Vector Algebra PYQs Set 15 (PDF)

Set 15 of 16 · Questions 421-450 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 15 (Q421-450)
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421. $Let the position vectors of three vertices of a triangle be 4 p + q - 3 r, - 5 p + q + 2 r and 2 p - q + 2 r . If the position vectors of the orthocenter and the circumcenter of the triangle are \frac{p + q + r}{4} and α p + β q + γ r respectively, then α + 2 β + 5 γ is equal to:$ (1) $3$ (2) $4$ (3) $1$ (4) $6$ 422. $The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l 2 = m2 + n2 is$ (1) $\frac{π}{6}$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{4}$ 423. $If the vector p i^+ j^+ k^, i^+ q j^+ k^and i^+ j^+ r k^(p \neq q \neq r \neq 1) are coplanar, then the value of pqr - (p+q+r) is-$ (1) $2$ (2) $0$ (3) $-1$ (4) $-2$ 424. $If (1, 0, 3), (2, 1, 5), (- 2, 3, 6) are the mid-points of the sides of a triangle, then the centroid of the triangle is$ (1) $(\frac{1}{3}, \frac{4}{3}, - \frac{14}{3})$ (2) $(\frac{1}{3}, \frac{4}{3}, \frac{14}{3})$ (3) $(\frac{1}{3}, - \frac{4}{3}, \frac{14}{3})$ (4) $(- \frac{1}{3}, \frac{4}{3}, \frac{14}{3})$ 425. $Let x = i^+ j^and y = 3 i^- 2 k^. Then, the vector r of magnitude 21 satisfying r \times x = y \times x and r \times y = x \times y is$ (1) $- i + 4 j - 2 k$ (2) $- i - 4 j - 2 k$ (3) $4 i + j - 2 k$ (4) $4 i - j - 2 k$ 426. $If a, b and c are unit vectors such that a + b + c = 0 and (a, b) = \frac{π}{3}, then \| a \times b \| + \| b \times c \| + \| c \times a \| =$ (1) $\frac{3}{2}$ (2) $0$ (3) $\frac{3 3}{2}$ (4) $3$ 427. $Let α = (λ - 2) a + b and β = (4 λ - 2) a + 3 b be two given vectors where vectors a and b are non -collinear . The value of λ for which vectors α and β are collinear, is :$ (1) $-4$ (2) $-3$ (3) $4$ 428. $Let a = 2 i + j + k, and b and c be two non-zero vectors, such that ∣ a + b + c ∣ = ∣ a + b - c ∣ and b \cdot c = 0 . Consider the following two statements: (A) ∣ a + λ c ∣ \geq ∣ a ∣ for all λ \in R . (B) a and c are always parallel.$ (1) $Only (B) is correct$ (2) $Neither (A) nor (B) is correct$ (3) $Only (A) is correct$ (4) $Both (A) and (B) are correct$ 429. $If [(a + 2 b + 3 c) \times (b + 2 c + 3 a)] \cdot (c + 2 a + 3 b) = 54, then value of [a b c] is$ (1) $0$ (2) $1$ (3) $3$ (4) $2$ 430. $If a = 2 i - 3 j + k, b = i - j + 2 k and c = 2 i + j + k are three vectors, then ∣ (a \times b) \times c ∣ =$ (1) $\| a \times (b \times c) \|$ (2) $\frac{39}{11} ∣ a \times (b \times c) ∣$ (3) $\frac{11}{39} ∣ a \times (b \times c) ∣$ (4) $11 \| a \times (b \times c) \|$ 431. $If a = i + 2 k, b = i + j + k, c = 7 i - 3 j + 4 k, r \times b + b \times c = 0 and r \cdot a = 0, then r \cdot c is equal to$ (1) $34$ (2) $12$ (3) $36$ (4) $30$ 432. $The vector equation of the plane passing through the intersection of the planes r \cdot (i^+ j^+ k^) = 1 and r \cdot (i^- 2 j^) = - 2 and the point (1, 0, 2) is :$ (1) $r \cdot (i^- 7 j^+ 3 k^) = \frac{7}{3}$ (2) $r \cdot (i^+ 7 j^+ 3 k^) = 7$ (3) $r \cdot (3 i^+ 7 j^+ 3 k^) = 7$ (4) $r \cdot (i^+ 7 j^+ 3 k^) = \frac{7}{3}$ 433. $Let: a = i + 2 j + 3 k, b = i - j + 2 k and c = 5 i - 3 j + 3 k be three vectors. If r is a vector such that, r \times b = c \times b and r \cdot a = 0 . Then 25∣ r ∣^{2} is equal to$ (1) $449$ (2) $336$ (3) $339$ (4) $560$ 434. $If origin is the ortho-center of an equilateral triangle whose vertices are a, b, c then$ (1) $a + b = c$ (2) $a + b = - c$ (3) $\| a \|^{2} = \| b \|^{2} = \| c \|^{2}$ (4) $a = b = c$ 435. $Let a = sin^{2} x i^+ cos^{2} x j^+ k^, (x \in R) . If the pairs of vectors a . i^; a . j^and a . k^are adjacent sides of 3 distinct parallelograms and A is the sum of the squares of areas of these parallelograms, then A lies in the interval$ (1) $(0, 1)$ (2) $[3, 4]$ (3) $[0, 2]$ (4) $[1, 2]$ 436. $Let a, b and c be three unit vectors such that a + b + c = 0 . If λ = a \cdot b + b \cdot c + c \cdot a and d = a \times b + b \times c + c \times a then the ordered pair, (λ, d) is equal to$ (1) $(\frac{3}{2}, 3 a \to \times b \to)$ (2) $(\frac{3}{2}, 3 a \to \times c \to)$ (3) $(- \frac{3}{2}, 3 a \to \times c \to)$ (4) $(- \frac{3}{2}, 3 a \to \times b \to)$ 437. $Let a = 4 i^+ 3 j^and b = 3 i^- 4 j^+ 5 k^and c is a vector such that c \cdot (a \times b) = - 25, c \cdot (i^+ j^+ k^) = 4 and projection of c on a is 1, then the projection of c on b equals$ (1) $\frac{5}{2}$ (2) $\frac{1}{5}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2}$ 438. $Vectors a, b, c, d are such that (a \times b) \times (c \times d) = 0 . P_{1} and P_{2} are two planes determined by vectors a, b and c, d respectively. Then the angle between the planes P_{1} and P_{2} is$ (1) $0$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 439. $The position vectors of the points A, B, C, D are 7 i - 4 j + 7 k, i - 6 j + 10 k, - i - 3 j + 4 k, 5 i - j + 5 k respectively, then A B C D is$ (1) $a parallelogram but not rhombus$ (2) $a square$ (3) $a quadrilateral which is not a parallelogram$ (4) $a rectangle$ 440. $Let a = i^+ j^+ 2 k^, b = b_{1} i^+ b_{2} j^+ 2 k^and c = 5 i^+ j^+ 2 k^be three vectors such that the projection vector of b on a is a . If a + b is perpendicular to c, then ∣ b ∣ is equal to :$ (1) $32$ (2) $6$ (3) $22$ (4) $4$ 441. $The distance of the point (1, 1, 9) from the point of intersection of the line \frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2} and the plane x + y + z = 17 is:$ (1) $38$ (2) $192$ (3) $219$ (4) $38$ 442. $If a + 2 b + 3 c = 0, then a \times b + b \times c + c \times a =$ (1) $0$ (2) $6 (b \times c)$ (3) $2 (b \times c)$ (4) $3 (c \times a)$ 443. $If a, b and c be three non-coplanar vectors and p, q and r be defined by p = \frac{b \times c}{a \cdot ( b \times c )}, q = \frac{c \times a}{b \cdot ( c \times a )}, r = \frac{a \times b}{c \cdot ( a \times b )} such that α = (a + b) \cdot p + (b + c) \cdot q + (c + a) \cdot r and β = \frac{( a + b ) \cdot ( b + c ) \times ( a + b + c )}{b \cdot ( a \times c )}, then α + β =$ (1) $2$ (2) $3$ (3) $4$ (4) $0$ 444. $If a, b, c are the position vectors of the points A, B, C respectively, then match the items of List-I with those of List-II. List-I A . a = 2 i + 3 j + 4 k b = 3 i + 4 j + 2 k c = 4 i + 2 j + 3 k B . a = i + 2 j + 3 k b = 3 i + 4 j + 7 k c = - 3 i - 2 j - 5 k C . a = 2 i - j + k b = i - 3 j - 5 k c = - 3 i - 4 j - 4 k D . a = i + j + k b = i + 2 j + 3 k c = 2 i - j + k List-II I. A, B, C are Collinear II. △ ABC is an isosceles triangle III. △ ABC is a right-angled triangle IV. △ ABC is a right-angled isosceles triangle V. △ ABC is an equilateral triangle The correct match is A B C D$ (1) $I IV III II$ (2) $I II III IV$ (3) $V I IV II$ (4) $V I III II$ 445. $A (1, 1, 1), B (1, - 4, 3), C (2, - 2, 0) and D (8, 1, 4) are the vertices of a tetrahedron. G_{1}, G_{2}, G_{3} and G_{4} are the centroids of the faces ABC, BCD, CDA and DAB. Then the centroid of the tetrahedron having G_{1}, G_{2}, G_{3}, G_{4} as its vertices is$ (1) $(12, - 4, 8)$ (2) $(4, \frac{- 4}{3}, \frac{8}{3})$ (3) $(2, \frac{- 2}{3}, \frac{4}{3})$ (4) $(3, - 1, 2)$ 446. $Let a = 2 i + α j + k, b = - i + k, c = β j - k, where α and β are integers and αβ = - 6 . Let the values of the ordered pair (α, β), for which the area of the parallelogram of diagonals a + b and b + c is \frac{21}{2}, be (α_{1}, β_{1}) and (α_{2}, β_{2}) . Then α_{1}^{2} + β_{1}^{2} - α_{2} β_{2} is equal to$ (1) $19$ (2) $17$ (3) $24$ (4) $21$ 447. $If a = i + 2 j - 3 k and b = 2 i - 3 j - 5 k, then$ (1) $\| a - b \| > \| a \| + \| b \|$ (2) $\| a - b \| > \| b \| - \| a \|$ (3) $\| a + b \| < \| a - b \|$ (4) $\|\| a \| - \| b \| > \| a - b ∣$ 448. $Consider a △ A B C where A (1, 3, 2), B (- 2, 8, 0) and C (3, 6, 7) . If the angle bisector of ∠ B A C meets the line B C at D, then the length of the projection of the vector A D on the vector A C is:$ (1) $\frac{37}{2 38}$ (2) $\frac{38}{2}$ (3) $\frac{39}{2 38}$ (4) $19$ 449. $Let 3 i^+ j^, i^+ 3 j^and β i^+ (1 - β) j^respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is \frac{3}{2}, then the sum of all possible values of β is :$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 450. $Let X be a vector in the plane containing vectors a = 2 i - j + k and b = i + 2 j - k . If the vector X is perpendicular to (3 i + 2 j - k) and its projection on a is \frac{17 6}{2}, then the value of ∣ x ∣^{2} is equal to$