**Chapter 2 Mathematical Methods**

**1. Choose the correct option.**

**Question 1.The resultant of two forces 10 N and 15 N acting along + x and – x-axes respectively, is**

(A) 25 N along + x-axis

(B) 25 N along – x-axis

(C) 5 N along + x-axis

(D) 5 N along – x-axis

**Answer:**

(D) 5 N along – x-axis

**Question 2.For two vectors to be equal, they should have the**

(A) same magnitude

(B) same direction

(C) same magnitude and direction

(D) same magnitude but opposite direction

**Answer:**

(C) same magnitude and direction

**Question 3.The magnitude of scalar product of two unit vectors perpendicular to each other is**

(A) zero

(B) 1

(C) -1

(D) 2

**Answer:**

(A) zero

**Question 4.The magnitude of vector product of two unit vectors making an angle of 60° with each other is**

**2. Answer the following questions.**

Solution:

Answer:

Answer:

Substituting for w_{x} in (i) using equation (ii),

Using equation (ii),

Alternate method:

When two vectors are parallel, one vector is scalar multiple of another,

Answer:

Let angle between two vectors be θ.

⇒ Two vectors are parallel.

Alternate method:

**3. Solve the following problems.**

Answer:

Using determinant for vectors in two dimensions,

Solution:

As dot product of two perpendicular vectors is zero. Taking dot product of a⃗ and b⃗

Solution:

Answer:

Value of a is .

**Question 5.Obtain derivatives of the following functions:**

(i) x sin x

(ii) x

^{4}+ cos x

(iii) x/sin x

**Answer:**

**(i) x sin x**

**(ii) x ^{4} + cos xSolution:**

**(iii) Solution:**

[Note: As derivative of (sin x) is cos x, negative sign that occurs in rule for differentiation for quotient of two functions gets retained in final answer]

**Question 6.Using the rule for differentiation for quotient of two functions, prove that = sec ^{2}xSolution:**

**Question 7.Evaluate the following integral:(i) ∫π/20sinxdx(ii) ∫51xdxAnswer:**

(i) ∫π/20sinxdx

Solution:

(ii) ∫51xdx

Solution:

**Intext Questions and Answers**

Can you recall? (Textbook Page No. 16)

**Question 1.Define scalars and vectors.Answer:**

- Physical quantities which can be completely described b their magnitude (a number and unit) are called scalars.
- Physical quantities which need magnitude, as well as direction for their complete description, are called vectors.

**Question 2.Which of the following are scalars or vectors?Displacements, distance travelled, velocity, speed, force, work done, energyAnswer:**

- Scalars: Distance travelled, speed, work done, energy.
- Vectors: Displacement, velocity, force.

**Question 3.What is the difference between a scalar and a vector?Answer:**

No. | Scalars | Vectors |

i. | It has magnitude only | It has magnitude as well as direction. |

ii. | Scalars can be added or subtracted according to the rules of the algebra. | Vectors are added or subtracted by the geometrical (graphical) method or vector algebra. |

iii. | It has no specific representation. | It is represented by the symbol (→) arrow. |

iv. | The division of a scalar by another scalar is valid. | The division of a vector by another vector is not valid. |

Example: Length, mass, time, volume, etc. | Example: Displacement, velocity, acceleration, force, etc. |

Internet my friend (Textbook page no. 28)

- hyperphysics.phy-astr.gsu.edu/hbase/vect. html#veccon
- hyperphysics.phy-astr.gsu.edu/hbase/ hframe.html

**Answer:**[Students can use links given above as a reference and collect information about mathematical methods]