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SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 2 - Vectors and Three Dimensional Geometry [Latest edition]

Chapters

12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 - Shaalaa.com
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Chapter 2: Vectors and Three Dimensional Geometry

MCQVery Short AnswersShort Answers IShort Answers IILong Answers III
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MCQ

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 2 Vectors and Three Dimensional Geometry MCQ

2 marks each

MCQ | Q 1

If |a̅| = 3, |b̅| =4, then the value of λ for which a̅ + λ b̅ is perpendicular to a̅ − λ b̅ is ______ 

  • `9/16`

  • `3/4`

  • `3/2`

  • `4/3`

MCQ | Q 2

`(hat"i" + hat"j" - hat"k")*(hat"i" - hat"j" + hat"k")` = ______

  • `hat"i" - hat"j" - hat"k"`

  • 1

  • −1

  • `−hat"j" + hat"k"`

MCQ | Q 3

The angle θ between two non-zero vectors `bar("a")` and `bar("b")` is given by cos θ = ______

  • `(bar"a"*bar"b")/(|bar("a")||bar("b")|)`

  • `bar("a")*bar("b")`

  • `|bar("a")||bar("b")|`

  • `(|bar("a")||bar("b")|)/(bar("a")*bar("b"))`

MCQ | Q 4

If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______ 

  • `sqrt(2)`

  • `sqrt(3)`

  • 1

  • 2

MCQ | Q 5

Select the correct option from the given alternatives:

If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______

  • 30° or 90°

  • 45° or 60°

  • 90° or 30°

  • 60° or 120°

MCQ | Q 6

Select the correct option from the given alternatives:

The distance of the point (3, 4, 5) from the Y-axis is ______ 

  • 3

  • 5

  • `sqrt(34)`

  • `sqrt(41)`

MCQ | Q 7

Select the correct option from the given alternatives:

If cos α, cos β, cos γ are the direction cosines of a line, then the value of sin2α + sin2β + sin2γ  is ______ 

  • 1

  • 2

  • 3

  • 4

MCQ | Q 8

If `|bar("a")|` = 2, `|bar("b")|` = 5, and `bar("a")*bar("b")` = 8 then `|bar("a") - bar("b")|` = ______ 

  • 13

  • 12

  • `sqrt(13)`

  • `sqrt(21)`

MCQ | Q 9

If `bar("AB") = 2hat"i" + hat"j" - 3hat"k"`, and A(1, 2 ,−1) is given point then coordinates of B are ____

  • (3, 3, −4)

  • (−3, 3 −2)

  • (3, 3, 2)

  • (−3, 3, 4)

MCQ | Q 10

Select the correct option from the given alternatives:

If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______ 

  • null vector

  • the unit vector along the line

  • any vector along the line

  • a vector perpendicular to the line

MCQ | Q 11

The values of c that satisfy `|"c"  bar("u")|` = 3, `bar("u") = hat"i" + 2hat"j" + 3hat"k"` is ______ 

  • `sqrt(14)`

  • `3sqrt(14)`

  • `3/sqrt(14)`

  • 3

MCQ | Q 12

The value of `hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j")`

  • 0

  • −1

  • 1

  • 3

MCQ | Q 13

Select the correct option from the given alternatives:

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC respectively of a ΔABC. The length of the median through A is

  • `sqrt(34)/2`

  • `sqrt(48)/2`

  • `sqrt(18)`

  • of the median through A is

Very Short Answers

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 2 Vectors and Three Dimensional Geometry Very Short Answers

1 Mark

Very Short Answers | Q 1

Find the magnitude of a vector with initial point : (1, −3, 4); terminal point : (1, 0, −1)

Very Short Answers | Q 2

Find the coordinates of the point which is located three units behind the YZ-plane, four units to the right of XZ-plane, and five units above the XY-plane.

Very Short Answers | Q 3

A(2, 3), B(−1, 5), C(−1, 1) and D(−7, 5) are four points in the Cartesian plane, Check if, `bar("CD")` is parallel to `bar("AB")`

Very Short Answers | Q 4

Find a unit vector in the opposite direction of `bar("u")`. Where `bar("u") = 8hat"i" + 3hat"j" - hat"k"`

Very Short Answers | Q 5

The non zero vectors `bar("a")` and `bar("b")` are not collinear find the value of `lambda` and `mu`: if `bar("a") + 3bar("b") = 2lambdabar("a") - mubar("b")`

Very Short Answers | Q 6

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

Very Short Answers | Q 7

Find the distance from (4, −2, 6) to the XZ- plane

Very Short Answers | Q 8

If the vectors `2hat"i" - "q"hat"j" + 3hat"k"` and `4hat"i" - 5hat"j" + 6hat"k"` are collinear then find the value of q

Very Short Answers | Q 9

Find `bar("a")*(bar("b") xx bar("c"))`, if `bar("a") = 3hat"i" - hat"j" + 4hat"k", bar("b") = 2hat"i" + 3hat"j" - hat"k", bar("c") = -5hat"i" + 2hat"j" + 3hat"k"`

Very Short Answers | Q 10

If a line makes angle 90°, 60° and 30° with the positive direction of X, Y and Z axes respectively, find its direction cosines

Short Answers I

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 2 Vectors and Three Dimensional Geometry Short Answers I

2 Marks

Short Answers I | Q 1

The vector `bar"a"` is directed due north and |a| = 24. The vector `bar"b"` is directed due west and `|bar"b"| = 7`. Find `|bar"a" + bar"b"|`.

Short Answers I | Q 2

Show that following points are collinear P(4, 5, 2), Q(3, 2, 4), R(5, 8, 0)

Short Answers I | Q 3

If a vector has direction angles 45° and 60°, find the third direction angle.

Short Answers I | Q 4

If `bar("c") = 3bar("a") - 2bar("b")` then prove that `[(bar("a"), bar("b"), bar("c"))]` = 0

Short Answers I | Q 5

If `|bar("a")*bar("b")| = |bar("a") xx bar("b")|` and `bar("a")*bar("b") < 0`, then find the angle between `bar("a")` and `bar("b")`

Short Answers I | Q 6

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2

Short Answers I | Q 7

If `bar("a"), bar("b")` and `bar("c")` are position vectors of the points A, B, C respectively and `5bar("a") - 3bar("b") - 2bar("c") = bar(0)`, then find the ratio in which the point C divides the line segement BA

Short Answers I | Q 8

If `bar("a")` and `bar("b")` are two vectors perpendicular each other, prove that `(bar("a") + bar("b"))^2 = (bar("a") - bar("b"))^2`

Short Answers I | Q 9

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `-5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3:2
(i) internally
(ii) externally

Short Answers I | Q 10

Find a unit vector perpendicular to the vectors `hat"j" + 2hat"k"`  and  `hat"i" + hat"j"`.

Short Answers II

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 2 Vectors and Three Dimensional Geometry Short Answers II

3 Marks

Short Answers II | Q 1

If two of the vertices of a triangle are A (3, 1, 4) and B(− 4, 5, −3) and the centroid of the triangle is at G (−1, 2, 1), then find the coordinates of the third vertex C of the triangle

Short Answers II | Q 2

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Short Answers II | Q 3

If a line has the direction ratios 4, −12, 18, then find its direction cosines

Short Answers II | Q 4

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Short Answers II | Q 5

Using properties of scalar triple product, prove that `[(bar"a" + bar"b",  bar"b" + bar"c",  bar"c" + bar"a")] = 2[(bar"a",  bar"b",  bar"c")]`.

Short Answers II | Q 6

The direction ratios of `bar"AB"` are −2, 2, 1. If A = (4, 1, 5) and l(AB) = 6 units, Then find B.

Short Answers II | Q 7

If G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c

Short Answers II | Q 8

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

Short Answers II | Q 9

Prove by vector method, that the angle subtended on semicircle is a right angle.

Short Answers II | Q 10

Prove that medians of a triangle are concurrent

Short Answers II | Q 11

Prove that altitudes of a triangle are concurrent

Long Answers III

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 Chapter 2 Vectors and Three Dimensional Geometry Long Answers III

4 Marks

Long Answers III | Q 1

Express `- hat"i" - 3hat"j" + 4hat"k"` as the linear combination of the vectors `2hat"i" + hat"j" - 4hat"k", 2hat"i" - hat"j" + 3hat"k"` and `3hat"i" + hat"j" - 2hat"k"`

Long Answers III | Q 2

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Long Answers III | Q 3

Prove that the angle bisectors of a triangle are concurrent

Long Answers III | Q 4

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

Long Answers III | Q 5

Find the angles between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0

Long Answers III | Q 6

Let `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Long Answers III | Q 7

D and E divides sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE

Long Answers III | Q 8

If `bar"u" = hat"i" - 2hat"j" + hat"k", bar"r" = 3hat"i" + hat"k"` and `bar"w" = hat"j", hat"k"` are given vectors , then find `[bar"u" + bar"w"]*[(bar"w" xx bar"r") xx (bar"r" xx bar"w")]`

Long Answers III | Q 9

Find the volume of a tetrahedron whose vertices are A (- 1, 2, 3), B (3, - 2, 1), C (2, 1, 3) and D (- 1, 2, 4).

Long Answers III | Q 10

If four points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d")` are coplanar, then show that `[(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`.

Chapter 2: Vectors and Three Dimensional Geometry

MCQVery Short AnswersShort Answers IShort Answers IILong Answers III
12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 - Shaalaa.com

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 2 - Vectors and Three Dimensional Geometry

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 2 (Vectors and Three Dimensional Geometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. SCERT Maharashtra Question Bank textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 chapter 2 Vectors and Three Dimensional Geometry are Representation of Vector, Vectors and Their Types, Algebra of Vectors, Coplanar Vectors, Vector in Two Dimensions (2-D), Three Dimensional (3-D) Coordinate System, Components of Vector, Position Vector of a Point P(X, Y, Z) in Space, Component Form of a Position Vector, Vector Joining Two Points, Section Formula, Scalar Product of Vectors (Dot), Vector Product of Vectors (Cross), Scalar Triple Product of Vectors, Vector Triple Product, Addition of Vectors.

Using SCERT Maharashtra Question Bank 12th Board Exam solutions Vectors and Three Dimensional Geometry exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in SCERT Maharashtra Question Bank Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer SCERT Maharashtra Question Bank Textbook Solutions to score more in exam.

Get the free view of chapter 2 Vectors and Three Dimensional Geometry 12th Board Exam extra questions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2021 and can use Shaalaa.com to keep it handy for your exam preparation

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