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RD Sharma Solution for Class 12 Science Mathematics Chapter 27 - Direction Cosines and Direction Ratios

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 27 - Direction Cosines and Direction Ratios.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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Exercise/Page

RD Sharma Solution for Class 12 Science Mathematics Chapter 27 - Direction Cosines and Direction Ratios Page/Excercise 27.1

Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

 

begin mathsize 12px style text m end text equals text cos   60 end text degree equals 1 half end style


begin mathsize 12px style straight n equals cos text   end text 30 degree equals fraction numerator square root of 3 over denominator 2 end fraction end style

begin mathsize 12px style therefore straight T text he   direction   cosines   of   the   line   are   0 , end text 1 half comma fraction numerator square root of 3 over denominator 2 end fraction. end style

Solution 2

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text the text    end text direction text    end text cosines text    end text of text    end text the text    end text line text    end text be text    end text straight l comma straight m comma straight n. end cell row cell Here comma end cell row cell straight a equals 2 comma straight b equals negative 1 comma straight c equals negative 2 text    end text are text    end text the text    end text direction text    end text ratios text    end text of text    end text the text    end text line. end cell row cell straight l equals plus-or-minus fraction numerator straight a over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight m equals plus-or-minus fraction numerator straight b over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight n equals plus-or-minus fraction numerator straight c over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 9 end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 9 end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 9 end fraction end cell row cell straight l equals 2 over 3 comma straight m equals negative 1 third comma straight n equals negative 2 over 3 end cell row cell therefore The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text are text    end text 2 over 3 comma negative 1 third comma negative 2 over 3. end cell end table end style

Solution 3

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text joining text    end text left parenthesis negative 2 comma 4 comma negative 5 right parenthesis text   end text and text   end text left parenthesis 1 comma text   end text 2 comma text   end text 3 right parenthesis text   are end text comma end cell row cell left parenthesis 1 plus 2 comma 2 minus 4 comma 3 plus 5 right parenthesis equals left parenthesis 3 comma negative 2 comma 8 right parenthesis end cell row cell Here comma straight a equals 3 comma straight b equals negative 2 comma straight c equals 8 end cell row cell Direction text    end text cosines text    end text are end cell row cell fraction numerator 3 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator negative 2 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator 8 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction end cell row cell equals fraction numerator 3 over denominator square root of 77 end fraction comma fraction numerator negative 2 over denominator square root of 77 end fraction comma fraction numerator 8 over denominator square root of 77 end fraction end cell end table end style

Solution 4

begin mathsize 12px style table attributes columnalign left end attributes row cell Here text    end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis. end cell row cell Direction text    end text ratios text    end text of text    end text AB equals left parenthesis 1 minus 2 comma negative 2 minus 3 comma 3 plus 4 right parenthesis equals left parenthesis negative 1 comma negative 5 comma 7 right parenthesis end cell row cell Direction text    end text ratios text    end text of text    end text BC equals left parenthesis 3 minus 1 comma 8 plus 2 comma negative 11 minus 3 right parenthesis equals left parenthesis 2 comma 10 comma negative 14 right parenthesis end cell row blank row cell Here comma text    end text the text    end text respective text    end text direction text    end text cosines text    end text of text    end text AB text    end text and text    end text AC comma end cell row cell fraction numerator negative 1 over denominator 2 end fraction equals fraction numerator negative 5 over denominator 10 end fraction equals fraction numerator 7 over denominator negative 14 end fraction text    end text are text    end text proportional. end cell row blank row cell Also comma text   end text straight B text    end text is text    end text the text    end text common text    end text point text    end text between text    end text the text    end text two text    end text lines comma end cell row cell therefore The text    end text points text     end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis text    end text are text    end text collinear. text   end text end cell row blank end table end style

Solution 5

begin mathsize 12px style table attributes columnalign left end attributes row cell straight A left parenthesis 3 comma 5 comma negative 4 right parenthesis comma straight B left parenthesis negative 1 comma 1 comma 2 right parenthesis text    end text and text    end text straight C left parenthesis negative 5 comma negative 5 comma negative 2 right parenthesis end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AB equals left parenthesis negative 1 minus 3 comma 1 minus 5 comma 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 4 comma 6 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AB text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator 6 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator 3 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text BC equals left parenthesis negative 5 plus 1 comma negative 5 minus 1 comma negative 2 minus 2 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 6 comma negative 4 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text BC text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 6 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 3 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AC equals left parenthesis negative 5 minus 3 comma negative 5 minus 5 comma negative 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 8 comma negative 10 comma 2 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AC text    end text will text    end text be end cell row cell fraction numerator negative 8 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator negative 10 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator 2 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction end cell row cell equals fraction numerator negative 8 over denominator square root of 168 end fraction comma fraction numerator negative 10 over denominator square root of 168 end fraction comma fraction numerator 2 over denominator square root of 168 end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 42 end fraction comma fraction numerator negative 5 over denominator square root of 42 end fraction comma fraction numerator 1 over denominator square root of 42 end fraction end cell end table end style

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16(i)

Solution 16(ii)

Solution 16(iii)

Solution 16(iv)

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 27 - Direction Cosines and Direction Ratios Page/Excercise MCQ

Solution 1

Correct option: (c)

Every point on xy-plane z co-ordinate is always zero.

 

Solution 2

Correct option: (c)

Point (x, y, z) is on the x-axis. Hence, y and z co-ordinate will be zero except the origin.

Solution 3

Correct option: (d)

Coordinates of the points given are diagonally opposite vertices of a parallelepiped. Hence, edges of parallelepiped can be 5-2, 7-3, 9-7 3,4,2.

Solution 4

Correct option : (a)

  

 

Solution 5

Correct option: (b)

  

Solution 6

Correct option: (c)

  

Solution 7

Correct option: (a)

  

Solution 8

Correct option: (b)

  

Solution 9

Correct option: (b)

  

 

Solution 10

Correct option: (a)

  

Solution 11

Correct option: (a)

Directions of OP from the origin

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

 

Solution 12

Correct option: (d)

  

Solution 13

Correct option: (c)

  

 

RD Sharma Solution for Class 12 Science Mathematics Chapter 27 - Direction Cosines and Direction Ratios Page/Excercise 27VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text cosines text    end text of text    end text the text    end text vector text    end text are end cell row cell straight l equals cos straight pi over 3 equals 1 half end cell row cell straight m equals cos straight pi over 4 equals fraction numerator 1 over denominator square root of 2 end fraction end cell row cell straight n equals cosθ end cell row blank row cell We text    end text know comma end cell row cell straight l squared plus straight m squared plus straight n squared equals 1 end cell row cell rightwards double arrow left parenthesis 1 half right parenthesis squared plus left parenthesis fraction numerator 1 over denominator square root of 2 end fraction right parenthesis squared plus cos squared straight theta equals 1 end cell row cell rightwards double arrow 1 fourth plus 1 half plus cos squared straight theta equals 1 end cell row cell rightwards double arrow cos squared straight theta equals 1 minus 1 half minus 1 fourth end cell row cell rightwards double arrow cos squared straight theta equals 1 fourth end cell row cell rightwards double arrow cosθ equals 1 half end cell row cell therefore straight theta equals straight pi over 3 end cell end table end style

Solution 20

begin mathsize 12px style table attributes columnalign left end attributes row cell Any text    end text point text    end text on text    end text the text    end text straight x minus axis text    end text can text    end text be text    end text written text    end text as text   end text left parenthesis straight a comma 0 comma 0 right parenthesis end cell row cell therefore Distance text    end text of text    end text the text    end text point text    end text from text    end text the text    end text straight x minus axis end cell row cell equals square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root end cell row cell equals square root of left parenthesis straight a minus straight a right parenthesis squared plus left parenthesis straight b minus 0 right parenthesis squared plus left parenthesis straight c minus 0 right parenthesis squared end root end cell row cell equals square root of 0 squared plus straight b squared plus straight c squared end root end cell row cell equals square root of straight b squared plus straight c squared end root end cell row cell therefore Distance text    end text of text    end text the text    end text point text    end text from text    end text the text    end text straight x minus axis text    end text is text    end text square root of straight b squared plus straight c squared end root. end cell end table end style

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 27 - Direction Cosines and Direction Ratios for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 27 - Direction Cosines and Direction Ratios.

Text Book Solutions

CBSE XII Science - Mathematics

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