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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.5

Choose the correct or the most suitable answer from the given four alternatives:

Question 1.

The value of \(\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}+\overrightarrow{\mathrm{DA}}+\overrightarrow{\mathrm{CD}}\) is ………………

Solution:

(c) \(\overrightarrow{0}\)

Question 2.

If \(\vec{a}+2 \vec{b}\) and \(3 \vec{a}+m \vec{b}\) are parallel, then the value of m is ………………

(a) 3

(b) \(\frac{1}{3}\)

(c) 6

(d) \(\frac{1}{6}\)

Solution:

(c) 6

Question 3.

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2 \hat{j}+\hat{k}\) is ………………

Solution:

Question 4.

A vector \(\overrightarrow{O P}\) makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between \(\overrightarrow{O P}\) and the z-axis is …………….

(a) 45°

(b) 60°

(c) 90°

(d) 30°

Solution:

(b) 60°

α = 60°, β = 45°

We know cos^{2}α + cos^{2}β + cos^{2}γ = 1

(i.e.,) \(\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}\) + cos^{2}γ = 1

cos^{2}γ = 1 – \(\frac{1}{4}-\frac{1}{2}=\frac{1}{4}\)

cos γ = \(\frac{1}{2}\) ⇒ y = π/3 = 60°

Question 5.

If \(\overrightarrow{B A}=3 \hat{i}+2 \hat{j}+\hat{k}\) and the position vector of B is \(\hat{i}+3 \hat{j}-\hat{k}\), then the position vector A is …………………

Solution:

Question 6.

A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to …………..

Solution:

Question 7.

The vectors \(\vec{a}-\vec{b}, \vec{b}-\vec{c}, \vec{c}-\vec{a}\) are ……………

(a) parallel to each other

(b) unit vectors

(c) mutually perpendicular vectors

(d) coplanar vectors

Solution:

(d) coplanar vectors

Question 8.

If ABCD is a parallelogram, then \(\overrightarrow{A B}+\overrightarrow{A D}+\overrightarrow{C B}+\overrightarrow{C D}\) is equal to ……………

Solution:

Question 9.

One of the diagonals of parallelogram ABCD with \(\vec{a}\) and \(\vec{b}\) as adjacent sides is \(\vec{a}+\vec{b}\). The other diagonal \(\overrightarrow{\mathrm{BD}}\) is ……………

Solution:

Question 10.

If \(\vec{a}\), \(\vec{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is ………….

Solution:

(c) \(\frac{2 \vec{a}+\vec{b}}{3}\)

Question 11.

If \(\vec{a}, \vec{b}, \vec{c}\) are the position vectors of three collinear points, then which of the following is true?

Solution:

(b) \(2 \vec{a}=\vec{b}+\vec{c}\)

Question 12.

If \(\vec{r}=\frac{9 \vec{a}+7 \vec{b}}{16}\), then the point P whose position vector \(\vec{r}\) divides the line joining the points with position vectors \(\vec{a}\) and \(\vec{b}\) in the ratio ………………

(a) 7 : 9 internally

(b) 9 : 7 internally

(c) 9 : 7 externally

(d) 7 : 9 externally

Solution:

Question 13.

If \(\lambda \hat{i}+2 \lambda \hat{j}+2 \lambda \hat{k}\) is a unit vector, then the value of λ is ……………..

(a) \(\frac{1}{3}\)

(b) \(\frac{1}{4}\)

(c) \(\frac{1}{9}\)

(d) \(\frac{1}{2}\)

Solution:

Question 14.

Two vertices of a triangle have position vectors \(3 \hat{i}+4 \hat{j}-4 \hat{k}\) and \(2 \hat{i}+3 \hat{j}+4 \hat{k}\) .If the position vector of the centroid is \(\hat{i}+2 \hat{j}+3 \hat{k}\), then the position vector of the third vertex is ………………….

Solution:

Question 15.

(a) 42

(b) 12

(c) 22

(d) 32

Solution:

(c) 22

Question 16.

If \(\vec{a}\) and \(\vec{b}\) having same magnitude and angle between them is 60° and their scalar product is \(\frac{1}{2}\) then \(|\vec{a}|\) is ……………

(a) 2

(b) 3

(c) 7

(d) 1

Solution:

(d) 1

Question 17.

The value of θ ∈ (0, \(\frac{\pi}{2}\)) for which the vectors are perpendicular, is equal to …………………

Solution:

Question 18.

(a) 15

(b) 35

(c) 45

(d) 25

Solution:

(d) 25

Question 19.

Vectors \(\vec{a}\) and \(\vec{b}\) are inclined at an angle θ = 120°

is equal to ………….

(a) 225

(b) 275

(c) 325

(d) 300

Solution:

(d) 300

Question 20.

If \(\vec{a}\) and \(\vec{b}\) are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between \(\vec{a}\) and \(\vec{a}+\vec{b}\) is ………………

(a) 30°

(b)60°

(c) 45 °

(d) 90°

Solution:

(a) 30°

Question 21.

If the projection of \(5\hat{i} -\hat{j}-3 \hat{k}\) on the vector \(\hat{i}+3 \hat{j}+\lambda \hat{k}\) is same as the projection of \(\hat{i}+3 \hat{j}+\lambda \hat{k}\) on \(5\hat{i}- \hat{j}-3 \hat{k}\) then λ is equal to ………………

(a) ±4

(b) ±3

(c) ±5

(d) ±1

Solution:

(c) ±5

Question 22.

If (1, 2, 4) and (2, – 3λ, – 3) are the initial and terminal points of the vector \(\hat{i}+5 \hat{j}-7 \hat{k}\), then the value of λ is equal to ……………..

(a) \(\frac{7}{3}\)

(b) \(-\frac{7}{3}\)

(c) \(-\frac{5}{3}\)

(d) \(\frac{5}{3}\)

Solution:

Question 23.

If the points whose position vector \(10 \hat{i}+3 \hat{j}, 12 \hat{i}-5 \hat{j}\) and \(\vec{a} \hat{i}+11 \hat{j}\) are collinear then a is equal to ………………

(a) 6

(b) 2

(c) 5

(d) 8

Solution:

(d) 8

equating \(\hat{j}\) components

⇒ -8 = 8t ⇒ t = -1

equation \(\hat{i}\) components

t(a – 10) = 2

(i.e.,) (-1) (a – 10) = 2

a – 10 = -2

a = – 2 + 10 = -8

Question 24.

If then x is equal to …………..

(a) 5

(b) 7

(c) 26

(d) 10

Solution:

(c) 26

Question 25.

If \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k},|\vec{b}|=5\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\), then the area of the triangle formed by these two vectors as two sides, is …………….

(a) \(\frac{7}{4}\)

(b) \(\frac{15}{4}\)

(c) \(\frac{3}{4}\)

(d) \(\frac{17}{4}\)

Solution: