Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 10 Vector Algebra.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

Plus Two Maths Vector Algebra 3 Marks Important Questions

Question 1.
(i) With help of a suitable figure for any three vectorsa,bandc show that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\vec{b}+\bar{c})\)
(ii) If \(\bar{a}\) = i – j + k and \(\bar{b}\) = 2i – 2j – k. What is the projection of a on b? (March – 2011)
Answer:
(i) Answered in previous years questions
No. 1(ii) (6 Mark question)
(ii) Projection of \(\bar{a} \text { on } \bar{b}=\frac{\bar{a} \cdot \bar{b}}{|\bar{b}|}=\frac{2+2-1}{\sqrt{4+4+1}}=1\)

Question 2.
(i) If \(\bar{a}\) = 3i – j – 5k and \(\bar{b}\) = i – 5j + 3k Show that \(\bar{a}\) + \(\bar{b}\) and a bare perpendicular.
(ii) Given the position vectors of three points as A(i – j + k); B(4i + 5j + 7k) C(3i + 3j + 5k)
(a)Find \(\bar{AB}\) and \(\bar{BC}\)
(b) Prove that A,B and C are collinear points. (March – 2011)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 1

Question 3.
(i) Write the unit vector in direction of i + 2j – 3k.
(ii) If \(\overline{P Q}\) = 31 + 2j — k and the coordinate of P are(1, -1,2) , find the coordinates of Q. (May – 2012)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 2

Question 4.
(a) The angle between the vectors \(\bar{a}\) and \(\bar{b}\) such that \(|\vec{a}|=|\bar{b}|=\sqrt{2}\)
\(\bar{a}\).\(\bar{b}\) = 1 is
\(\begin{array}{lll}
\text { (i) } \frac{\pi}{2} & \text { (ii) } \frac{\pi}{3} & \text { (iii) } \frac{\pi}{4} & \text { (iv) } 0
\end{array}\)
(b) Find the unit vector along \(\bar{a}-\bar{b}\) where \(\bar{a}\) = i + 3j – k and \(\bar{b}\) 3i + 2j + k (March -2016)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 3

Plus Two Maths Vector Algebra 4 Marks Important Questions

Question 1.
Consider the vectors \(\bar{a}\) = 21+ j – 2k and \(\bar{b}\) = 6i – 3j + 2k.
(i) Find \(\bar{a} \bar{b}\) and \(\bar{a} \times \bar{b}\).
(ii) Verity that \(|\bar{a} \times \bar{b}|=|\vec{a}|^{2}|\bar{b}|^{2}-(\bar{a} \cdot \bar{b})^{2}\) (March – 2012)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 4

Question 2.
(i) For any three vectors \(\bar{a}, \bar{b}, \bar{c}\), show that \(\bar{a} \times(\bar{b}+\bar{c})+\bar{b} \times(\bar{c}+\bar{a})+\bar{c} \times(\bar{a}+\bar{b})=0\)
(ii) Given A (1, 1, 1), B (1, 2, 3), C (2, 3, 1) are the vertices of MBCa triangle. Find the area of the ∆ABC (May – 2012)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 5

Question 3.
Consider A (2, 3, 4) , B (4, 3, 2) and C (5, 2, -1) be any three points
(i) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(ii) Find the area of triangle ABC (March – 2013)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 6

Question 4.
(i) Find the angle between the vectors \(\bar{a}\) =3i + 4j + k and \(\bar{b}\) = 2i + 3j – k
(ii) The adjacent sides of a parallelogram are \(\bar{a}\) = 3i + λj + 4k and \(\bar{b}\) = i – λj + k
(a) Find \(\bar{a} \times \bar{b}\)
(b) If the area of the parallelogram is square units, find the value of A (May – 2013)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 7

Question 5.
Let \(\bar{a}\) = 2i – j + 2k and \(\bar{b}\) = 6i + 2j + 3k
(i) Find a unit vector in the direction of \(\bar{a}\) + \(\bar{b}\)
(ii) Find the angle between a and b (March – 2014)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 8

Question 6.
Consider the triangle ABC with vertices A(1, 1, 1) , B (1, 2, 3) and C (2, 3, 1)
(i) Find \(\overline{A B}\) and \(\overline{A C}\)
(ii) Find \(\overline{A B}\) x \(\overline{A C}\)
(iii) Hence find the area of the triangle (March – 2014)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 9

Question 7.
Consider the vectors \(\bar{a}\) = i – 7j + 7k; \(\bar{b}\) = 3i – 2j + 2k
(a) Find \(\bar{a b}\).
(b) Find the angle between \(\bar{a}\) and \(\bar{b}\).
(c) Find the area of parallelogram with adjacent sides \(\bar{a}\) and \(\bar{b}\). (May – 2014)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 10

Question 8.
(a) If the points A and B are (1, 2, -1) and (2, 1, -1) respectively, then is
(i) i + J
(ii) i – J
(iii) 2i + j – k
(iv) i + j + k
(b) Find the value of for which the vectors 2i – 4j + 5k, i – λj + k and 3i + 2j – 5k are coplanar.
(c) Find the angle between the vectors a = 2i + j – k and b = i – j + k (March – 2016)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 11

Question 9.
(i) \((\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})\) is equaito
\(\begin{array}{lll}
\text { (a) } \bar{a} & \text { (b) }|\bar{a}|^{2}-|\bar{b}|^{2} & \text { (c) } \bar{a} \times \bar{b} \text { (d) } 2(\bar{a} \times \bar{b})
\end{array}\)

(ii) If \(\bar{a}\) and \(\bar{b}\) are any two vectors, then
\((\bar{a} \times \bar{b})^{2}=\left|\begin{array}{ll}
\bar{a} \cdot \bar{a} & \bar{a} \cdot \bar{b} \\
\bar{a} \cdot \bar{b} & \bar{b} \bar{b}
\end{array}\right|\)

(iii) Using vectors, show that the points A(1, 2, 7), B(2, 6, 3), C(3, 10, -i) are collinear. (May – 2016)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 12

Plus Two Maths Vector Algebra 6 Marks Important Questions

Question 1.
(i) Find a vector in the direction of \(\bar{r}\) = 3E – 4j that has a magnitude of 9.
(ii) For any three vectors \(\bar{a,b}\) and \(\bar{c}\), and Prove that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\bar{b}+\bar{c})\).
(iii) Find a unit vector perpendicular to \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}\) = i – 3j + 3k and \(\bar{b}\) and \(/bar{c}\) = 3E—3j+2k. (March – 2010)
Answer:
(i) Unit vector of magnitude 9
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 13

Question 2.
Let A(2, 3, 4), B(4, 3, 2) and C(5, 2, -1) be three points
(i) Find \(\overline{A B}\) and \(\overline{B C}\)
(ii) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(iii) Fiñd the area of the triangle ABC. (May – 2010)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 14

Question 3.
ABCD s a parallelogram with A as the origin, \(\bar{b}\) and \(\bar{d}\) are the position vectors of B and D respectively.
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 15
(i) What is the position vector of C?
(ii) What is the angle between \(\bar{AB}\) and \(\bar{AD}\)?
(iii) If \(|\overrightarrow{A C}|=|\overrightarrow{B D}|\), show that ABCD is a rectangle. (May – 2011)
Answer:
(i) Since ABCD is a parallelogram with A as the
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 16

Question 4.
(a) If \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) respectively are the position vectors representing the vertices A,B,C,D of a parallelogram, then write \(/bar{d}\) in terms of \(\bar{a}, \bar{b}, \bar{c}\).
(b) Find the projection vector of \(/bar{b}\) = i + 2j + k along the vector \(/bar{a}\) = 21 – i – j + 2k. Also write \(/bar{b}\) as the sum of a vector along \(/bar{a}\) and a perpendicular to \(/bar{a}\).
(C) Find the area of a parallelogram for which the vectors 21 + j, 31 + j +4k are adjacent sides. (March – 2015)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 17

Question 5.
(a) Write the magnitude of a vector \(/bar{a}\) in terms of dot product.
(b) If \(\bar{a}, \bar{b}, \bar{a}+\bar{b}\) are unit vectors, then prove that the angle between \(/bar{a}\) and \(/bar{b}\) is \(\frac{2 \pi}{3}\)
(c) If 2i + j – 3k and mi + 3j – k are perpendicular to each other, then find ‘m’.
Also find the area of the rectangle having these two vectors as sides. (March – 2015)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 18

Question 6.
Consider the triangle ABC with vertices A(1, 2, 3), B(-1, 0, 4), C(0, 1, 2)
(a) Find \(\overline{A B}\) and \(\overline{A C}\)
(b) Find \(\angle A\)
(c) Find the area of triangle ABC. (May – 2015)
Answer:
Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra 19