## 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

Answer/Explanation

Answer: b
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

Answer/Explanation

Answer: c
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

Answer/Explanation

Answer: c
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

Answer/Explanation

Answer: d
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

Answer/Explanation

Answer: b
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

Answer/Explanation

Answer: d
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) –$$\frac{2}{7}$$

Answer/Explanation

Answer: c
Explaination:

8. Distance between planes

Answer/Explanation

Answer: c
Explaination:

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

Answer/Explanation

Answer:
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.

Answer/Explanation

Answer:
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 ________ .

Answer/Explanation

Answer:
Explaination:

12. The vector equation of the line

State true or false.

Answer/Explanation

Answer:
Explaination:

13. The Cartesian equation of a line AB is

Find the direction cosines of a line parallel to AB.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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:

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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.

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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 $$\overrightarrow{P Q}$$.

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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 ________ .

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

29. Cartesian equation of the plane

State true or false.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
Explaination:

33. Find the angle between the planes

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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

Answer/Explanation

Answer:
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

Answer/Explanation

Answer: a
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

Answer/Explanation

Answer: d
Explaination:

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

Answer/Explanation

Answer: d
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]

Answer/Explanation

Answer: c
Explaination:

5.

(a) 30°
(b) 45°
(c) 60°
(d) 90°

Answer/Explanation

Answer: a
Explaination:

6.

Answer/Explanation

Answer: c
Explaination:

7.

Answer/Explanation

Answer: b
Explaination:

8.

Answer/Explanation

Answer: a
Explaination:

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

Answer/Explanation

Answer:
Explaination: True

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

Answer/Explanation

Answer:
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 _________ .

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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?

Answer/Explanation

Answer:
Explaination:

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

(ii) co-initial vectors

Answer/Explanation

Answer:
Explaination:

16.

Answer/Explanation

Answer:
Explaination:

17.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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]

Answer/Explanation

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

20. ABCDEF is a regular hexagon,

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

24. Write a unit vector in the direction of the

Answer/Explanation

Answer:
Explaination:

25. The value of X for which vectors

are orthogonal is __________ .

Answer/Explanation

Answer:
Explaination:

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

State true or false.

Answer/Explanation

Answer:
Explaination:

27.
are perpendicular. State true or false.

Answer/Explanation

Answer:
Explaination:

28. Find the angle between the vectors

Answer/Explanation

Answer:
Explaination:

29.

Answer/Explanation

Answer:
Explaination:

30.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

32.

Answer/Explanation

Answer:
Explaination:

33.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

35.

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
Explaination:

37.

Answer/Explanation

Answer:
Explaination:

38.

Answer/Explanation

Answer:
Explaination:

39.

Answer/Explanation

Answer:
Explaination:

40.

Answer/Explanation

Answer:
Explaination:

41.

Answer/Explanation

Answer:
Explaination:

42.

Answer/Explanation

Answer:
Explaination:

43.

Answer/Explanation

Answer:
Explaination:

44.

Answer/Explanation

Answer:
Explaination:

45.

Answer/Explanation

Answer:
Explaination:

46.

Answer/Explanation

Answer:
Explaination:

47.

Answer/Explanation

Answer:
Explaination:

48.

Answer/Explanation

Answer:
Explaination:

49.

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

52.

Answer/Explanation

Answer:
Explaination: coplanr

53.

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

Answer/Explanation

Answer: b
Explaination:

54.

Answer/Explanation

Answer:
Explaination:

55.

Answer/Explanation

Answer:
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 ______ .

Answer/Explanation

Answer:
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 ______ .

Answer/Explanation

Answer:
Explaination: One, as there is one arbitrary constant.

3. The degree of differential equation

is not defined. State true or false.

Answer/Explanation

Answer:
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

Answer/Explanation

Answer: b
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

Answer/Explanation

Answer: c
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

Answer/Explanation

Answer: d
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

Answer/Explanation

Answer: b
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.

Answer/Explanation

Answer:
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

Answer/Explanation

Answer:
Explaination: Degree 1

10. Write the degree of the differential equation

Answer/Explanation

Answer:
Explaination: Degree 3.

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

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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

Answer/Explanation

Answer: c
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
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 _______ .

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
Explaination:

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

Answer/Explanation

Answer:
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]

Answer/Explanation

Answer:
Explaination:

We hope the given Maths MCQs for Class 12 with Answers Chapter 9 Differential Equations will help you. If you have any query regarding CBSE Class 12 Maths Differential Equations MCQs Pdf, drop a comment below and we will get back to you at the earliest.