Vector Algebra PYQs Set 2 (PDF)

Set 2 of 16 · Questions 31-60 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 2 (Q31-60)
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31. $The image of the line \frac{x - 1}{3} = \frac{y - 3}{1} = \frac{z - 4}{- 5} in the plane 2x - y + z + 3 = 0 is the line :$ (1) $\frac{x - 3}{3} = \frac{y + 5}{1} = \frac{z - 2}{- 5}$ (2) $\frac{x - 3}{- 3} = \frac{y + 5}{- 1} = \frac{z - 2}{5}$ (3) $\frac{x + 3}{3} = \frac{y - 5}{1} = \frac{z - 2}{- 5}$ (4) $\frac{x + 3}{- 3} = \frac{y - 5}{- 1} = \frac{z + 2}{5}$ 32. $Let a = i + j + k, b = i + 3 j + 5 k and c = 7 i + 9 j + 11 k, then the area of parallelogram having diagonals a + b and b + c is$ (1) $46 sq units$ (2) $26 sq units$ (3) $6 sq units$ (4) $66 sq units$ 33. $Let A (2, 3, - 1), B (4, 1, 0), C (- 1, - 1, 11) be the vertices of a triangle ABC . Let D be the point where the bisector of BAC! meet the side BC . Then the direction ratio of AD are$ (1) $(35, - 19, 49)$ (2) $(17, - 14, 49)$ (3) $(17, - 38, 49)$ (4) $(17, - 38, 23)$ 34. $Let a = i - 2 j + 2 k, b = 6 i + 2 j - 3 k and c = 3 i - 4 j - 12 k be three vectors. If p is the projection of b on a and q is the projection of c on a, then 13 p =$ (1) $4 q$ (2) $5 q$ (3) $6 q$ (4) $7 q$ 35. $Let a, b and c be three non zero vectors such that b . c = 0 and a \times (b \times c) = \frac{b - c}{2} . If d be a vector such that b . d = a . b, then (a \times b) . (c \times d) is equal to$ (1) $\frac{3}{4}$ (2) $\frac{1}{2}$ (3) $- \frac{1}{4}$ (4) $\frac{1}{4}$ 36. $Let a = a_{i} i + a_{2} j + a_{3} k and b = b_{1} i + b_{2} j + b_{3} k be two vectors such that \| a \| = 1; a \cdot b = 2 and \| b \| = 4 . If c = 2 (a \times b) - 3 b, then the angle between b and c is equal to :$ (1) $cos^{- 1} (\frac{2}{3})$ (2) $cos^{- 1} (- \frac{1}{3})$ (3) $cos^{- 1} (- \frac{3}{2})$ (4) $cos^{- 1} (\frac{2}{3})$ 37. $If the vectors a = λi + μ j + 4 k, b = 2 i + 4 j - 2 k and c = 2 i + 3 j + k are coplanar and the projection of a on the vector b is 54 units, then the sum of all possible values of λ + μ is equal to$ (1) $0$ (2) $6$ (3) $24$ (4) $18$ 38. $If the vector a = 3 j + 4 k is the sum of two vectors a_{1} and a_{2}, vector a_{1} is parallel to b = i + j and vector a_{2} is perpendicular to b, then a_{1} =$ (1) $\frac{1}{2} (i + j)$ (2) $\frac{1}{3} (i + j)$ (3) $\frac{2}{3} (i + j)$ (4) $\frac{3}{2} (i + j)$ 39. $If a, b, c are unit vectors such that a + b + c = 0, then a \cdot b + b \cdot c + c \cdot a =$ (1) $\frac{3}{2}$ (2) $- \frac{3}{2}$ (3) $\frac{2}{3}$ (4) $\frac{1}{2}$ 40. $If a, b and c are nonzero coplanar vectors, then [2 a - b 3 b - c 4 c - a] =$ (1) $0$ (2) $25$ (3) $9$ (4) $27$ 41. $Let b = 3 i - 2 j + k and c = i - j - k be two vectors. If a is a vector such that a + b + c = 0, then \| a \times b + b \times c + c \times a \| =$ (1) $15$ (2) $261$ (3) $234$ (4) $33$ 42. $If a is vector perpendicular to both b and c, then$ (1) $a + (b + c) = 0$ (2) $a \times (b + c) = 0$ (3) $a \times (b \times c) = 0$ (4) $a \cdot (b \times c) = 0$ 43. $A vector perpendicular to the plane containing the points A (1, - 1, 2), B (2, 0, - 1), C (0, 2, 1) is$ (1) $4 i + 8 j - 4 k$ (2) $8 i + 4 j + 4 k$ (3) $3 i + j + 2 k$ (4) $i + j - k$ 44. $Let a = i + 2 j + 3 k, b = 2 i + 3 j - 5 k and c = 3 i - j + λk be three vectors. Let r be anit vector along b + c . If r \cdot a = 3, then 3 λ is equal to:$ (1) $21$ (2) $30$ (3) $25$ (4) $27$ 45. $If a, b, c are distinct real numbers and P, Q, R are three points whose position vectors are respectively ai + bj + c k, bi + c j + ak and c i + aj + bk, then ∠ QRP =$ (1) $cos^{- 1} (a + b + c)$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $cos^{- 1} (\frac{a ^{2} + b ^{2} + c ^{2}}{abc})$ 46. $The vectors 2 i - 3 j + k, i - 2 j + 3 k and 3 i + j - 2 k$ (1) $are linearly dependent$ (2) $are linearly independent$ (3) $form sides of a triangle$ (4) $are coplanar$ 47. $If a = i + j + k, c = j - k are given vectors, then a vector b satisfying the equations a \times b = c and a \cdot b = 3 is$ (1) $5 i + 2 j + 2 k$ (2) $\frac{5}{2} i + j + k$ (3) $\frac{5}{3} i + \frac{2}{3} j + \frac{2}{3} k$ (4) $i + \frac{2}{5} j + \frac{2}{5} k$ 48. $Let a = i + 2 j + 3 k, b = 3 i + j - k and c be three vectors such that c is coplanar with a and b . If the vector C is perpendicular to b and a \cdot c = 5, then \| c \| is equal to$ (1) $\frac{11}{6}$ (2) $\frac{1}{3 2}$ (3) $16$ (4) $18$ 49. $Let u, v and w be the vector such that \| u \| = 1; \| v \| = 2; \| w \| = 3 . If the projection v along u is equal to that of w along u and v, w are perpendicular to each other, then \| u - v + w \| is equal to$ (1) $2$ (2) $7$ (3) $14$ (4) $14$ 50. $A unit vector perpendicular to both i + j + k and 2 i + j + 3 k is$ (1) $(2 i - j - k) 6$ (2) $\frac{( 2 i - j - k )}{6}$ (3) $2 i + j + k$ (4) $\frac{3 i + j - 2 k}{6}$ 51. $Let a = 2 i - 7 j + 5 k, b = i + k and c = i + 2 j - 3 k be three given vectors. If r is a vector such that r \times a = c \times a and r . b = 0, then (r) is equal to$ (1) $\frac{11}{7} 2$ (2) $\frac{11}{7}$ (3) $\frac{11}{5} 2$ (4) $\frac{914}{7}$ 52. $Let a and b be the vectors of the same magnitude such that \frac{\| a + b \| + \| a - b \|}{\| a + b \| - \| a - b \|} = 2 + 1 . Then \frac{\| a + b \| ^{2}}{\| a \| ^{2}} is:$ (1) $2 + 2$ (2) $2 + 42$ (3) $4 + 22$ (4) $1 + 2$ 53. $If a = i + j + k, b = 2 i + λj + k, c = i - j + 4 k and a \cdot (b \times c) = 10, then λ is equal to$ (1) $6$ (2) $7$ (3) $9$ (4) $10$ 54. $If a, b, c are non-zero vectors such that (a \times b) \times c = \frac{1}{3} \| b \| \| c \| a, c ⊥ a and θ is the angle between the vectors b and c, then sin θ is equal to$ (1) $\frac{2 2}{3}$ (2) $\frac{1}{3}$ (3) $\frac{2}{3}$ (4) $\frac{2}{3}$ 55. $\| a \times b \|^{2} + \| a \cdot b \|^{2} = 144 and \| a \| = 4 then \| b \| is equal to$ (1) $8$ (2) $12$ (3) $4$ (4) $3$ 56. $The diagonals of a parallelogram are the vectors 3 i + 6 j - 2 k and - i - 2 j - 8 k, then the length of the shorter side of parallelogram is$ (1) $23$ (2) $14$ (3) $35$ (4) $29$ 57. $Let a = 3 i + 2 j + k, b = 2 i - j + 3 k and c be a vector such that (a + b) \times c = 2 (a \times b) + 24 j - 6 k and (a - b + i) \cdot c = - 3 . Then \| c \|^{2} is equal to______$ 58. $Let v = α i + 2 j - 3 k, w = 2 α i + j - k, and u be a vector such that ∣ u ∣ = α > 0 . If the minimum value of the scalar triple product [u v w] is - α 3401, and (u \cdot i)^{2} = \frac{m}{n} where m and n are coprime natural numbers, then m + n is equal to ________$ 59. $The least positive integral value of α, for which the angle between the vectors α i - 2 j + 2 k and α i + 2 α j - 2 k is acute, is_____$ 60. $Let a = i + j + k, b = 3 i + 2 j - k, c = λj + μ k and d be a unit vector such that a \times d = b \times d and c \cdot d = 1 . If c is perpendicular to a, then \| 3 λ d + μ c \|^{2} is equal to ____.$