Vector Algebra PYQs Set 12 (PDF)

Set 12 of 16 · Questions 331-360 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 12 (Q331-360)
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331. $If a = 2 i + 3 j - k, b = i + 2 j - 5 k, c = 3 i + 5 j - k, then a vector perpendicular to a and in the plane containing b and c is$ (1) $- 17 i + 21 j - 97 k$ (2) $17 i + 21 j - 123 k$ (3) $- 17 i - 21 j + 97 k$ (4) $- 17 i - 21 j - 97 k$ 332. $If a & b are perpendicular vactors, then a \times (a \times (a \times (a \times b))) is equal to$ (1) $12 \| a \|^{4} b$ (2) $a \times b$ (3) $\| a \|^{4} b$ (4) $0$ 333. $Let a and b be two non-zero vectors perpendicular to each other and a \| = \| b If \| a \times b \| = \| a \|, thenthe angle between the vectors (a + b + (a \times b)) and a is equal to:$ (1) $sin^{- 1} (\frac{1}{3})$ (2) $cos^{- 1} (\frac{1}{3})$ (3) $cos^{- 1} (\frac{1}{2})$ (4) $sin^{- 1} (\frac{1}{6})$ 334. $Let u and v be two vectors. Then \| u - v \| = ∥ u \| - \| v ∥ if and only if$ (1) $\| u \| = \| v \|$ (2) $u and v have the same direction$ (3) $u and v have the opposite direction$ (4) $u = v$ 335. $Three vectors a, b and c satisfy a + b + c = 0 . If \| a \| = 3, \| b \| = 4, \| c \| = 2, then a \cdot b + b \cdot c + c \cdot a + 2 (\| a \| + \| b \| + \| c \|) is equal to$ (1) $\frac{- 7}{2}$ (2) $\frac{7}{2}$ (3) $\frac{- 11}{2}$ (4) $\frac{11}{2}$ 336. $Let a, b, c be three vectors such that a \times b = 2 (a \times c) . If \| a \| = 1, \| b \| = 4, \| c \| = 2, and the angle between b and c is 60^{\circ}, then \| a \cdot c \| is equal to$ (1) $1$ (2) $2$ (3) $0$ (4) $4$ 337. $If a = i + j + k, b = i - j + 2 k, c = x i + (x - 2) j - k and c is linear combination of a and b, then x has the value$ (1) $-4$ (2) $0$ (3) $1$ (4) $-2$ 338. $Let α = 3 i + j and β = 2 i - j + 3 k . If β = β_{1} - β_{2}, where β_{1} is parallel to a and β_{2} is perpendicular to a then β_{1} \times β_{2} is equal to:$ (1) $- 3 i + 9 j + 5 k$ (2) $3 i - 9 j - 5 k$ (3) $\frac{1}{2} (- 3 i + 9 j + 5 k)$ (4) $\frac{1}{2} (3 i - 9 j + 5 k)$ 339. $Let ABCDEF be a regular hexagon with the vertices A, B, C, D, E and F counter clockwise. Then, the vector AB + BC is equal parallel to$ (1) $BC + CD$ (2) $CD + DE$ (3) $AF + FE$ (4) $FE + ED$ 340. $If the area of triangle with vertices i + y j, i + 2 k and 3 j + k is 6 sq. units, then the values of y are$ (1) $2, - 4$ (2) $2,4$ (3) $- 2, 4$ (4) $3,4$ 341. $Let a = 3 i + 2 j + x k and b = i - j + k, for some real x . Then (a \times b) = r is possible if :$ (1) $32 < r \leq 332$ (2) $r \geq 5 \frac{3}{2}$ (3) $0< r \leq \frac{3}{2}$ (4) $3 \frac{3}{2} < r<532$ 342. $Let a, b, c be the position vectors of the vertices of a triangle ABC . Through the vertices, lines are drawn parallel to the sides to form the triangle A^{’} B^{’} C^{’} . Then the centroid of Δ A^{’} B^{’} C^{’} is$ (1) $\frac{a + b + c}{9}$ (2) $\frac{a + b + c}{6}$ (3) $\frac{a + b + c}{3}$ (4) $\frac{2 ( a + b + c )}{3}$ 343. $Let a, b, c be three distinct real numbers, none equal to one. If the vectors a i^+ j^+ k^, i^+ b j^+ k^and i^+ j^+ c k^are coplanar, then \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} is equal to$ (1) $2$ (2) $-1$ (3) $-2$ (4) $1$ 344. $If the vectors BC = 2 i + j + k and CD = i + 2 j - 2 k represent two adjacent sides of a parallelogram ABCD and θ is the angle between its diagonals AC and BD then tan θ =$ (1) $\frac{- 3}{209}$ (2) $\frac{- 10 2}{3}$ (3) $\frac{10 2}{209}$ (4) $- \frac{3}{10 2}$ 345. $If a + b + c = 0 and \| a \| = 7, \| b \| = 5, \| c \| = 3 then the angle between b and c is$ (1) $30^{\circ}$ (2) $45^{\circ}$ (3) $60^{\circ}$ (4) $90^{\circ}$ 346. $Let a = 2 i^+ 7 j^- k^, b^= 3 i^+ 5 k^and c = i^- j^+ 2 k^Let d be a vector which is perpendicular to both a and b, and c . d = 12 . Then is equal to$ (1) $24$ (2) $44$ (3) $42$ (4) $48$ 347. $L and M are two points with position vectors 2 a - b and a + 2 b respectively. The position vector of the point N which divides the line segment LM in the ratio 2 : 1 externally is$ (1) $3 b$ (2) $4 b$ (3) $5 b$ (4) $3 a + 4 b$ 348. $Statement-1 : The point A(1, 0, 7) is the mirror image of the point B(1, 6, 3) in the line : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} Statement-2 : The line : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} bisects the line segment joining A(1, 0, 7) and B(1, 6, 3).$ (1) $Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.$ (2) $Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.$ (3) $Statement-1 is true, Statement-2 is false.$ (4) $Statement-1 is false, Statement-2 is true.$ 349. $The position vectors of vertices of △ ABC are 4 i - 2 j, i + 4 j - 3 k and - i + 5 j + k respectively, then m ∠ ABC =$ (1) $π /6$ (2) $π /3$ (3) $π /4$ (4) $π /2$ 350. $Let PQRS be a quadrilateral. If M and N are midpoints of the sides PQ and RS respectively then PS + QR =$ (1) $3 MN$ (2) $4 MN$ (3) $2 MN$ (4) $2 NM$ 351. $If the position vectors of the vertices of △ ABC are 3 i + 4 j - k, i + 3 j + k and 5 (i + j + k), respectively. Then, the magnitude of the altitude from A onto the side BC is$ (1) $\frac{4}{3} 5$ (2) $\frac{5}{3} 5$ (3) $\frac{7}{3} 5$ (4) $\frac{8}{3} 5$ 352. $Let S be the set of all real values of λ such that a plane passing through the points (- λ^{2}, 1, 1), (1, - λ^{2}, 1) and (1, 1, - λ^{2}) also passes through the point (-1, -1, 1). Then S is equal to -$ (1) ${1, -1}$ (2) ${3}$ (3) ${3, - 3}$ (4) ${3, -3}$ 353. $Let a = i^+ j^+ k^and b = j^- k^. If c is a vector such that a \times c = b and a . c = 3, then a . (b \times c) is equal to :$ (1) $-2$ (2) $-6$ (3) $6$ (4) $2$ 354. $ABC is a right-angled triangle in which max {AB, BC, AC} = BC . If the position vectors of B and C are respectively 3 i - 2 j + k and 5 i + j - 3 k, then AB \cdot AC + BA \cdot BC + CA \cdot CB =$ (1) $28$ (2) $29$ (3) $27$ (4) $25$ 355. $Let the position vectors of the points A, B, C and D be 5 i^+ 5 j^+ 2 λ k^, i^+ 2 j^+ 3 k^, - 2 i^+ λ j^+ 4 k^and - i^+ 5 j^+ 6 k^. Let the set S = {λ \in R: the points A, B, C and D are coplanar}. The \sum_{λ \in S} (λ + 2)^{2} equal to$ (1) $25$ (2) $\frac{37}{2}$ (3) $14$ (4) $41$ 356. $The sum of all values of a, for which the points whose position vectors are i^- 2 j^+ 3 k^, 2 i^- 3 j^+ 4 k^,_{(} α + 1_{)} i^+ 2 k^and 9 i^+_{(} α - 8_{)} j^+ 6 k^are coplanar, is equal to$ (1) $-2$ (2) $2$ (3) $6$ (4) $4$ 357. $If a = \frac{1}{10} (3 i^+ k^) and b = \frac{1}{7} (2 i^+ 3 j^- 6 k^), then the value of (2 a - b) . [(a \times b) \times (a + 2 b)] is:$ (1) $- 5$ (2) $-3$ (3) $5$ (4) $3$ 358. $Let a = i + j + k, b = 2 i + 2 j + k and d = a \times b . If c is a vector such that a \cdot c = ∣ c ∣, ∣ c - 2 a ∣^{2} = 8 and the angle between d and c is \frac{π}{4}, then ∣10 - 3 b \cdot c ∣ + ∣ d \times c ∣^{2} is equal to$ 359. $Let a = i^- α j^+ β k^, b = 3 i^+ β j^- α k^and c = - α i^- 2 j^+ k^, where α and β are integers. If a . b = - 1 and b . c = 10, then (a \times b) . c is equal to$ 360. $If the projection of the vector i^+ 2 j^+ k^on the sum of the two vectors 2 i^+ 4 j^- 5 k^and - λ i^+ 2 j^+ 3 k^is 1, then λ is equal to .$