Vector Algebra PYQs Set 6 (PDF)

Set 6 of 16 · Questions 151-180 of 461

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Vector Algebra · Set 6 (Q151-180)
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151. $If 2 a \cdot b = \| a \| \cdot \| b \| then the angle between a & b is$ (1) $30^{\circ}$ (2) $0^{\circ}$ (3) $90^{\circ}$ (4) $60^{\circ}$ 152. $Find the co-ordinates of the point which divides the join of the points (2, - 1, 3) and (4, 3, 1) in the ratio 3 : 4 internally.$ (1) $(\frac{20}{7}, \frac{5}{7}, \frac{15}{7})$ (2) $(\frac{13}{7}, \frac{12}{7}, \frac{5}{7})$ (3) $(\frac{9}{7}, \frac{17}{7}, \frac{21}{7})$ (4) $(\frac{11}{7}, \frac{30}{7}, \frac{2}{7})$ 153. $If a and b are two vectors such that \| a \| = 2, \| b \| = 3 and a + t b and a - t b are perpendicular, where t is a positive scalar, then$ (1) $t = \pm \frac{2}{3}$ (2) $t = \frac{4}{9}$ (3) $t = \frac{2}{3}$ (4) $t = \frac{2}{9}$ 154. $The vectors a = x i - 2 j + 5 k and b = i + y j - z k are collinear, if$ (1) $x = 1, y = - 2, z = - 5$ (2) $x = 1/2, y = - 4, z = - 10$ (3) $x = - 1/2, y = 4, z = 10$ (4) $All of these$ 155. $Let u = i - 2 j and v = - 3 i + 5 j . Consider three points P, Q and R having position vectors \frac{- 1}{7} i, \frac{- 1}{4} j and - 2 i + 3 j respectively. Among these, the points in the line segment passing through u and v are$ (1) $Only P and Q$ (2) $Only P and R$ (3) $Only Q and R$ (4) $All P, Q and R$ 156. $Which of the following is NOT equal to w \cdot (u \times v) ?$ (1) $u \cdot (v \times w)$ (2) $v \cdot (w \times u)$ (3) $(u \times v) \cdot w$ (4) $v \cdot (u \times w)$ 157. $If a = i + 2 j + 3 k, b = - i + 2 j + k, c = i + 2 j - 2 k, n is perpendicular to both a and b and 0 is the angle between c and n then sin θ =$ (1) $\frac{2}{3}$ (2) $\frac{2}{3 3}$ (3) $\frac{2}{3}$ (4) $\frac{3}{2}$ 158. $If a, b, c and d are vectors in which \| d \| = 1 and given a + b + c = s d, b + c + d = a, a \cdot d = 4, then s is equal to$ (1) $7$ (2) $8$ (3) $-1$ (4) $4$ 159. $The acute angle between r = (- i + 3 k) + λ (2 i + 3 j + 6 k) and r \cdot (10 i + 2 j - 11 k) = 3, is$ (1) $sin^{- 1} (\frac{8}{21})$ (2) $cos^{- 1} (\frac{8}{21})$ (3) $sin^{- 1} (\frac{5}{21})$ (4) $cos^{- 1} (\frac{5}{21})$ 160. $Let a = i and b = j, the point of intersection of the lines r \times a = b \times a and r \times b = a \times b is$ (1) $r = i + j$ (2) $r = i - j$ (3) $r = k$ (4) $r = 2 i + j$ 161. $Let a, b and c be three non-zero vectors such that b and c are non-collinear if a + 5 b is collinear with c, b + 6 c is collinear with a and a + α b + β c = 0, then α + β is equal to$ (1) $35$ (2) $30$ (3) $-30$ (4) $-25$ 162. $Let a = - i + 2 j + 2 k, b = 8 i + 7 j - 3 k and c be vector such that a \times c = b . If c \cdot (i + j + k) = 4, then \| a + c \|^{2} is equal to:$ (1) $27$ (2) $33$ (3) $35$ (4) $30$ 163. $If a + lb + l^{2} c = 0 and a \times b + b \times c + c \times a = 3 (b \times c), then the minimum value of such l is$ (1) $1$ (2) $-2$ (3) $\frac{- 9}{4}$ (4) $0$ 164. $Let a, b \in R . If the mirror image of the point P (a, 6, 9) with respect to the line \frac{x - 3}{7} = \frac{y - 2}{5} = \frac{z - 1}{- 9} is (20, b, - a - 9), then \| a + b \| is equal to:$ (1) $86$ (2) $88$ (3) $84$ (4) $90$ 165. $The sum of the lengths of projections of p i + q j + r k on the co-ordinate axes, where p = 4, q = - 5, r = 7 is$ (1) $28 units$ (2) $6 units$ (3) $16 units$ (4) $20 units$ 166. $If a = i + j - 2 k, b = 2 i - j + k and c = 3 i - k and c = ma + nb, then m + n =$ (1) $0$ (2) $2$ (3) $1$ (4) $-1$ 167. $[a + 2 b - c, a - b, a - b - c] =$ (1) $0$ (2) $2 [a, b, c]$ (3) $[a, b, c]$ (4) $3 [a, b, c]$ 168. $Let A = 2 i + j - 2 k and B = i + j . If C is a vector such that A \cdot C = \| C \|, \| C - A \| = 22 and the angle between A \times B and C is 30^{\circ}, then the value of \| (A \times B) \times C \| is$ (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $3$ (4) $2$ 169. $The value of α, so that the volume of the parallelopiped formed by i + α j + k, j + α k and α i + k becomes maximum, is$ (1) $\frac{- 1}{3}$ (2) $\frac{1}{3}$ (3) $- 3$ (4) $3$ 170. $If the vectors a = i + j + k, b = i - j + 2 k and c = x i + (x - 2) j - k are coplanar, then x =$ (1) $1$ (2) $2$ (3) $0$ (4) $-2$ 171. $Let a = x i + y j + z k and x = 2 y .If \| a \| = 52 and a makes and angle of 135^{\circ} with the z -axis then a =$ (1) $2 3 i + 3 j - 3 k$ (2) $2 6 i + 6 j - 6 k$ (3) $2 5 i + 5 j + 5 k$ (4) $2 5 i + 5 j - 5 k$ 172. $If a = i - 2 j - 2 k and b = 2 i - 3 j + k, then the component of b perpendicular to a is$ (1) $\frac{1}{3} (4 i - 5 j + 7 k)$ (2) $\frac{1}{3} (8 i - 13 j - k)$ (3) $\frac{2}{3} (i - 2 j - 2 k)$ (4) $\frac{1}{7} (i - 5 j - 17 k)$ 173. $If a = i + λj + 2 k; b = μ i + j - k are orthogonal and \| a \| = \| b \|, then (λ, μ) =$ (1) $(\frac{1}{4}, \frac{7}{4})$ (2) $(\frac{7}{4}, \frac{1}{4})$ (3) $(\frac{1}{4}, \frac{9}{4})$ (4) $(\frac{- 1}{4}, \frac{9}{4})$ 174. $If a = (p, - 2, 5) and b = (1, q, - 3) are collinear vectors then$ (1) $p = \frac{5}{3}, q = \frac{6}{5}$ (2) $p = \frac{- 5}{3}, q = \frac{- 6}{5}$ (3) $p = \frac{5}{3}, q = \frac{- 6}{5}$ (4) $p = \frac{- 5}{3}, q = \frac{6}{5}$ 175. $If P is a point lying on the line passing through the point A (i - j + 3 k) and parallel to the vector 2 i + j - 2 k such that \| AP \| = 18, then a position vector of P is$ (1) $- 13 i - 5 j + 9 k$ (2) $11 i - 7 j - 15 k$ (3) $13 i - 5 j + 9 k$ (4) $13 i + 5 j - 9 k .$ 176. $Let a be a unit vector perpendicular to the vectors b = i - 2 j + 3 k and c = 2 i + 3 j - k, and makes an angle of cos^{- 1} (- \frac{1}{3}) with the vector i + j + k . If a makes an angle of \frac{π}{3} with the vector i + α j + k, then the value of α is:$ (1) $6$ (2) $- 6$ (3) $- 3$ (4) $3$ 177. $Let A = (1, 2, 0), B = (2, 0, - 1), C = (0, - 2, 3) and D = (- 1, 2, - 3) be four points in the space. Let G_{1} be the centroid of triangle ABC and G_{2} be the centroid of tetrahedron ABCD . If P divides G_{1} G_{2} in the ratio 4 : 3 internally then P =$ (1) $(\frac{5}{7}, \frac{2}{7}, \frac{1}{7})$ (2) $(\frac{1}{7}, \frac{2}{7}, \frac{3}{7})$ (3) $(\frac{4}{7}, \frac{- 2}{7}, \frac{1}{7})$ (4) $(\frac{1}{7}, \frac{- 3}{7}, \frac{5}{7})$ 178. $If a, b and c are non-coplanar, then the value of a \cdot (\frac{b \times c}{3 b \cdot ( c \times a )}) - b \cdot (\frac{c \times a}{2 c \cdot ( a \times b )}) is$ (1) $\frac{- 1}{2}$ (2) $\frac{- 1}{3}$ (3) $\frac{- 1}{6}$ (4) $\frac{1}{6}$ 179. $Let a, b \to and c \to be three unit vectors such that \| a \to - b \to \|^{2} + \| a \to - c \to \|^{2} = 8 then \| a \to + 2 b \to \|^{2} + \| a \to + 2 c \to \|^{2} is equal to$ 180. $If a = 2 i^+ j^+ 2 k^then the value of \| i^\times (a \times i^) \|^{2} + \| j^\times (a \times j^) \|^{2} + \| k^\times (a \times k^) \|^{2} is equal to$