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Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 - Vectors [Latest edition]

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Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Chapter 5: Vectors

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Miscellaneous exercise 5

Exercise 5.1 [Pages 151 - 152]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.1 [Pages 151 - 152]

Exercise 5.1 | Q 1 | Page 151

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"|`.

Exercise 5.1 | Q 2 | Page 151

In the triangle PQR, `bar"PQ" = bar"2a", bar"QR" = bar"2b"`. The midpoint of PR is M. Find the following vectors in terms of `bar"a"` and `bar"b"`:

(i) `bar"PR"` (ii) `bar"PM"` (iii) `bar"QM"`.

Exercise 5.1 | Q 3 | Page 151

OABCDE is a regular hexagon. The points A and B have position vectors `bar"a"` and `bar"b"` respectively referred to the origin O. Find, in terms of `bar"a"` and `bar"b"` the position vectors of C, D and E.

Exercise 5.1 | Q 4 | Page 151

ABCDEF is a regular hexagon. Show that `bar"AB" + bar"AC" + bar"AD" + bar"AE" + bar"AF" = 6bar"AO"`, where O is the centre of the hexagon.

Exercise 5.1 | Q 5 | Page 151

Check whether the vectors `2hat"i" + 2hat"j" + 3hat"k",  - 3hat"i" + 3hat"j" + 2hat"k"` and `3hat"i" + 4hat"k"` form a triangle or not.

Exercise 5.1 | Q 6 | Page 151

In the given figure express `bar"c"` and `bar"d"` in terms of `bar"a"` and `bar"b"`.

Exercise 5.1 | Q 7 | Page 151

Find a vector the direction of `bar"a" = hat"i" - 2hat"j"` that has magnitude 7 units.

Exercise 5.1 | Q 8 | Page 151

Find the distance of (4, - 2, 6) from each of the following:
(a) The XY-plane
(b) The YZ-plane
(c) The XZ-plane
(d) The X-axis
(e) The Y-axis
(f) The Z-axis.

Exercise 5.1 | Q 9.1 | Page 152

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.

Exercise 5.1 | Q 9.2 | Page 152

Find the coordinates of the point which is located in the YZ-plane, one unit to the right of the XZ- plane, and six units above the XY-plane.

Exercise 5.1 | Q 10 | Page 152

Find the area of the traingle with vertices (1, 1, 0), (1, 0, 1) and (0, 1, 1).

Exercise 5.1 | Q 11 | Page 152

If `bar"AB" = 2hat"i" - 4hat"j" + 7hat"k"` and initial point A(1, 5, 0). Find the terminal point B.

Exercise 5.1 | Q 12.1 | Page 152

Show that the following points are collinear:

A = (3, 2, -4), B = (9, 8, -10), C = (-2, -3, 1)

Exercise 5.1 | Q 12.2 | Page 152

Show that the following points are collinear:

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).

Exercise 5.1 | Q 13 | Page 152

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

Exercise 5.1 | Q 14 | Page 152

Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer.

Exercise 5.1 | Q 15 | Page 152

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"`.

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Exercise 5.2 [Page 160]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.2 [Page 160]

Exercise 5.2 | Q 1.1 | Page 160

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 is internally.

Exercise 5.2 | Q 1.2 | Page 160

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 is externally.

Exercise 5.2 | Q 2 | Page 160

Find the position vector of midpoint M joining the points L(7, - 6, 12) and N(5, 4, - 2).

Exercise 5.2 | Q 3 | Page 160

If the points A (3, 0, p), B (- 1, q, 3) and C (- 3, 3, 0) are collinear, then find
(i) the ratio in which the point C divides the line segment AB
(ii) the values of p and q.

Exercise 5.2 | Q 4 | Page 160

The position vector of points A and B are `6bar"a" + 2bar"b"` and `bar"a" - 3bar"b"`. If the point C divides AB in the ratio 3 : 2, show that the position vector of C is `3bar"a" - bar"b"`.

Exercise 5.2 | Q 5 | Page 160

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Exercise 5.2 | Q 6 | Page 160

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

Exercise 5.2 | Q 7 | Page 160

Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.

Exercise 5.2 | Q 8 | Page 160

Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of the lengths of the parallel sides.

Exercise 5.2 | Q 9 | Page 160

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.

Exercise 5.2 | Q 10 | Page 160

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Exercise 5.2 | Q 11 | Page 160

If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.

Exercise 5.2 | Q 12 | Page 160

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

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Exercise 5.3 [Pages 169 - 170]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.3 [Pages 169 - 170]

Exercise 5.3 | Q 1 | Page 169

Find two unit vectors each of which is perpendicular to both `bar"u"` and `bar"v"` where `bar"u" = 2hat"i" + hat"j" - 2hat"k"`,  `bar"v" = hat"i" + 2hat"j" - 2hat"k"`.

Exercise 5.3 | Q 2 | Page 169

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

Exercise 5.3 | Q 3 | Page 169

Find the values of c so that for all real x, the vectors `"xc"hat"i" - 6hat"j" + 3hat"k"` and `"x"hat"i" + 2hat"j" + 2"cx"hat"k"` make an obtuse angle.

Exercise 5.3 | Q 4 | Page 169

Show that the sum of the length of projections of `"p"hat"i" + "q"hat"j" + "r"hat"k"` on the coordinate axes, where p = 2, q = 3 and r = 4 is 9.

Exercise 5.3 | Q 5 | Page 169

Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

Exercise 5.3 | Q 6.1 | Page 169

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = - 9hat"i" + 6hat"j" + 15hat"k"` , `bar"b" = 6hat"i" - 4hat"j" - 10hat"k"`

Exercise 5.3 | Q 6.2 | Page 169

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = 2hat"i" + 3hat"j" - hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`

Exercise 5.3 | Q 6.3 | Page 169

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = -3/5hat"i" + 1/2hat"j" + 1/3hat"k"` , `bar"b" = 5hat"i" + 4hat"j" + 3hat"k"`

Exercise 5.3 | Q 6.4 | Page 169

Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:

`bar"a" = 4hat"i" - hat"j" + 6hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`

Exercise 5.3 | Q 7 | Page 169

Find the angle P of the triangle whose vertices are P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1).

Exercise 5.3 | Q 8.1 | Page 169

If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"q".`

Exercise 5.3 | Q 8.2 | Page 169

If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"r".`

Exercise 5.3 | Q 9 | Page 169

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

Exercise 5.3 | Q 10 | Page 169

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

Exercise 5.3 | Q 11 | Page 169

If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.

Exercise 5.3 | Q 12 | Page 170

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

Exercise 5.3 | Q 13 | Page 170

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

Exercise 5.3 | Q 14 | Page 170

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

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Exercise 5.4 [Pages 178 - 179]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.4 [Pages 178 - 179]

Exercise 5.4 | Q 1 | Page 178

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

Exercise 5.4 | Q 2 | Page 178

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

Exercise 5.4 | Q 3 | Page 178

If `bar"a".bar"b" = sqrt3` and `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`, find the angle between `bar"a"` and `bar"b"`.

Exercise 5.4 | Q 4 | Page 178

If `bar"a" = 2hat"i" + hat"j" - 3hat"k"` and  `bar"b" = hat"i" - 2hat"j" + hat"k"`, find a vector of magnitude 5 perpendicular to both `bar"a"` and `bar"b"`.

Exercise 5.4 | Q 5.1 | Page 178

Find `bar"u".bar"v"` if `|bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8`

Exercise 5.4 | Q 5.2 | Page 178

Find `|bar"u" xx bar"v"|` if `|bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12`

Exercise 5.4 | Q 6 | Page 178

Prove that `2(bar"a" - bar"b") xx 2(bar"a" + bar"b") = 8(bar"a" xx bar"b")`

Exercise 5.4 | Q 7 | Page 178

If `bar"a" = hat"i" - 2hat"j" + 3hat"k"`  , `bar"b" = 4hat"i" - 3hat"j" + hat"k"` , `bar"c" = hat"i" - hat"j" + 2hat"k"` verify that `bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"`

Exercise 5.4 | Q 8 | Page 178

Find the area of the parallelogram whose adjacent sides are `bar"a" = 2hat"i" - 2hat"j" + hat"k"` and `bar"b" = hat"i" - 3hat"j" - 3hat"k"`

Exercise 5.4 | Q 9 | Page 178

Show that vector area of a parallelogram ABCD is `1/2 (bar"AC" xx bar"BD")` where AC and BD are its diagonals.

Exercise 5.4 | Q 10 | Page 179

Find the area of parallelogram whose diagonals are determined by the vectors `bar"a" = 3hat"i" - hat"j" - 2hat"k"` and `bar"b" = - hat"i" + 3hat"j" - 3hat"k"`.

Exercise 5.4 | Q 11 | Page 179

If `bar"a", bar"b", bar"c", bar"d"` are four distinct vectors such that `bar"a" xx bar"b" = bar"c" xx bar"d"` and `bar"a" xx bar"c" = bar"b" xx bar"d"` prove that `bar"a" - bar"d"` is parallel to `bar"b" - bar"c"`.

Exercise 5.4 | Q 12 | Page 179

If `bar"a" = hat"i" + hat"j" + hat"k"  "and"  bar"c" = hat"j" - hat"k"`, find `bar"a"` vector `bar"b"` satisfying `bar"a" xx bar"b" = bar"c"  "and"  bar"a".bar"b" = 3`

Exercise 5.4 | Q 13 | Page 179

Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`

Exercise 5.4 | Q 14 | Page 179

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"`.

Exercise 5.4 | Q 15 | Page 179

Prove, by vector method, that sin (α + β) = sin α . cos β + cos α . sin β

Exercise 5.4 | Q 16.1 | Page 179

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are - 2, 1, - 1 and - 3, - 4, 1

Exercise 5.4 | Q 16.2 | Page 179

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

Exercise 5.4 | Q 17 | Page 179

Prove that the two vectors whose direction cosines are given by relations al  + bm + cn = 0 and fmn  + gnl + hlm = 0 are perpendicular, if `"f"/"a" + "g"/"b" + "h"/"c" = 0`

Exercise 5.4 | Q 18 | Page 179

If A(1, 2, 3) and B(4, 5, 6) are two points, then find the foot of the perpendicular from the point B to the line joining the origin and the point A.

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Exercise 5.5 [Pages 183 - 184]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Exercise 5.5 [Pages 183 - 184]

Exercise 5.5 | Q 1 | Page 183

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"` and `bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"` 

Exercise 5.5 | Q 2 | Page 183

If the vectors `3hat"i" + 5hat"k", 4hat"i" + 2hat"j" - 3hat"k"` and `3hat"i" + hat"j" + 4hat"k"`  are the coterminus edges of the parallelopiped, then find the volume of the parallelopiped.

Exercise 5.5 | Q 3 | Page 183

If the vectors `- 3hat"i" + 4hat"i" - 2hat"k" , hat"i" + 2hat"k"` and `hat"i" - "p"hat"j"` are coplanar, then find the value of p.

Exercise 5.5 | Q 4.1 | Page 184

Prove that `[bar"a"  bar"b" + bar"c"  bar"a" + bar"b" + bar"c"] = 0`

Exercise 5.5 | Q 4.2 | Page 184

Prove that `(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a" bar"b" bar"c"]`.

Exercise 5.5 | Q 5 | Page 184

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

Exercise 5.5 | Q 6 | Page 184

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"u" xx bar"r") xx (bar"r" xx bar"w")]`

Exercise 5.5 | Q 7 | Page 184

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).

Exercise 5.5 | Q 8 | Page 184

If `bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j"` and `bar"c" = 2hat"i" + hat"j" + 3hat"k"`, then verify that `bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`

Exercise 5.5 | Q 9 | Page 184

If `bar"a" = hat"i" - 2hat"j"`, `bar"b" = hat"i" + 2hat"j" , bar"c" = 2hat"i" + hat"j" - 2hat"k"`, then find (i) `bar"a" xx (bar"b" xx bar"c")` (ii) `(bar"a" xx bar"b") xx bar"c"` Are the results same? Justify.

Exercise 5.5 | Q 10 | Page 184

Show that `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"`

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Miscellaneous exercise 5 [Pages 187 - 189]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Miscellaneous exercise 5 [Pages 187 - 189]

Miscellaneous exercise 5 | Q 1.01 | Page 187

Select the correct option from the given alternatives:

If `|bar"a"| = 2, |bar"b"| = 3, |bar"c"| = 4` then `[bar"a" + bar"b"    bar"b" + bar"c"    bar"c" - bar"a"]` is equal to

  • 24

  • - 24

  • 0

  • 48

Miscellaneous exercise 5 | Q 1.02 | Page 188

Select the correct option from the given alternatives:

If `|bar"a"| = 3, |bar"b"| = 4,` then the value of λ for which `bar"a" + lambdabar"b"`, is perpendicular to `bar"a" - lambdabar"b"`, is

  • `9/16`

  • `3/4`

  • `3/2`

  • `4/3`

Miscellaneous exercise 5 | Q 1.03 | Page 188

Select the correct option from the given alternatives:

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

  • `sqrt2`

  • `sqrt3`

  • 1

  • 2

Miscellaneous exercise 5 | Q 1.04 | Page 188

Select the correct option from the given alternatives:

If `|bar"a"| = 3, |bar"b"| = 5, |bar"c"| = 7 and bar"a" + bar"b" + bar"c" = bar0`, then the angle between `bar"a"  "and"  bar"b"` is

  • `pi/2`

  • `pi/3`

  • `pi/4`

  • `pi/6`

Miscellaneous exercise 5 | Q 1.05 | Page 188

Select the correct option from the given alternatives:

The volume of tetrahedron whose vectices are (1,-6,10), (-1, -3, 7), (5, -1, λ) and (7, -4, 7) is 11 cu units, then the value of λ is

  • 7

  • 2

  • 1

  • 5

Miscellaneous exercise 5 | Q 1.06 | Page 188

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°

Miscellaneous exercise 5 | Q 1.07 | Page 188

Select the correct option from the given alternatives:

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

  • 3

  • 5

  • `sqrt34`

  • `sqrt41`

Miscellaneous exercise 5 | Q 1.08 | Page 188

Select the correct option from the given alternatives:

The line joining the points (2, 1, 8) and (a, b, c) is parallel to the line whose direction ratios are 6, 2, 3. The value of a, b, c are

  • 4, 3, - 5

  • 1, 2, `(- 13)/2`

  • 10, 5, -2

  • 3, 5, 11

Miscellaneous exercise 5 | Q 1.09 | Page 188

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

Miscellaneous exercise 5 | Q 1.1 | Page 188

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

Miscellaneous exercise 5 | Q 1.11 | Page 188

Select the correct option from the given alternatives:

If `|bar"a"| = 3` and - 1 ≤ k ≤ 2, then `|"k"bar"a"|` lies in the interval

  • [0, 6]

  • [-3, 6]

  • [3, 6]

  • [1, 2]

Miscellaneous exercise 5 | Q 1.12 | Page 188

Select the correct option from the given alternatives:

Let α, β, γ be distinct real numbers. The points with position vectors `alphahat"i" + betahat"j" + gammahat"k",  betahat"i" + gammahat"j" + alphahat"k",   gammahat"i" + alphahat"j" + betahat"k"`

  • are collinear

  • form an equilateral triangle

  • form a scalene triangle

  • form a right angled triangle

Miscellaneous exercise 5 | Q 1.13 | Page 189

Select the correct option from the given alternatives:

Let `bar"p"  "and"  bar"q"` be the position vectors of P and Q respectively, with respect to O and `|bar"p"| = "p", |bar"q"| = "q"`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular; then

  • 9p2 = 4q2 

  • 4p2 = 9q2 

  • 9p = 4q

  • 4p = 9q 

Miscellaneous exercise 5 | Q 1.14 | Page 189

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 Δ ABC. The length of the median through A is

  • `sqrt34/2`

  • `sqrt48/2`

  • `sqrt18`

  • of the median through A is

Miscellaneous exercise 5 | Q 1.14 | Page 189

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 Δ ABC. The length of the median through A is

  • `sqrt34/2`

  • `sqrt48/2`

  • `sqrt18`

  • of the median through A is

Miscellaneous exercise 5 | Q 1.15 | Page 189

Select the correct option from the given alternatives:

If `bar"a"  "and"  bar"b"` are unit vectors, then what is the angle between `bar"a"` and `bar"b"` for `sqrt3bar"a" - bar"b"` to be a unit vector?

  • 30°

  • 45°

  • 60°

  • 90°

Miscellaneous exercise 5 | Q 1.16 | Page 189

Select the correct option from the given alternatives:

If θ be the angle between any two vectors `bar"a"  "and"  bar"b"` then `|bar"a" . bar"b"| = |bar"a" xx bar"b"|`, when θ is equal to

  • 0

  • `pi/4`

  • `pi/2`

  • `pi`

Miscellaneous exercise 5 | Q 1.17 | Page 189

Select the correct option from the given alternatives:

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

  • 0

  • - 1

  • 1

  • 3

Miscellaneous exercise 5 | Q 1.18 | Page 189

Select the correct option from the given alternatives:

Let a, b, c be distinct non-negative numbers. If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k" , hat"i" + hat"k"  "and"  "c"hat"i" + "c"hat"j" + "b"hat"k"` lie in a plane, then c is

  • the arithmetic mean of a and b

  • the geometric mean of a and b

  • the harmonic man of a and b

  • 0

Miscellaneous exercise 5 | Q 1.19 | Page 189

Select the correct option from the given alternatives:

Let `bar"a" = hat"i" - hat"j", bar"b" = hat"j" - hat"k", bar"c" = hat"k" - hat"i".` If `bar"d"` is a unit vector such that `bar"a". bar"d" = 0 = [bar"b" bar"c" bar"d"]`, then `bar"d"` equals

  • `+- (hat"i" + hat"j" - 2hat"k")/sqrt6`

  • `+- (hat"i" + hat"j" + hat"k")/sqrt3`

  • `+-(hat"i" + hat"j" - hat"k")/sqrt3`

  • `+-  hat"k"`

Miscellaneous exercise 5 | Q 1.2 | Page 189

Select the correct option from the given alternatives:

If `bar"a", bar"b", bar"c"` are non-coplanar unit vectors such that `bar"a"xx (bar"b"xxbar"c") = (bar"b"+bar"c")/sqrt2`, then the angle between `bar"a"  "and"  bar"b"` is 

  • `(3pi)/4`

  • `pi/4`

  • `pi/2`

  • `pi`

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Miscellaneous exercise 5 [Pages 190 - 193]

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 5 Vectors Miscellaneous exercise 5 [Pages 190 - 193]

Miscellaneous exercise 5 | Q 1 | Page 190

ABCD is a trapezium with AB parallel to DC and DC = 3AB. M is the midpoint of DC. `bar"AB" = bar"p", bar"BC" = bar"q"`.
Find in terms of `bar"p" and bar"q"`:

(i) `bar"AM"` (ii) `bar"BD"` (iii) `bar "MB"` (iv) `bar"DA"`

Miscellaneous exercise 5 | Q 2 | Page 190

The points A, B, C have position vectors `bar"a", bar"b" and bar"c"` respectively. The point P is the midpoint of AB. Find the vector `bar"PC"` in terms of `bar"a", bar"b", bar"c"`.

Miscellaneous exercise 5 | Q 3 | Page 190

In a pentagon ABCDE, show that `bar"AB" + bar"AE" + bar"BC" + bar"DC" + bar"ED" = 2bar"AC"`

Miscellaneous exercise 5 | Q 4 | Page 190

In a parallelogram ABCD, diagonal vectors are `bar"AC" = 2hat"i" + 3hat"j" + 4hat"k" and bar"BD" = - 6hat"i" + 7hat"j" - 2hat"k"`, then find the adjacent side vectors `bar"AB" and bar"AD"`.

Miscellaneous exercise 5 | Q 5 | Page 190

If two sides of a triangle are `hat"i" + 2hat"j" and hat"i" + hat"k"`, find the length of the third side.

Miscellaneous exercise 5 | Q 6 | Page 190

If `|bar"a"| = |bar"b"| = 1,  bar"a".bar"b" = 0, bar"a" + bar"b" + bar"c" = bar"0", "find"  |bar"c"|`.

Miscellaneous exercise 5 | Q 7.1 | Page 190

Find the lengths of the sides of the triangle and also determine the type of a triangle:

A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)

Miscellaneous exercise 5 | Q 7.2 | Page 190

Find the lengths of the sides of the triangle and also determine the type of a triangle:

L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).

Miscellaneous exercise 5 | Q 8.1 | Page 190

Find the component form of `bar"a"` if it lies in YZ-plane makes 60° with positive Y-axis and `|bar"a"| = 4`.

Miscellaneous exercise 5 | Q 9 | Page 190

Two sides of a parallelogram are `3hat"i" + 4hat"j" - 5hat"k"` and  `-2hat"j" + 7hat"k"`. Find unit vectors parallel to the diagonals.

Miscellaneous exercise 5 | Q 10 | Page 190

If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar"AD" + bar"BE" + bar"CF" = bar0`.

Miscellaneous exercise 5 | Q 11 | Page 190

Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).

Miscellaneous exercise 5 | Q 12 | Page 190

Express `hat"i" + 4hat"j" - 4hat"k"` as the linear combination of the vectors `2hat"i" - hat"j" + 3hat"k", hat"i" - 2hat"j" + 4hat"k"` and `- hat"i" + 3hat"j" - 5hat"k"`.

Miscellaneous exercise 5 | Q 13 | Page 190

If `bar"OA" = bar"a" and bar"OB" = bar"b",` then show that the vector along the angle bisector of ∠AOB is given by `bar"d" = lambda(bar"a"/|bar"a"| + bar"b"/|bar"b"|).`

Miscellaneous exercise 5 | Q 15 | Page 191

A point P with position vector `(- 14hat"i" + 39hat"j" + 28hat"k")/5` divides the line joining A (1, 6, 5) and B in the ratio 3 : 2, then find the point B.

Miscellaneous exercise 5 | Q 16 | Page 191

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Miscellaneous exercise 5 | Q 17 | Page 191

ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.

Miscellaneous exercise 5 | Q 18 | Page 191

If ABC is a triangle whose orthocentre is P and the circumcentre is Q, prove that `bar"PA" + bar"PB" + bar"PC" = 2bar"PQ".`

Miscellaneous exercise 5 | Q 19 | Page 191

If P is orthocentre, Q is the circumcentre and G is the centroid of a triangle ABC, then prove that `bar"QP" = 3bar"QG"`.

Miscellaneous exercise 5 | Q 20 | Page 191

In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Miscellaneous exercise 5 | Q 21 | Page 191

Dot product of a vector with vectors `3hat"i" - 5hat"k",  2hat"i" + 7hat"j" and hat"i" + hat"j" + hat"k"` are respectively -1, 6 and 5. Find the vector.

Miscellaneous exercise 5 | Q 22 | Page 191

If `bar"a", bar"b", bar"c"` are unit vectors such that `bar"a" + bar"b" + bar"c" = bar0,` then find the value of `bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".`

Miscellaneous exercise 5 | Q 23 | Page 191

If a parallelogram is constructed on the vectors `bar"a" = 3bar"p" - bar"q", bar"b" = bar"p" + 3bar"q" and |bar"p"| = |bar"q"| = 2` and angle between `bar"p" and bar"q"` is `pi/3,` and angle between lengths of the sides is `sqrt7 : sqrt13`.

Miscellaneous exercise 5 | Q 24 | Page 191

Express the vector `bar"a" = 5hat"i" - 2hat"j" + 5hat"k"` as a sum of two vectors such that one is parallel to the vector `bar"b" = 3hat"i" + hat"k"` and other is perpendicular to `bar"b"`.

Miscellaneous exercise 5 | Q 25 | Page 191

Find two unit vectors each of which makes equal angles with bar"u", bar"v" and bar"w" where bar"u" = 2hat"i" + hat"j" - 2hat"k", bar"v" = hat"i" + 2hat"j" - 2hat"k", bar"w" = 2hat"i" - 2hat"j" + hat"k".

Miscellaneous exercise 5 | Q 26 | Page 191

Find the acute angle between the curves at their points of intersection, y = x2, y = x3.

Miscellaneous exercise 5 | Q 27.1 | Page 191

Find the direction cosines and direction angles of the vector `2hat"i" + hat"j" + 2hat"k"`

Miscellaneous exercise 5 | Q 28 | Page 191

Let bar"b" = 4hat"i" + 3hat"j" and bar"c" be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along bar"b" and bar"c" respectively.

Miscellaneous exercise 5 | Q 29 | Page 192

Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.

Miscellaneous exercise 5 | Q 30 | Page 192

Find the angle between the lines whose direction cosines are given by the equations 6mn - 2nl + 5lm = 0, 3l + m + 5n = 0.

Miscellaneous exercise 5 | Q 31 | Page 192

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.

Miscellaneous exercise 5 | Q 32 | Page 192

Show that the vector area of a triangle ABC, the position vectors of whose vertices are `bar"a", bar"b" and bar"c"` is `1/2[bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"]`.

Miscellaneous exercise 5 | Q 33 | Page 192

Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0) and (0, 0, c). What is the area of the triangle with these vertices?

Miscellaneous exercise 5 | Q 34.01 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b" xx bar"c")`

Miscellaneous exercise 5 | Q 34.02 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a" xx (bar"b".bar"c")`

Miscellaneous exercise 5 | Q 34.03 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a" xx(bar"b" xx bar"c")`

Miscellaneous exercise 5 | Q 34.04 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b".bar"c")`

Miscellaneous exercise 5 | Q 34.05 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b") xx (bar"c".bar"d")`

Miscellaneous exercise 5 | Q 34.06 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a" xx bar"b").(bar"c"xxbar"d")`

Miscellaneous exercise 5 | Q 34.07 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b").bar"c"`

Miscellaneous exercise 5 | Q 34.08 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`(bar"a".bar"b")bar"c"`

Miscellaneous exercise 5 | Q 34.09 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`|bar"a"|(bar"b".bar"c")`

Miscellaneous exercise 5 | Q 34.1 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a".(bar"b" + bar"c")`

Miscellaneous exercise 5 | Q 34.11 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`bar"a". bar"b" + bar"c"`

Miscellaneous exercise 5 | Q 34.12 | Page 192

State whether the expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar:

`|bar"a"|. (bar"b" + bar"c")`

Miscellaneous exercise 5 | Q 35 | Page 192

For any vectors `bar"a", bar"b", bar"c"` show that `(bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a" = 2bar"a" xx bar"c"`

Miscellaneous exercise 5 | Q 36.1 | Page 192

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c"`, then is `bar"b" = bar"c"` ?

Miscellaneous exercise 5 | Q 36.2 | Page 192

Suppose `bar"a" = bar"0"`:

If `bar"a" xx bar"b" = bar"a" xx bar"c"`, then is `bar"b" = bar"c"` ?

Miscellaneous exercise 5 | Q 36.3 | Page 192

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c" and bar"a" xx bar"b" = bar"a" xx bar"c"`,  then is `bar"b" = bar"c"`?

Miscellaneous exercise 5 | Q 37.1 | Page 192

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then verify that the points are the vertices of a parallelogram.

Miscellaneous exercise 5 | Q 37.2 | Page 192

If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then find its area.

Miscellaneous exercise 5 | Q 38 | Page 193

Let A, B, C, D be any four points in space. Prove that `|bar"AB" xx bar"CD" + bar"BC" xx bar"AD" + bar"CA" + bar"BD"|` = 4 (area of triangle ABC).

Miscellaneous exercise 5 | Q 39 | Page 192

Let hat"a", hat"b", hat"c" be unit vectors such that hat"a".hat"b" = hat"a".hat"c" = 0 and 6  the angle between hat"b" and hat"c" is pi/6. Prove that hat"a" = +- 2(hat"b" xx hat"c").

Miscellaneous exercise 5 | Q 40 | Page 192

Find the value of ‘a’ so that the volume of parallelopiped formed by hat"i" + "a"hat"j" + hat"k", hat"j" + "a"hat"k" and "a"hat"i" + hat"k" becomes minimum.

Miscellaneous exercise 5 | Q 41 | Page 193

Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.

Miscellaneous exercise 5 | Q 42 | Page 192

If `bar"a", bar"b", bar"c"` are three non-coplanar vectors show that `(bar"a".(bar"b" xx bar"c"))/((bar"c" xx bar"a").bar"b") + (bar"b".(bar"a" xx bar"c"))/((bar"c" xx bar"a").bar"b") = 0`

Miscellaneous exercise 5 | Q 43 | Page 193

Prove that `(bar"a" xx bar"b").(bar"c" xx bar"d")` =
`|bar"a".bar"c"    bar"b".bar"c"|`
`|bar"a".bar"d"    bar"b".bar"d"|.`

Miscellaneous exercise 5 | Q 44 | Page 193

Find the volume of a parallelopiped whose coterimus edges are represented by the vectors `hat"i" + hat"k", hat"i" + hat"k", hat"i" + hat"j"`. Also find volume of tetrahedron having these coterminus edges.

Miscellaneous exercise 5 | Q 45 | Page 193

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"]`.

Miscellaneous exercise 5 | Q 46 | Page 193

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"]`.

Miscellaneous exercise 5 | Q 47 | Page 193

If `bar"a", bar"b", bar"c"` are three non-coplanar vectors, then `(bar"a" + bar"b" + bar"c").[(bar"a" + bar"b") xx (bar"a" + bar"c")] = - [bar"a"  bar"b" bar"c"]`

Miscellaneous exercise 5 | Q 48 | Page 193

If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair is also perpendicular.

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Chapter 5: Vectors

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Miscellaneous exercise 5
Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 - Vectors

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 (Vectors) 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 Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 5 Vectors 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, Dot/Scalar Product of Vectors, Cross/Vector Product of Vectors, Scalar Triple Product of Vectors, Vector Triple Product, Addition of Vectors.

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