^{\prime}+b b^{\prime}=c c^{\prime}+d d^{\prime}\)

26.

The equation of straight line passing through the point (a, b, c) and parallel to Z-axis is

(a)

\(\frac{x-a}{1}=\frac{y-b}{1}=\frac{z-c}{0}\)

(b)

\(\frac{x-a}{0}=\frac{y-b}{1}=\frac{z-c}{1}\)

(c)

\(\frac{x-a}{1}=\frac{y-b}{0}=\frac{z-c}{0}\)

(d)

\(\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\)

27.

If the direction cosines of a line are k, k and k, then

(a)

\(k>0\)

(b)

\(0<k<1\)

(c)

\(k=1\)

(d)

\(k=\frac{1}{\sqrt{3}} \text { or }-\frac{1}{\sqrt{3}}\)

28.

If a line in the ZX-plane makes an angle 30o with Z-axis, the direction cosines of this line are:

(a)

\(\frac { \sqrt { 3 } }{ 2 } ,0,\frac { 1 }{ 2 } \)

(b)

\(0,\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ 2 } \)

(c)

\(\frac { 1 }{ 2 } ,0,\frac { \sqrt { 3 } }{ 2 } \)

(d)

\(\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ 2 } ,0\)

29.

If a line in the ZX-plane makes an angle 60o with Z-axis, the direction cosines of this line are:

(a)

\(\frac { \sqrt { 3 } }{ 2 } ,0,\frac { 1 }{ 2 } \)

(b)

\(\frac { 1 }{ 2 } ,0,\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ 2 } ,0\)

(d)

\(0,\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ 2 } \)

30.

If a line makes angles 45°, 150°, 135°, with x, y and z-axes respectively, find its direction cosines.

(a)

\(\frac { 1 }{ \sqrt { 2 } } ,-\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ \sqrt { 2 } } \)

(b)

\(\frac { 1 }{ \sqrt { 2 } } ,\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ \sqrt { 2 } } \)

(c)

\(\frac { 1 }{ \sqrt { 2 } } ,-\frac { 1 }{ 2 } ,-\frac { \sqrt { 3 } }{ \sqrt { 2 } } \)

(d)

\(\frac { 1 }{ \sqrt { 2 } } ,\frac { \sqrt { 3 } }{ 2 } ,-\frac { 1 }{ \sqrt { 2 } } \)

31.

If l, m , n are the direction cosines of any line, then sum of the squares of the direction cosines of the line is always

(a)

-1

(b)

\(\sqrt3\)

(c)

(d)

32.

What are direction ratios of a line

(a)

numbers which are proportional to the direction angles of a line

(b)

numbers which are proportional to the direction cosines of a line

(c)

numbers which are same direction angles of a line

(d)

numbers which are proportional to the direction cosines of a line

33.

The direction cosines of the line joining the points (2, -1, 8) and (-4, -3, 5) are:

(a)

\(\frac { 6 }{ 7 } ,\frac { -2 }{ 7 } ,\frac { 3 }{ 7 } \)

(b)

\(\frac { 6 }{ 7 } ,\frac { -2 }{ 7 } ,\frac { 3 }{ 7 } \)

(c)

\(\frac { -6 }{ 7 } ,\frac { 2 }{ 7 } ,\frac { -3 }{ 7 } \)

(d)

\(\frac { 6 }{ 7 } ,\frac { 2 }{ 7 } ,\frac { 3 }{ 7 } \)

34.

Find the direction cosines of the x axis.

(a)

1, 0, 0

(b)

0, 0, 0

(c)

0, 1, 0

(d)

0, 0, 1

35.

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

(a)

\(\frac { 2 }{ 3 } ,\frac { -2 }{ 3 } ,\frac { 1 }{ 3 } \)

(b)

\(\frac { -2 }{ 3 } ,\frac { 2 }{ 3 } ,\frac { -1 }{ 3 } \)

(c)

\(\frac { 6 }{ 3 } ,\frac { -6 }{ 3 } ,\frac { -1 }{ 3 } \)

(d)

\(\frac { 2 }{ 9 } ,\frac { -2 }{ 9 } ,\frac { 1 }{ 9 } \)

36.

If the direction cosines of a line from the positive X-axis and Y-axis are \(\frac { 1 }{ 2 } ,\frac { 1 }{ \sqrt { 2 } } \) . The angle of the line through Z-axis is:

(a)

30°

(b)

45°

(c)

135°

(d)

60°

37.

The co-ordinates of the vertices of the triangle are A(-2, 3, 6), B(-4, 4, 9) and C(0, 5, 8). The direction cosines of the median BE are:

(a)

1 , 0 , -2/3

(b)

3/\(\sqrt{13}\), 0, -2/\(\sqrt{13}\)

(c)

3/4, 0, -2/4

(d)

-3/\(\sqrt{12}\), 0, -2/\(\sqrt{13}\)

38.

The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are

(a)

Perpendicular

(b)

Parallel

(c)

intersect y-axis

(d)

passes through \((0,0,\frac{5}{4})\)

39.

Distance of plane \(\overrightarrow { r } .(2\widehat { i } +3\widehat { j } -6\widehat { k } )+2=0\) from origin is

(a)

(b)

(c)

\(\frac27\)

(d)

-\(\frac27\)

40.

Direction ratios of a line are 2, 3, -6. Then direction cosines of a line making obtuse angle with the y-axis are

(a)

\(\frac { 2 }{ 7 } ,\frac { -3 }{ 7 } ,\frac { -6 }{ 7 } \)

(b)

\(\frac { -2 }{ 7 } ,\frac { 3 }{ 7 } ,\frac { -6 }{ 7 } \)

(c)

\(\frac { -2 }{ 7 } ,\frac { -3 }{ 7 } ,\frac { 6 }{ 7 } \)

(d)

\(\frac { -2 }{ 7 } ,\frac { -3 }{ 7 } ,\frac { -6 }{ 7 } \)

Assertion and reason

10 x 1 = 10

41.

Assertion (A) The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by \(\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}\) is \(\frac{x+3}{-2}=\frac{y-4}{4}=\frac{z+8}{-5}\).
Reason (R) If the cartesian equation of a line is \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\), then its vector form is \(\vec{r}=5\hat{i}-4\hat{j}+6\hat{k}+\lambda (3\hat{i}+7\hat{j}+2\hat{k})\)
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

42.

Assertion (A) The points (1, 2, 3), (-2, 3, 4) and (7, 0, 1) are collinear.
Reason (R) If a line makes angles \(\frac{\pi}{2}, \frac{3\pi}{4}\) and \(\frac{\pi}{4}\)with X, Y and Z - axes respectively, then its direction cosines are 0, \(\frac{-1}{\sqrt{2}}\) and \(\frac{1}{\sqrt{2}}\).
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

43.

Assertion (A) The acute angle between the line \(\vec{r}=\hat{i}+\hat{j}+2 \hat{k}+\lambda(\hat{i}-\hat{j})\) and the X - axis is \(\frac{\pi}{4}\).
Reason (R) The acute angle \(\theta\) between the lines \(\vec{r}=x_1 \hat{i}+y_1 \hat{j}+z_1 \hat{k}+\lambda\left(a_1 \hat{i}+b_1 \hat{j}+c_1 \hat{k}\right)\) and \(\vec{r}=x_2 \hat{i}+y_2 \hat{j}+z_2 \hat{k}+\mu\left(a_2 \hat{i}+b_2 \hat{j}+c_2 \hat{k}\right)\) is given by \(\cos \theta=\frac{\left|a_1 a_2+b_1 b_2+c_1 c_2\right|}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\)
(a) Both (A) and (R) are correct and (R) is the correct explanation of (A).
(b) Both (A) and (R) are correct but (R) is not the correct explanation of (A).
(c) (A) is correct but (R) is incorrect.
(d) Both (A) and (R) are incorrect.

44.

Assertion (A) The lines \(\vec{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1} \text { and } \vec{r}=\vec{a}_1+\mu \overrightarrow{b_2}\) are perpendicular, when \(\overrightarrow{b_1} \cdot \overrightarrow{b_2}=0\).
Reason (R) The angle \(\theta\) between the lines \(\vec{r}=\vec{a}_1+\lambda \vec{b}_1 \text { and } \vec{r}=\vec{b}_2+\mu \overrightarrow{b_2}\) is given by \(\cos \theta=\frac{\overrightarrow{b_1} \cdot \overrightarrow{b_2}}{\left|\overrightarrow{b_1}\right|\left|\overrightarrow{b_2}\right|}\)
(a) Both (A) and (R) are correct and (R) is the correct explanation of (A).
(b) Both (A) and (R) are correct but (R) is not the correct explanation of (A).
(c) (A) is correct but (R) is incorrect.
(d) Both (A) and (R) are incorrect.

45.

Consider three planes
P₁:x - y + z = 1
P₂:x + y - z = 1
P₃:x - 3y + 3z = 2
Let L₁, L₂, L₃ be the lines of intersection of the planes P₂ and P₃, P₃ and P₁, P₁ and P₂, respectively.
Assertion: At least two of the lines L₁, L₂ and L₃ are nonparallel
Reason: The three planes does not have a common point.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

46.

Consider the planes 3x - 6y - 2z = 15 and 2x + y - 2z = 5.
Assertion: The parametric equations of the line of intersection of the given planes are x = 3 + 14t, y = 1 + 2t, z = 15t.
Reason: The vector \(14\hat{i}+2\hat{j}+15\hat{k}\) is parallel to the line of intersection of given planes.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

47.

Assertion: Distance of a point with position vector a from a plane r. N = d is given by |a. N - d|.
Reason: The length of perpendicular from origin O to the plane r.N = d is \(\frac{|d|}{|N|}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

48.

Consider the lines
\(L_{1}:\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2},L_{2}:\frac{x-2}{2}=\frac{y-2}{2}=\frac{z-3}{3}\)
Assertion: The distance of point (1,1,1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines L₁ and L₂ is \(\frac{13}{5\sqrt{3}}\)
Reason: The unit vector perpendicular to both the lines L₁ and L₂ is \(\frac{-i-7j+5\hat{k}}{5\sqrt{3}}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

49.

Assertion: The pair of lines given by \(\overrightarrow{r}=\hat{i}-\hat{j}+\lambda (2i+k)\) and \(\overrightarrow{r}=2\hat{i}-\hat{k}+\mu (i+\hat{j}-k)\)intersect.
Reason: Two lines intersect each other, if they are not parallel and shortest distance = 0.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

50.

Assertion: If a variable line in two adjacent positions has direction cosines l, m, n, and l + \(\delta\)l, m + \(\delta\)m, n + \(\delta\)n, then the small angle \(\delta\)\(\theta\) between the two positions is given by \(\delta\)\(\theta\)=\(\delta\)l² + \(\delta\)m² + \(\delta\)n²
Reason: If O is the origin and A is (a, b, c), then the equation of plane through at right angle to OA is given by ax + by + cz = a² + b² + c².
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

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Answers Key:

Multiple Choice Question

40 x 1 = 40

(a)
\(-\frac{2}{3}\)

(b)
\(-\frac{2}{3},-\frac{2}{3},-\frac{1}{3}\)

(a)
\(0,-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\)

(d)
3 : 1 externally

(d)
\(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)

(b)
-2

(b)
\(\frac{6}{7},-\frac{3}{7},-\frac{2}{7}\)

10.

11.

(d)
none of these

12.

(b)
\(\cos ^{-1}\left(-\frac{4}{21}\right)\)

13.

(b)
1 ,1, -1

14.

15.

(b)
parallel

16.

(b)
\(\left(\frac{56}{17}, \frac{43}{17} \cdot \frac{111}{17}\right)\)

17.

(d)
y = 0, z = 0

18.

19.

20.

(d)
\(\frac{2}{\sqrt{29}} \text { units }\)

21.

(d)
a pair of perpendicular planes

22.

(a)
1, 4, 5

23.

(a)
1

24.

(a)
(-1,-1,-1

25.

26.

27.

(d)
\(k=\frac{1}{\sqrt{3}} \text { or }-\frac{1}{\sqrt{3}}\)

28.

(a)
\(\frac { \sqrt { 3 } }{ 2 } ,0,\frac { 1 }{ 2 } \)

29.

(a)
\(\frac { \sqrt { 3 } }{ 2 } ,0,\frac { 1 }{ 2 } \)

30.

(b)
\(\frac { 1 }{ \sqrt { 2 } } ,\frac { \sqrt { 3 } }{ 2 } ,\frac { 1 }{ \sqrt { 2 } } \)

31.

(d)
0

32.

(b)
numbers which are proportional to the direction cosines of a line

33.

(b)
\(\frac { 6 }{ 7 } ,\frac { -2 }{ 7 } ,\frac { 3 }{ 7 } \)

34.

(a)
1, 0, 0

35.

(a)
\(\frac { 2 }{ 3 } ,\frac { -2 }{ 3 } ,\frac { 1 }{ 3 } \)

36.

(d)
60°

37.

(b)
3/\(\sqrt{13}\), 0, -2/\(\sqrt{13}\)

38.

(b)
Parallel

39.

As normal form of plane is
\(\overrightarrow { r } .(-\frac { 2 }{ 7 } \widehat { i } -\frac { 3 }{ 7 } \widehat { j } +\frac { 9 }{ 7 } \widehat { k } )\) = \(\frac27\)
∴ distance = \(\frac27\)
∴ p = \(\frac27\)

40.

As direction cosines of a line whose direction ratio are 2,3, -6 are
\(\frac { -2 }{ 7 } ,\frac { 3 }{ 7 } ,\frac { 6 }{ 7 } \)
As angle with the y-axis is obtuse,
∴ cos β < 0,
Therefore direction ratios are \(\frac { -2 }{ 7 } ,\frac { -3 }{ 7 } ,\frac { 6 }{ 7 } \)

Assertion and reason

10 x 1 = 10

41.

(d) R is correct; A is incorrect

42.

(b) Both A and R are correct; R is not the correct explanation of A

43.

(a) The equation of the X-axis may be written as \(\vec{r}=t \hat{i} .\)
Now, the acute angle \(\theta\) between the line
\(\begin{aligned} \vec{r} & =\hat{i}+\hat{j}+2 \hat{k}+\lambda(\hat{i}-\hat{j}) \text { and } \vec{r}=t \hat{i} \end{aligned}\)
\(\begin{aligned} \therefore \cos \theta & =\frac{|1 \times 1+(-1) \times 0+0 \times 0|}{\sqrt{1^2+(-1)^2+0^2} \sqrt{1^2+0^2+0^2}}=\frac{1}{\sqrt{2}} \end{aligned}\)
\(\begin{aligned} \Rightarrow \quad \theta & =\frac{\pi}{4} \end{aligned}\)
Hence, both Assertion and Reason are true and Reason is a correct explanation of Assertion.

44.

(a) Assertion The given lines are \(\vec{r}=\vec{a}_1+\lambda \overrightarrow{b_1}\) and \(\vec{r}=\vec{a}_2+\lambda \vec{b}_2\)
Let \(\theta\) be the angle between these lines
\(\therefore \cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|| \vec{b}_2 \mid}\)
When \(\overrightarrow{b_1} \cdot \overrightarrow{b_2}=0\)
Then, \(\cos \theta=\frac{0}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}=0\)
\(\Rightarrow \quad \cos \theta=\cos \frac{\pi}{2} \Rightarrow \theta=\frac{\pi}{2}\)
Hence, the given lines are perpendicular.
Both Assertion and Reason are true and Reason is a correct explanation of Assertion.