## Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 11 Three Dimensional Geometry. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Three Dimensional Geometry MCQs Pdf with Answers to know their preparation level.

## Three Dimensional Geometry Class 12 Maths MCQs Pdf

1. The distance of point (2, 5, 7) from the x-axis is
(a) 2
(b) √74
(c) √29
(d) √53

Explaination:
(b), as distance of point (2, 5, 7) from the x-axis is

2. P is a point on the line segment joining the points (3, 5, -1) and (6, 3, -2). If y-coordinate of point P is 2, then its x-coordinate will be
(a) 2
(b) $$\frac{17}{3}$$
(c) $$\frac{15}{2}$$
(d) -5

Explaination:
(c), as let P divides the join of (3, 5, -1) and (6, 3, -2) in the ratio k : 1

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

Explaination:
(c), 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 $$\frac{-2}{7}, \frac{-3}{7}, \frac{6}{7}$$.

4. A line makes angle α, β, γ with x-axis, y-axis and z-axis respectively then cos 2α + cos 2β + cos 2γ is equal to
(a) 2
(b) 1
(c) -2
(d) -1

Explaination:

5. The equations of y-axis in space are
(a) x = 0, y = 0
(b) x = 0, z = 0
(c) y = 0, z = 0
(d) y = 0

Explaination: (b), as on the y-axis, x-coordinate and z-coordinate are zeroes.

6. If the direction cosines of a line are $$\frac{k}{3}, \frac{k}{3}, \frac{k}{3}$$, then value of k is
(a) k > 0
(b) 0 < k < 1.
(c) k = $$\frac{1}{3}$$
(d) k = ± 73

Explaination:

7. Distance of plane $$\vec{r} \cdot(2 \hat{i}+3 \hat{i}-6 \hat{k})+2=0$$, from origin is
(a) 2
(b) 14
(c) $$\frac{2}{7}$$
(d) –$$\frac{2}{7}$$

Explaination:

8. Distance between planes

Explaination:

9. The line joining the points (0, 5, 4) and (1, 3, 6) meets XY-plane at the point ________ .

Explaination:
(-2, 9, 0), as line is $$\frac{x-1}{1}=\frac{y-3}{-2}=\frac{z-6}{2}=\lambda$$
General point on line is (λ + 1, -2λ + 3, 2λ + 6)
If it meets AT-plane, then 2λ + 6 = 0
⇒ λ = – 3
∴ Point is (-2, 9, 0)

10. A line makes angles α, β, γ with z-axis, x-axis and y-axis respectively. Then direction cosines of line are cos β, cos γ, cos α. State true or false.

Explaination: True, as direction cosines of a line are cosines of the angles which a line makes with x, y and z-axes respectively.

11. A line makes angles $$\frac{\pi}{4}, \frac{3 \pi}{4}$$ with x-axis and y-axis respectively. Then the angle which it makes with z-axis can be ________ .

Explaination:

12. The vector equation of the line

State true or false.

Explaination:

13. The Cartesian equation of a line AB is

Find the direction cosines of a line parallel to AB.

Explaination:

14. Find the direction cosines of the line passing through the following points: (-2, 4, -5), (1, 2, 3) [NCERT]

Explaination:

15. Find the Cartesian equation of the line which passes through the point (-2,4, -5) and is parallel to the line $$\frac { x+3 }{ 3 } =\frac { 4-y }{ 5 } =\frac { z+8 }{ 6 }$$ [Delhi 2013]

Explaination:

16. Write the vector equation of the following line: $$\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { 6-z }{ 2 }$$

Explaination:
The line passes through the point (5, -4, 6) and dr’s of the line are 3, 7, – 2.
∴ vector equation is

17. Write the Cartesian equation of the following line given in vector form:

Explaination:
Point through which line passes is (2, 1, -4) and dr’s: 1, – 1, – 1.
∴ Cartesian equation of line

18. What are the direction cosines of a line, which makes equal angles with the coordinate axes? [NCERT; Foreign 2011]

Explaination:

19. If the direction cosines of a given line are $$\frac{1}{k}, \frac{1}{k}, \frac{1}{k}$$ then, find the value of k.

Explaination:

20. If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction ofz-axis. [Delhi 2017]

Explaination:
Let angle with z-axis be γ.
cos²90° + cos²60° + cos² γ = 1
⇒ 0 + $$\frac{1}{4}$$ + cos² γ = 1
⇒ cos² γ = $$\frac{3}{4}$$
cos γ = $$\pm \frac{\sqrt{3}}{2}$$
γ = 30°, 150°

21. Find the vector equation of the line passing through the point A(1, 2,-1) and parallel to the line 5x – 25 = 14 – 7y = 35z. [Delhi 2017]

Explaination:
Given line is 5x – 25 = 14 – 7y = 35z
⇒ 5(x – 5) = – 7(y – 2) = 35z

DR’s of line are 7, – 5 and 1
dr’s of line parallel to the given line are 7,-5, 1.
vector equation of line through the point (1, 2, – 1) and having dr’s 7,-5 and 1 is

22. Write the distance of the point (3, – 5, 12) from the x-axis. [Foreign 2017]

Explaination:
Distance of the point (3, – 5, 12) from the x-axis

23. Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular. [Delhi 2011]

Explaination:
DR’s of lines are 2, 7, – 3 and – 1, 2, 4
As 2 × (- 1) + 7 × 2 – 3 × 4 = 0, so lines are perpendicular. Angle = 90°

24. The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its z-coordinate. [AI2017]

Explaination:
Let point R(4, y, z) lies on the line joining P(2, 2, 1) and Q(5, 1, -2). Let R divides PQ in ratio k: 1

25. If P(1, 5, 4) and Q(4, 1, – 2), find the direction ratios of $$\overrightarrow{P Q}$$.

Explaination:
Direction ratios of $$\overrightarrow{P Q}$$ =4 – 1, 1 – 5 and -2 -4, i.e. 3, -4 and – 6.

26. The equations of a line are 5x – 3 = 15y + 7 = 3 – 10z. Write the direction cosines of the line. [All India]

Explaination:
The equation of a line are 5x – 3 = 15y + 7 = -10z + 3

27. Equation of the perpendicular drawn from the point with position vector $$2 \hat{i}- \hat{j}+ \hat{k}$$ to the plane $$\vec{r} \cdot(\hat{i}-3 \hat{k})=5$$ is ________ .

Explaination:

28. General equation of a plane passing through the intersection of two given

Explaination:

29. Cartesian equation of the plane

State true or false.

Explaination:

30. Find the distance of the point (2,3,4) from the plane

Explaination:

31. Write the intercept cut off by the plane 2x + y – z = 5 on the x-axis. [Delhi 2011]

Explaination:
For intercept on the x-axis, put y = 0 and z = 0
⇒ 2x = 5
⇒ x = $$\frac{5}{2}$$
∴ x-intercept = $$\frac{5}{2}$$

32. Find the distance of the plane 3x – 4y + 12z = 3 from the origin. [AI 2012]

Explaination:

33. Find the angle between the planes

Explaination:

34. Find the distance between the planes 2x – y + 2z – 5 and 5x – 2.5y + 5z = 20. [AI 2017]

Explaination:
Planes are 2x – y + 2z = 5
⇒ 2x – y + 2z – 5 = 0
and 5x – 2.5y + 5z = 20
⇒ 2x – y + 2z – 8 = 0

35. A line passes through the point with position vector $$2 \hat{i}-3 \hat{j}+4 \hat{k}$$ and is perpendicular to the plane $$\vec{r} \cdot(3 \hat{i}+4 \hat{j}-5 \hat{k})=7$$. Find the

Explaination:

We hope the given Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry will help you. If you have any query regarding CBSE Class 12 Maths Three Dimensional Geometry MCQs Pdf, drop a comment below and we will get back to you at the earliest.

## Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 10 Vector Algebra. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Vector Algebra MCQs Pdf with Answers to know their preparation level.

## Vector Algebra Class 12 Maths MCQs Pdf

1. A vector equally inclined to axes is

Explaination:
(a), as direction ratios are 1, 1, 1 and
direction cosines $$\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$$
⇒ cos α = cos β = cos γ
⇒ α = β = γ

2. The position vector of a point which divides the join of points with position vectors $$\vec{a}+\vec{b}$$ and $$2 \vec{a}-\vec{b}$$ in the ratio 1:2 internally is

Explaination:

3. A vector in the direction of vector $$\hat{i}-2 \hat{j}+2 \hat{k}$$ that has magnitude 15 is

Explaination:

4. If $$|\vec{a}|$$ = 4 and -3 ≤ λ ≤ 2 then the range of $$|\lambda \vec{a}|$$ is
(a) [0, 8]
(b) [-12, 8]
(c) [0, 12]
(d) [8, 12]

Explaination:

5.

(a) 30°
(b) 45°
(c) 60°
(d) 90°

Explaination:

6.

Explaination:

7.

Explaination:

8.

Explaination:

9. Mathematically a vector is defined as a “directed line segment.” State true or false.

Explaination: True

10. If vectors are equal then their magnitudes are equal but the converse may not be true. State true or false.

Explaination:

11. Given vector $$\overrightarrow{P Q}=2 \hat{i}+\hat{j}-3 \hat{k}$$ and position vector of point P is $$3 \hat{j}-2 \hat{k}$$, then position vector of point Q is _________ .

Explaination:

12. Find a unit vector in the direction of $$\overrightarrow{a}=3 \hat{i}-2 \hat{j}+6 \hat{k}$$

Explaination:

13. For what value of p, is ($$\hat{i}+ \hat{j}- \hat{k}$$)p a irnit vector?

Explaination:

14. If $$\overrightarrow{A B}=2 \hat{i}+\hat{j}-2 \hat{k}$$ and $$\overrightarrow{B C}=6 \hat{i}+3\hat{j}-6 \hat{k}$$, can we say that the points A, B, C are collinear?

Explaination:

15. In the given figure, find
(i) parallel or collinear vectors

(ii) co-initial vectors

Explaination:

16.

Explaination:

17.

Explaination:

18. Find a vector in the direction of $$\vec{a}=\vec{i}-2 \hat{j}$$ whose magnitude is 7. [NCERT]

Explaination:

19. For what value of ‘a’ the vectors $$2 \hat{i}-3 \hat{j}+4 \hat{k}$$ and $$\hat{a} \hat{i}+\hat{6} \hat{j}-8 \hat{k}$$ are collinear?[Delhi2011]

Explaination:
For vectors to be collinear $$\frac{2}{a}=\frac{-3}{6}=\frac{4}{-8}$$
⇒ a = -4

20. ABCDEF is a regular hexagon,

Explaination:

21. The position vectors A, B, C, D are

Explaination:

22. Write the direction cosines of the vector $$-2 \hat{i}+\hat{j}-5 \hat{k}$$ [Delhi 2011]

Explaination:

23. Show that the vector $$\hat{i}+\hat{j}+ \hat{k}$$ is equally inclined to axes.

Explaination:

24. Write a unit vector in the direction of the

Explaination:

25. The value of X for which vectors

are orthogonal is __________ .

Explaination:

26. For any non zero vector $$\vec{a}$$,

State true or false.

Explaination:

27.
are perpendicular. State true or false.

Explaination:

28. Find the angle between the vectors

Explaination:

29.

Explaination:

30.

Explaination:

31. If $$\vec{p}$$ is a unit vector and

Explaination:

32.

Explaination:

33.

Explaination:

34.
$$\vec{b}$$ are along adjacent sides of a rectangle.

Explaination:

35.

Explaination:

36. Find the magnitude of each of the two vectors $$\vec{a}$$ and $$\vec{b}$$, having the same magnitude such that the angle between them is 60° and their scalar product is $$\frac{9}{2}$$. [CBSE 2018]

Explaination:

37.

Explaination:

38.

Explaination:

39.

Explaination:

40.

Explaination:

41.

Explaination:

42.

Explaination:

43.

Explaination:

44.

Explaination:

45.

Explaination:

46.

Explaination:

47.

Explaination:

48.

Explaination:

49.

Explaination:

50. Write the value of $$(\hat{i} \times \hat{j}) . \hat{k}+\hat{i} \hat{j}$$ [AI 2012]

Explaination:

51. Write a unit vector perpendicular to both the
[AI 2015]

Explaination:

52.

Explaination: coplanr

53.

(a) 4
(b) 0
(c) -3
(d) 2

Explaination:

54.

Explaination:

55.

Explaination:

We hope the given Maths MCQs for Class 12 with Answers Chapter 10 Vector Algebra will help you. If you have any query regarding CBSE Class 12 Maths Vector Algebra MCQs Pdf, drop a comment below and we will get back to you at the earliest.

## Maths MCQs for Class 12 with Answers Chapter 9 Differential Equations

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 9 Differential Equations. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Differential Equations MCQs Pdf with Answers to know their preparation level.

## Differential Equations Class 12 Maths MCQs Pdf

1. Order of differential equation correspon- ding to family of curves y = Ae2x + Be2x is ______ .

Explaination: 2, as there are two arbitrary constants and we have to differentiate twice.

2. The order of the differential equation corresponding to the family of curves y = c(x – c)², c is constant is ______ .

Explaination: One, as there is one arbitrary constant.

3. The degree of differential equation

is not defined. State true or false.

Explaination: Three, as equation cannot be represented as polynomial of derivatives.

4. If p and q are the degree and order of the differential equation

then the value of 2p – 3q is
(a) 7
(b) -7
(c) 3
(d) -3

Explaination:
(b), as degree p = 1 and order q = 3
∴ 2p – 3q = 2 – 9 = -7

5. The degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) 4

Explaination:
(c), as differential equation is

Exponent of highest order derivative is 3.

6. The degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) not defined

Explaination:
(d), as equation cannot be represented as a polynomial of derivatives.

7. The order of the differential equation of all the circles of given radius 4 is
(a) 1
(b)2
(c) 3
(d) 4

Explaination:
(b), as centre is arbitrary (h, k), two arbitrary constants so we have to differentiate twice to eliminate h, k
∴ order is 2.

8. Degree of the differential equation

is not defined. State true or false.

Explaination:
False, as equation can be written as

Further it can be written as a polynomial of derivatives. 9.

9. Write the degree of the differential equation

Explaination: Degree 1

10. Write the degree of the differential equation

Explaination: Degree 3.

11. Find the value of m and n, where m and n are order and degree of differential equation

Explaination:

Order of differential equation (m) = 3
Degree of differential equation (n) = 2

12. Write the order and degree of the differential equation $$\frac{d y}{d x}+\sin \left(\frac{d y}{d x}\right)$$ = 0. [HOTS]

Explaination:
Highest order derivative is $$\frac{dy}{dx}$$. Hence, order of differential equation is 1. Equation cannot be written as a polynomial’ in derivatives. Hence, degree is not defined.

13. The differential equation of the family of lines passing through ongrn is
(a) y = mx
(b) $$\frac{dy}{dx}$$ = m
(c) x dy – y dx = 0
(d) $$\frac{dy}{dx}$$ = 0

Explaination:
(c), as general equation of line through origin is
y = mx
⇒ $$\frac{dy}{dx}$$ = m
Substituting in (i), we get dy
y = $$\frac{dy}{dx}$$.x
⇒ x dy – y dx = 0

14. Find the differential equation representing the family of curves y = aebx + 5, where a and b are arbitrary constants. [CBSE 2018]

Explaination:
Consider y = aebx + 5
.On differentiating both sides, w.r.t, x
$$\frac{dy}{dx}$$ = abebx + 5 = by …..(i)
Again differentiating w.r.t. x, we get
$$\frac{d^{2} y}{d x^{2}}=b \cdot \frac{d y}{d x}$$ …..(ii)
From (i) and (ii), eliminating b, we get
$$y \cdot \frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}$$ as required equation.

15. Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of the x-axis. [NCERT; DoE]

Explaination:
General equation of parabola is y² = 4ax …..(i)
Differentiating, we get 2yy’ = 4a
⇒ yy’ = 2a.
Substituting in (i), we gety2 = 2xyy’.

16. Form the differential equation of the family of parabolas having vertex at the origin and axis along positive y-axis. [Delhi 2011]

Explaination:
x² = 4 ay
⇒ 2x = 4 ay’
⇒ $$\frac{x^{2}}{2 x}=\frac{4 a y}{4 a y^{\prime}}$$
⇒ xy’ – 2y = 0 is the required equation.

17. General solution of the differential equation log$$\frac{dy}{dx}$$ = 2x +y is _______ .

Explaination:

18. Solve the differential equation $$\frac{dy}{dx}$$ = ex – y + x3e-y.

Explaination:

19. Find the particular solution of the differential equation $$\frac{dy}{dx}$$ =y tanx, given that y= 1 when x = 0.

Explaination:
∫ $$\frac{dy}{y}$$ = ∫tan x dx
⇒ log |y| = log|sec x| + log C
⇒ y = C sec x ….(i)
Given y = 1, x = 0
⇒ 1 = C sec 0
⇒ C = 1
∴ solution is y = sec x [from (i)]

20. Find the general solution of the differential equation $$\frac{dy}{dx}$$ = $$\frac{x+1}{2-y}$$, (y ≠ 2). [NCERT]