Vector Algebra PYQs Set 5 (PDF)

Set 5 of 16 · Questions 121-150 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 5 (Q121-150)
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121. $If a = i + 2 j + 2 k, \| b \| = 5 and the angle between a and b is \frac{π}{6}, then the area of the triangle formed by these two vectors as two sides is$ (1) $15$ (2) $\frac{15 3}{2}$ (3) $\frac{15}{2}$ (4) $\frac{15}{4}$ 122. $If a = 2 i + j - k, b = i - j + 3 k, x = (\frac{a \cdot b}{\| b \| ^{2}}) b, y = (\frac{a \cdot b}{\| a \| ^{2}}) a and θ is angle between a and b, then x^{2} + y^{2} =$ (1) $17 cos^{2} θ$ (2) $(6 + 11) cos^{2} θ$ (3) $17 cos 2 θ$ (4) $17 sin^{2} θ$ 123. $Suppose a + b + c = 0, \| a \| = 3, \| b \| = 5, \| c \| = 7, then the angle between a & b is$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{4}$ 124. $If a + xb + yc = 0, a \times b + b \times c + c \times a = 6 (b \times c), then the locus of the point (x, y) is$ (1) $x^{2} + y^{2} = 1$ (2) $x + y - 5 = 0$ (3) $2 x + 6 y = 5$ (4) $x + y + 6 = 0$ 125. $If a and b are vectors such that \| a \| = 3, \| b \| = 2 and a \cdot b = 5, then \| a - b \| =$ (1) $3$ (2) $5$ (3) $23$ (4) $3$ 126. $Let a = 2 i + j - 3 k and b = i + 3 j + 2 k . Then the volume of the parallelopiped having coterminous edges as a, b and c, where c is the vector perpendicular to the plane of a, b and \| c \| = 2 is$ (1) $2195$ (2) $24$ (3) $200$ (4) $195$ 127. $Let OA = - 4 i + 3 k, OB = 14 i + 2 j - 5 k \cdot OD bisects ∠ AOB and \| OD \| = 6, then OD =$ (1) $\pm (i + j + 2 k)$ (2) $\pm (i + 2 j + k)$ (3) $\pm (2 i + j + k)$ (4) $\pm \frac{1}{2} (2 i + j + 7 k)$ 128. $Let a, b, c be co-initial vectors and a = 2 i - j + 5 k and b = 3 i + 7 j - k . Let (a, b) = θ be an acute angle and c be the vector along the bisector of the angle θ . If λ, x, y \in R, then c =$ (1) $λ (5 i + 6 j + 4 k)$ (2) $λ (- i - 8 j + 6 k)$ (3) $(2 x + 3 y) i + (7 y - x) j + (5 x - y) k$ (4) $(2 x + 3 y) i + (x + 7 y) j + (5 x + y) k$ 129. $If 3 i - 5 j + 2 k, 7 i + 2 j - 4 k, i - 3 j + 4 k and - 7 i - 17 j + 16 k are position vectors of the points A, B, C and D respectively, then the angle between AB and CD is$ (1) $0$ (2) $\frac{π}{4}$ (3) $\frac{π}{2}$ (4) $π$ 130. $a = i - j + k, b = i - 2 j + k, c = p i + 2 j + q k and d = p i + q j + 2 k are given vectors. If the projection of c on a is 53 units and if a, b and c form a parallelopiped of volume 5 cubic units, then tan^{- 1} (b . d) =$ (1) $\frac{π}{2}$ (2) $\frac{π}{6}$ (3) $\frac{π}{4}$ (4) $π3$ 131. $If [a \times b b \times c c \times a] = λ [a b c]^{2}, then λ is equal to$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 132. $If the four points, whose position vectors are 3 i - 4 j + 2 k, i + 2 j - k, - 2 i - j + 3 k and 5 i - 2 α j + 4 k are coplanar, then α is equal to$ (1) $\frac{73}{17}$ (2) $- \frac{107}{17}$ (3) $- \frac{73}{17}$ (4) $\frac{107}{17}$ 133. $If vectors a_{1} = x i - j + k and a_{2} = i + y j + z k are collinear, then a possible unit vector parallel to thevector x i + y j + z k is$ (1) $\frac{1}{2} (- j + k)$ (2) $\frac{1}{2} (i - j)$ (3) $\frac{1}{3} (i - j + k)$ (4) $\frac{1}{3} (i + j - k)$ 134. $If (α, β, γ) is a triad of real numbers satisfying i - 2 j + 5 k = α (i + j + k) + β (i + 2 j + 3 k) + γ (2 i - j + k), then α^{2} - β^{2} + γ^{2} =$ (1) $23$ (2) $31$ (3) $40$ (4) $-6$ 135. $If direction cosines of a vector of magnitude 3 are \frac{2}{3}, - \frac{1}{3}, \frac{2}{3} and a > 0, then vector is$ (1) $2 i - j + 2 k$ (2) $i + 2 j + 2 k$ (3) $2 i + j + 2 k$ (4) $i - 2 j + 2 k$ 136. $a, b are two vectors such that \| a \| = 3, \| b \| = 2 . If x is a unit vector satisfying x \times a = b then x =$ (1) $\frac{1}{2} [(x \cdot a) a - b \times a]$ (2) $\frac{1}{2} [\pm (x \cdot a) a + (b \times a)]$ (3) $\frac{1}{3} [(x \cdot a) a + b \times a]$ (4) $\frac{1}{3} [a \times b \pm a]$ 137. $If P is a point on the line parallel to the vector 2 i - 3 j - 6 k and passing through the point A whose position vector i + 2 j - 2 k and AP = 21, then the position vector of P can be$ (1) $6 i - 9 j - 18 k$ (2) $6 i + 9 j - 18 k$ (3) $- 5 i + 11 j + 16 k$ (4) $5 i - 11 j + 16 k$ 138. $Let AB = 2 i + 4 j - 5 k and AD = i + 2 j + λk, λ \in R . Let the projection of the vector v = i + j + k on the diagonal AC of the parallelogram ABCD be of length one unit. If α, β, where α > β, be the roots of the equation λ^{2} x^{2} - 6 λx + 5 = 0, then 2 α - β is equal to$ (1) $3$ (2) $4$ (3) $1$ (4) $6$ 139. $Let a = 2 i - j + k be the position vector of a point A . Let b = i + 2 j - k and c = i + j - 2 k be two vectors and r be a vector passing through the point A (a) and parallel to the vector b . If the projection of r on c is \frac{9}{6} then \| r \| =$ (1) $26$ (2) $5$ (3) $5$ (4) $34$ 140. $Let a = 2 i + λ_{1} j + 3 k, b = 4 i + (3 - λ_{2}) j + 6 k and c = 3 i + 6 j + (λ_{3} - 1) k be three vectors such that b = 2 a and a is perpendicular to c Then a possible value of (λ_{1}, λ_{2}, λ_{3}) is :$ (1) $(1, 3, 1)$ (2) $(- \frac{1}{2}, 4, 0)$ (3) $(\frac{1}{2}, 4, - 2)$ (4) $(1, 5, 1)$ 141. $If 4 i + 7 j + 8 k, 2 i + 3 j + 4 k, 2 i + 5 j + 7 k are respectively the position vectors of the vertices A, B, C of △ ABC, then the position vector of the point where the bisector of angle A meet BC is$ (1) $2 i + \frac{13}{3} j + 2 k$ (2) $2 i - \frac{13}{3} j + 6 k$ (3) $2 i + 13 j + 6 k$ (4) $2 i + \frac{13}{3} j + 6 k$ 142. $Let a = i + 2 j - k and b = i + j + k .If p is a unit vector such that [ab p] is maximum, then p =$ (1) $\frac{1}{6} (i - 2 j + k)$ (2) $\frac{1}{3} (i + j - k)$ (3) $\frac{1}{14} (3 i - 2 j - k)$ (4) $\frac{1}{14} (i + 2 j + 3 k)$ 143. $A new tetrahedron is formed by joining the centroids of the faces of a given tetrahedron QABC . Then the ratio of the volume of the new tetrahedron to that of the given tetrahedron is$ (1) $\frac{3}{25}$ (2) $\frac{1}{27}$ (3) $\frac{5}{62}$ (4) $\frac{1}{162}$ 144. $The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points A, B, C is (2, a, 4), a ϵ N . If the volume of the tetrahedron OABC is 144 u ni t s^{3}, then which of the following points is NOT on P?$ (1) $(2, 2, 4)$ (2) $(0, 4, 4)$ (3) $(3, 0, 4)$ (4) $(0, 6, 3)$ 145. $If the line, \frac{x -3}{2} = \frac{y + 2}{-1} = \frac{z + 4}{3} lies in the plane, lx + my - z = 9, then l2 + m2 is equal to$ (1) $18$ (2) $5$ (3) $2$ (4) $26$ 146. $If (1, 5, 35), (7, 5, 5), (1, λ, 7) and (2 λ, 1, 2) are coplanar, then the sum of all possible values of λ is:$ (1) $- \frac{44}{5}$ (2) $\frac{39}{5}$ (3) $- \frac{39}{5}$ (4) $\frac{44}{5}$ 147. $Let the line \frac{x -2}{3} = \frac{y -1}{-5} = \frac{z + 2}{2} lies in the plane x + 3y - αz + β = 0. Then (α, β) equals$ (1) $(6, - 17)$ (2) $(- 6, 7)$ (3) $(5, - 15)$ (4) $(- 5, 15)$ 148. $If a = α i + β j + 3 k b = - β i - α j - k and c = i - 2 j - k such that a \cdot b = 1 and b \cdot c = - 3 then \frac{1}{3} ((a \times b) \cdot c) is equal to\dots\dots\dots$ 149. $If a and b \to are unit vectors, then the greatest value of 3 \| a \to + b \to \| + \| a \to - b \to \| is$ 150. $Let a = i + 2 j + λk, b = 3 i - 5 j - λk, a \cdot c = 7, 2 b \cdot c + 43 = 0, a \times c = b \times c then ∣ a \cdot b ∣ is equal to$