Vector Algebra PYQs Set 8 (PDF)

Set 8 of 16 · Questions 211-240 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 8 (Q211-240)
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211. $The component of i in the direction of the vector i + j + 2 k is$ (1) $66$ (2) $6$ (3) $\frac{6}{6}$ (4) $6$ 212. $Angle made by the position vector of the point (5, - 4, - 3) with the positive direction of X -axis is$ (1) $\frac{π}{2}$ (2) $\frac{π}{6}$ (3) $\frac{π}{4}$ (4) $\frac{π}{3}$ 213. $If the origin and the points P (2, 3, 4), Q (1, 2, 3) and R (x, y, z) are co-planar, then$ (1) $x - 2 y - z = 0$ (2) $x + 2 y + z = 0$ (3) $x - 2 y + z = 0$ (4) $2 x - 2 y + z = 0$ 214. $A space vector makes the angles 150^{\circ} and 60^{\circ} with the positive direction of X and Y -axes. The angle made by the vector with the positive direction of Z -axis is$ (1) $60^{\circ}$ (2) $90^{\circ}$ (3) $120^{\circ}$ (4) $180^{\circ}$ 215. $a is perpendicular to both b and c . The angle between b and c is \frac{2 π}{3} . If \| a \| = 2, \| b \| = 3, \| c \| = 4, then c \cdot (a \times b) is equal to$ (1) $183$ (2) $123$ (3) $83$ (4) $63$ 216. $Between the following two statements: Statement I : Let a = i + 2 j - 3 k and b = 2 i + j - k . Then the vector r satisfying a \times r = a \times b and a \cdot r = 0 is of magnitude 10 . Statement II : In a triangle ABC, cos 2 A + cos 2 B + cos 2 C \geq - \frac{3}{2} .$ (1) $Statement I is incorrect but Statement II is correct.$ (2) $Both Statement I and Statement II are correct.$ (3) $Statement I is correct but Statement II is incorrect.$ (4) $Both Statement I and Statement II are incorrect.$ 217. $If \| a \| = 2, \| b \| = 3, \| c \| = 5 and cach of the angles between the vector a and b, b and c, c and a is 60^{\circ}, then the value of \| a + b + c \| is$ (1) $69$ (2) $70$ (3) $80$ (4) $39$ 218. $Let a = i - 2 j + 3 k, b = 2 i + j - k, c = λi + j + k and v = a \times b . If v \cdot c = 11 and the length of the projection of b on c is p, then 9 p^{2} is equal to$ (1) $12$ (2) $6$ (3) $4$ (4) $9$ 219. $If 7 i - 4 j + 5 k is the position vector of the vertex A of a tetrahedron ABCD and - i + 4 j - 3 k is the position vector of the centroid of the triangle BCD , then the position vector of the centroid of the tetrahedron ABCD is$ (1) $- i + 4 j - 3 k$ (2) $- \frac{1}{2} i + 4 j - 3 k$ (3) $i + 2 j - k$ (4) $- i - 2 j + k$ 220. $Let a = 2 i + j - 2 k, b = i + j and c = a \times b . Let d be a vector such that \| d - a \| = 11, \| c \times d \| = 3 and the angle between c and d is \frac{π}{4} . Then a \cdot d is equal to$ (1) $3$ (2) $11$ (3) $0$ (4) $1$ 221. $Let a = - i - j + k, a . b = 1 and a \times b = i - j, then a - 6 b is equal to$ (1) $3 (i - j - k)$ (2) $3 (i + j + k)$ (3) $3 (i - j + k)$ (4) $3 (i + j - k)$ 222. $If the angle between a and b is \frac{2 π}{3} and the projection of a in the direction of b is -2 , then \| a \| =$ (1) $4$ (2) $2$ (3) $3$ (4) $1$ 223. $If a and b are vectors such that \| a + b \| = \| a - b \|, then the angle between a and b is$ (1) $120^{\circ}$ (2) $60^{\circ}$ (3) $90^{\circ}$ (4) $30^{\circ}$ 224. $Let a = 2 i - j + k and b = λj + 2 k, λ \in Z be two vectors. Let c = a \times b and d be a vector of magnitude 2 in yz - plane. If \| c \| = 53, then the maximum possible value of (c \cdot d)^{2} is equal to:$ (1) $26$ (2) $52$ (3) $208$ (4) $104$ 225. $If a, b, c are unit vectors satisfying the relation a + b + 3 c = 0, then the angle between a and b is$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 226. $If a and b are unit vectors such that a + b is also a unit vector, then the angle between a and b is\dots\dots\dots..$ (1) $60^{\circ}$ (2) $90^{\circ}$ (3) $30^{\circ}$ (4) $120^{\circ}$ 227. $If OA = 2 i + 2 j + k, OB = 2 i + 4 j + 4 k and the length of the internal bisector of ∠ BOA of triangle AOB is k, then 9 k^{2} =$ (1) $225$ (2) $136$ (3) $712$ (4) $20$ 228. $Let a = i + 2 j - 2 k and b = 2 i - j - 2 k . If the orthogonal projection vector of a on b be x and the orthogonal projection vector of b on a be y, then \| x - y \| =$ (1) $\frac{4}{9} 26$ (2) $\frac{8}{9} 10$ (3) $\frac{4}{9} 10$ (4) $\frac{8}{9} 26$ 229. $M and N are the mid points of the sides BC and CD of a parallelogram ABCD respectively then AM + AN =$ (1) $\frac{1}{3} AC$ (2) $\frac{2}{3} AC$ (3) $\frac{3}{4} AC$ (4) $\frac{3}{2} AC$ 230. $If a, b and c are three vectors such that \| a \| = 1, \| b \| = 2, \| c \| = 3 and a \cdot b = b \cdot c = c \cdot a = 0, then \| [abc] \| is equal to$ (1) $2$ (2) $3$ (3) $4$ (4) $6$ 231. $If a = 2 i + λj + k and b = i + 2 j + 3 k are orthogonal, then value of λ is$ (1) $\frac{3}{2}$ (2) $1$ (3) $- \frac{5}{2}$ (4) $0$ 232. $If a and b are two unit vectors inclined at an angle \frac{π}{3}, then the value of \| a + b \| is$ (1) $equal to 1$ (2) $greater than 1$ (3) $equal to 0$ (4) $less than 1$ 233. $For a non-zero real number x, if the points with position vectors (x - u) i + x j + x k, x i + (x - v) j + x k, x i + x j + (x - w) k and (x - 1) i + (x - 1) j + (x - 1) k are coplanar, then$ (1) $u + v + w = 1$ (2) $uvw = 1$ (3) $\frac{1}{u} + \frac{1}{v} + \frac{1}{w} = 1$ (4) $uv + vw + uw = 1$ 234. $Let O be the origin. Let OP = x i + y j - k and OQ = - i + 2 j + 3 x k, x, y \in R, x > 0, be such that \| PQ \| = 20 and the vector OP is perpendicular to OQ . If OR = 3 i + z j - 7 k, z \in R, is coplanar with OP and OQ, then the value of x^{2} + y^{2} + z^{2} is equal to$ (1) $7$ (2) $9$ (3) $2$ (4) $1$ 235. $For three unit vectors a, b, c satisfying \| a - b \|^{2} + \| b - c \|^{2} + \| c - a \|^{2} = 9 and \| 2 a + k b + k c \| = 3, the positive value of k is$ (1) $4$ (2) $3$ (3) $5$ (4) $6$ 236. $The equation of the plane containing the line 2x - 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1, is$ (1) $2x + 6y + 12z = 13$ (2) $x + 3y + 6z = -7$ (3) $x + 3y + 6z = 7$ (4) $2x + 6y + 12z = -13$ 237. $Let a vector a = 2 i - j + λk, λ > 0, make an obtuse angle with the vector b = - λ^{2} i + 42 j + 42 k and an angle θ, \frac{π}{6} < θ < \frac{π}{2}, with the positive z -axis. If the set of all possible values of λ is (α, β) - {γ}, then α + β + γ is equal to____。$ 238. $Let the vectors a, b \to, c \to be such that \| a \to \| = 2, \| b \to \| = 4 and \| c \to \| = 4 . If the projection of b on a is equal to the projection of c on a and b is perpendicular to c, then the value of \| a \to + b \to - c \to \| is$ 239. $Let PQR be a triangle such that PQ = - 2 i - j + 2 k and PR = ai + bj - 4 k, a, b \in Z . Let S be the point on QR, which is equidistant from the lines PQ and PR. If \| PR \| = 9 and PS = i - 7 j + 2 k, then the value of 3 a - 4 b is____$ 240. $If P \to= (a + 1) i^+ a j^+ a k^Q \to= a i^+ (a + 1) j^+ a k^R \to= a i^+ a j^+ (a + 1) k^And P \to, Q \to, R \to are coplanar vectors and 3 (P \to \cdot Q \to)^{2} - λ \| R \to \times Q \to \|^{2} = 0 then value of λ is$