13%20-%20Vector%20Algebra

13%20-%20Vector%20Algebra - 13 VECTOR ALGEBRA Page 1...

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13 - VECTOR ALGEBRA Page 1 ( Answers at the end of all questions ) ( 1 ) If C is the midpoint of AB and P is any point outside AB, then ( a ) PA + PB = 2 PC ( b ) + = ( c ) + + 2 = 0 ( d ) + + = [ AIEEE 2005 ] ( 2 ) For any vector a , the value of ( a × ^ i) 2 + ( ^ k ) 2 is equal to ( a ) 3 a 2 ( b ) a 2 ( c ) 2 a 2 ( d ) 4 a 2 [ AIEEE 2005 ] ( 3 ) Let a, b and c be distinct non-negative numbers. If the vectors a ^ i + a ^ j + ^ k , ^ + ^ k and c ^ + c ^ + b ^ lie in a plane, then c is ( a ) the Geometric Mean of a and b ( b ) the Arithmetic Mean of a and b ( c ) equal to zero ( d ) the Harmonic Mean of a and b [ AIEEE 2005 ] ( 4 ) If a, b, c are non-coplanar vectors and λ is a real number , then λ ( a + b) · [ λ 2 b × λ c] = a · [ ( b + c) × b ] for ( a ) exactly one value of λ ( b ) no value of λ ( c ) exactly three values of λ ( d ) exactly two values of λ [ AIEEE 2005 ] ( 5 ) Let = ^ - ^ k, = x ^ + ^ + ( 1 - x ) ^ and = y ^ + x ^ + ( 1 + x - y ) ^ k. Then [ c b a ] depends on ( a ) only y ( b ) only x ( c ) both x and y ( d ) neither x nor y [ AIEEE 2005 ] ( 6 ) Let , c , b , a be three non-zero vectors such that no two of these are collinear. If the vector b 2 a + is collinear with c , and c 3 b + is collinear with a , then b 2 a + c 6 + , for some non-zero scalar λ equals ( a ) λ ( b ) λ ( c ) λ ( d ) 0 [ AIEEE 2004 ]

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13 - VECTOR ALGEBRA Page 2 ( Answers at the end of all questions ) ( 7 ) A particle is acted upon by constant forces ^ ^ ^ k 3 j i 4 - + and ^ ^ ^ k j i 3 - + which displace it from a point ^ ^ ^ k 3 j 2 i + + to the point ^ ^ ^ k j 4 i 5 + + . The work done in standard units by the forces is given by ( a ) 40 ( b ) 30 ( c ) 25 ( d ) 15 [ AIEEE 2004 ] ( 8 ) If c , b , a are non-coplanar vectors and λ is a real number, then the vectors c 3 b 2 a + + , c 4 b + λ and ( 2 λ - 1 ) c are non-coplanar for ( a ) all values of λ ( b ) all except one value of λ ( c ) all except two values of λ ( d ) no value of λ [ AIEEE 2004 ] ( 9 ) Let w , v , u be such that l u l = 1, l v l = 2 and l w l = 3. If the projection of v along u is equal to that of w along and v , are perpendicular to each other, then l - + equals ( a ) 2 ( b ) 7 ( c ) 14 ( d ) 14 [ AIEEE 2004 ] ( 10 ) Let c , b , a be non-zero vectors such that ( c ) b a × × = a l c l l b l 3 1 . If θ is the acute angle between the vectors c and b , then sin θ equals ( a ) 3 1 ( b ) 3 2 ( c ) 3 2 ( d ) 3 2 2 [ AIEEE 2004 ] ( 11 ) If 2 2 2 2 2 2 c 1 c c b 1 b b a 1 a a + + + = 0 and vectors ( 1, a, a 2 ), ( 1, b, b 2 ) and ( 1, c, c 2 ) are non-coplanar, then the product abc equals ( a ) 2 ( b ) - 1 ( c ) 1 ( d ) 0 [ AIEEE 2003 ] ( 12 ) c , b , a are three vectors, such that + + c b a = 0, l a l = 1, l b l = 2, l c l = 3, then b a + c b + a c is equal to ( a ) 0 ( b ) - 7 ( c ) 7 ( d ) 1 [ AIEEE 2003 ]
13 - VECTOR ALGEBRA Page 3 ( Answers at the end of all questions ) ( 13 ) A particle acted on by constant forces 4 i + j - 3 k and 3 j - is displaced from the point 5 4 j + k . The total work done by the forces is ( a ) 20 units ( b ) 30 units ( c ) 40 units ( d ) 50 units [ AIEEE 2003 ] ( 14 ) If u, v and w are three non-coplanar vectors, then ( u + - w) . ( - v) × ( - w ) equals ( a ) 0 ( b ) u.

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13%20-%20Vector%20Algebra - 13 VECTOR ALGEBRA Page 1...

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