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NCERT solution class 12 chapter 4 Differential Equations exercise 4.2 mathematics part 2

EXERCISE 4.2


Page No 440:

Question 1:

Compute the magnitude of the following vectors:

Answer:

The given vectors are:

Question 2:

Write two different vectors having same magnitude.

Answer:

Hence, are two different vectors having the same magnitude. The vectors are different because they have different directions.

Question 3:

Write two different vectors having same direction.

Answer:

The direction cosines of are the same. Hence, the two vectors have the same direction.

Question 4:

Find the values of x and y so that the vectors are equal

Answer:

The two vectors will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

Question 5:

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Answer:

The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

Hence, the required scalar components are –7 and 6 while the vector components are 

Question 6:

Find the sum of the vectors.

Answer:

The given vectors are.

Question 7:

Find the unit vector in the direction of the vector.

Answer:

The unit vector in the direction of vector is given by.

Question 8:

Find the unit vector in the direction of vector, where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively.

Answer:

The given points are P (1, 2, 3) and Q (4, 5, 6).

Hence, the unit vector in the direction of  is

.

Question 9:

For given vectors, and , find the unit vector in the direction of the vector 

Answer:

The given vectors are and.

Hence, the unit vector in the direction of is

a→+b→a→+b→=i^+k^2=12i⏜+12k⏜.

Question 10:

Find a vector in the direction of vector which has magnitude 8 units.

Answer:

Hence, the vector in the direction of vector which has magnitude 8 units is given by,

Question 11:

Show that the vectorsare collinear.

Answer:

.

Hence, the given vectors are collinear.

Question 12:

Find the direction cosines of the vector 

Answer:

Hence, the direction cosines of 

Question 13:

Find the direction cosines of the vector joining the points A (1, 2, –3) and

B (–1, –2, 1) directed from A to B.

Answer:

The given points are A (1, 2, –3) and B (–1, –2, 1).

Hence, the direction cosines of are 

Question 14:

Show that the vector is equally inclined to the axes OX, OY, and OZ.

Answer:

Therefore, the direction cosines of 

Now, let αβ, and γbe the angles formed by with the positive directions of xy, and z axes.

Then, we have

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

Question 15:

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  respectively, in the ration 2:1

(i) internally

(ii) externally

Answer:

The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: is given by:

  1. Internally:

  1. Externally:

Position vectors of P and Q are given as:

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,

Page No 441:

Question 16:

Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

Answer:

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,

Question 17:

Show that the points A, B and C with position vectors,respectively form the vertices of a right angled triangle.

Answer:

Position vectors of points A, B, and C are respectively given as:

 

AB→2+CA→2=35+6=41=BC→2Hence, ABC is a right-angled triangle.

Question 18:

In triangle ABC which of the following is not true:

A. 

B. 

C. 

D. 

Answer:

On applying the triangle law of addition in the given triangle, we have:

From equations (1) and (3), we have:

Hence, the equation given in alternative C is incorrect.

The correct answer is C.

Question 19:

If are two collinear vectors, then which of the following are incorrect:

A. , for some scalar λ

B. 

C. the respective components of are proportional

D. both the vectors have same direction, but different magnitudes

Answer:

If are two collinear vectors, then they are parallel.

Therefore, we have:

 (For some scalar λ)

If λ = ±1, then .

Thus, the respective components of are proportional.

However, vectors can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.


 

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