Vector Algebra PYQs Set 11 (PDF)

Set 11 of 16 · Questions 301-330 of 461

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 11 (Q301-330)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
301. $If a, b, c are unit vectors such that a is perpendicular to both b, c and angle between bc is \frac{2 π}{3}, then \| a + 3 b - 4 c \|^{2} =$ (1) $6$ (2) $14$ (3) $38$ (4) $26$ 302. $If e is the unit vector perpendicular to the plane determined by the point 2 i + j + k, i - j + k and - i + j - k If a = 2 i - 3 j + 6 k, then the projection vector of a on e is$ (1) $\frac{11}{14} (- 2 i + j + 3 k)$ (2) $\frac{1}{3} (i - 2 j + 2 k)$ (3) $\frac{1}{7} (2 i - 3 j + 6 k)$ (4) $\frac{1}{14} (2 i - j + 3 k)$ 303. $If A (2 i + j - k), B (λi + 5 j + 4 k), C (- 4 i + 3 j + 2 k) and D (- i - 2 j + 3 k) are four points in space such that AB = x AC + y AD for some real numbers x \neq = 0, y \neq = 0 then 17 (λ + 9) =$ (1) $5$ (2) $3$ (3) $7$ (4) $9$ 304. $Let x_{0} be the point of local maxima of f (x) = a \cdot (b \times c) Where a = x i^- 2 j^+ 3 k^, b = - 2 i^+ x j^- k^and c = 7 i^- 2 j^+ x k^Then the value of a \cdot b \to + b \to \cdot c \to + c \to \cdot a \to at x = x_{0} is:$ (1) $14$ (2) $- 30$ (3) $- 4$ (4) $- 22$ 305. $If angle between the vectors a = 2 λ^{2} i + 4 λj + k and b = 7 i - 2 j + λk is obtuse, then the values of λ is$ (1) $(0, \frac{1}{2})$ (2) $(\frac{1}{2}, \infty)$ (3) $(- \infty, 0)$ (4) $(- 1, 0)$ 306. $If a is a unit vector, then \| a \times i \|^{2} + \| a \times j \|^{2} + \| a \times k \|^{2} =$ (1) $2$ (2) $4$ (3) $1$ (4) $0$ 307. $Let a = i + α j + β k, α, β \in R . Let a vector b be such that the angle between a and b is \frac{π}{4} and \| b \|^{2} = 6, If a \cdot b = 32, then the value of (α^{2} + β^{2}) \| a \times b \|^{2} is equal to$ (1) $90$ (2) $75$ (3) $95$ (4) $85$ 308. $Find \| a \times b \|^{2}, if \| a \| = 2, \| b \| = 3 and (a, b) = \frac{π}{6}$ (1) $-9$ (2) $9$ (3) $3$ (4) $-3$ 309. $If a \cdot i = a \cdot (i + j) = a \cdot (i + j + k) = 1 then a =$ (1) $i$ (2) $i + j - k$ (3) $i + j$ (4) $i - k$ 310. $If 2 i + 4 j - 5 k, i + j + k, j + 2 k are the position vectors of the vertices A, B, C of a triangle respectively, then a unit vector along the median drawn through the vertex A is$ (1) $\frac{1}{174} (5 i + 10 j - 7 k)$ (2) $\frac{1}{214} (3 i + 6 j - 13 k)$ (3) $\frac{1}{66} (i + j - 8 k)$ (4) $\frac{1}{7} (3 i + 6 j - 2 k)$ 311. $If the volume of a parallelopiped, whose coterminous edges are given by the vectors a = i^+ j^+ n k^, b \to= 2 i^+ 4 j^- n k^and c \to= i^+ n j^+ 3 k^(n \geq 0) is 158 cu .units, then:$ (1) $n = 7$ (2) $b \to \cdot c \to= 10$ (3) $n = 9$ (4) $a \cdot c = 17$ 312. $ABCD is a parallelogram and P is a point on the segment AD dividing it internally in the ratio 3 : 1 . If the line BP meets the diagonal AC in Q, then AQ : QC equals$ (1) $3 : 4$ (2) $4 : 3$ (3) $3 : 2$ (4) $2 : 3$ 313. $If \| a \| = 13, \| b \| = 5 and a \cdot b = 60 then \| a \times b \| =$ (1) $15$ (2) $20$ (3) $30$ (4) $25$ 314. $If b, c are non collinear vectors, \| c \| \neq = 0, a \times (b \times c) + (a . b) b = (4 - 2 β - sin α) b + (β^{2} - 1) c and (c . c) a = c, then the scalars α and β are$ (1) $α = \frac{π}{2} + \frac{nπ}{3}, n \in Z; β = 1$ (2) $α = \frac{π}{2} + 2 nπ, n \in Z; β = 1$ (3) $α = \frac{π}{2} + (2 n + 1) \frac{π}{2}, n \in Z, β = 2$ (4) $α = (2 n + 1) \frac{π}{2}, n \in Z, β = \frac{3}{2}$ 315. $Let a, bc \in R such that a^{2} + b^{2} + c^{2} = 1 . If a cos θ = b cos (θ + \frac{2 π}{3}) = cos (θ + \frac{4 π}{3}) where θ = \frac{π}{9} then the angle between the vectors and a i^+ b j^+ c k^and b i^+ c j^+ a k^is:$ (1) $0$ (2) $\frac{π}{9}$ (3) $\frac{π}{2}$ (4) $\frac{2 π}{3}$ 316. $Let a, b, c be three vectors such that a is perpendicularto b and b is perpendicular to c . If \| a \| = 2, \| b \| = 3, \| c \| = 5 and \| a + b + c \| = 43, then the angle between a and c is$ (1) $cos^{- 1} \frac{2}{5}$ (2) $\frac{π}{3}$ (3) $cos^{- 1} (\frac{2}{3})$ (4) $\frac{π}{6}$ 317. $Let ABC be a triangle and a, b, c be the position vectors of A, B, C respectively. If D divides BC in the ratio 2 : 3 internally and E divides CA in the ratio 2 : 1 internally then the position vector of the point P which divides DE in the ratio 3 : 5 internally is$ (1) $\frac{1}{8} (2 a + 3 b + 3 c)$ (2) $\frac{1}{8} (3 a + 2 b + 3 c)$ (3) $\frac{1}{8} (3 a + 3 b + 2 c)$ (4) $\frac{3}{8} (a + b + c)$ 318. $If a, b and c are unit vectors such that a + b + c = 0 then angle between a and b is$ (1) $\frac{π}{2}$ (2) $\frac{π}{3}$ (3) $\frac{2 π}{3}$ (4) $π$ 319. $If a = i + j + k, a \cdot b = 1 and a \times b = j - k, then b =$ (1) $2 i$ (2) $2 j - k$ (3) $i - j + k$ (4) $i$ 320. $The set of real values of λ for which the vectors λi - 3 j + 5 k and 2 λi - λj + k are perpendicular to each other is$ (1) ${0, 1}$ (2) ${- 2}$ (3) ${2, - 1}$ (4) $ϕ$ 321. $If p, q and r are nonzero, noncoplanar vectors, then [p + q - r p - q q - r] =$ (1) $3 [pq r]$ (2) $0$ (3) $[pq r]$ (4) $2 [pq r]$ 322. $The scalar product of the vector i + j + k with a unit vector along the sum of the vectors 2 i + 4 j - 5 k and λi + 2 j + 3 k is equal to 1 , then value of λ is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 323. $If the mirror image of the point (1, 3, 5) with respect to the plane 4 x - 5 y + 2 z = 8 is (α, β, γ) then 5 (α + β + γ) equals:$ (1) $47$ (2) $39$ (3) $43$ (4) $41$ 324. $If a = \frac{1}{10} (3 i + k) and b = \frac{1}{7} (2 i + 3 j - 6 k), then the value of (2 a - b) \cdot [(a \times b) \times (a + 2 b)] is$ (1) $5$ (2) $3$ (3) $-5$ (4) $-3$ 325. $If a and b are unit vectors and θ is the angle between a and b, then sin \frac{θ}{2} is$ (1) $\frac{∣ a + b ∣}{2}$ (2) $\frac{∣ a - b ∣}{2}$ (3) $\| a - b \|$ (4) $\| a + b \|$ 326. $If a and b are unit vectors and ∣ a + b ∣ = 1 then ∣ a - b ∣ is equal to$ (1) $5$ (2) $3$ (3) $2$ (4) $1$ 327. $If a = 2 i + 3 j, b = 3 j + 4 k and c = 5 i + 4 k are three vectors, then a vector which is perpendicular to a and b \times c is$ (1) $45 i - 30 j + 15 k$ (2) $3 i - 2 j + k$ (3) $- 30 i + 20 j + 4 k$ (4) $- 45 i + 30 j + 4 k$ 328. $Let a = i - j, b = i + j + k and c be a vector such that a \times c + b = 0 and a . c = 4 , then (c)^{2} is equal to :$ (1) $\frac{19}{2}$ (2) $9$ (3) $8$ (4) $\frac{17}{2}$ 329. $Let a = i - 3 j + 7 k, b = 2 i - j + k and c be a vector such that (a + 2 b) \times c = 3 (c \times a) . If a \cdot c = 130, then b \cdot c is equal to_______$ 330. $A (2, 6, 2), B (- 4, 0, λ), C (2, 3, - 1) and D (4, 5, 0), ∣ λ ∣ \leq 5 are the vertices of a quadrilateral ABCD. If its area is 18 square units, then 5 - 6 λ is equal to _______$