Chapter 24 : Scalar Or Dot Product - Rd Sharma Solutions for Class 12-science Maths CBSE

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

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Chapter 24 - Scalar Or Dot Product Exercise Ex. 24.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4
Solution 4
Question 5
Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10

Question 11

Solution 11

Question 12

begin mathsize 12px style table attributes columnalign left end attributes row cell text Dot   products   of   a   vector   with   vectors   end text straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma text   2 end text space straight i with hat on top plus straight j with hat on top minus 3 straight k with hat on top text   end text end cell row cell text and   end text straight i with hat on top plus straight j with hat on top plus straight k with hat on top text   are   respectively   4 , 0   and   2 .  Find   the   vector. end text end cell end table end style

Solution 12

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text the text    end text unknown text    end text vector text    end text be text    end text apostrophe straight a with rightwards arrow on top equals straight a subscript 1 straight i with hat on top plus straight b subscript 1 straight j with hat on top plus straight c subscript 1 straight k with hat on top apostrophe end cell row cell straight b with rightwards arrow on top equals straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma text    end text straight c with rightwards arrow on top equals 2 straight i with hat on top plus straight j with hat on top minus 3 straight k with hat on top comma straight d with rightwards arrow on top equals straight i with hat on top plus straight j with hat on top plus straight k with hat on top end cell row cell It text    end text is text    end text given text    end text that text    end text straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 4 end cell row cell straight a subscript 1 minus straight b subscript 1 plus straight c subscript 1 equals 4..... left parenthesis straight i right parenthesis end cell row cell straight a with rightwards arrow on top. straight c with rightwards arrow on top equals 0 end cell row cell 2 straight a subscript 1 plus straight b subscript 1 minus 3 straight c subscript 1 equals 0....... left parenthesis ii right parenthesis end cell row cell straight a with rightwards arrow on top. straight d with rightwards arrow on top equals 2 end cell row cell straight a subscript 1 plus straight b subscript 1 plus straight c subscript 1 equals 2........ left parenthesis iii right parenthesis end cell row blank row cell Solving text    end text left parenthesis straight i right parenthesis comma left parenthesis ii right parenthesis text    end text and text    end text left parenthesis iii right parenthesis comma end cell row cell straight a subscript 1 equals 2 comma straight b subscript 1 equals negative 1 comma straight c subscript 1 equals 1 end cell row blank row cell therefore the text    end text vector text    end text straight a with rightwards arrow on top equals 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top end cell end table end style

Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16

Solution 16
Question 17
Solution 17

Question 18
Solution 18

Question 19
Solution 19

Question 20

begin mathsize 12px style for space any space two space vectors space straight a with rightwards arrow on top and space straight b with rightwards arrow on top comma end style

Solution 20

Question 21

  

Solution 21

  

Question 22

begin mathsize 12px style table attributes columnalign left end attributes row cell If space straight p with rightwards arrow on top equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top space and space straight q with rightwards arrow on top  = straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top ,  then space find space the space value space of space straight lambda straight comma end cell row cell so space that space straight p with rightwards arrow on top plus straight q with rightwards arrow on top space and space straight p with rightwards arrow on top minus straight q with rightwards arrow on top space are space perpendicular space vectors. end cell end table end style

Solution 22

begin mathsize 12px style table attributes columnalign left end attributes row cell straight p with rightwards arrow on top equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top text   end text and text   end text straight q with rightwards arrow on top text   end text equals straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top end cell row cell straight p with rightwards arrow on top plus straight q with rightwards arrow on top text   end text end cell row cell equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top plus straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top end cell row cell equals 6 straight i with hat on top plus left parenthesis straight lambda plus 3 right parenthesis straight j with hat on top minus 8 straight k with hat on top end cell row blank row cell straight p with rightwards arrow on top minus straight q with rightwards arrow on top text   end text end cell row cell equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top minus straight i with hat on top minus 3 straight j with hat on top plus 5 straight k with hat on top end cell row cell equals 4 straight i with hat on top plus left parenthesis straight lambda minus 3 right parenthesis straight j with hat on top plus 2 straight k with hat on top end cell row blank row cell left parenthesis straight p with rightwards arrow on top plus straight q with rightwards arrow on top right parenthesis. left parenthesis straight p with rightwards arrow on top minus straight q with rightwards arrow on top right parenthesis equals 0 end cell row cell rightwards double arrow left square bracket 6 straight i with hat on top plus left parenthesis straight lambda plus 3 right parenthesis straight j with hat on top minus 8 straight k with hat on top right square bracket. left square bracket 4 straight i with hat on top plus left parenthesis straight lambda minus 3 right parenthesis straight j with hat on top plus 2 straight k with hat on top right square bracket equals 0 end cell row cell rightwards double arrow 24 plus left parenthesis straight lambda squared minus 9 right parenthesis minus 16 equals 0 end cell row cell rightwards double arrow straight lambda squared minus 9 plus 8 equals 0 end cell row cell rightwards double arrow straight lambda squared minus 1 equals 0 end cell row cell therefore straight lambda equals plus-or-minus 1 end cell end table end style

Question 23

Solution 23

Question 24

Solution 24

Question 25
Solution 25
Question 26

Solution 26

Question 27
Solution 27

Question 28

Solution 28

Question 29
Solution 29
Question 30

Solution 30


Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38
Solution 38
Question 39

Solution 39

Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44

Solution 44

Question 45
Solution 45
Question 46
Solution 46

 

Question 47
Solution 47
Question 48
Solution 48

Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

begin mathsize 12px style table attributes columnalign left end attributes row cell text If   end text straight a with rightwards arrow on top comma text   end text straight b with rightwards arrow on top text   are   two   vectors   such   that   end text vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line equals vertical line straight b with rightwards arrow on top vertical line comma text   then   prove   end text end cell row cell text that   end text straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top text   is   perpendicular   to   end text straight a with rightwards arrow on top. end cell end table end style

Solution 61

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals vertical line straight b with rightwards arrow on top vertical line squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 0 end cell row cell rightwards double arrow straight a with rightwards arrow on top. left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis equals 0 end cell row cell therefore straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top text    end text is text    end text perpendicular text    end text to text   end text straight a with rightwards arrow on top. end cell end table end style

Chapter 24 - Scalar Or Dot Product Exercise Ex. 24.2

Question 1

  

Solution 1

  

Question 2

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Solution 2

 

  

 

 

  

Question 3

(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other rum sides.

Solution 3

 

  

  

Question 4

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Solution 4

 

  

  

Question 5

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

Solution 5

 

  

Question 6

Prove that the diagonals of a rhombus are perpendicular bisectors of each other.

Solution 6

  

Question 7

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

Solution 7

 

  

Question 8

If AD is the median of D ABC, using vectors, prove that

AB2 + AC2 = 2 (AD2 + CD2).

Solution 8

 

  

  

Question 9

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.

Solution 9

  

Question 10

In a quadrilateral ABCD, prove that AB2 + BC2 + CD2+ DA2 = AC2 + BD2 + 4 PQ2, where P and Q are middle points of diagonals AC and BD. 

Solution 10

 

  

 

 

  

Chapter 24 - Scalar Or Dot Product Exercise MCQ

Question 1

Solution 1

Correct option: (c)

  

Question 2

Solution 2

Correct option: (b)

  

Question 3

Solution 3

Correct option: (d)

  

 

Question 4

Solution 4

Correct option: (c)

  

 

Question 5

(a) Null vector

(b) Unit vector

(c) Constant vector

(d) None of these

Solution 5

Correct option: (b)

  

Question 6

Solution 6

Correct option: (c)

  

 

Question 7

Solution 7

Correct option: (a)

  

Question 8

  1. A circle
  2. An ellipse
  3. A hyperbola
  4. None of these

 

Solution 8

Correct option: (b)

  

Question 9

Solution 9

Correct option: (b)

  

 

Question 10

Solution 10

Correct option: (c)

  

Question 11

Solution 11

Correct option: (d)

  

 

Question 12

Solution 12

Correct option: (b)

  

 

Question 13

Solution 13

Correct option: (b)

  

Question 14

Solution 14

Correct option: (c)

  

Question 15

Solution 15

Correct option: (c)

  

Question 16

  1. 1
  2. 0
  3. 2
  4. -1
  5. -2
Solution 16

Correct option: (a)

  

 

Question 17

Solution 17

Correct option: (b)

  

Question 18

  1. Positive
  2. Negative
  3. 0
  4. None of these
Solution 18

Correct option: (c)

  

 

Question 19

Solution 19

Correct option: (a)

  

Question 20

Solution 20

Correct option: (c)

  

Question 21

  1. (-4,7)
  2. [-4,7]
  3. R - [-4,7]
  4. R - (4,7)
Solution 21

Correct option: (c)

 

NOTE: Answer not matching with back answer.

 

 

Question 22

Solution 22

Correct option: (d)

 

NOTE: Answer not matching with back answer.

Question 23

Solution 23

Correct option: (d)

  

Question 24

Solution 24

Correct option: (b)

  

Question 25

Solution 25

Correct option: (d)

  

 

Question 26

Solution 26

Correct option: (d)

  

Chapter 24 - Scalar Or Dot Product Exercise Ex. 24VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13


Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

begin mathsize 12px style text Write   the   projection   of   the   vector   end text straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top text   on   the   vector   2 end text straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top. end style

Solution 33

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text straight a with rightwards arrow on top equals straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top end cell row cell straight b with rightwards arrow on top equals 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top end cell row blank row cell Projection text    end text of text    end text straight a with rightwards arrow on top text    end text on text    end text straight b with rightwards arrow on top end cell row cell equals straight a with rightwards arrow on top. fraction numerator straight b with rightwards arrow on top over denominator vertical line straight b with rightwards arrow on top vertical line end fraction end cell row cell equals left parenthesis straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top right parenthesis. fraction numerator left parenthesis 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis over denominator vertical line 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top vertical line end fraction end cell row cell equals fraction numerator left parenthesis straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top right parenthesis. left parenthesis 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis over denominator vertical line 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top vertical line end fraction end cell row cell equals fraction numerator 1 cross times 2 plus 3 cross times left parenthesis negative 3 right parenthesis plus 7 cross times 6 over denominator vertical line square root of 2 squared plus left parenthesis negative 3 right parenthesis squared plus 6 squared end root vertical line end fraction end cell row cell equals fraction numerator 33 over denominator square root of 49 end fraction end cell row cell equals 33 over 7 end cell end table end style

Question 34

Solution 34

Question 35

Solution 35

Question 36

begin mathsize 12px style text Write   the   projection   of   the   vector   7 end text straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top text   on   the   vector   2 end text straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top. end style

Solution 36

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text straight a with rightwards arrow on top equals 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top comma straight b with rightwards arrow on top equals 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top end cell row cell Projection text    end text of text    end text straight a with rightwards arrow on top text    end text on text    end text straight b with rightwards arrow on top equals straight a with rightwards arrow on top. fraction numerator straight b with rightwards arrow on top over denominator vertical line straight b with rightwards arrow on top vertical line end fraction end cell row cell equals left parenthesis 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis. fraction numerator left parenthesis 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top right parenthesis over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left parenthesis 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis. left parenthesis 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top right parenthesis over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left square bracket 7 cross times 2 plus 1 cross times 6 plus left parenthesis negative 4 right parenthesis cross times 3 right square bracket over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left square bracket 14 plus 6 minus 12 right square bracket over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator 8 over denominator vertical line square root of 49 vertical line end fraction equals 8 over 7 end cell end table end style

Question 37

begin mathsize 12px style table attributes columnalign left end attributes row cell text Write   the   value   of   end text straight lambda text   so   that   the   vectors   end text straight a with rightwards arrow on top equals text 2 end text straight i with hat on top plus straight lambda straight j with hat on top plus straight k with hat on top text   and   end text end cell row cell straight b with rightwards arrow on top equals straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top text   are   perpendicular   to   each   other. end text end cell end table end style

Solution 37

begin mathsize 12px style table attributes columnalign left end attributes row cell straight a with rightwards arrow on top equals text 2 end text straight i with hat on top plus straight lambda straight j with hat on top plus straight k with hat on top end cell row cell straight b with rightwards arrow on top equals straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top end cell row cell Since text    end text straight a with rightwards arrow on top text    end text and text    end text straight b with rightwards arrow on top text    end text are text    end text perpendicular text    end text to text    end text each text    end text other end cell row cell straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 0 end cell row cell rightwards double arrow left parenthesis text 2 end text straight i with hat on top plus straight lambda straight j with hat on top plus straight k with hat on top right parenthesis. left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top right parenthesis equals 0 end cell row cell rightwards double arrow 2 cross times 1 plus straight lambda cross times left parenthesis negative 2 right parenthesis plus 1 cross times 3 equals 0 end cell row cell rightwards double arrow 2 minus 2 straight lambda plus 3 equals 0 end cell row cell rightwards double arrow 5 equals 2 straight lambda end cell row cell therefore straight lambda equals 5 over 2 end cell end table end style

Question 38

begin mathsize 12px style table attributes columnalign left end attributes row cell Write text   end text the text   end text projection text   end text of text   end text straight b with rightwards arrow on top plus straight c with rightwards arrow on top text   end text on text   end text straight a with rightwards arrow on top comma text   end text when text   end text end cell row cell straight a with rightwards arrow on top equals 2 straight i with hat on top minus 2 straight j with hat on top plus straight k with hat on top comma text   end text straight b with rightwards arrow on top equals straight i with hat on top plus 2 straight j with hat on top minus 2 straight k with hat on top text   end text and text   end text straight c with rightwards arrow on top equals 2 straight i with hat on top minus straight j with hat on top plus 4 straight k with hat on top. end cell end table end style

Solution 38

Question 39

begin mathsize 12px style table attributes columnalign left end attributes row cell text If   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top text   are   perpendicular   vectors ,  end text end cell row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line equals 13 text   and   end text vertical line straight a with rightwards arrow on top vertical line equals 5 text ,  find   the   value   of   end text vertical line straight b with rightwards arrow on top vertical line. end cell end table end style

Solution 39

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma straight a with rightwards arrow on top text    end text and text    end text straight b with rightwards arrow on top text    end text are text    end text perpendicular text    end text to text    end text each text    end text other. end cell row cell straight a with rightwards arrow on top text   end text. straight b with rightwards arrow on top text   end text equals 0 end cell row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals 13 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals 169 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals 169 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line squared plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top plus vertical line straight b with rightwards arrow on top vertical line squared equals 9 end cell row cell rightwards double arrow 5 squared plus 2 cross times 0 plus vertical line straight b with rightwards arrow on top vertical line squared equals 169 end cell row cell rightwards double arrow vertical line straight b with rightwards arrow on top vertical line squared equals 169 minus 25 end cell row cell rightwards double arrow vertical line straight b with rightwards arrow on top vertical line squared equals 144 end cell row cell therefore vertical line straight b with rightwards arrow on top vertical line equals 12 end cell end table end style

Question 40

begin mathsize 12px style table attributes columnalign left end attributes row cell text If   vectors   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top text   are   such   that   end text vertical line straight a with rightwards arrow on top vertical line equals 3 text ,  end text vertical line straight b with rightwards arrow on top vertical line equals 2 over 3 text   and   end text end cell row cell straight a with rightwards arrow on top cross times straight b with rightwards arrow on top text   is   a   unit   vector ,  then   write   the   angle   between   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top. end cell end table end style

Solution 40

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 3 text ,  end text vertical line straight b with rightwards arrow on top vertical line equals 2 over 3 text   end text end cell row cell vertical line straight a with rightwards arrow on top cross times straight b with rightwards arrow on top vertical line equals vertical line vertical line straight a with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line sinθ. vertical line straight n with hat on top vertical line vertical line end cell row cell rightwards double arrow 1 equals vertical line 3 cross times 2 over 3 cross times sinθ.1 vertical line end cell row cell rightwards double arrow 1 equals 3 cross times 2 over 3 cross times sinθ end cell row cell rightwards double arrow sinθ equals 1 half end cell row cell therefore straight theta equals straight pi over 6 end cell end table end style

Question 41

begin mathsize 12px style table attributes columnalign left end attributes row cell text If   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top text   are   two   unit   vectors   such   that   end text straight a with rightwards arrow on top plus straight b with rightwards arrow on top text   is   also   end text end cell row cell text a   unit   vector ,  then   find   the   angle   between   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top. end cell end table end style

Solution 41

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 1 comma vertical line straight b with rightwards arrow on top vertical line equals 1 comma end cell row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals 1 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line squared plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus vertical line straight b with rightwards arrow on top vertical line squared equals 1 end cell row cell rightwards double arrow 1 plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 equals 1 end cell row cell rightwards double arrow 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 1 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 1 half end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line. cosθ equals negative 1 half end cell row cell rightwards double arrow 1 cross times 1 cross times cosθ equals negative 1 half end cell row cell rightwards double arrow cosθ equals cos fraction numerator 2 straight pi over denominator 3 end fraction end cell row cell therefore straight theta equals fraction numerator 2 straight pi over denominator 3 end fraction end cell end table end style

Question 42

begin mathsize 12px style table attributes columnalign left end attributes row cell text If   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top text   are   unit   vectors ,  then   find   the   angle   between   end text straight a with rightwards arrow on top text   and   end text straight b with rightwards arrow on top text ,  end text end cell row cell text given   that  ( end text square root of text 3 end text end root straight a with rightwards arrow on top minus straight b with rightwards arrow on top text )  is   a   unit   vector end text. end cell end table end style

Solution 42

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 1 comma vertical line straight b with rightwards arrow on top vertical line equals 1 comma end cell row cell vertical line straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top vertical line equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top vertical line squared equals 1 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow 3 straight a with rightwards arrow on top. straight a with rightwards arrow on top minus square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top minus square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals 1 end cell row cell rightwards double arrow 3 cross times 1 squared minus 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 squared equals 1 end cell row cell rightwards double arrow 3 minus 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 equals 1 end cell row cell rightwards double arrow negative 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 3 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight b with rightwards arrow on top equals fraction numerator 3 over denominator 2 square root of 3 end fraction end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line. vertical line straight b with rightwards arrow on top vertical line cosθ equals fraction numerator square root of 3 over denominator 2 end fraction end cell row cell rightwards double arrow cosθ equals fraction numerator square root of 3 over denominator 2 end fraction end cell row cell rightwards double arrow cosθ equals cos straight pi over 6 end cell row cell therefore straight theta equals straight pi over 6 end cell end table end style

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Why CBSE Class 12 Science Maths solutions are important?

Maths is a subject which requires practising a variety of problems to understand concepts clearly. By solving as many problems as you can, you’ll be able to train your brain in thinking the logical way to solve maths problems. For practising problems, study materials such as sample papers, previous year papers, and NCERT solutions are needed.

Some of the best Maths experts work with us to give you the best solutions for Maths textbook questions and sample paper questions. Chapter-wise NCERT solutions for Class 12 Science Maths can be easily accessible on TopperLearning. Use these solutions to practise problems based on concepts such as direction ratios, probability, area between lines, inverse trigonometric functions, and more.

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