#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrics

Chapter 1.3: Trigonometric Functions

Chapter 1.4: Pair of Lines

Chapter 1.5: Vectors and Three Dimensional Geometry

Chapter 1.6: Line and Plane

Chapter 1.7: Linear Programming Problems

Chapter 2.1: Differentiation

Chapter 2.2: Applications of Derivatives

Chapter 2.3: Indefinite Integration

Chapter 2.4: Definite Integration

Chapter 2.5: Application of Definite Integration

Chapter 2.6: Differential Equations

Chapter 2.7: Probability Distributions

Chapter 2.8: Binomial Distribution

## 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 MCQ

#### 2 marks each

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`

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

`hat"i" - hat"j" - hat"k"`

1

−1

`−hat"j" + hat"k"`

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"))`

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

**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°

**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)`

**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 sin^{2}α + sin^{2}β + sin^{2}γ is ______

1

2

3

4

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

13

12

`sqrt(13)`

`sqrt(21)`

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)

**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

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

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

**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

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

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.

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")`

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

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")`

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

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

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

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"`

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

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"|`.

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

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

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

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")`

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

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

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

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

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

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

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

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

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

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")]`.

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

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

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

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

Prove that medians of a triangle are concurrent

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

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"`

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

Prove that the angle bisectors of a triangle are concurrent

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)

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

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") `

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

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")]`

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).

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

## 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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