Vector Algebra PYQs Set 3 (PDF)

Set 3 of 16 · Questions 61-90 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 3 (Q61-90)
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61. $If a and b are mutually perpendicular unit vectors, then (3 a + 2 b) \cdot (5 a - 6 b) =$ (1) $5$ (2) $3$ (3) $6$ (4) $12$ 62. $If a, b and c are unit vectors such that a + b + c = 0, then the a \cdot b + b \cdot c + c \cdot a is equal to$ (1) $\frac{3}{2}$ (2) $- \frac{3}{2}$ (3) $\frac{1}{2}$ (4) $- \frac{1}{2}$ 63. $If a \cdot b = 0 and a + b makes an angle 60^{\circ} with a then$ (1) $\| a \| = 2 \| b \|$ (2) $2 \| a \| = \| b \|$ (3) $\| a \| = 3 \| b \|$ (4) $3 \| a \| = \| b \|$ 64. $Let a be a vector in the plane containing vectors b = i + 2 j + k and c = 2 i - j + k . If a is perpendicular to i + j + 3 k and its projection on b is 36, then ∣ a ∣^{2} is equal to$ (1) $186$ (2) $36$ (3) $128$ (4) $264$ 65. $Let α = 3 i + 2 j + 2 k and b = i + 2 j - 2 k be two vectors. If a vector perpendicular to both the vectors a + b and a - b has the magnitude 12 then one such vector is:$ (1) $4(2 i + 2 j + 2 k)$ (2) $4(2 i - 2 j - k)$ (3) $4(2 i + 2 j - k)$ (4) $) 4(-2 i - 2 j + k)$ 66. $If θ is the angle between the vectors 2 i - j + 2 k and ai + 4 j + bk and cos θ = \frac{2}{3} then 2 (a + b + 3) =$ (1) $a^{2} + b^{2}$ (2) $a^{2}$ (3) $b^{2}$ (4) $ab$ 67. $If a, b, c are three vectors such that \| a + b + c \| = 1, c = λ (a \times b) and \| a \| = \frac{1}{3}, \| b \| = \frac{1}{2}, \| c \| = \frac{1}{6}, then the angle between a and b is$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 68. $If a = 2 i + k, b = i + j + k, c = 4 i - 3 j + 7 k, then the vector r satisfying r \times b = c \times b and r . a = 0 is$ (1) $i + 8 j + 2 k$ (2) $i - 8 j + 2 k$ (3) $i - 8 j - 2 k$ (4) $- i - 8 j + 2 k$ 69. $If a = \frac{3}{2} k and b = \frac{2 i + 2 j - k}{2}, then angle between a + b and a - b is$ (1) $45^{\circ}$ (2) $90^{\circ}$ (3) $30^{\circ}$ (4) $60^{\circ}$ 70. $If \| a \| = 1, \| b \| = 2, \| a - b \|^{2} + \| a + 2 b \|^{2} = 20 then (a, b) =$ (1) $\frac{π}{3}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{2 π}{3}$ 71. $The value of [a - b b - c c - a] is equal to$ (1) $0$ (2) $1$ (3) $2 [ab c]$ (4) $2$ 72. $For three vectors p, q and r, if r = 3 p + 4 q and 2 r = p - 3 q, then$ (1) $\| r \| < 2 \| q \| and r, q have the same direction$ (2) $\| r \| > 2 \| q \| and r, q have opposite directions$ (3) $\| r \| < 2 \| q \| and r, q have opposite directions$ (4) $\| r \| > 2 \| q \| and r, q have the same direction$ 73. $If i^is the position vector of the centroid G of triangle ABC and 2 i^+ j^+ k^, 2 i^+ 4 j^- 4 k^are respectively the position vectors of its vertices A and B, then A G^{2} + B G^{2} + C G^{2} =$ (1) $77$ (2) $74$ (3) $86$ (4) $83$ 74. $Let x \in R and log_{2} x > 0 . Then, the vectors A = (2, log_{2} x, s) and B = (log_{2} x, s, log_{2} x) include an acute angle if$ (1) $s > 1$ (2) $s > - 1$ (3) $s = - 1$ (4) $s < - 1$ 75. $Let a = j^- k^and c = i^- j^- k^. Then the vector b satisfying a \times b + c = 0 and a . b = 3 is$ (1) $2 i^- j^+ 2 k^$ (2) $i^- j^-2 k^$ (3) $i^+ j^-2 k^$ (4) $- i^+ j^-2 k^$ 76. $Let p = i + 2 j - k, q = 2 i - j + k . If a and b are two vectors such that p = a - 2 b . q = 2 a + b, then the angle between a and b is$ (1) $cos^{- 1} (\frac{3}{2 221})$ (2) $\frac{π}{2}$ (3) $cos^{- 1} \frac{7}{143}$ (4) $\frac{π}{3}$ 77. $If the points with position vectors i - 2 j + 3 k, 2 i + 3 j - 4 k, - 3 i + j - 5 k and ai - 2 j + 4 k are coplanar then a =$ (1) $\frac{- 4}{19}$ (2) $\frac{42}{19}$ (3) $\frac{- 42}{19}$ (4) $\frac{4}{19}$ 78. $If a + 2 b + 3 c = 0 and (a \times b) + (b \times c) + (c \times a) = λ (b \times c) then the value of λ is equal to$ (1) $4$ (2) $2$ (3) $6$ (4) $3$ 79. $If p = i + j, q = 4 k - j and r = i + k then the unit vector in the direction of 3 p + q - 2 r is$ (1) $\frac{1}{3} (i^+ 2 j^+ 2 k^)$ (2) $\frac{1}{3} (2 i + j + 2 k)$ (3) $\frac{1}{3} (i - 2 j + 2 k)$ (4) $i + 2 j + 2 k$ 80. $The position vector of a point that divides the line segment joining P \equiv (1, 2, - 1) and Q \equiv (- 1, 1, 1) externally in the ratio 1 : 2, is$ (1) $- 3 i + 3 k$ (2) $3 i + 3 j - 3 k$ (3) $3 i + j + 3 k$ (4) $3 i - 3 k$ 81. $If a ⊥ b and (a + b) ⊥ (a + mb), then m =$ (1) $-1$ (2) $1$ (3) $\frac{- \| a \| ^{2}}{\| b \| ^{2}}$ (4) $0$ 82. $If a \cdot b = - \| a \| \| b \|, then the angle between a and b is$ (1) $90^{\circ}$ (2) $60^{\circ}$ (3) $45^{\circ}$ (4) $180^{\circ}$ 83. $Let a = i - 2 j, b = 2 j + 3 k, c = p i + q j and d = p j - q k be four vectors. If (a \times b) \cdot c = 3 = (a \times b) \cdot d, then 3 p + q =$ (1) $0$ (2) $3$ (3) $-2$ (4) $6$ 84. $Let a and b be two unit vectors such that the angle between them is \frac{π}{3} . If λa + 2 b and 3 a - λb are perpendicular to each other, then the number of values of λ in [- 1, 3] is:$ (1) $2$ (2) $1$ (3) $0$ (4) $3$ 85. $If \| a \times b \|^{2} + \| a \cdot b \|^{2} = 144 and \| a \| = 6, then \| b \| is equal to$ (1) $3$ (2) $2$ (3) $4$ (4) $6$ 86. $The equation of the plane passing through the point (1, 2, - 3) and perpendicular to the planes 3 x + y - 2 z = 5 and 2 x - 5 y - z = 7, is:$ (1) $3 x - 10 y - 2 z + 11 = 0$ (2) $6 x - 5 y - 2 z - 2 = 0$ (3) $11 x + y + 17 z + 38 = 0$ (4) $6 x - 5 y + 2 z + 10 = 0$ 87. $Let A, B, C be three points in xy -plane, whose position vector are given by 3 i + j, i + 3 j and a i + (1 - a) j respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors OA and OB is \frac{9}{2}, then the sum of all the possible values of a is:$ (1) $2$ (2) $9/2$ (3) $1$ (4) $0$ 88. $Let A (2, 3, 5) and C (- 3, 4, - 2) be opposite vertices of a parallelogram ABCD if the diagonal BD = i + 2 j + 3 k then the area of the parallelogram is equal to$ (1) $\frac{1}{2} 410$ (2) $\frac{1}{2} 474$ (3) $\frac{1}{2} 586$ (4) $\frac{1}{2} 306$ 89. $Let a = i + 2 j + k, b = 3 i - 3 j + 3 k, c = 2 i - j + 2 k and d be a vector such that b \times d = c \times d and a \cdot d = 4 . Then \| (a \times d) \|^{2} is equal to _______.$ 90. $The projection of the line segment joining the points (1, - 1, 3) and (2, - 4, 11) on the line joining the points (- 1, 2, 3) and (3, - 2, 10) is:$