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

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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 cos²α + cos²β + cos²γ = 1
(i.e.,) \(\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{\sqrt{2}}\right)^{2}\) + cos²γ = 1
cos²γ = 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: