Vector Algebra PYQs Set 1 (PDF)

Set 1 of 16 · Questions 1-30 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 1 (Q1-30)
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1. $If the volume of the tetrahedron formed by the coterminous edgcs a, b and c is 4 , then the volume of the parallelopiped formed by the coterminous edges a \times b, b \times c and c \times a is$ (1) $576$ (2) $48$ (3) $16$ (4) $144$ 2. $The set of all α, for which the vectors a = αt i + 6 j - 3 k and b = t i - 2 j - 2 αt k are inclined at an obtuse angle for all t \in R, is$ (1) $(- \frac{4}{3}, 1)$ (2) $[0, 1)$ (3) $(- \frac{4}{3}, 0)$ (4) $(- 2, 0]$ 3. $Let a, b, c be three-unit vectors satisfying \| a - b \|^{2} + \| a - c \|^{2} = 10. Then Statement (I) : \| a + 2 b \|^{2} + \| 2 a + c \|^{2} = 2 . Statement (II) : \| 2 a + 3 b \|^{2} + \| 3 a + 2 c \|^{2} = 10 . Which of the above statements is (are) true?$ (1) $Statement I is true, but Statement II is false$ (2) $Statement II is true but Statement I is false$ (3) $Both Statement I and Statement II are true$ (4) $Both Statement I and Statement II are false$ 4. $Let a = i + x j + k, b = i + j + k, and \| a + b \| = \| a \| + \| b \| then$ (1) $x = 1$ (2) $x = - 1$ (3) $x = 0$ (4) $No such Real x exits$ 5. $If cos α, cos β, cos γ are the direction cosines of a vector a, then cos 2 α + cos 2 β + cos 2 γ is equal to$ (1) $2$ (2) $3$ (3) $-1$ (4) $0$ 6. $Let a = 2 i + j - 2 k and b = i + j . If c is a vector such that a \cdot c = \| c \|, \| c - a \| = 22 and the angle between a \times b and c is \frac{2 π}{3}, then \| (a \times b) \times c \| =$ (1) $\frac{3}{2}$ (2) $\frac{3 3}{2}$ (3) $33$ (4) $43$ 7. $If the magnitudes of a, b and a + b are respectively 3,4 and 5 , then the magnitude of (a - b) is$ (1) $3$ (2) $4$ (3) $6$ (4) $5$ 8. $If a = 2 i^+ j^+ 3 k^, b = i^+ 3 j^- k^and c = 3 i^- j^- 2 k^, then the value of a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c is equal to$ (1) $2020$ (2) $2025$ (3) $2030$ (4) $1849$ 9. $If the area of the parallelogram with a and b as two adjacent sides is 15 sq. units, then the area of the parallelogram having 3 a + 2 b and a + 3 b as two adjacent sides in sq. units is$ (1) $120$ (2) $105$ (3) $75$ (4) $45$ 10. $Let a = 2 i + 3 j + k, b = 4 i + j, c = i - 3 j - 7 k If r = x i + y j + z k, r \cdot a = 9, r \cdot b = 7, r \cdot c = 6 then (x, y, z) =$ (1) $(1, - 3, 2)$ (2) $(- 1, 3, - 2)$ (3) $(1, 3, 2)$ (4) $(1, 3, - 2)$ 11. $If a and b are unit vectors and α is the angle between them, then a + b is a unit vector when cos α =$ (1) $- \frac{1}{2}$ (2) $\frac{1}{2}$ (3) $- \frac{3}{2}$ (4) $\frac{3}{2}$ 12. $Let a unit vector which makes an angle of 60^{\circ} with 2 i + 2 j - k and angle 45^{\circ} with i - k be C . Then C + (- \frac{1}{2} i + \frac{1}{3 2} j - \frac{2}{3} k) is:$ (1) $\frac{2}{3} i - \frac{1}{2} k$ (2) $(\frac{1}{3} + \frac{1}{2}) i + (\frac{1}{3} - \frac{1}{3 2}) j + (\frac{1}{3} + \frac{2}{3}) k$ (3) $\frac{2}{3} i + \frac{1}{3 2} j - \frac{1}{2} k$ (4) $- \frac{2}{3} i + \frac{2}{3} j + (\frac{1}{2} + \frac{2 2}{3}) k$ 13. $If a = (2 x + y) i + 3 j + 9 k and b = 2 i + j - (x - y) k are two collinear vectors, then x^{3} + 27 y^{3} =$ (1) $1241$ (2) $1512$ (3) $1072$ (4) $1729$ 14. $If (k, 1, 5), (1, 0, 3), (7, - 2, m) are collinearr then (k, m) =$ (1) $(- 2, - 1)$ (2) $(2, 1)$ (3) $(- 2, 1)$ (4) $(2, - 1)$ 15. $If a = (1, 2, 3), b = (2, - 1, 1), c = (3, 2, 1) and a \times (b \times c) = α a + β b + γ c, then$ (1) $α = 1, β = 10, γ = 3$ (2) $α = 0, β = 10, γ = - 3$ (3) $α + β + γ = 8$ (4) $α = β = γ = 0$ 16. $Let a = i + 2 j + k, b = 3 (i - j + k) . Let c be the vector such that a \times c = b and a \cdot c = 3 . Then a \cdot ((c \times b) - b - c) is equal to :$ (1) $32$ (2) $24$ (3) $20$ (4) $36$ 17. $The triad (x, y, z) of real number such that (3 i - j + 2 k) = (2 i + 3 j - k) x + (i - 2 j + 2 k) y + (- 2 i + j - 2 k) z is$ (1) $(- 2, 5, 3)$ (2) $(2, - 5, 3)$ (3) $(2, 5, 3)$ (4) $(2, 5, - 3)$ 18. $If ABCD is a cyclic quadrilateral with R as the radius of the circumcircle and (AB)^{2} + (CD)^{2} = 4 R^{2} then$ (1) $b \cdot c - a \cdot d = 0$ (2) $a \cdot c - b \cdot d = 0$ (3) $a \cdot b + c \cdot d = 0$ (4) $a \cdot c + b \cdot d = 0$ 19. $Let P be a point in the plane of the vectors AB = 3 i + j - k and AC = i - j + 3 k such that P is equidistant from the lines AB and AC . If \| AP \| = \frac{5}{2}, then the area of the triangle ABP is:$ (1) $2$ (2) $\frac{30}{4}$ (3) $\frac{3}{2}$ (4) $\frac{26}{4}$ 20. $Let a = 2 i - 3 j + 4 k, b = 7 i + 2 j - 3 k, c = i + j + k . The vector x such that x, c = 60 and perpendicular to both a, b is$ (1) $14 i - 6 j - 12 k$ (2) $i + 34 j + 25 k$ (3) $4 i - 21 j - 12 k$ (4) $6 i - 6 j + 28 k$ 21. $If α is the angle between two vector p = 3 i + 4 j - k and q = 2 i - j + k, then sin (α) =$ (1) $\frac{145}{156}$ (2) $\frac{135}{156}$ (3) $\frac{155}{156}$ (4) $\frac{165}{156}$ 22. $If P, Q are two points on the curve y = 2^{x + 2} in the rectangular Cartesian coordinate system such that OP \cdot i = - 1, OQ \cdot i = 2 then OQ - 4 OP =$ (1) $3 i + 8 j$ (2) $4 i + 6 j$ (3) $6 i + 8 j$ (4) $4 i + 3 j$ 23. $If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :$ (1) $ab' + bc' + 1 = 0$ (2) $bb' + cc' + 1 = 0$ (3) $cc' + a + a' = 0$ (4) $aa' + c + c' = 0$ 24. $Let m be a vector of magnitude 3 and perpendicular to the vectors i + j and j - k . Let n be another vector of magnitude 26 and perpendicular to the vectors 2 i - j and j + 2 k . The area (in sq. units) of the triangle formed with m and n as sides is$ (1) $2$ (2) $6$ (3) $23$ (4) $32$ 25. $Let a = - i + j + 2 k, b = i - j - 3 k, c = a \times b and d = c \times a . Then (a - b) \cdot d is equal to:$ (1) $-4$ (2) $4$ (3) $2$ (4) $-2$ 26. $If ∣ a \times b ∣^{2} + ∣ a \cdot b ∣^{2} = 36 and ∣ a ∣ = 3 then ∣ b ∣ is equal to$ (1) $9$ (2) $4$ (3) $36$ (4) $2$ 27. $If a, b and c are mutually perpendicular vectors of the same magnitude, then the cosine of the angle between a and a + b + c is$ (1) $\frac{1}{2}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2}$ 28. $If the vectors a = i^- j^+ 2 k^, b = 2 i^+ 4 j^+ k^and c = λ i^+ j^+ µ k^are mutually orthogonal, then (λ , µ) =$ (1) $(2, - 3)$ (2) $(-2, 3)$ (3) $(3, - 2)$ (4) $(-3, 2)$ 29. $Let a and b be two vectors such that \| a \| = 1, \| b \| = 4 and a \cdot b = 2 . If c = (2 a \times b) - 3 b and the angle between b and c is α, then 192 sin^{2} α is equal to_____$ 30. $Let three vectors a, b and c be such that c is coplanar with a and b, a \cdot c = 7 and b is perpendicular to c where a = - i + j + k and b = 2 i + k then the value of 2 \| a + b + c \|^{2} is$