Vector Algebra PYQs Set 7 (PDF)

Set 7 of 16 · Questions 181-210 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 7 (Q181-210)
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181. $If two vectors a and b which are perpendicular to each other are such that \| a \| = 8 and \| b \| = 3, then \| a - 2 b \| =$ (1) $10$ (2) $2$ (3) $6$ (4) $12$ 182. $Let ABC be a triangle. Let u = AB and v = AC . If D is a middle point of BC, then AD =$ (1) $\frac{u - v}{2}$ (2) $\frac{v - u}{2}$ (3) $\frac{u + v}{2}$ (4) $u + v$ 183. $Let a = 4 i - j + k, b = 11 i - j + k and c be a vector such that (a + b) \times c = c \times (- 2 a + 3 b) . If (2 a + 3 b) \cdot c = 1670, then \| c \|^{2} is equal to :$ (1) $1609$ (2) $1618$ (3) $1600$ (4) $1627$ 184. $If the volume of the parallelopiped with a, b and c as coterminous edges is 40 cubic units, then the volume of the parallelopiped having b + c, c + a and a + b as coterminous edges in cubic units is$ (1) $160$ (2) $40$ (3) $80$ (4) $120$ 185. $Let a, b and c be three unit vectors , out of which vectors b and c are non-parallel. If α and β are the angles which vector a makes with vectors b And c respectively and a \times (b \times c) = \frac{1}{2} b, then (α - β) is equal to :$ (1) $30^{0}$ (2) $90^{0}$ (3) $60^{0}$ (4) $45^{0}$ 186. $If the angle between the vectors 2 α^{2} i + 4 α j + k and 7 i - 2 j + α k is obtuse, then$ (1) $α > \frac{1}{2}$ (2) $0 < α < \frac{1}{2}$ (3) $α < 0$ (4) $\| α \| < \frac{1}{2}$ 187. $If α, β, γ are real numbers such that (\frac{7}{3} + β) i - j + (α + γ) k = \frac{5}{3} (α i + j - k) + β (2 j + k) + (i + γ j + 3 k), then 5 α - 9 β + 13 γ =$ (1) $4$ (2) $12$ (3) $0$ (4) $15$ 188. $If P, Q, R and S are the points with position vectors i + j - k, 2 i - j + 3 k, 2 i - 3 k and 3 i - 2 j + k respectively, then the angle between PQ and RS is$ (1) $0$ (2) $\frac{π}{2}$ (3) $π$ (4) $\frac{π}{3}$ 189. $Let a = i + 2 j - 3 k and b = 2 i - 3 j + 5 k If r \times a = b \times r, r \cdot (α i + 2 j + k) = 3 and r \cdot (2 i + 5 j - α k) = - 1, α \in R, then the value of α + \| r \|^{2} is equal to:$ (1) $9$ (2) $15$ (3) $13$ (4) $11$ 190. $Let α = 4 i + 3 j + 5 k and β = i + 2 j - 4 k . Let β_{1} be parallel to α and β_{2} be perpendicular to α . If β = β_{1} + β_{2},. then the value of 5 β_{2} . (i + j + k) is$ (1) $6$ (2) $8$ (3) $7$ (4) $9$ 191. $The vector x is perpendicular to the vectors a = 3 i + 2 j + 2 k, b = 18 i - 22 j - 5 k and makes an obtuse angle with j . If \| x \| = 14, then x =$ (1) $8 i + 12 j + 24 k$ (2) $- 8 i + 6 j + 24 k$ (3) $8 i - 12 j - 24 k$ (4) $- 8 i - 12 j + 24 k$ 192. $The projection of a = 3 i - j + 5 k on b = 2 i + 3 j + k is$ (1) $\frac{8}{14}$ (2) $14$ (3) $\frac{8}{35}$ (4) $\frac{8}{39}$ 193. $For a triangle ABC, let p = BC, q = CA and r = BA . If \| p \| = 23, \| q \| = 2 and cosθ = \frac{1}{3}, where θ is the angle between p and q, then \| p \times (q - 3 r) \|^{2} + 3 \| r \|^{2} is equal to :$ (1) $410$ (2) $340$ (3) $200$ (4) $220$ 194. $The vector a = - i + 2 j + k is rotated through a right angle, passing through the y -axis in its way and the resulting vector is b . the the projection of 3 a + 2 b on c = 5 i + 4 j + 3 k is$ (1) $32$ (2) $1$ (3) $6$ (4) $23$ 195. $The incentre of the triangle formed by the points (0, 0, 0), (3, 0, 0) and (0, 4, 0) is$ (1) $(1, 1, 0)$ (2) $(- 1, - 1, 0)$ (3) $(1, 0, 1)$ (4) $(- 1, 0, - 1)$ 196. $Let ABC be a triangle and a, b, c be the position vectors of A, B, C respectively. Let D divide BC in the ratio 3 : 1 internally and E divide AD in the ratio 4 : 1 internally. Let BE meet AC in F . If E divides BF in the ratio 3 : 2 internally then the position vector of F is$ (1) $\frac{a + b + c}{3}$ (2) $\frac{a - 2 b + 3 c}{2}$ (3) $\frac{a + 2 b + 3 c}{2}$ (4) $\frac{a - b + 3 c}{3}$ 197. $Let a = 2 i - 5 j + 5 k and b = i - j + 3 k . If c is a vector such that 2 (a \times c) + 3 (b \times c) = 0 and (a - b) \cdot c = - 97, then \| c \times k \|^{2} is equal to$ (1) $233$ (2) $218$ (3) $193$ (4) $205$ 198. $Let a and b be two vectors. Let ∣ a ∣ = 1, ∣ b ∣ = 4 and a \cdot b = 2 . If c = (2 a \times b) - 3 b, then the value of b \cdot c is$ (1) $- 24$ (2) $- 48$ (3) $- 84$ (4) $- 60$ 199. $If the volume of the parallelopiped with coterminous edges - p j + 5 k, i - j + q k and 3 i - 5 j is 8 , then$ (1) $3 pq + 2 = 0$ (2) $3 pq - 2 = 0$ (3) $pq + 2 = 0$ (4) $pq - 2 = 0$ 200. $Let a = i - 2 j + 3 k . If b is a vector such that a \cdot b = \| b \|^{2} and \| a - b \| = 7, then \| b \| =$ (1) $14$ (2) $21$ (3) $7$ (4) $7$ 201. $A unit vector perpendicular to both the vectors i + j and j + k is$ (1) $\frac{i + j + k}{3}$ (2) $\frac{i - j + k}{3}$ (3) $\frac{- i - j + k}{3}$ (4) $\frac{i + j - k}{3}$ 202. $If a, b, c are mutually perpendicular vectors having magnitudes 1, 2, 3 respectively, then [a + b + c b - a c] =$ (1) $0$ (2) $6$ (3) $12$ (4) $18$ 203. $Let λ ϵ R, a = λi + 2 j - 3 k, b = i - λj + 2 k . If ((a + b) \times (a \times b)) \times (a - b) = 8 i - 40 j - 24 k, then (λ (a + b) \times (a - b))^{2} is equal to$ (1) $140$ (2) $132$ (3) $144$ (4) $136$ 204. $Let a = 6 i + j - k and b = i + j . If c is a is vector such that \| c \| \geq 6, a \cdot c = 6 \| c \|, \| c - a \| = 22 and the angle between a \times b and c is 60^{\circ}, then \| (a \times b) \times c \| is equal to:$ (1) $\frac{9}{2} (6 - 6)$ (2) $\frac{3}{2} 6$ (3) $\frac{9}{2} (6 + 6)$ (4) $\frac{3}{2} 3$ 205. $Let the position vectors of two points A and B be a + b + c and a - 2 b + 3 c, respectively. If the points P and Q divide AB in the ratio 1 : 3 internally and externally respectively, then 3 \| AB \| =$ (1) $4 \| PQ \|$ (2) $3 \| PQ \|$ (3) $1 2 \| PQ \|$ (4) $2 \| PQ \|$ 206. $Let a = 3 i + j - 2 k, b = 4 i + j + 7 k and c = i - 3 j + 4 k be three vectors. If a vectors p satisfies p \times b = c \times b and p \cdot a = 0, then p \cdot (i - j - k) is equal to$ (1) $24$ (2) $36$ (3) $28$ (4) $32$ 207. $A, B, C, D are any 4 points and \| AB \times CD + BC \times AD + CA \times BD \| = λ (Area of △ ABC) then λ =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 208. $If line joining points A and B having position vectors 6 a - 4 b + 4 c and - 4 c respectively, and the line joining the points C and D having position vectors - a - 2 b - 3 c and a + 2 b - 5 c intersect, then their point of intersection is$ (1) $B$ (2) $C$ (3) $D$ (4) $A$ 209. $Let a, b, c be three vectors such that \| a \| = 3, \| b \| = 5, b \cdot c = 10 and the angle between b and c is \frac{π}{3} . If a is perpendicular to the vector b \times c, then \| a \times (b \times c) \| is equal to$ (1) $103$ (2) $53$ (3) $60$ (4) $30$ 210. $Let c be the projection vector of b = λi + 4 k, λ > 0, on the vector a = i + 2 j + 2 k . If \| a + c \| = 7, then the area of the parallelogram formed by the vectors b and c is .$