Vector Algebra PYQs Set 9 (PDF)

Set 9 of 16 · Questions 241-270 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 9 (Q241-270)
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241. $If a + b, b + c and c + a are coterminous edges of a parallelepiped, then its volume is$ (1) $3 [a c b]$ (2) $0$ (3) $2 [ab c]$ (4) $4 [ba c]$ 242. $If two vertices of a triangle are A (3, 1, 4) and B (- 4, 5, - 3) and the centroid of the triangle is G (- 1, 2, 1), then the third vertex C of the triangle is$ (1) $(2, 0, 2)$ (2) $(- 2, 0, 2)$ (3) $(0, - 2, 2)$ (4) $(2, - 2, 0)$ 243. $The angle between the vectors 2 k - 3 j and i - 2 k is$ (1) $cos^{- 1} (\frac{8}{65})$ (2) $cos^{- 1} (\frac{- 4}{65})$ (3) $cos^{- 1} (\frac{2}{65})$ (4) $cos^{- 1} (\frac{3}{13})$ 244. $The projections of a vector on the three coordinate axes are 6, -3, 2 respectively. The direction cosines of the vector are$ (1) $6, -3, 2$ (2) $\frac{6}{5}, - \frac{3}{5}, \frac{2}{5}$ (3) $\frac{6}{7}, - \frac{3}{7}, \frac{2}{7}$ (4) $- \frac{6}{7}, - \frac{3}{7}, \frac{2}{7}$ 245. $If a = 2 i + 3 j - 5 k, b = mi + nj + 12 k and a \times b = 0, then (m, n) is equal to$ (1) $(\frac{- 24}{5}, \frac{- 36}{5})$ (2) $(\frac{- 24}{5}, \frac{36}{5})$ (3) $(\frac{24}{5}, \frac{- 36}{5})$ (4) $(\frac{24}{5}, \frac{36}{5})$ 246. $Let a \to= i^- 2 j^+ k^and b \to= i^- j^+ k^be two vectors. If c \to is a vector such that b \to \times c \to= b \to \times a \to and c \to \cdot a \to= 0 then c \to \cdot b \to is equal to$ (1) $\frac{1}{2}$ (2) $\frac{1}{3}$ (3) $- \frac{1}{2}$ (4) $- \frac{1}{3}$ 247. $If the components of a = α i + β j + γ k along and perpendicular to b = 3 i + j - k respectively, are \frac{16}{11} (3 i + j - k) and \frac{1}{11} (- 4 i - 5 j - 17 k), then α^{2} + β^{2} + γ^{2} is equal to:$ (1) $26$ (2) $18$ (3) $23$ (4) $16$ 248. $If the vectors x i - 3 j + 7 k and i + y j - z k are collinear then the value of \frac{x y ^{2}}{z} is equal to$ (1) $9/7$ (2) $- 9/7$ (3) $- 7/9$ (4) $7/9$ 249. $Let the volume of a parallelepiped whose coterminous edges are given by u \to= i^+ j^+ λ k^, v \to= i^+ j^+ 3 k^and w \to= 2 i^+ j^+ k^be 1 cu unit. If θ be the angle betweenthe edges u \to and w \to then cos θ can be:$ (1) $\frac{7}{3 10}$ (2) $\frac{7}{6 3}$ (3) $\frac{5}{3 3}$ (4) $\frac{5}{7}$ 250. $Let the angle θ, 0 < θ < \frac{π}{2} between two unit vectors a and b be sin^{- 1} (\frac{65}{9}) . If the vector c = 3 a + 6 b + 9 (a \times b), then the value of 9 (c \cdot a) - 3 (c \cdot b) is$ (1) $31$ (2) $29$ (3) $24$ (4) $27$ 251. $Let a = i + j + k and b be two vectors such that a \cdot b = 1, cos (a, b) = \frac{1}{3} and the components of b w.r.t (i, j, k) be integers. Then the number of possible vectors that represent b is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 252. $If G (g), H (h) and P (p) are centroid, orthocenter and circumcenter of a triangle and x p + y h + z g = 0 then (x, y, z) \equiv$ (1) $1, 1, - 2$ (2) $2, 1, - 3$ (3) $1, 3, - 4$ (4) $2, 3, - 5$ 253. $OA and BO are two vectors of magnitudes 5 and 6 respectively, If ∠ BOA = 60^{\circ} then OA \cdot OB is equal to$ (1) $0$ (2) $15$ (3) $-15$ (4) $153$ 254. $Let a = 2 i + 5 j - k, b = 2 i - 2 j + 2 k and c be three vectors such that (c + i) \times (a + b + i) = a \times (c + i) . If a \cdot c = - 29, then c \cdot (- 2 i + j + k) is equal to:$ (1) $15$ (2) $12$ (3) $10$ (4) $5$ 255. $If a and b are the two vectors such that \| a \| = 33, \| b \| = 4 and \| a + b \| = 7, then the angle between a and b is$ (1) $120^{\circ}$ (2) $60^{\circ}$ (3) $30^{\circ}$ (4) $150^{\circ}$ 256. $If \| a \| = 3, \| b \| = 4 and the angle between a and b is 120^{\circ}, then \| 4 a + 3 b \| is equal to$ (1) $25$ (2) $7$ (3) $13$ (4) $12$ 257. $a, b and c are three vectors such that \| a \| = 1, \| b \| = 2, \| c \| = 3 and b, c are perpendicular. If projection of b on a is the same as the projection of c on a, then \| a - b + c \| is equal to$ (1) $2$ (2) $7$ (3) $14$ (4) $21$ 258. $Let a, b and c be three non-coplanar vectors. The vector equation of a line which passes through the point of intersection of two lines, one joining the points a + 2 b - 5 c, - a - 2 b - 3 c and the other joining the points - 4 c, 6 a - 4 b + 4 c is$ (1) $r = 2 a - 4 b + 3 c + μ (a - 6 b + 4 c)$ (2) $r = 3 a + 6 b - c + μ (a + 2 b + c)$ (3) $r = 2 a + 3 b - c + μ (a + b - c)$ (4) $r = - 2 b + 3 c + μ (a - 4 b + 3 c)$ 259. $If a = x i + y j + z k, then (a \times i) \cdot (i + j) + (a \times j) \cdot (j + k) + (a \times k) \cdot (k + i) =$ (1) $x - y + z$ (2) $x + y + z$ (3) $x + y - z$ (4) $- x + y + z$ 260. $(a + 2 b - c) \cdot {(a - b) \times (a - b - c)} is equal to$ (1) $2 [abc]$ (2) $[abc]$ (3) $3 [abc]$ (4) $[abc]^{2}$ 261. $If a = 2 i - 3 j + 5 k, b = 3 i - 4 j + 5 k and c = 5 i - 3 j - 2 k, then the volume of the parallelopiped with coterminous edges a + b, b + c, c + a, is$ (1) $1$ (2) $5$ (3) $8$ (4) $16$ 262. $If a = 2 i - 5 j + 8 k, b = 7 i - 5 j + 3 k are two vectors and (2 a - 3 b) \times (4 a + b) = x i + y j + z k, then x + y + z =$ (1) $-1000$ (2) $1400$ (3) $1000$ (4) $-1400$ 263. $If the vectors ai + j + k, i + bj + k and i + j + c k are coplanar (a \neq = b \neq = c \neq = 1), then the value of abc - (a + b + c) =$ (1) $2$ (2) $-2$ (3) $0$ (4) $-1$ 264. $If 35 i + 14 j - 77 k, 2 i + 7 j + 5 k and 5 i + 2 j + λk are coplanar, then λ is equal to$ (1) $11$ (2) $-11$ (3) $-10$ (4) $10$ 265. $If the points having the position vectors 3 i - 2 j - k, 2 i + 3 j - 4 k, - i + j + 2 k and 4 i + 5 j + λk are coplanar, then λ =$ (1) $- \frac{146}{17}$ (2) $8$ (3) $-8$ (4) $\frac{146}{17}$ 266. $Let a = i + j + k, b = 2 i + 4 j - 5 k and c = x i + 2 j + 3 k, x \in R . If d is the unit vector in the direction of b + c such that a \cdot d = 1, then (a \times b) \cdot c is equal to$ (1) $11$ (2) $3$ (3) $9$ (4) $6$ 267. $A (2, 3, 4), B (4, 5, 7), C (2, - 6, 3) and D (4, - 4, k) are four points. If the line AB is parallel to CD , then k is equal to$ (1) $2$ (2) $4$ (3) $5$ (4) $6$ 268. $If \| a \| = 16, \| b \| = 4, then \| a \times b \|^{2} + \| a \cdot b \|^{2} =$ (1) $4$ (2) $16$ (3) $8$ (4) $64$ 269. $Let a = i + j + k, b = - i - 8 j + 2 k and c = 4 i + c_{2} j + c_{3} k be three vectors such that b \times a = c \times a . If the angle between the vector c and the vector 3 i + 4 j + k is θ, then the greatest integer less than or equal to tan^{2} θ is :$ 270. $Let a, b, c be three vectors such that (a) = 31, 4 (b) = (c) = 2 and 2 (a \times b) = 3 (c \times a) If the angle between b and c is \frac{2 π}{3}, then (\frac{a \times c}{a . b}) is equal to ________$