Vector Algebra PYQs Set 13 (PDF)

Set 13 of 16 · Questions 361-390 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 13 (Q361-390)
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361. $The equation of the line through the point (0, 1, 2) and perpendicular to the line \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{- 2} is:$ (1) $\frac{x}{- 3} = \frac{y - 1}{4} = \frac{z - 2}{3}$ (2) $\frac{x}{3} = \frac{y - 1}{4} = \frac{z - 2}{3}$ (3) $\frac{x}{3} = \frac{y - 1}{- 4} = \frac{z - 2}{3}$ (4) $\frac{x}{3} = \frac{y - 1}{4} = \frac{z - 2}{- 3}$ 362. $The sum of the distinct real values of μ, for which the vectors µ i + j + k, i + μ j + k, i + j + μ k, are co-planer, is :$ (1) $-1$ (2) $0$ (3) $1$ (4) $2$ 363. $Let a = i^+ j^+ 2 k^and b = - i^+ 2 j^+ 3 k^. Then the vector product (a + b) \times ((a \times ((a - b) \times b)) \times b) is equal to :$ (1) $5 (34 i^- 5 j^+ 3 k^)$ (2) $7 (34 i^- 5 j^+ 3 k^)$ (3) $7 (30 i^- 5 j^+ 7 k^)$ (4) $5 (30 i^- 5 j^+ 7 k^)$ 364. $Let a be a non-zero vector parallel to the line of intersection of the two planes described by i^+ j^, i^+ k^and i^- j^, i^- k^. If θ is the angle between the vector a and the vector b = 2 i^- 2 j^+ k^and a . b = 6, then the ordered pair (θ, ∣ a \times b ∣) is equal to$ (1) $(\frac{π}{3}, 36)$ (2) $(\frac{π}{4}, 36)$ (3) $(\frac{π}{3}, 6)$ (4) $(\frac{π}{4}, 6)$ 365. $Let A, B, C be three points whose position vectors respectively are: a = i^+ 4 j^+ 3 k^b = 2 i^+ α j^+ 4 k^, α \in R c = 3 i^- 2 j^+ 5 k^If α is the smallest positive integer for which a, b, c are non-collinear, then the length of the median, in ∆ ABC, through A is:$ (1) $\frac{82}{2}$ (2) $\frac{62}{2}$ (3) $\frac{69}{2}$ (4) $\frac{66}{2}$ 366. $A (a), B (b), C (c), D (d) are four concyclic points, such that xa + yb + zc + td = 0, x + y + z + t = 0 where x, y, z, t are constants not all zero. If the chords AB and CD intersect at P, then$ (1) $x y ∣ a + c ∣^{2} = ∣ z t ∣∣ b + d ∣^{2}$ (2) $x y ∣ a - b ∣^{2} = ∣ z t ∣∣ c - d ∣^{2}$ (3) $x t ∣ a - c ∣^{2} = ∣ y z ∣∣ b - c ∣^{2}$ (4) $x z ∣ b + d ∣^{2} = ∣ y t ∣∣ a + c ∣^{2}$ 367. $If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is:$ (1) $\frac{21}{2}$ (2) $\frac{59}{8}$ (3) $\frac{57}{8}$ (4) $\frac{95}{8}$ 368. $Statement -1 : The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x - y + z = 5. Statement -2 : The plane x - y + z = 5 bisects the line segment joining A(3, 1,6) and B(1, 3, 4).$ (1) $Statement -1 is true, Statement-2 is true ; Statement -2 is not a correct explanation for Statement -1.$ (2) $Statement-1 is true, Statement-2 is false.$ (3) $Statement -1 is false, Statement -2 is true.$ (4) $Statement -1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1$ 369. $If a and b are two non collinear vectors, then \frac{a \times ( b \times a )}{\| a \| ^{2}} represent$ (1) $a vector perpendicular to the plane of a, b$ (2) $projection of b along a vector perpendicular to the vector a .$ (3) $projection of a along the vector perpendicular to b .$ (4) $a vector on the plane of a,b whose magnitude is equal to \| a \| + \| b \| .$ 370. $Let OA = 2 a, OB = 6 a + 5 b and OC = 3 b, where O is the origin. If the area of the parallelogram with adjacent sides OA and OC is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :$ (1) $32$ (2) $40$ (3) $38$ (4) $35$ 371. $Points A (3, 2, 4), B (\frac{33}{5}, \frac{28}{5}, \frac{38}{5}) and C (9, 8, 10) are given. The ratio in which B divides AC is$ (1) $5 : 3$ (2) $2 : 1$ (3) $1 : 3$ (4) $3 : 2$ 372. $Let three vectors a = α i + 4 j + 2 k, b = 5 i + 3 j + 4 k, c = x i + y j + z k form a triangle such that c = a - b and the area of the triangle is 56 . If α is a positive real number, then \| c \|^{2} is equal to:$ (1) $16$ (2) $14$ (3) $12$ (4) $10$ 373. $P is the point of intersection of the diagonals of the parallelogram ABCD . If S is any point in space and SA + SB + SC + SD = λSP, then λ equals$ (1) $2$ (2) $4$ (3) $6$ (4) $8$ 374. $If M and N are the mid-points of the sides BC and CD respectively of a parallelogram ABCD, then AM + AN equals$ (1) $\frac{4}{3} AC$ (2) $\frac{5}{3} AC$ (3) $\frac{3}{2} AC$ (4) $\frac{6}{5} AC$ 375. $In a quadrilateral ABCD, the point P divides DC in the ratio 1 : 3 internally and Q is the mid-point of AC . If AB + AD + BC - 2 DC = λPQ, then the value of λ is$ (1) $-2$ (2) $2$ (3) $4$ (4) $-4$ 376. $A tetrahedron has vertices P(1,2,1), Q(2,1,3), R(-1,1,2) and O(0,0,0) the angle between the faces OPQ and PQR is :$ (1) $cos^{- 1} (\frac{19}{35})$ (2) $cos^{- 1} (\frac{7}{31})$ (3) $cos^{- 1} (\frac{17}{31})$ (4) $cos^{- 1} (\frac{9}{35})$ 377. $Let L be a line obtained from the intersection of two planes x + 2 y + z = 6 and y + 2 z = 4 . If point P (α, β, γ) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21 (α + β + γ) equals :$ (1) $142$ (2) $68$ (3) $136$ (4) $102$ 378. $If the points with position vectors α i^+ 10 j^+ 13 k^, 6 i^+ 11 j^+ 11 k^, \frac{9}{2} i^+ β j^- 8 k^are collinear, then (19 α - 6 β)^{2} is equal to$ (1) $36$ (2) $25$ (3) $49$ (4) $16$ 379. $In △ OAC, if B is the mid-point of side AC and OA = a, OB = b, then OC is equal to$ (1) $2 b - a$ (2) $b - 2 a$ (3) $a - 2 b$ (4) $a - b$ 380. $ABCD is a parallelogram and P is the mid-point of the side AD . The line BP meets the diagonal AC in Q . Then, the ratio of AQ : QC is equal to$ (1) $1 : 2$ (2) $2 : 1$ (3) $1 : 3$ (4) $3 : 1$ 381. $Let a = α i^+ j^+ β k^and b = 3 i^- 5 j^+ 4 k^be two vectors, such that a \times b = - i^+ 9 j^+ 12 k . Then the projection of b - 2 a on b + a is equal to$ (1) $2$ (2) $\frac{39}{5}$ (3) $9$ (4) $\frac{46}{5}$ 382. $A vector a has components 3 p and 1 with respect to a rectangular cartesian system. This system is rotatedthrough a certain angle about theorigin in the counter clockwise sense. If, with respect to new system, a has components p + 1 and 10, then avalue of p is equal to:$ (1) $1$ (2) $- \frac{5}{4}$ (3) $\frac{4}{5}$ (4) $- 1$ 383. $If the lines \frac{x -2}{1} = \frac{y -3}{1} = \frac{z -4}{- k} and \frac{x -1}{k} = \frac{y -4}{2} = \frac{z -5}{1} are coplanar, then k can have$ (1) $any value$ (2) $exactly one value$ (3) $exactly two values$ (4) $exactly three values$ 384. $In a triangle ABC if \| BC \| = 8, \| CA \| = 7 ∣ AB ∣ = 10, then the projection of the vector AB on AC is equal to:$ (1) $\frac{25}{4}$ (2) $\frac{85}{14}$ (3) $\frac{127}{20}$ (4) $\frac{115}{16}$ 385. $M and N are the midpoints of the diagonals AC and BD respectively of quadrilateral ABCD, then AB + AD + CB + CD =$ (1) $2 MN$ (2) $2 NM$ (3) $4 MN$ (4) $4 NM$ 386. $The magnitude of the projection of the vector 2 i + 3 j + k and on the vector perpendicular to the plane containing the vectors i + j + k and i + 2 j + 3 k, is :$ (1) $52$ (2) $6$ (3) $8$ (4) $\frac{3}{2}$ 387. $Let the position vectors of points ’ A ’ and ’ B ’ be i^+ j^+ k^and 2 i^+ j^+ 3 k^respectively. A point P^{’} divides the line segment AB internally in the ratio λ : 1 (λ > 0) of O is the origin and OB \to \cdot OP \to - 3 \| OA \to \times OP \to \|^{2} = 6 then λ is equal to$ 388. $Let ABC be a triangle of area 152 and the vectors AB = i + 2 j - 7 k, BC = ai + bj + ck and AC = 6 i + d j - 2 k, d > 0 . Then the square of the length of the largest side of the triangle ABC is_______$ 389. $Let b = i^+ j^+ λ k^, λ \in R . If a is a vector such that a \times b = 13 i^- j^- 4 k^and a . b + 21 = 0, then (b - a) . (k^- j^) + (b + a) . (i^- k^) is equal to$ 390. $Let a = i^+ j^+ k^, b and c = j^- k^be three vectors such that a \times b = c and a . b = 1 . If the length of projection vector of the vector b on the vector a \times c is l, then the value of 3 l^{2} is equal to$