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RD Sharma Solutions is a pretty popular name that specializes in creating NCERT solutions. The RD Sharma class 12th exercise MCQ book especially has garnered a lot of praise for being a lifesaver for students in India. Students who are in class 12 should definitely practice the RD Sharma class 12 chapter 26 exercise MCQ to prepare for exams and test their maths skills at home.

RD Sharma class 12 solutions Direction Cosines & Direction Ratios MCQ should be utilized by all students attending maths for their board exams. Chapter 26 of the NCERT is titled Direction Cosines & Direction Ratios. RD Sharma Solutions The main concept covered in this chapter is finding the directional ratio of lines, direction cosines of line, Coordinates of the foot on the y-axis and collinear points. Exercise MCQ of this chapter has a total of 18 questions. The RD Sharma class 12th exercise MCQ will help the students in finding the right options for the multiple-choice questions.

Chapter 26- Directions Cosines and Directions Ratios - Ex-26.1

Chapter 26- Directions Cosines and Directions Ratios - Ex-FBQ

Chapter 26- Directions Cosines and Directions Ratios - Ex-VSA

Given: P(x,y,z) lie on xy-plane

Solution:

z-coordinate of every point on xy plane is zero

Therefore option (c) z=0 is correct for point P(x,y,z) on xy plane

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 2

Answer:Given: P(x,y,z) lie on x-axis except origin

Solution:

Both y and z coordinate on each point of the x-axis are equal to zero. The x-coordinate at origin is also equal to zero.

Therefore, y and z coordinates on each point of the x-axis, except the origin are equal to zero, hence option

is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 3

Answer: (d) All of theseHint: Distance between plane passing through points is the length of edge

Given: Points (5,7,9) and (2,3,7)

Solution: In geometry, parallelopiped is a three dimension figure with six parallelogram

Let P and Q be the points. So, all the plane passes through these points. Now distance between planes joining (a,0,0) and (b,0,0) is |b-a|. Similarly, we can generate 3 pairs of plane from each coordinate from these 2 points.

For x-coordinate, value of a is

a=|5-2|=3

For y-coordinate, value of b is

b=|7-3|=4

For z-coordinate value of c is

c=|9-7|=2

Therefore, the edges of the parallelopoid are 3,4,2. Hence option (d) all of these is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 4

Answer: (a) 7Hint: Distance between planes passing through points is the length of edge

Given: Points (2,3,5) and (5,9,7)

Solution: In geometry parallelopiped is a three dimension figure with six parallelogram

z

Let P and Q be the points. So, all the plane passes through these points. Now distance between planes joining (a,0,0) and (b,0,0) is |b-a|. Similarly, we can generate 3 pairs of plane from each coordinate from these 2 points.

For x-coordinate, value of a is

a=|2-5|=3

For y-coordinate, value of b is

b=|3-9|=6

For z-coordinate value of c is

c=|5-7|=2

The edges of parallelopiped are 3,6,2

Length of diagonal=

Therefore, the length of diagonal is 7. Hence option (a) 7 is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 5

Answer: (b) Externally in ratio 2:3Hint: Use section formula

Given: Point(-1,3,4) and (2,-5,6)

To Find: Ratio in which xy plane divide the points

Solution:

Suppose xy plane divide the line segment joining the points O(-1,3,4) and Q(2,-5,6) in ratio k:1

Using section formula coordinates of the point of intersection are

The z-coordinate of any point of the xy-plane is zero.

Therefore, the xy plane divide the given points in 2:3 externally. Hence option(b) externally in the ratio 2:3 is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 6

Answer: (c) -1Hint: Use section formula

Given: x-coordinate of point P(x,y,z) on the join of Q(2,2,1) and R(5,1,-2) is 4

To Find: z-coordinate of point P(x,y,z)

Solution:

Suppose point P divide the line segment joining the points Q(2,2,1) and R(5,1,-2) in ratio t:1

Using section formula, coordinates of point P

x-coordinate of point P is 4

Now z-coordinate will be

Therefore z coordinate of P is -1. Hence option (c) -1 is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 7

Answer:Hint: Use distance formula

Given: Point P(a,b,c)

To Find: Distance of point P from x-axis

Solution:

The distance of the point (a,b,c) from x-axis will be perpendicular distance from point (a,b,c) to x-axis whose coordinates are (a,0,0)

Distance formula is given by

where

Therefore, option

is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 8

Answer: (b) 3:1 externally

Hint: Use of section formulaGiven: Points (1,2,3) and (4,2,1)

To find: Ratio in which xy plane divides the point

Solution: Let the xy plane divides the given point in the ratio t:1

Using section formula the point of intersection are:

The z-coordinate of any point of xy plane is zero.

Therefore, the xy plane divide the given points in 3:1 externally. Hence option (b) 3:1 externally is correct.

Directions Cosines and Direction Ratio Exercise Multiple Choice Question, question 9

Final Answer: (a) 3:2 externallyHint: Use section formula

Given: Points P(3,2,-4), Q(5,4,-6) and R(9,8,-10)

To Find: R divides PQ in what ratio

Solution:

Let R divides the line PQ in the ratio k:1

Using section formula

comparing x-coordinate

Therefore R divides the line PQ in ratio 3:2 externally. Hence option (b) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 10.

Final Answer:(a)

Hint: Bisection divides triangle in ratios

Given: A(3,2,0), B(5,3,2) and C(-9,6,-3) are vertices of triangle ABC

To find: Point D coordinates which meets BC at bisector of

Solution:

As bisector AD meets BC at D, therefore

Using distance formula

and,

Therefore,

D divides BC in the ratio 3:13 internally

Using section formula coordinates of D are

Hence coordinate of D are

Therefore option (a) is correct

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 11

Final answer: (a) (-1,2,-2)Hint: Use direction to find direction cosine

Given:

To Find: Point P

Solution: Direction cosine is related to direction ratio as

Now,

=Position of P- Position of O

where (x,y,z) are the coordinates of P

Now,

Comparing both,

(x,y,z)=(-1,2,-2)

Therefore coordinates of point P are (-1,2,-2). Hence option (a) is correct

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 12.

Final Answer:

(d)Given: A cube

To Find: Angle between its diagonals

Solution: Let us consider a cube OABCDEFG with vertices as shown

O(0,0,0), A(a,0,0), B(a,a,0), C(0,a,0), D(0,a,a), E(0,0,a), F(a,0,a), G(a,a,a)

There are four diagonals OG, CF, AD, and BE

Let us consider OG and AD

= Position vector of G- Position vector of O

Also,

=Position vector of D - Position vector of A

Now using dot product

Therefore angle between diagonals of cube is

Hence option (d) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 13.

Final Answer:(c)

Hint: Use direction cosines

Given: Line makes angles

with diagonals of cube.

To Find:

Solution: Let us consider a cube as shown

Direction ratio of OP are (a-0,a-0,a-0)

=(a,a,a)

Direction cosine of OP are

Similarly, direction cosine of all diagonals

Let (l,m,n) are the direction cosine of line which is inclined at angle

with diagonals. Now, using dot product

Squaring and adding above equations, we get

\

Therefore, value of

Hence option(c) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 14.

Final Answer: (a) (2,0,0)Given: Point (2,5,7)

To Find: Foot of perpendicular on x-axis

Solution: As the foot of perpendicular of the point (2,5,7) lie on x-axis, the x-coordinate at the foot is 2, while y and z coordinates on the x axis are always zero. So, the coordinates at foot of perpendicular are (2,0,0)

Therefore, opton (a) (2,0,0) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 15.

Final Answer: (a) 2Hint: Use section formula

Given: Points (3,2,-1) and (6,2,-2) and P(5,y,z)

To Find: y-coordinate of P(5,y,z)

Solution: Let P divide the line joining (3,2,-1)and (6,2,-2) in t:1

Using section formula

Comparing x-coordinate

Now comparing y-coordinate

Therefore, y-coordinate of P(5,y,z) is 2. hence option(a) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 16.

Final Answer:(d)

Hint: Use distance formula

Given:

Point P

To Find: Distance of P from y-axis

Solution: The distance of the point

from y-axis whose coordinates are

Distance formula is

where

Therefore, option(d) is correct.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 17.

Final Answer:(d)

Hint: Use property of direction cosines

Given: Direction cosine of line are k,k,k

To Find: value of k

Solution: Since direction cosine of line are k,k,k

We know that

Therefore, option (d) is correct

i.e.

Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 18.

Final Answer: (c) (0,-3,0)Given: Point (2,-3,4)

To Find: Foot of the perpendicular on y-axis

Solution: As the foot of the perpendicular of point (2,-3,4) lie on y-axis, the y-coordinate at the foot is -3, while x and z coordinate are zero on y-axis. So, the coordinate of foot of perpendicular are (0,-3,0)

Therefore, option (c) is correct.

The Class 12 RD Sharma Chapter 26 Exercise MCQ Solution book has become the most trusted reference guide among the students and the educational institutions. The following points explain why the RD Sharma Class 12 Solutions Direction Cosines & Direction Ratios Ex MCQ is an excellent guide:

A team of experts have dedicated their knowledge and skill in developing the best reference guide, the RD Sharma Class 12th exercise MCQ for the welfare of the students.

The RD Sharma Class 12 Solutions Direction Cosines & Direction Ratios ex MCQ is available according to the recent updated NCERT syllabus. Therefore, the students need not worry thinking if it is out-dated.

Most of the previous batch students have benefitted as the questions asked in public exams were picked from the RD Sharma books.

When the students use these books for completing their homework and assignments, they get a good practice to face their board exams as well.

The career 360 website provides the RD Sharma Class 12th Exercise MCQ books for free of cost to be accessed and downloaded from its website. Using the PDF format can be easier as it is easily accessible without carrying heavy solution guides wherever needed.

Students can use the RD Sharma Class 12 Solutions Chapter 26 ex MCQ book to compare with the solutions that they have initially tried. When they get to know the source of their mistakes, it will be easy to rectify them.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. What is the simplest way to download the RD Sharma class 12th exercise MCQ book?

The RD Sharma Class 12th Exercise MCQ solution can be accessed without any prior technical knowledge. Everyone can acquire this set of books easily by downloading it from the Career 360 website.

2. How many questions are there in Direction Cosines & Direction Ratios ex MCQ?

There are a total of 18 questions in Direction Cosines & Direction Ratios ex MCQ book. The chapter is pretty concise hence the questions are less in number.

3. Which are the best NCERT Solutions for Class 12?

The best NCERT solutions for class 12 are the RD Sharma solutions. Their maths solutions are exceptional and should be practiced by all students.

4. How do the RD Sharma Class 12th exercise MCQ Solution books help the students for their exams?

By owning the RD Sharma Class 12th exercise MCQ solution book, the students can prepare for their school tests, public exams, JEE mains and other entrance exams too. This saves a lot of money as no separate study material is required other than the RD Sharma books.

5. How are MCQ, VSA, FBQ, and RE sections important for exam preparations?

The MCQ, VSA, FBQ, and RE sections are essential for all exams. These parts contain questions from the entire chapter. If students practice this part thoroughly, they will have a clear understanding of the chapter in general.

Mar 22, 2023

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