Vector Algebra PYQs Set 10 (PDF)

Set 10 of 16 · Questions 271-300 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 10 (Q271-300)
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271. $The area of the quadrilateral ABCD when A (0, 4, 1) B (2, 3, - 1) C (4, 5, 0) and D (2, 6, 2) is equal to$ (1) $9 sq . units$ (2) $18 squnits$ (3) $27 sq . units$ (4) $81 sq units$ 272. $Let a = i + 2 j - k and b = i + j - k be two vectors. If c is a vector such that b \times c = b \times a and c \cdot a = 0, then c \cdot b is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{- 3}{2}$ (4) $\frac{- 1}{2}$ 273. $If a and b are unit vectors then what is the angle between a and b for 3 a - b to be unit vector?$ (1) $30^{\circ}$ (2) $45^{\circ}$ (3) $60^{\circ}$ (4) $90^{\circ}$ 274. $a, b, c are three vectors such that \| a \| = 1, \| b \| = 2, \| c \| = 3 and b . c = 0 . If the projection of b along a is equal to the projection of c along a, then \| 2 a + 3 b - 3 c \| =$ (1) $3$ (2) $22$ (3) $9$ (4) $11$ 275. $If the vector 19 i + 22 j + 5 k bisects an angle between the vectors a and 6 i + 8 j, then the unit vector in the direction of a is$ (1) $\frac{1}{5} (4 i + 3 k)$ (2) $\frac{1}{3} (2 i + 2 j + k)$ (3) $\frac{1}{3} (i + 2 j + 2 k)$ (4) $\frac{1}{3} (2 i + 2 j - k)$ 276. $If α = i - 3 j, β = i + 2 j - k then, express β in the form β = β_{1} + β_{2} where β_{1} is parallel to α and β_{2} is perpendicular to α then β_{1} is given by$ (1) $\frac{5}{8} (i - 3 j)$ (2) $i - 3 j$ (3) $\frac{5}{8} (i + 3 j)$ (4) $- \frac{1}{2} (i - 3 j)$ 277. $A vector a \to= α i^+ 2 j^+ β k^(α, β, \in R) lies in the plane of the vectors, b \to= i^+ j^and c \to= i^- j^+ 4 k^. If a \to bisects the angle between b \to and c \to, then:$ (1) $a \cdot k + 2 = 0$ (2) $a \cdot k + 4 = 0$ (3) $a \cdot k - 2 = 0$ (4) $a \cdot k + 5 = 0$ 278. $Let u = i - j - 2 k, v = 2 i + j - k, v . w = 2 and v \times w = u + λ v . then u . w is equal to$ (1) $1$ (2) $\frac{3}{2}$ (3) $2$ (4) $- \frac{2}{3}$ 279. $If the volume of the parallelopiped formed by the vectors i + aj + k, j + ak and ai + k becomes minimum, then a is equal to$ (1) $\frac{1}{3}$ (2) $\frac{1}{3}$ (3) $\frac{2}{3}$ (4) $\frac{2}{3}$ 280. $If a and b are two vectors such that a = 2 i + 2 j + p k, \| b \| = 7, a \cdot b = 4 and \| a \times b \| = 517 then p =$ (1) $\pm 5$ (2) $\pm 6$ (3) $\pm 1$ (4) $\pm 3$ 281. $Let u = 2 i + 3 j + k, v = - 3 i + 2 j and w = i - j + 4 k . Then which of the following statement is true?$ (1) $u is perpendicular to v but not w$ (2) $v is perpendicular to w but not u$ (3) $w is perpendicular to u but not v$ (4) $u is perpendicular to both v and w$ 282. $Let a = i + 2 j + k and b = 2 i + j - k . Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a . Then such a vector c is:$ (1) $\frac{1}{2} (- i + k)$ (2) $\frac{1}{5} (j - 2 k)$ (3) $\frac{1}{3} (i - j + k)$ (4) $\frac{1}{3} (- i + j - k)$ 283. $If a (α \times β) + b (β \times γ) + c (γ \times α) = 0 and atleast one of the scalars a, b, c is non-zero, then the vectors α, β, γ are$ (1) $parallel$ (2) $non coplanar$ (3) $coplanar$ (4) $mutually perpendicular$ 284. $If a = i - 2 j - 3 k, b = 2 i + j - k, c = i + 3 j - 2 k, then [(a \times b) \times (b \times c) (b \times c) \times (c \times a) (c \times a) \times (a \times b)] =$ (1) $160000$ (2) $-8000$ (3) $400$ (4) $-40$ 285. $If i, j, k are unit vectors along the positive direction of X, Y and Z - axes, then a FALSE statement in the following is$ (1) $\sum i \times (j \times k) = 0$ (2) $\sum i \times (j + k) = 0$ (3) $\sum i \cdot (j + k) = 0$ (4) $\sum i \cdot (j \times k) = 0$ 286. $If a, b and c are unit vectors, such that a + b + c = 0, then 3 a \cdot b + 2 b \cdot c + c \cdot a =$ (1) $1$ (2) $-1$ (3) $3$ (4) $-3$ 287. $For λ > 0, let θ be the angle between the vectors a = i + λj - 3 k and b = 3 i - j + 2 k . If the vectors a + b and a - b are mutually perpendicular, then the value of (14 cos θ)^{2} is equal to$ (1) $50$ (2) $40$ (3) $25$ (4) $20$ 288. $If a is a nonzero vector such that its projections on the vectors 2 i - j + 2 k, i + 2 j - 2 k and k are equal, then a unit vector along a is:$ (1) $\frac{1}{155} (- 7 i + 9 j + 5 k)$ (2) $\frac{1}{155} (- 7 i + 9 j - 5 k)$ (3) $\frac{1}{155} (7 i + 9 j - 5 k)$ (4) $\frac{1}{155} (7 i + 9 j + 5 k)$ 289. $If a + 2 b + 3 c = 0 and (a \times b) + (b \times c) + (c \times a) = λ (b \times c), then λ has the value$ (1) $6$ (2) $5$ (3) $3$ (4) $4$ 290. $The value of x if x (i + j + k) is a unit vector is$ (1) $\pm \frac{1}{3}$ (2) $\pm 3$ (3) $\pm 3$ (4) $\pm \frac{1}{3}$ 291. $For a positive real number λ, if the vector a = λi - 5 j + 6 k satisfies the equation [i \times (a \times i) + j \times (a \times j) + k \times (a \times k)]^{2} = 440, then λ =$ (1) $3$ (2) $4$ (3) $7$ (4) $11$ 292. $If e_{1}, e_{2} are two non-collinear unit vectors such that ∣ e_{1} + e_{2} ∣ = 3 then (2 e_{1} - 5 e_{2}) \cdot (3 e_{1} + e_{2}) =$ (1) $\frac{11}{2}$ (2) $\frac{- 11}{2}$ (3) $\frac{9}{2}$ (4) $\frac{- 9}{2}$ 293. $Let a unit vector u = x i + y j + z k make angles \frac{π}{2}, \frac{π}{3} and \frac{2 π}{3} with the vectors \frac{1}{2} i + \frac{1}{2} k, \frac{1}{2} j + \frac{1}{2} k and \frac{1}{2} i + \frac{1}{2} j respectively. If v = \frac{1}{2} i + \frac{1}{2} j + \frac{1}{2} k, then \| u - v \|^{2} is equal to$ (1) $\frac{11}{2}$ (2) $\frac{5}{2}$ (3) $9$ (4) $7$ 294. $A vector a of length 2 units is making an angle 60^{\circ} with each of the X -axis and Y -axis. If another vector b of length 2 units is making an angle 45^{\circ} with each of the Y -axis and Z -axis, then a \times b =$ (1) $(1 - 2) i - j + k$ (2) $i - 2 j + k$ (3) $2 i - j + 2 k$ (4) $i - 2 j + (1 - 2) k$ 295. $A unit vector perpendicular to the plane contain ing the vectors i + 2 j + k and - 2 i + j + 3 k is$ (1) $\frac{i + j + k}{3}$ (2) $\frac{- i + j - k}{3}$ (3) $\frac{i + j - k}{3}$ (4) $\frac{- i - j - k}{3}$ 296. $If \| a \| = 2 and \| b \| = 3 and the angle between a and b is 120^{\circ}, then the length of the vector (\frac{1}{2} a - \frac{1}{3} b)^{2} is$ (1) $2$ (2) $\frac{1}{6}$ (3) $3$ (4) $1$ 297. $If \| a + b \| = \| a - b \| then$ (1) $a and b are coincident.$ (2) $a and b are perpendicular.$ (3) $Inclined to each other at 60^{\circ} .$ (4) $a and b are parallel.$ 298. $If a unit vector a makes angles \frac{π}{3} with i, \frac{π}{4} with j and θ \in (0, π) with k, then a value of θ is:$ (1) $\frac{5 π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{5 π}{12}$ (4) $\frac{2 π}{3}$ 299. $Let c be a vector perpendicular to the vectors a = i + j - k and b = i + 2 j + k If c . (i + j + 3 k) = 8 then the value of c \cdot (a \times b) is equal to$ 300. $Let a = 9 i - 13 j + 25 k, b = 3 i + 7 j - 13 k and c = 17 i - 2 j + k be three given vectors. If r is a vector such that r \times a = (b + c) \times a and r \cdot (b - c) = 0, then \frac{\| 593 r + 67 a \| ^{2}}{( 593 ) ^{2}} is equal to______$