Tiêu đề 23.VECTOR ALGEBRA.pdf ✅

23. VECTOR ALGEBRA-MCQS TYPE (1.) Let ˆ ˆ ˆ ˆ u i j 2k, v 2i j k, v w 2 = − − = + −  = and v w u v  = + . Then u w is equal to [24 Jan 2023(Evening)] (a.) 1 (b.) 3 2 (c.) 2 (d.) 2 3 − (2.) Let PQR be a triangle. The points A,B and C are on the sides QR,RP and PQ respectively such that QA RB PC 1 AR BP CQ 2 = = = . Then ( ) ( ) Area PQR Area ABC is equal to [24 Jan 2023(Evening)] (a.) 4 (b.) 3 (c.) 2 (d.) 5 2 (3.) Let ˆ ˆ ˆ  = + + 4 3 5 i j k and ˆ ˆ ˆ  = + − i j k 2 4 . Let 1 be parallel to  and 2 be perpendicular to  . If    = +1 2 , then the value of ˆ ˆ ˆ 2 5 i j k    + +     is [24 Jan 2023(Morning)] (a.) 6 (b.) 11 (c.) 7 (d.) 9 (4.) Let a and b be two vectors. Let a b = = 1, 4 and a b = 2 . If c a b b =  − (2 3 ) , then the value of b c is [30 Jan 2023(Evening)] (a.) -24 (b.) -48 (c.) -84 (d.) -60 (5.) If the vectors ˆ ˆ ˆ ˆ ˆ ˆ a i j k b i j k = + + = − + −   4 , 2 4 2 and ˆ ˆ ˆ c i j k = + + 2 3 are coplanar and the projection of a on the vector b is 54 units, then the sum of all possible values of  + is equal to [29 Jan 2023(Morning)] (a.) 0 (b.) 6 (c.) 24 (d.) 18 (6.) If the four points, whose position vectors are ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 4 2 , 2 , 2 3 i j k i j k i j k − + + − − − + and ˆ ˆ ˆ 5 2 4 i j k − +  are coplanar, then  is equal to [25 Jan 2023(Evening)] (a.) 73 17 (b.) 107 17 − (c.) 73 17 − (d.) 107 17 (7.) Let ˆ ˆ ˆ ˆ ˆ a 2i 7 j 5k , b i k = − + = + and ˆ ˆ ˆ c i 2 j 3k = + − be three given vectors. If r is a vector such that r a c a  =  and r b 0  = , then r is equal to : [01 Feb 2023(Evening)] (a.) 11 2 7 (b.) 11 7 (c.) 11 2 5 (d.) 914 7
(8.) Let ˆ ˆ ˆ a i j k = − − 5 3 and ˆ ˆ ˆ b i j k = + + 3 5 be two vectors. Then which one of the following statements is TRUE? [01 Feb 2023(Evening)] (a.) Projection of a on b is 17 35 and the direction of the p (b.) Projection of a on b is 17 35 − and the direction of the p (c.) Projection of a on b is 17 35 and the direction of the projection vector is opposite to the direction of b (d.) Projection of a on b is 17 35 − and the direction of the projection vector is opposite to the direction of b (9.) If a b c , , are three non-zero vectors and ˆ n is a unit vector perpendicular to C such that ( ) ˆ a b n, 0 = −    and b c 12,  = then c a b   ( ) is equal to :[30 Jan 2023(Morning)] (a.) 15 (b.) 9 (c.) 12 (d.) 6 (10.) Let a unit vector OP make angle    , , with the positive directions of the co- ordinate axes OX,OY , OZ respectively, where 0, 2 OP         is perpendicular to the plane through points (1, 2,3) , (2,3, 4) and (1,5,7) , then which one of the following is true ?[30 Jan 2023(Morning)] (a.) , 2          and , 2          (b.) 0, 2         and 0, 2         (c.) , 2          and 0, 2         (d.) 0, 2         and , 2          (11.) The projections of a vector on the three coordinate axis are 6, 3,2 − respectively. The direction cosines of the vector are [AIEEE-2009] (a.) 6 3 2 , , 5 5 5 − (b.) 6 3 2 , , 777 − (c.) 6 3 2 , , 7 7 7 − − (d.) 6, 3,2 − (12.) If u v w , , are non-coplanar vectors and p q, are real numbers, then the equality 3 , , , , 2 , , 0 u pv pw pv w qu w qv qu       − − =   holds for [AIEEE-2009] (a.) Exactly two values of ( p q, ) (b.) More than two but not all values of ( p q, )
(c.) All values of ( p q, ) (d.) Exactly one value of ( p q, ) (13.) Let ˆ ˆ a j k = − and ˆ ˆ ˆ c i j k = − − . Then the vector b satisfying a b c  + = 0 and a b = 3 is [AIEEE-2010] (a.) ˆ ˆ ˆ − + − i j k 2 (b.) ˆ ˆ ˆ 2 2 i j k − + (c.) ˆ ˆ ˆ i j k − − 2 (d.) ˆ ˆ ˆ i j k + − 2 (14.) If the vectors ˆ ˆ ˆ a i j k = − + 2 . ˆ ˆ ˆ b i j k = + + 2 4 and ˆ ˆ ˆ c i j k = + +   are mutually orthogonal, then ( , ) = [AIEEE- 2010] (a.) (−3, 2) (b.) (2, 3− ) (c.) (−2,3) (d.) (3, 2− ) (15.) Let a b c , , be three non-zero vectors which are pairwise non-collinear. If a b +3 is collinear with c and b c + 2 is collinear with a , then a b c + + 3 6 is [AIEEE-2011] (a.) 0 (b.) a c + (c.) a (d.) c (16.) If the vectors ˆ ˆ ˆ ˆ ˆ ˆ pi j k i q j k + + + + , and ( ) ˆ ˆ ˆ i j r k p q r + +   1 are coplanar, then the value of pqr p q r − + + ( ) is [AIEEE-2011] (a.) -1 (b.) -2 (c.) 2 (d.) 0 (17.) Let ˆ a and ˆ b be two unit vectors. If the vectors ˆ ˆ c a b = + 2 and ˆ ˆ d a b = − 5 4 are perpendicular to each other, then the angle between ˆ a and ˆ b is [AIEEE-2012] (a.) 4  (b.) 6  (c.) 2  (d.) 3  (18.) Let ABCD be a parallelogram such that AB q AD p = = , and  BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD , then ' ? is given by [AIEEE-2012] (a.) p q r q p p p    = − +      (b.) p q r q p p p    = −     (c.) ( ) ( ) 3 3 p q r q p p p  = − +  (d.) ( ) ( ) 3 3 p q r q p p p  = −  (19.) If the vectors ˆ ˆ AB i k = + 3 4 and ˆ ˆ ˆ AC i j k = − + 5 2 4 are the sides of a triangle ABC , then the length of the median through A is [JEE (Main)-2013] (a.) 18 (b.) 72 (c.) 33 (d.) 45
(20.) The angle between the lines whose direction consines satisfy the equations I m n + + = 0 and 2 2 2 R m n = + is [JEE (Main)-2014] (a.) 6  (b.) 2  (c.) 3  (d.) 4  (21.) If   2 [ ] a bb cc a abc    =  then  is equal to [JEE (Main)-2014] (a.) 0 (b.) 1 (c.) 2 (d.) 3 (22.) Let a b, and c be three non-zero vectors such that no two of them are collinear and ( ) 1 3 a b c b c a   = . If  is the angle between vectors b and ' c , then a value of sin is [JEE (Main)-2015] (a.) 2 2 3 (b.) 2 3 − (c.) 2 3 (d.) 2 3 3 − (23.) Let a b, and c be three unit vectors such that ( ) ( ) 3 2 a b c b c   = + . If b is not parallel to c , then the angle between a and b is [JEE (Main)-2016] (a.) 2  (b.) 2 3  (c.) 5 6  (d.) 3 4  (24.) Let ˆ ˆ ˆ a i j k = + − 2 2 and ˆ ˆ b i j = + . Let c be a vector such that c a a b c − =   = 3, 3 ( ) and the angle between c and a b be 30 . Then a c is equal to [JEE (Main)-2017] (a.) 2 (b.) 5 (c.) 1 8 (d.) 25 8 (25.) Let u be a vector coplanar with the vectors ˆ ˆ ˆ a i j k = + − 2 3 and ˆ ˆ b j k = + . If u is perpendicular to a and u b = 24 , then 2 | | a is equal to [JEE (Main)-2018] (a.) 336 (b.) 315 (c.) 256 (d.) 84 (26.) Let ˆ ˆ ˆ ˆ ˆ a i j b i j k = − = + + , and c be a vector such that a c b  + = 0 and a c = 4 , then 2 | | c is equal to [JEE (Main)-2019] (a.) 17 2 (b.) 19 2 (c.) 9 (d.) 8 (27.) Let ˆ ˆ ˆ ˆ ˆ ˆ a i j k b b i b j k = + + = + + 2 , 2 1 2 and ˆ ˆ ˆ c i j k = + + 5 2 be three vectors such that the projection vector of b on a is a . If a b + is perpendicular to c , then b is equal to [JEE (Main)-2019] (a.) 22 (b.) 32 (c.) 4 (d.) 6