Vector Algebra PYQs Set 4 (PDF)

Set 4 of 16 · Questions 91-120 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 4 (Q91-120)
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91. $The position vector of a point P is 2 i + j + 3 k and a = - i - 2 k, b = i + j + 2 k are two vectors which determine a plane π . The equation of a line through P normal to b and lying on the plane π is$ (1) $r = 2 i + j + 3 k + λ (- i + 5 j - 2 k)$ (2) $r = 2 i + j + 3 k + λ (i + j + k)$ (3) $r = 2 i + j + 3 k + λ (- 2 i - j + 3 k)$ (4) $r = 2 i + j + 3 k + λ (- 3 i + 4 j - 5 k)$ 92. $Let a, b, c be three non-zero vectors which are pairwise non-collinear. If a + 3 b is collinear with c and b + 2 c is collinear with a, then a + 3 b + 6 c is :$ (1) $a$ (2) $c$ (3) $0$ (4) $a + c$ 93. $Let p, q, r be three non-coplanar vectors and [pqr] a = q \times r, [pqr] b = r \times p, [pqr] c = p \times q . If a, b, c denote the coterminous edges of a parallelopiped, then its height with the base having a and c is$ (1) $\| p \|$ (2) $\frac{1}{\| a \|}$ (3) $\frac{1}{\| b \|}$ (4) $\frac{1}{\| q \|}$ 94. $Let a = - 5 i + j - 3 k, b = i + 2 j - 4 k and c = (((a \times b) \times i) \times i) \times i . Then c + (- i + j + k) is equal to$ (1) $-12$ (2) $-10$ (3) $-13$ (4) $-15$ 95. $p = 2 i - 3 j + k, q = i + j - k . If the vectors a and b are the orthogonal projections of p on q and q on p respectively, then \frac{a \times b}{a . b} =$ (1) $\frac{2 i + 3 j + 5 k}{19 2}$ (2) $\frac{2 i + 3 j + 5 k}{38}$ (3) $\frac{2 i + 3 j + 5 k}{2}$ (4) $\frac{3 i - 2 j}{13}$ 96. $If r = (2 - λ + μ) i + (1 - μ) j + (2 - 3 λ + 2 μ) k is the vector equation of a plane, then the equivalent cartesian equation of the plane is$ (1) $3 x + y - z = 5$ (2) $3 x - y + z = 5$ (3) $- 3 x + y + z = 5$ (4) $3 x - y - z = 5$ 97. $Consider two vectors u = 3 i - j and v = 2 i + j - λk, λ > 0 . The angle between them is given by cos^{- 1} (\frac{5}{2 7}) . Let v = v_{1} + v_{2}, where v_{1} is parallel to u and v_{2} is perpendicular to u . Then the value (v_{1})^{2} + (v_{2})^{2} is equal to$ (1) $\frac{23}{2}$ (2) $\frac{25}{2}$ (3) $10$ (4) $14$ 98. $The vectors 3 a - 5 b and 2 a + b are mutually perpendicular and the vectors a + 4 b and - a + b are also mutually perpendicular then the acute angle between a and b is$ (1) $cos^{- 1} (\frac{19}{5 43})$ (2) $cos^{- 1} (\frac{9}{5 43})$ (3) $π - cos^{- 1} (\frac{19}{5 43})$ (4) $π - cos^{- 1} (\frac{9}{5 43})$ 99. $If 2 i + 3 j, i + j + k and λi + 4 j + 2 k taken in an order are coterminous edges of a parallelopiped of volume 2 cu units, then value of λ is$ (1) $-4$ (2) $2$ (3) $3$ (4) $4$ 100. $Let a = i + j + k, b = i - 2 j + k, c = i + 3 j - 2 k, d = 2 i + j - k be four vectors and let l = b \cdot c and m = c \cdot a . Then [mb + l a b d] =$ (1) $79$ (2) $-63$ (3) $0$ (4) $1$ 101. $Let a = 3 i - j + 2 k, b = a \times (i - 2 k) and c = b \times k . Then the projection of c - 2 j on a is:$ (1) $214$ (2) $14$ (3) $37$ (4) $27$ 102. $Let A = (α, 1, 2 α), B = (3, 1, 2) and C = 4 i - j + 3 k . If AB \times C = 6 i + 9 j - 5 k, then α^{2} + α + 5 =$ (1) $11$ (2) $7$ (3) $9$ (4) $5$ 103. $If a, b, c are three vectors with magnitudes 3, 1, 2 respectively, such that a \times (a \times c) + 3 b = 0, if θ is the angle between a and c, then sec^{2} θ is$ (1) $1$ (2) $\frac{3}{2}$ (3) $\frac{4}{3}$ (4) $\frac{2}{3}$ 104. $Let a = 2 i - 3 j + 4 k and b = 7 i + j - 6 k .If r \times a = r \times b, r \cdot (i + 2 j + k) = - 3, then r \cdot (2 i - 3 j + k) is equal to:$ (1) $12$ (2) $8$ (3) $13$ (4) $10$ 105. $If (a \times b)^{2} + (a \cdot b)^{2} = 144 and \| a \| = 4, then \| b \| =$ (1) $12$ (2) $16$ (3) $8$ (4) $3$ 106. $If \| a \| = 4, \| b \| = 5, \| a - b \| = 3 and θ is the angle between the vectors a and b, then cot^{2} θ =$ (1) $\frac{9}{16}$ (2) $\frac{4}{3}$ (3) $\frac{3}{4}$ (4) $\frac{16}{9}$ 107. $Let a = 2 i - j + 3 k, b = 3 i - 5 j + k and c be a vector such that a \times c = c \times b and (a + c) (b + c) = 168 . Then the maximum value of \| c \|^{2} is:$ (1) $462$ (2) $77$ (3) $154$ (4) $308$ 108. $If a = 3, b = 4, c = 5 each one of a, b & c is perpendicular to the sum of the remaining then \| a + b + c \| is equal to$ (1) $\frac{5}{2}$ (2) $\frac{2}{5}$ (3) $52$ (4) $5$ 109. $If 3 i - 2 j - k, 2 i + 3 j - 4 k, - i + j + 2 k and 4 i + 5 j + λk are respectively the position vectors of four coplanar points P, Q, R and S, then λ =$ (1) $\frac{46}{17}$ (2) $- \frac{46}{17}$ (3) $\frac{146}{17}$ (4) $- \frac{146}{17}$ 110. $A vector makes equal angles α with X and Y -axis, and 90^{\circ} with Z -axis. Then, α is equal to$ (1) $60^{\circ} or 120^{\circ}$ (2) $30^{\circ} and 150^{\circ}$ (3) $45^{\circ} and 135^{\circ}$ (4) $90^{\circ}$ 111. $Let a = 2 i + j - k, b = ((a \times (i + j)) \times i) \times i . Then the square of the projection of a on b is :$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $2$ (4) $\frac{1}{5}$ 112. $If M_{1}, M_{2}, M_{3} and M_{4}, are respectively the magnitudes of the vectors a_{1} = 2 i - j + k, a_{2} = - 3 i - 4 j - 4 k, a_{3} = - i + j - k, a_{4} = - i + 3 j + k, then the correct order of M_{1}, M_{2}, M_{3} and M_{4} is$ (1) $M_{3} < M_{1} < M_{4} < M_{2}$ (2) $M_{3} < M_{1} < M_{2} < M_{4}$ (3) $M_{3} < M_{4} < M_{1} < M_{2}$ (4) $M_{3} < M_{4} < M_{2} < M_{1}$ 113. $Let a = 2 i - 3 j + k, b = 3 i + 2 j + 5 k and a vector c be such that (a - c) \times b = - 18 i - 3 j + 12 k and a \cdot c = 3 . If b \times c = d, then \| a \cdot d \| is equal to$ (1) $15$ (2) $18$ (3) $12$ (4) $9$ 114. $If a = α i + 3 j - 6 k and b = 2 i - j + β k, then the value of α, β so that a and b may be collinear are$ (1) $(5, 3)$ (2) $(6, 2)$ (3) $(2, - 6)$ (4) $(- 6, 2)$ 115. $The area of the parallelogram whose adjacent sides are i + k and 2 i + j + k is$ (1) $3$ (2) $2$ (3) $4$ (4) $3$ 116. $If a, b, c are unit vectors and θ is angle between a and c and a + 2 b + 2 c = 0, then \| a \times c \| =$ (1) $\frac{15}{2}$ (2) $\frac{15}{4}$ (3) $15$ (4) $\frac{15}{3}$ 117. $If a, b and c are three non-coplanar vectors and p, q and r are vectors defined by p = \frac{b \times c}{[ ab c ]}, q = \frac{c \times a}{[ ab c ]} and r = \frac{a \times b}{[ ab c ]}, then the value of (a + b) \cdot p + (b + c) \cdot q + (c + a) \cdot r =$ (1) $3$ (2) $0$ (3) $1$ (4) $2$ 118. $Let the three sides of a triangle ABC be given by the vectors 2 i - j + k, i - 3 j - 5 k and 3 i - 4 j - 4 k . Let G be the centroid of the triangle ABC . Then 6 ((AG)^{2} + \| BG \|^{2} + \| CG \|^{2}) is equal to____.$ 119. $Let a = 2 i - 3 j + 4 k, b = 3 i + 4 j - 5 k and a vector c be such that a \times (b + c) + b \times c = i + 8 j + 13 k . If a \cdot c = 13, then (24 - b \cdot c) is equal to$ 120. $Let a and b be two vectors such that ∣ a ∣ = 14, ∣ b ∣ = 6 and ∣ a \times b ∣ = 48 . Then (a \cdot b)^{2} is equal to$