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UNIT 4. VECTOR ALGEBRA [CBSE BOARD PAPERS FROM 1992- ]

01. If a, b and c are the lengths of the side opposite respectively to the angles A,B & C of ΔABC, show that with the help of vectors that Cos C = a²+b²-c² [ CBSE 2005 ] 2ab

02. Find the volume of the parallelopiped where sides are given by 2i-3j+4k, i+2j-k and 3i-j+2k.

03.In any ΔABC, show with the help of vectors that Sin A = Sin B = Sin C a b c 04. Find λ so that the vectors 2i-4j+5k,i- λj +k and 3i+2j-5k are coplanar. 05.Prove by means of vectors, that an angle subtended at any point on the circumference of a circle by a diameter is a right angle.

06.Find the volume of a parallelepiped, whose sides are given by –3i+7j+5k, 5i+7j-3k and 7i-5j-3k.

07.Given |a|= 10, |b| = 2 and a. b = 12, find |a X b |.

08.Find a unit vector perpendicular to both the vectors 4i-j+3k and –2i+j-2k.

09.Find the magnitude of a :a = (I+3j-2k)X(-I+3k) [Ans:91]

10.Show that the diagonals of a quadrilateral bisect each other if and only if it is a parallelogram, using vector method.

11.Find the magnitude of a: a = (3k+4j)X(i + j- k) 12.If |a|= 13, |b| = 5 and a. b = 60, find |a X b|. 13.Let a= I+j+2k and b= 3i+2j-k, then find (a+3b).(2a-b) 14.Find the magnitude of a X b if a= 2i+j+k and b= i-2j+k. 15. If a= 2i-3j+4k, b= 3i+2j-4k and c= 4i-3j+5k, state which of the following are meaningful and evaluate those that are meaningful: (a. b)X c, ax(b X c),a.(b X c).

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16.If a, b and c are any three vectors, prove that

a X (b + c)+b X (c + a)+ c X (a + b) = 0. 17.For what value of λ are the vectors λi+2j+k and 4i-9j+2k perpendicular to each other? 18.Prove that [a + b, b + c, c +a] = 2[a, b, c].

19.Dot product of a vector with vectors I+j-3k,I+3j-2k and 2i+j+4k are 0,5 and 8 respectively. Find the vector? 20. Find the value of λ so that the vectors a= 2i+ λj +k and b= i-2j+3k are perpendicular to each other?

21.Four points A,B,C and D with position vectors a, b, c and d respectively are such that 3a-b+2c-4d =0. Show that the four points are coplanar. Also, find the position vector of the point of intersection of lines AC and BC.

22.Simplify: [a-b, b-c, c-a] 23.Find the cosine of an acute angle between the vectors 2i-3j+k and i+j-2k.

24.Find the work done by the force F= 2i+j+k acting on a particle if the particle is displaced from the point with position vector 2i+2j+2k to the point with position vector 3i+4j+5k.

25.Prove that (a X b)X(c X d) = [a, b, d]c- [a, b, c]d. 26.Find a unit vector perpendicular to the plane containing the vectors a=2i+j+k and b= I+2j+k.

27.If a, b and c represent the vectors BC, CA and AB of a ΔABC, show that a X b = b X c = c X a. Hence deduce the sine formula for a triangle.

28.For any vector a , prove that i X (a X i)+j X(a X j)+ k X(a X k) = 2a.

29.If a= I-2j+3k and b= 2i+3j-5k , then find a X b . verify that a and a X b are perpendicular to each other.

30.Prove that [(b + c)X(c + a)].(a +b) = 2[a, b, c]. 31.Find the value of λ so that the vectors d = I+2j-3k,b=3i+ λj +k and c=i+2j+2k are coplanar.

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32.If a X b and c are vectors such that a-b = a. c , a X b = a X c and a0, then

prove that b=c. 33.Show that the vectors i-2j+3k, -2i+3j-4k and i-3j+5k are coplanar.

34.If a X b= c X d and a X c = b X d, prove that a-d is parallel to b-c provided

ad and bc. 35.Define a X b and prove that |a X b| = (a. b) tan θ, where θ is the angle between the vectors a and b.

36.Show that the points A,B,C and D with position vectors a, b, c and d respectively such that 5a-3b+4c-6d =0 are coplanar. 37.Using vectors, prove that the altitudes of a triangle are concurrent. 38.Find the work done in moving a particle from the point A with position vector 2i-6j+7k to the point B with position vector 3i-j+5k by a force F= i+3j-k. 39.Find a unit vector perpendicular to both a= 3i+j-2k and b=2i+3j-k. 40.Find λ if a= 4i-j+k and b = λi-2j+2k are perpendicular to each other. 41.Show that the points with position vectors 6i-7j,16i-19j-4k,3j-6k and 2i-5j+10k are coplanar. 42.Find the projection of a= 2i-j+k on b= i-2j+k. 43.Find the moment about the point i+2j-k of a force represented by i+2j+k acting through the point 2i+3j+k. 44.Find the moment of F about the point (2,-1,3) when the force F= 3i+2j-4k is acting at the point (1,-1,2). 45.Find a vector whose magnitude is 3 units and which is perpendicular to the following two vectors a and b ; a= 3i+j-4k and b= 6i+5j-2k. 46.For any three vectors a, b and c , show that a-b, b-c and c-a are coplanar. 47.If a= 4i+3j+k and b= i-2k , find |2bXa|.

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48.If a = I+2j+3k, b= 2i-j+k and c= I+j-2k, verify that a X(b X c) = (a. c)b – (a. b)c. 49.If three vectors a, b and c are such that a+ b+ c=0, prove that a X b= b X c = c X a. 50.Using vectors, prove that a parallelogram whose diagonals are equal is a rectangle. 51.If a, b and c are position vectors of points A,B and C, then prove that (a X b + b X c + c X a) is a vector perpendicular to the plane of ΔABC. 52.IF a= i+j+2k and b= 3i+2j-k, find (a+3b).(2a-b). 53.If a= i+2j-3k and b= 3i-j+2k , show that (a+3b).(2a-b). 54.If |a| = 5, |b| = 13 and |a X b| = 25, find a. b. 55.For any two vectors a and b, show that (1+|a|²)(1+|b|²) = (1- a. b)² +|a +b+(a + b)|². 56.Find the area of a parallelogram whose adjacent sides are given by the vectors i- 3j + k and i + j + k. 57.Prove that (a X b)² = |a|²|b|² - (a. b)². 58.If a, b and c are respectively the lengths of the sides opposite to the angles A, B & C of a ΔABC, prove that a= b cos C + c cos B. 59.If a, b and c are three vectors, prove that (a X b)X c = a X (b XC) iff b X(c X a) = 0. 60.In ΔOAB, OA= 3i+2j-k and OB= i+3j+k. Find the area of the triangle. 61.Using vectors, prove that the midpoint of the hypotenuse of a right angled triangle is equidistant from its vertices. 62.If a = i + j, b= j-3k and c= i+4k, verify a X(b X c) = (a. c)b- (a. b)c. 63. Using vectors, show that points A(-1,4,-3), B(3,2,-5),C(-3,8,-5) and D(-3,2,1) are coplanar.

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64. If the vectors a = 2i – j +k , b= i + 2j+ 3k and c = 3i + λ j + 5k are coplanar, find the value of λ. [CBSE 04] 65. Prove that ( aXb)² = |a|² |b|² - (a.b)² [CBSE 04] 66. The volume of the parallelopiped whose edges are –12i+ λk, 3j-k and 2i+j-15k is 546 cubic units. Find the value of λ. [CBSE 04] 67. Show that the four points whose position vectors are 6i-7j, 16i-29j-4k, 3j-6k and 2i+5j+10k are coplanar. [CBSE 04] 68. If a, b and c are three mutually perpendicular vectors of equal magnitude, find the angle between a and (a + b + c ). [ CBSE 2005 ] 69. Show that the four points A, B, C and D , whose position vectors are 6i – 7j, 16i- 19j- 4k, 3j-6k and 2i – 5j + 10k respectively are coplanar. [CBSE 05] 70. Using vectors, prove that if the diagonals of a parallelogram are equal in length, then it is a rectangle. [ CBSE 2005 ] 71. Find the angle between the vectors a + b and a- b if a = 2i –j + 3k and b = 3i + j -2k. [CBSE 06] 72. Using vectors, prove that in a triangle ABC, a = b = c Sin A Sin B Sin C where a, b and c are the lengths of the sides opposite, respectively, to the angles A, B and C of triangle ABC. [CBSE 06] 73. If a = i+2j-3k, b= 3i-j+2k, show that [a+b] and [a-b] are perpendicular to each other.[ CBSE D 06] 74. Using vectors, prove that the line segment joining the mid-points of non- parallel sides of a trapezium is parallel to the base and is equal to half the sum of the parallel sides.[ CBSE D 06] 75. If a = 3i+2j+9k and b= i+λj+3k, find the value of λ so that a+b is perpendicular to a-b. [ CBSE D ‘06] 76. Find the projection of the b+c on a where a = i + 2j +k, b= i+3j+k and c= i+k.[ CBSE ‘07]

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