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#### Applications of Vector Algebra Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. If a vector $\vec { \alpha }$ lies in the plane of $\vec { \beta }$ and $\vec { \gamma }$ , then

(a)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$ = 1

(b)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= -1

(c)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= 0

(d)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= 2

2. If $\vec { a } ,\vec { b } ,\vec { c }$ are non-coplanar, non-zero vectors such that $[\vec { a } ,\vec { b } ,\vec { c } ]$ = 3, then ${ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }$ is equal to

(a)

81

(b)

9

(c)

27

(d)

18

3. The coordinates of the point where the line $\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(\hat { i } +4\hat { j } )$ meets the plane $\vec { r } =(\hat { i } +\hat { j } -\hat { k } )$ = 3 are

(a)

(2,1,0)

(b)

(7,1,7)

(c)

(1,2,6)

(d)

(5,1,1)

4. 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)

5. 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

6. 5 x 2 = 10
7. The volume of the parallelepiped whose coterminus edges are $7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k }$$-3\hat { i } +7\hat { j } +5\hat { k }$ is 90 cubic units. Find the value of λ.

8. Show that the straight line passing through the points (6,7,5)A and (8,10,6)B is perpendicular to the straight line passing through the points C(10, 2, -5) and D(8, 3, -4)

9. Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line ${ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right)$

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p=-1

10. Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3.as direction ratios of normal to the plane.

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2

11. Let $\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c }$ be unit vectors such $\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0$ and the angle between $\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$ is $\frac { \pi }{ 6 }$Prove that $\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right)$

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Type I even degree reciprocal equation

12. 5 x 3 = 15
13. If D is the midpoint of the side BC of a triangle ABC, then show by vector method that ${ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })$

14. Prove by vector method that an angle in a semi-circle is a right angle.

15. If $\vec { a } =\hat { i } -\hat { k } ,\vec { b } =x\hat { i } +\hat { j } +(1-x)\hat { k } ,\vec { c } =y\hat { i } +x\hat { j } +(1+x+y)\hat { k }$show that $[\vec { a } ,\vec { b } ,\vec { c } ]$ depends on neither x nor y.

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

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0

17. 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

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a, b, c

18. 4 x 5 = 20
19. Prove by vector method that sin(α + β )=sin α cos β + cos α sin β

20. If $\vec { a } =\vec { i } -\vec { j } ,\vec { b } =\hat { i } -\hat { j } -4\hat { k } ,\vec { c } =3\hat { j } -\hat { k }$ and $\vec { d } =2\hat { i } +5\hat { j } +\hat { k }$
(i) $(\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d }$
(ii) $(\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { c } ,\vec { d } ]\vec { b } -[\vec { b } ,\vec { c } ,\vec { d } ]\vec { a }$

21. 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.

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points on the plane

22. Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0