Commerce Exam  >  Commerce Notes  >  Mathematics (Maths) Class 11  >  Trigonometric Equations

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce PDF Download

TRIGONOMETRIC EQUATIONS

A. Solution of Trigonometric Equations 

A solution of trigonometric equation is the value of the unknown angle that satisfies the equation.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Thus, the trigonometric equation may have infinite number of solutions (because of their periodic nature) and can be classified as :

(1) Principal solution

(2) General solution.

(1) Principal solutions : The solutions of trigonometric equation which lie in the interval [0, 2π) are called principal solutions. 

(1) General solutions :  The solutions expressing all the values which would satisfy the given equation, and it is expressed in a generalized form in terms of ‘n’.

General Solutions of Trigonometric EquationsGeneral Solutions of Trigonometric Equations

Solved Examples:

Ex.1 Find the Principal solutions of the equation sinx = 1/2.

Sol. 

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

sin x = 1/2

there exists two values 

i.e.  π/6 and  5π/6 which lie in [0,2π) and whose sine is 1/2

Principal solutions of the equation sinx = 1/2 are π/6 and  5π/6

Ex.2 Solve  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.3 Solve tan θ = 2 

Sol.  tan θ = 2 ....(1)

Let 2 tan α ⇒  tan θ = tan α 

 ⇒ θ= n π  + α, where α = tan–1 (2), n ∈ I

Ex.4 Solve Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.5 Solve Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - CommerceTrigonometric Equations | Mathematics (Maths) Class 11 - Commerce        

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Solutions of equations by factorising 

Ex.6 Solve the equation sin3 x cos x -sin x cos3x = 1/4.

Sol.
The equation can be written as 4 sin x cos x (sinx – cos2 x) = 1,

= –2 sin 2x cos 2x = –sin 4x = 1

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.7 Find the general solution of the equation

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.8  Find the general solution of the equation sin3x(1 + cot x) + cos3x(1 + tan x) = cos 2x.

Sol.

sin2x(cos x + sin x) + cos2x (cos x + sin x) = cos 2x

(cos x + sin x)(cos2x + sin2x) = (cos x + sin x)(cos x – sin x)

either cos x + sin x = 0 ....(1) or

cos x - sin x = 1 ....(2)

from (1) tan x = – 1 or 1 – sin 2x = 1 ⇒ sin 2x = 0

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

If sin 2x = 0 ⇒ 2x = nπ ⇒  x = nπ/2 this is to be rejected because of the tan x or cot x will not be defined so x = (nπ - π/4), n ∈ I

Ex.9Find the solutions of the equation, 
Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

2 sin2x = 1 + cos x ; 2 cos2x + cos x – 1 = 0

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

C. Solutions of equations reducible to quadratic equations 

Ex.10  Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. The given equation makes no sense when cos x = 0; therefore we can suppose that cos x ≠ 0. Noting that the right-hand member of the equation is equal to Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and dividing both members by Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

⇒ (tanx – 3) (tan x + 1) = 0

⇒  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.11 Find the general solution set of the equation logtan x(2 + 4 cos2x) = 2.

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.12 The equation cos2x -sin x + a = 0 has roots when x Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce find 'a'.

Sol. 1 – sin2x – sin x + a = 0 =  sin2x + sin x – (a + 1) = 0 (let sin x = t)

 t2 + t - (a + 1) = 0,

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.13  Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Using the identity (sin2 x + cos2 x)2 = 1 we get sin4 x + cos4 x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.14  Find all solutions of the equation (tan2 x -1)-1 = 1 + cos 2x, which satisfy the inequality Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Let us reduce the initial trigonometric equation to the form Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

The following values of x are solutions of this equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

By the hypothesis, we must choose those values of x which satisfy the inequalities

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.15 Determine all the values of a for which the equation sin4 x -2 cos2 x + a2 = 0 is solvable. Find the solutions.

Sol. Applying the formula sin4 x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce, cos2 x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and putting cos 2x = t

we rewrite the given equation in the form t2 -6t + 4a2 -3 = 0 ........(1)

The original equation has solutions for a given value of a if and only if, for his value of a, the roots t1 and t2 of the equation (1) are real and at least one of these roots does not exceed unity in its absolute value.

Solving equation (1), we find t1 = 3 -2 Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce, t2 = 3 + 2 Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce.

Hence the rotos of equation (1) are real if Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

If condition (2) is fulfilled, then t2 > 1 and, therefore, this root can be dicarded. Thus, the problem is reduced to finding the values of a satisfying condition (2), for which Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

From (3) we find Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Since the inequality Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce is fulfilled for Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce the system of inequalities (4) is reduced to the inequality Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Thus, the original equation is solvable if Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and its solutions are

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

D. Solving equations by introducing an Auxiliary argument 

Ex. 16 Solve sin x + cos x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce sin x + cos x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce              ....(i)           Here a = 1, b = 1

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.17  Solve the equation cos 7x -sin 5x = Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce (cos 5x -sin 7x).

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.18  Solve the equation 2 sin 17x + Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce cos 5x + sin 5x = 0

Sol. Dividing both sides of the equation by 2, we reduce it to the form Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

whence we obtain  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.19  Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Using the formula for the sum of cubes of two members we transform the left-hand side of the equation in the following way : Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Hence, the original equation takes the form Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

The expression in the first brackets is different from zero for all x. Therefore it is sufficient to consider the equation sin x + cos x -1 = 0. The latter is reduced to the form

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

E. Solving equations by Transforming a sum of Trigonometric functions into a product 

Ex.20 Solve cos 3x + sin 2x -sin 4x = 0

Sol. 

cos 3x + sin 2x – sin 4 x = 0 ⇒  cos 3x + 2 cos 3x . sin (–x) = 0 ⇒  cos 3x – 2 cos x . sin x = 0

⇒  cos 3x (1 – 2 sin x) = 0 ⇒  cos 3x = 0 or 1 – 2 sin x = 0

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

solution of given equation is Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

F. Solving equations by transforming a product of trigonometric functions into a sum 

Ex.21 Solve sin 5x . cos 3x = sin 6x . cos 2x

Sol.  

sin 5x . cos 3x = sin 6x . cos 2x

⇒  2sin 5x . cos 3x = 2sin 6x . cos 2x

⇒  sin 8x + sin 2x = sin 8x + sin 4x

⇒  sin 4x – sin 2x = 0

⇒  2 sin 2x . cos 2x – sin 2x = 0

⇒  sin 2x (2 cos 2x – 1) = 0

⇒  sin 2x = 0 or 2 cos 2x – 1 = 0 ⇒  2x = nπ, n ∈ I or cos2x = 1/2

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce  

G. Solving equations by a change of variable

(i)  Equations of the form  P (sin x ± cos x , sin x . cos x) = 0  ,  where  P (y , z) is  a  polynomial , can be solved by the change cos x ± sin x = t ⇒ 1 ± 2 sin x . cos x = t2.

(ii)  Equations  of  the form  of   a . sin x + b . cos x + d = 0 ,  where  a , b & d are  real  numbers  & a, b ≠ 0  can be solved by changing  sin x & cos x into their corresponding tangent of half the angle.

(iii) Many  equations  can  be  solved  by  introducing  a  new variable .  eg.  the equation sin4 2 x + cos4 2 x = sin 2 x . cos 2 x  changes to

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.22   Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.  

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.23  Solve the equation sin 2x -12 (sin x -cos x) + 12 = 0

 Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce


Ex.24 Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

Ex.25 Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

Ex.26     Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.27 Solve 3 cos x + 4 sin x = 5

Sol. 

3 cos x + 4 sin x = 5

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 equation (i) becomes  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

H. Solving equations with the use of the Boundness of the functions sin x & cos x 

Ex.28  Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol.

The equation makes no sense for x = π/2 + kπ and for x = -π/4 + kπ. For all the other values of x it is equivalent to the equation

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

After simple transformations we obtain sin x (3 + sin 2x + cos 2x) = 0.

It is obvious that the equation sin 2x + cos 2x + 3 = 0 has no solution, and therefore, the original equation is reduced to the equation sin x = 0  ⇒  x = kπ

Ex.29  Solve the equation (sin x + cos x) √2 = tan x + cot x.

Sol.

Let us transform the equation to the form 

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce  ....(1)

We have |sin a| ≤ 1, and therefore (1) holds

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

or  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

But the first two equations have no roots in common while the second two equations have the common roots x = π/4 + 2k π . Consequently the roots of the given equation are x = π/4 + 2k π

Ex.30 Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. 

Obviously no solution is possible if π/2 < x < 2π as LHS < 2.

If 0 < x < π/2 , then LHS = sin2n – 1 x + 2 cos2n – 1 x < sinx + 2 cosx = 1 + cos2 x < 2 when n ∈ N – {1}.

Obviously, a solution exists only when x = 0 ⇒  The general solution is x = 2mπ, m ∈ I.

When n = 1

sin x + 2 cos x = 2 

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

Ex.31 Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

Sol.Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

since square of the cosine of any argument doesn't exceed 1, the given equation holds true if and only if we have, simultaneously Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

so, equation (3) has no solution for k ≠ 0 for k = 0

sin x + √2 cosx = 0 or √2 sinx - sinx - √2 or, sinx = -1/√2 , √2

but sin x = √2 is not possible. so only solution to the equation (1) is

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce equation (2) becomes an identity but  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce doesn’t satisfy equation (2) so, solution to the original equation  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

 

Ex.32 Find the general solution of the equation, sin 3x + cos 4x - 4 sin 7x = cos 10x + sin 17x.

Sol. 

(sin 17x - sin 3x) - cos 10x - cos 4x + 4 sin 7x = 0 ⇒   2 cos 10x  sin 7x + 2 sin 7x  sin 3x + 4 sin 7x=0

⇒  sin 7x (cos 10x - sin 3x + 2) = 0 

 Hence sin 7x = 0 ⇒ x = nπ/7, n ∈ I

or  cos 10x - sin 3x + 2 = 0 ⇒ cos 10x = - 1    and    sin 3x = 1  given x = (4n + 1) π/6

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Those starred also satisfy  cos 10x = - 1 ,  the general term of which is

x = 3 (4k - 1) π/6 , k E I , Hence x = nπ/7 or 3(4k + 1) π/6 where, n, k ∈  I

I. Simultaneous equations 

Ex.33Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Transform the system to the Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Adding together the equations of system (1) and subtracting the first equation form the second we obtain the system

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce .....(2)

The first equation of system (2) can be rewritten as Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

If sin (x – y) = 0, then x – y = kπ. But from the second equation of system (2) we find    cos (x – y) = –1, x – y = (2n + 1)π.

Consequently, in this case we have an infinitude of solutions : x – y (2n + 1) π.

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce then 3x = y = 2kπ. But x – y = (2n + 1) π

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Ex.34 Solve the system of equations Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. Adding up the equations of the system, we arrive at an equation

sin x sin y + cos x cos y = √3/2 ⇔ cos (x-y) = √3/2

Subtracting the first equation of the system from the second. we arrive at an equation

cos x cosu y - sin x sin y = √3/2 ⇔ cos (x-y) = 0

Thus the initial system is equivalent to the system  Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

⇔ n, k ∈ Z, cos (x + y) = 0, 

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

J. Miscellaneous Questions 

Ex.35  Solve the equation 2 cot 2x – 3 cot 3x = tan 2x

Sol. The give equation can be rewritten in the form Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Note that this equation has sense if the condition sin 2x ≠0, sin3x ≠0, cos2x ≠0 holds. For the values of x satisfying this condition we have 3 sin x cos 2x = sin 3x. Transforming the last equation we obtain Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and thus arrive at the equation 2 sin3 x = 0, which is equivalent to the equation sin x = 0. Hence, due to the above note, the original equation has no solutions.

Ex.36  Solve the equation Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Sol. The right-hand side of the equation is not determined for Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce, because for Trigonometric Equations | Mathematics (Maths) Class 11 - Commercethe function Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce is not defined, for Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce the function tan x/2 is not defined and for

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce the denominator of the right member of the right member vanishes. For Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce we have
Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce.

Hence, for Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce (where k and m are arbitrary integers) the right member of the equation is equal to -2 sin x cos x.

The left member of the equation has no sense for Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce and for all the other values of x it is equal to -tan x because

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

Thus, Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce then the original equation is reduced to the form tan x = 2 sin x cos x.

This equation has the roots Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce It follows that the original equation has no roots.

The document Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
All you need of Commerce at this link: Commerce
85 videos|243 docs|99 tests

Up next

FAQs on Trigonometric Equations - Mathematics (Maths) Class 11 - Commerce

1. What are the eligibility criteria for Airforce X Y and Indian Navy SSR exams?
Ans. The eligibility criteria for the Airforce X Y exam require candidates to be between 17 and 21 years of age, have passed intermediate or equivalent examination with Mathematics, Physics, and English, and meet the required physical and medical standards. For the Indian Navy SSR exam, candidates must be between 17 and 20 years of age, have passed 10+2 examination with Mathematics and Physics, and meet the required physical and medical standards.
2. How can I apply for the Airforce X Y and Indian Navy SSR exams?
Ans. To apply for the Airforce X Y exam, candidates need to visit the official website of the Indian Air Force and fill out the online application form. The application process for the Indian Navy SSR exam is also online, and candidates can apply through the official website of the Indian Navy.
3. What is the selection process for Airforce X Y and Indian Navy SSR exams?
Ans. The selection process for both exams includes a written examination, followed by a physical fitness test, medical examination, and document verification. The written examination consists of multiple-choice questions and assesses the candidate's knowledge in subjects like Physics, Mathematics, English, and General Awareness.
4. What topics should I focus on while preparing for Airforce X Y and Indian Navy SSR exams?
Ans. For the Airforce X Y exam, candidates should focus on topics like Mathematics, Physics, English, and General Awareness. In the Indian Navy SSR exam, candidates should pay attention to subjects like Mathematics, Physics, and English. It is important to have a thorough understanding of the concepts and practice solving previous years' question papers.
5. What is the pay scale and career growth opportunities after qualifying in Airforce X Y and Indian Navy SSR exams?
Ans. After qualifying in the Airforce X Y and Indian Navy SSR exams, candidates receive a stipend during their training period. Once training is completed, they are appointed as Airmen in the Indian Air Force or Sailors in the Indian Navy, respectively. The pay scale and career growth opportunities vary based on the rank and experience, with opportunities for promotions and higher positions within the respective forces.
85 videos|243 docs|99 tests
Download as PDF

Up next

Explore Courses for Commerce exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Download the FREE EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

Objective type Questions

,

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

,

shortcuts and tricks

,

mock tests for examination

,

Important questions

,

Free

,

video lectures

,

Semester Notes

,

Viva Questions

,

Extra Questions

,

pdf

,

Exam

,

Summary

,

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

,

Previous Year Questions with Solutions

,

study material

,

ppt

,

MCQs

,

Sample Paper

,

practice quizzes

,

Trigonometric Equations | Mathematics (Maths) Class 11 - Commerce

,

past year papers

;