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NCERT solution class 12 chapter 4 Differential Equations exercise 4.4 mathematics part 2


Page No 454:

Question 1:

Find, if and.


We have,


Question 2:

Find a unit vector perpendicular to each of the vector and, where and.


We have,


Hence, the unit vector perpendicular to each of the vectors and is given by the relation,

Question 3:

If a unit vector  makes an angleswith with and an acute angle θ with, then find θ and hence, the compounds of.


Let unit vector  have (a1a2a3) components.


Since  is a unit vector, .

Also, it is given that  makes angleswith with , and an acute angle θ with

Then, we have:

Hence, and the components of  are.

Question 4:

Show that


Question 5:

Find λ and μ if .


On comparing the corresponding components, we have:


Question 6:

Given that  and. What can you conclude about the vectors?



(i) Either  or, or 

(ii) Either  or, or 

But,  and  cannot be perpendicular and parallel simultaneously.

Hence,  or.

Question 7:

Let the vectors given as . Then show that 


We have,

On adding (2) and (3), we get:

Now, from (1) and (4), we have:

Hence, the given result is proved.

Question 8:

If either  or, then. Is the converse true? Justify your answer with an example.


Take any parallel non-zero vectors so that.

It can now be observed that:

Hence, the converse of the given statement need not be true.

Question 9:

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and

C (1, 5, 5).


The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sidesand of ΔABC are given as:

Area of ΔABC 

Hence, the area of ΔABC

Page No 455:

Question 10:

Find the area of the parallelogram whose adjacent sides are determined by the vector .


The area of the parallelogram whose adjacent sides are is.

Adjacent sides are given as:

Hence, the area of the given parallelogram is.

Question 11:

Let the vectors and be such that and, then is a unit vector, if the angle between and is

(A)  (B)  (C)  (D) 


It is given that.

We know that, where  is a unit vector perpendicular to both and and θ is the angle between and.

Now,  is a unit vector if.

Hence,  is a unit vector if the angle between and is.

The correct answer is B.

Question 12:

Area of a rectangle having vertices A, B, C, and D with position vectors and  respectively is

(A)  (B) 1

(C) 2 (D) 


The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

The adjacent sides and  of the given rectangle are given as: 

⇒AB→×BC→=2Now, it is known that the area of a parallelogram whose adjacent sides are is.

Hence, the area of the given rectangle is

The correct answer is C.


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