12th Standard Maths English Medium Applications of Vector Algebra Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

Multiple Choice Questions

15 x 1 = 15
1. If $\vec{a}$ and $\vec{b}$ are parallel vectors, then $[\vec { a } ,\vec { c } ,\vec { b } ]$ is equal to

(a)

2

(b)

-1

(c)

1

(d)

0

2. The volume of the parallelepiped with its edges represented by the vectors $\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k }$ is

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { \pi }{ 3}$

(c)

${ \pi }$

(d)

$\frac { \pi }{ 4 }$

3. If $\vec { a }$ and $\vec { b }$ are unit vectors such that $[\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 }$, then the angle between $\vec { a }$ and $\vec { b }$ is

(a)

$\frac { \pi }{ 6 }$

(b)

$\frac { \pi }{ 4 }$

(c)

$\frac { \pi }{ 3 }$

(d)

$\frac { \pi }{ 2 }$

4. Consider the vectors $\vec { a } ,\vec { b } ,\vec { c } ,\vec { c }$ such that $(\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )$ = $\vec { 0 }$ Let ${ P }_{ 1 }$ and ${ P }_{ 2 }$ be the planes determined by the pairs of vectors $\vec { a } ,\vec { b }$ and $\vec { c } ,\vec { d }$ respectively. Then the angle between ${ P }_{ 1 }$ and ${ P }_{ 2 }$ is

(a)

(b)

45°

(c)

60°

(d)

90°

5. If $\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c }$ where $\vec { a } ,\vec { b } ,\vec { c }$ are any three vectors such that $\vec { a } ,\vec { b }$ $\neq$ 0 and  $\vec { a } .\vec { b }$ $\neq$ 0 then $\vec { a }$ and $\vec { c }$ are

(a)

perpendicular

(b)

parallel

(c)

inclined at an angle $\frac{\pi}{3}$

(d)

inclined at an angle  $\frac{\pi}{6}$

6. Let $\overset { \rightarrow }{ a }$,$\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$ be three non- coplanar vectors and let $\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r }$ be the vectors defined by the relations $\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] }$ Then the value of  $\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r }$=

(a)

0

(b)

1

(c)

2

(d)

3

7. The number of vectors of unit length perpendicular to the vectors $\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right)$ and $\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right)$is

(a)

1

(b)

2

(c)

3

(d)

8. If $\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right)$, then

(a)

$\left| \overset { \rightarrow }{ d } \right|$

(b)

$\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c }$

(c)

$\overset { \rightarrow }{ d } =\overset { \rightarrow }{ 0 }$

(d)

a, b, c are coplanar

9. If the vector $\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ 2k }$$\overset { \wedge }{ -i } +\overset { \wedge }{ 2k }$ and $2\overset { \wedge }{ i } +x\overset { \wedge }{ j } -y\overset { \wedge }{ k }$  are mutually orthogonal, then the values of x, y, z are

(a)

(10, 4, 1)

(b)

(-10, 4, 1)

(c)

(-10, -4, $\frac { 1 }{ 2 }$)

(d)

(-10, 4, $\frac { 1 }{ 2 }$)

10. If the work done by a force $\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k }$ in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m is

(a)

5

(b)

2

(c)

3

(d)

6

11. If $\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k }$ is a unit vector, then the value of λ is

(a)

土 $\frac { 1 }{ 3 }$

(b)

土 $\frac { 1 }{ 4 }$

(c)

土 $\frac { 1 }{ 9 }$

(d)

$\frac { 1 }{ 2 }$

12. For any three vectors $\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$$\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) \times \left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right)$ is

(a)

0

(b)

$\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right]$

(c)

2$\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right]$

(d)

${ \left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] }^{ 2 }$

13. Let $\overset { \rightarrow }{ u } ,\overset { \rightarrow }{ v } ,\overset { \rightarrow }{ w }$ be vectors such that $\overset { \rightarrow }{ u } +\overset { \rightarrow }{ v } +\overset { \rightarrow }{ w } =\overset { \rightarrow }{ 0 }$. If $\left| \overset { \rightarrow }{ u } \right|$= 3$\left| \overset { \rightarrow }{ v } \right|$= 4$\left| \overset { \rightarrow }{ w } \right|$=5  then $\overset { \rightarrow }{ u } .\overset { \rightarrow }{ v } +\overset { \rightarrow }{ v } .\overset { \rightarrow }{ w } +\overset { \rightarrow }{ w } .\overset { \rightarrow }{ u }$ is ______________

(a)

25

(b)

-25

(c)

5

(d)

$\sqrt { 5 }$

14. The length of the 丄r from the origin to plane $\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right)$26 is _____________

(a)

2

(b)

$\frac { 1 }{ 2 }$

(c)

26

(d)

$\frac { 26 }{ 169 }$

15. Let $\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,$ and $\overset { \rightarrow }{ c }$ be three vectors having magnitudes 1, 1, 2 respectively.
If $\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } =0$ then the acute angle between $\overset { \rightarrow }{ a }$ and $\overset { \rightarrow }{ c }$ is ____________________

(a)

0

(b)

$\frac { \pi }{ 3 }$

(c)

$\frac { \pi }{ 6 }$

(d)

$\frac { 2\pi }{ 3 }$

1. 2 Marks

10 x 2 = 20
16. If $\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k }$$\vec{b}=\hat{i}-\hat{2j}+\hat{k}$,  $\vec{c}=\hat{4i}-\hat{4k} \ and$ $find \ {\vec a } .(\vec { b } \times \vec { c } )$

17. If $\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k }$ are coplanar, find the value of m.

18. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors $-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k }$ and $2\hat { i } +4\hat { j } -2\hat { k }$

19. Determine whether the three vectors $2\hat { i } +3\hat { j } +\hat { k }$$\hat { i } -2\hat { j } +2\hat { k }$ and $\hat { 3i } +\hat { j } +2\hat { k }$ are coplanar.

20. Prove that $(\vec { a } .(\vec { b } \times \vec { c } ))\vec { a } =(\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )$

21. Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2,3, 4) and parallel to the straight line $\frac { x-1 }{ -4 } =\frac { y-3 }{ 5 } =\frac { z-8 }{ 6 }$

22. Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

23. A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point

24. Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65 . Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

25. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

()

-1

1. 3 Marks

10 x 3 = 30
26. With usual notations, in any triangle ABC, prove the following by vector method.
(i) a2=b2+c2−2bc cos A
(ii) b2=c2+a2−2ca cos B
(iii) c2= a2+b2−2ab cos C

27. If D is the midpoint of the side BC of a triangle ABC, then show by vector method that ${ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })$

28. In triangle, ABC the points, D, E, F are the midpoints of the sides, BC, CA and AB respectively. Using vector method, show that the area of ΔDEF is equal to $\frac{1}{4}$(area of ΔABC )

29. Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

30. Forces of magnit $5\sqrt { 2 }$ and $5\sqrt { 2 }$ units acting in the directions $\hat { 3i } +\hat { 4j } +\hat { 5k }$ and $\hat { 10i } +\hat { 6j } -\hat { 8k }$ respectively, act on a particle which is displaced from the point with position vector $\hat { 4i } +\hat { 3j } -\hat { 2k }$ to the point with position vector $\hat { 6i } +\hat { j } -\hat { 3k }$. Find the work done by the forces.

31. Find the torque of the resultant of the three forces represented by $-\hat { 3i } +\hat { 6j } +\hat { 3k }$$\hat { 4i } -\hat { 10j } +\hat { 12k }$ and $\hat { 4i } +\hat { 7j }$ acting at the point with position vector $\hat { 8i } -\hat { 6j } -\hat { 4k }$, about the point with position vector $\hat { 18i } +\hat { 3j } -\hat { 9k }$

32. Find the vector equation in parametric form and Cartesian equations of the line passing through (-4, 2, -3) and is parallel to the line  $\frac { -x-2 }{ 4 } =\frac { y+3 }{ -2 } =\frac { 2z-6 }{ 3 }$

33. Show that the straight lines x + 1=  2y = −12z and x = y + 2 = 6z − 6 are skew and hence find the shortest distance between them.

34. Prove that $\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right]$=$\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right]$

()

0

35. Show that the lines $\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 }$ and $\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 }$ do not intersect

()

a, b, c

1. 5 Marks

6 x 5 = 30
36. By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

37. With usual notations, in any triangle ABC, prove by vector method that $\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }$

38. Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

39. Using vector method, prove that cos(α − β )=cos α cos β +sin α sin β

40. Determine whether the pair of straight lines $\vec { r } (2\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )$$\vec { r } =(2\hat { j } -3\hat { k } )+s(\hat { i } +2\hat { j } +3\hat { k } )$ are parallel. Find the shortest distance between them.

41. Show that the points A, B, C with position vector $2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k }$ and $3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k }$ respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

()

points on the plane