Vector Algebra PYQs Set 14 (PDF)

Set 14 of 16 · Questions 391-420 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 14 (Q391-420)
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391. $Let a be a vector which is perpendicular to the vector 3 i^+ \frac{1}{2} j^+ 2 k^. If , then the projection of the vector a on the vector 2 i^+ 2 j^+ k^is$ (1) $\frac{1}{3}$ (2) $1$ (3) $\frac{5}{3}$ (4) $\frac{7}{3}$ 392. $If the orthocenter and the centroid of a triangle are (- 3, 5, 2) and (3, 3, 4) respectively, then its circumcenter is$ (1) $(6, 2, 5)$ (2) $(6, 2, - 5)$ (3) $(6, - 2, 5)$ (4) $(6, - 2, - 5)$ 393. $In quadrilateral ABCD, AB = a, BC = b, DA = a - b, M is the midpoint of BC and X is a point on DM such that DX = \frac{4}{5} DM . Then the points A, X and C$ (1) $form an equilateral triangle$ (2) $are collinear$ (3) $form an isosceles triangle$ (4) $form a right-angled triangle$ 394. $The two vectors i + j + k and i + 3 j + 5 k represents the two sides AB and AC respectively of a △ ABC . The length of the median though A is$ (1) $14$ (2) $7$ (3) $14$ (4) $\frac{14}{2}$ 395. $In △ ABC, L, M, N are points on BC, CA, AB respectively, dividing them in the ratio 1 : 2, 2 : 3, 3 : 5 . If the point K divides AB in the ratio 5 : 3, then \frac{\| AL + BM + CN \|}{\| CK \|}$ (1) $\frac{5}{8}$ (2) $\frac{2}{5}$ (3) $\frac{3}{5}$ (4) $\frac{1}{15}$ 396. $The vectors a and b are not perpendicular and c and d are two vectors satisfying : b \times c = b \times d and a . d = 0 . Then the vector d is equal to :$ (1) $b - (\frac{b \cdot c}{a \cdot d}) c$ (2) $c + (\frac{a \cdot c}{a \cdot b}) b$ (3) $b + (\frac{b \cdot c}{a \cdot b}) c$ (4) $c - (\frac{a \cdot c}{a \cdot b}) b$ 397. $An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If O P = u, O R = v and O Q = α u + β v, then α, β^{2}, are the roots of the equation$ (1) $x^{2} + x - 2 = 0$ (2) $x^{2} - x - 2 = 0$ (3) $3 x^{2} - 2x - 1 = 0$ (4) $3 x^{2} + 2x - 1 = 0$ 398. $If the vertices of the triangles are (1, 2, 3), (2, 3, 1,), (3, 1, 2) and if H, G, S and I respectively denote its orthocenter, centroid, circumcenter and incenter, then H + G + S + I is equal to$ (1) $(2, 2, 2)$ (2) $(4, 4, 4)$ (3) $(6, 6, 6)$ (4) $(8, 8, 8)$ 399. $Let a, b, c be three vectors mutually perpendicular to each other and have same magnitude. If a vector r satisfies. a \times {(r - b) \times a} + b \times {(r - c) \times b} + c \times {(r - a) \times c} = 0, then r is equal to :$ (1) $\frac{1}{3} (a + b + c)$ (2) $\frac{1}{3} (2 a + b - c)$ (3) $\frac{1}{2} (a + b + c)$ (4) $\frac{1}{2} (a + b + 2 c)$ 400. $Let a = i + 2 j + 4 k, b = i + λj + 4 k and c = 2 i + 4 j + (λ^{2} - 1) k be coplanar vectors. Then the non-zero vector a \times c is:$ (1) $-10 i - 5 j$ (2) $-14 i - 5 j$ (3) $-14 i + 5 j$ (4) $-10 i + 5 j$ 401. $If the line \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} and \frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1} intersect, then k is equal to :$ (1) $- 1$ (2) $\frac{2}{9}$ (3) $\frac{9}{2}$ (4) $0$ 402. $A equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the origin is :$ (1) $x - 2y + 2z - 3 = 0$ (2) $x - 2y + 2z + 1 = 0$ (3) $x - 2y + 2z - 1 = 0$ (4) $x - 2y + 2z + 5 = 0$ 403. $Let OA = a, OB = 12 a + 4 b and OC = b, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then \frac{area of the quadrilateral OABC}{area of S} is equal to_______$ (1) $6$ (2) $10$ (3) $7$ (4) $8$ 404. $D, E, F are respectively the points on the sides BC, CA and AB of a △ ABC dividing them in the ratio 2 : 3, 1 : 2, 3:1 internally. The lines BE and CF intersect on the line AD at P . If AP = x_{1} \cdot AB + y_{1} \cdot AC then x_{1} + y_{1} =$ (1) $\frac{5}{6}$ (2) $1$ (3) $\frac{3}{2}$ (4) $2$ 405. $ABCDEF is a regular hexagon whose centre is O . Then, AB + AC + AD + AE + AF is equal to$ (1) $2 AO$ (2) $3 AO$ (3) $5 AO$ (4) $6 AO$ 406. $If A (3, 1, - 1), B (\frac{5}{3}, \frac{7}{3}, \frac{1}{3}), C (2, 2, 1) and D (\frac{10}{3}, \frac{2}{3}, \frac{- 1}{3}) are the vertices of a quadrilateral ABCD, then its area is$ (1) $\frac{2 2}{3}$ (2) $\frac{5 2}{3}$ (3) $22$ (4) $\frac{4 2}{3}$ 407. $A plane passes through the points A (1, 2, 3), B (2, 3, 1) and C (2, 4, 2) . If O is the origin and P is (2, - 1, 1),then the projection of OP on this plane is of length:$ (1) $\frac{2}{5}$ (2) $\frac{2}{3}$ (3) $\frac{2}{11}$ (4) $\frac{2}{7}$ 408. $Let ABCD be a parallelogram such that A B = q, A D = p and ∠BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then is given by :$ (1) $r = 3 q - \frac{3 ( p . q )}{( p . p )} p$ (2) $r = - q + (\frac{p \cdot q}{p \cdot p}) p$ (3) $r = q - (\frac{p \cdot q}{p \cdot p}) p$ (4) $r = - 3 q + \frac{3 ( p . q )}{( p . p )} p$ 409. $In △ PQR, M is the mid-point of QR and C is the midpoint of PM . If QC when extended meets PR at N, then \frac{\| QN \|}{\| CN \|} =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 410. $Let the position vectors of the vertices A, B and C of a tetrahedron A B C D be i + 2 j + k, i + 3 j - 2 k and 2 i + j - k respectively. The altitude from the vertex D to the opposite face A B C meets the median line segment through A of the triangle A B C at the point E . If the length of A D is \frac{110}{3} and the volume of the tetrahedron is \frac{805}{6 2}, then the position vector of E is$ (1) $\frac{1}{12} (7 i + 4 j + 3 k)$ (2) $\frac{1}{2} (i + 4 j + 7 k)$ (3) $\frac{1}{6} (12 i + 12 j + k)$ (4) $\frac{1}{6} (7 i + 12 j + k)$ 411. $Let a = 3 i^+ j^and b = i^+ 2 j^+ k^.Let c be a vector satisfying a \times (b \times c) = b + λ c . If b and c are non-parallel, then the value of λ, is:$ (1) $- 5$ (2) $5$ (3) $1$ (4) $- 1$ 412. $Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is$ (1) $\frac{3}{2}$ (2) $\frac{5}{2}$ (3) $\frac{7}{2}$ (4) $\frac{9}{2}$ 413. $OABC is a tetrahedron. If D, E are the midpoints of OA and BC respectively, then DE =$ (1) $12(-OA+OB+OC)$ (2) $\frac{1}{2} (OA + OB - OC)$ (3) $\frac{1}{2} (OA - OB + OC)$ (4) $\frac{1}{2} (OA + OB + OC)$ 414. $Let a = α i^+ j^- k^and b = 2 i^+ j^- α k^, α > 0 . If the projection of a \times b on the vector - i^+ 2 j^- 2 k^is 30, then α is equal to$ (1) $\frac{15}{2}$ (2) $8$ (3) $\frac{13}{2}$ (4) $7$ 415. $The position vectors of the points A and B with respect to O are 2 i + 2 j + k and 2 i + 4 j + 4 k . The length of the internal bisector of ∠ BOA of △ AOB is (take proportionality constant is 2)$ (1) $\frac{136}{9}$ (2) $\frac{136}{3}$ (3) $\frac{20}{3}$ (4) $\frac{25}{3}$ 416. $Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and , then k is equal to$ (1) $4$ (2) $-2$ (3) $2$ (4) $-4$ 417. $Let a vector α i + β j be obtained by rotating the vector 3 i + j by an angle 45^{\circ} about the origin incounterclockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to$ (1) $\frac{1}{2}$ (2) $1$ (3) $\frac{1}{2}$ (4) $22$ 418. $If \| a \| = 4, \| b \| = 5, \| a - b \| = 3 and θ is the angle between the vectors a and b, then tan^{2} θ =$ (1) $\frac{4}{3}$ (2) $\frac{3}{4}$ (3) $\frac{16}{9} .$ (4) $\frac{9}{16}$ 419. $If a = 2 i^+ j^+ 3 k^, b = 3 i^+ 3 j^+ k^and c = c_{1} i^+ c_{2} j^+ c_{3} k^are coplanar vectors and a . c = 5, b ⊥ c then 122 (c_{1} + c_{2} + c_{3}) is equal to ___________ .$ 420. $Let (λ, 2, 1) be a point on the plane which passes through the point (4, - 2, 2) . If the plane is perpendicular to the line joining the point (- 2, - 21, 29) and (- 1, - 16, 23), then (\frac{λ}{11})^{2} - \frac{4 λ}{11} - 4 is equal to$