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:

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:

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 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:

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:

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:

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:

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:

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:

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:

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:

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 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

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:

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.

(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

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:

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:

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: