# 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]

## Chapter 2: Vectors and Three Dimensional Geometry

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

### 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

### 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".

### 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

### 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")].

## 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.

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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.

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