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CLASS – XII, SUB – MATH, , REVISION SET- 04, , TIME – 2 Hrs., F.M. - 100, , 1) Represent graphically a displacement of –, i) 50 km, 30o east of north., ii) 40 km, 20o east of south., iii) 20 km south-west., iv) 60 km, 40o north of west., 2) If a + 2b + 3c, 2a + b + 3c, 2a+5b-c and 5a + 2b - c be the position vectors of A,B,C and, D respectively, prove that AB and CD are parallel. Is ABCD a parallelogram?, 3) If OP = 2i + 3j – 4k and OQ = 5i+4j-3k. Find PQ and the direction cosines of PQ., 4) Write the direction ratios of the vector a = i+j+-2k and hence calculate its direction cosines., 5) Prove that the vectors 2i – j + k, i – 3j – 5k and 3i – 4j – 4k form a right-angled triangle., 6) If the points with position vectors 60i + 3j, 40i – 8j, ai – 52j are collinear then prove that A = -40, 7) Consider two points P and Q with position vectors OP = 3a – 2b and OQ = a + b. Find, the position vectors of a point R which divides the line joining P and Q in the ratio 2:1., i) Internally, ii) Externally, 8) Find the angle between two vectors a and b with magnitudes 1 and 2 respectively if a.b = 1., 9) If a = 3i + 2j + 9k and b = i + 𝜆j + 3k, find the value of 𝜆 so that a + b is perpendicular to a – b., 10) P,Q,R,S are the points i – j – k, -i+ j, 2i – 3k and 3i – 2j – k respectively. Show that the, Projection of PQ on RS is equal to that of RS on PQ and each being -4/3., 11) Find the angles between the following pairs of vectors. i + j - k and i – j + k., 12) Let a = i + 4j + 2k, b = 3i – 2j + 7k and c = 2i – j + 4k. Find a vector d which is perpendicular to, both a and b and c . d = 15., 13) Find |a| and |b| if (a + b).(a – b) = 8 and |a| = 8|b|., 𝑏 2 +𝑐 2 −𝑎 2, , 14) Prove by vector method than in any ABC. i) cos A =, ii) a = b cos C + c cos B, 2𝑏𝑐, 15) If a and b are two vectors such that |a| = 2, |b| = 7 and a x b = 3i + 2j + 6k, find the, angle between a and b., 16) If a and b are two vectors suct that |a| = 5, |b| = 4 and |a.b| = 10, find the angle between a, and b hence find |a x b|., 17) Find a unit vector perpendicular to each of the vectors (a+b) and (a-b), where, a = i + j + k and b = i + 2j + 3k., 18) If A,B,C are points (1,0,-1), (0,1,-1) and (-1,0,1) respectively, find the sine of the angle, between the lines AB and AC., 19) If a = 2i + 5j - 7k, and b = -3i+4j+k and c = i-2j=3k, show that (a x b) x c and a x(b x c) are not, Same., 20) Find the area of the parallelogram whose adjacent sides are a =i + 2j + 3k and b = 3i -2j + k., Or,, 21) Show that the area of the triangle whose two adjacent sides are determined by the vectors, 1, a = 3i + 4j, b = -5i + 7j is 20 square units., 2