Indian Air Force X Group Mathematics Model Paper PDF 2022

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Indian Air Force X Group Math Model Papers PDF files

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AirForce X&Y Group Syllabus

Mathematics:-

Sets, Relations, functions
त्रिकोणमिति[Trigonometry(Inverse Trigonometric functions)]
सम्मिश्र संख्या(Complex Numbers)
द्विघातीय समीकरण(Quadratic Equations)
अनुक्रम और श्रृंखला[Sequence & Series (AP & GP)]
क्रमचय और संचय)Permutation&Combination
द्विपद प्रमेय(Binomial Theorem)
Coordinate geometry – straight line, Circles, Parabola, Ellipse, Hyperbola
Exponential&Logarithmic Series
आव्यूह और सारणिक(Matrices & Determinants)
Limit & Continuity
Differentiation
Application of Differentiation
Indefinite&Definite Integrals
Application of Integration
Differential Equations
प्रायिकता(Probability)
सांख्यिकी(Statistics)

Other Competitive Exams PDF

Maths Question and Answer

Q.1. A student has 6 pants and 7 shuts. The number of ways in which he can wear the dress in different
combinations are
I. 6! x 7!
2. 13
3. 42
4. None of these
Answer
Correct Option – 3

Q. 2. From a group of 10 men and 8 women, a team of 3 men or 2 women is to be made. How many teams can be
made?
I. 148
2. 130
3. 150
4. 145
Answer
Correct Option – 1

Q. 3. Find the value of tan (19n / 3)?
I. ✓3
2. 1 / ✓3
3. 1
4. 0
Answer
Correct Option – 1

Q.4. If sin 0 – cos 0 = 0, and O < 0 < 90°
, then 0 =
1. 30°
2. 45°
3. 60°
4. 90°
Answer
Correct Option – 2

Q. 5. A kite is flying with the string inclined 30° with the horizontal. If the string is 60 m long, then what is the height of the kite above the ground?
1. 45
2. 30
3. 40
4. 20
Answer
Correct Option – 2

Q. No.1: Determine order and degree (if defined) of differential equation (y′′′)² + (y″)³ + (y′)⁴ + y⁵ = 0

Solution:

Given differential equation is (y′′′)² + (y″)³ + (y′)⁴ + y⁵ = 0

The highest order derivative present in the differential equation is y′′′.

Therefore, its order is 3.

The given differential equation is a polynomial equation in y′′′, y′′, and y′.

The highest power raised to y′′′ is 2.

Hence, its degree is 2.

Q. No. 2: Verify that the function y = a cos x + b sin x, where, a, b ∈ R is a solution of the differential equation d²y/dx²+ y=0.

Solution:

The given function is y = a cos x + b sin x … (1)

Differentiating both sides of equation (1) with respect to x,

dy/dx = – a sinx + b cos x

d²y/dx² = – a cos x – b sinx

LHS = d²y/dx²+ y

= – a cos x – b sinx + a cos x + b sin x

= 0

= RHS

Hence, the given function is a solution to the given differential equation.

Q. No. 3: The number of arbitrary constants in the general solution of a differential equation of fourth order is:

(A) 0 (B) 2 (C) 3 (D) 4

Solution:

We know that the number of constants in the general solution of a differential equation of order n is equal to its order.

Therefore, the number of constants in the general equation of the fourth-order differential equation is four.

Hence, the correct answer is D.

Note: The number of constants in the general solution of a differential equation of order n is equal to zero.

Q. No. 4: Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.

Solution:

Given,

y = a sin (x + b) … (1)

Differentiating both sides of equation (1) with respect to x,

dy/dx = a cos (x + b) … (2)

Differentiating again on both sides with respect to x,

d²y/dx² = – a sin (x + b) … (3)

Eliminating a and b from equations (1), (2) and (3),

d²y/ dx²+ y = 0 … (4)

The above equation is free from the arbitrary constants a and b.

This the required differential equation.

Q. No. 5: Find the differential equation of the family of lines through the origin.

Solution:

Let y = mx be the family of lines through the origin.

Therefore, dy/dx = m

Eliminating m, (substituting m = y/x)

y = (dy/dx) . x

x. dy/dx – y = 0

Q. No. 6: Form the differential equation of the family of circles having a centre on y-axis and radius 3 units.

Solution:

The general equation of the family of circles having a centre on the y-axis is x² + (y – b)² = r²

Given the radius of the circle is 3 units.

The differential; equation of the family of circles having a centre on the y-axis and radius 3 units is as below:

x² + (y – b)² = 3²

x² + (y – b)² = 9 ……(i)

Differentiating (i) with respect to x,

2x + 2(y – b).y′ = 0

⇒ (y – b). y′ = -x

⇒ (y – b) = -x/y′ …….(ii)

Substituting (ii) in (i),

x² + (-x/y′)² = 9

⇒ x²[1 + 1/(y′)²] = 9

⇒ x²[(y′)² + 1) = 9 (y′)²

⇒ (x² – 9) (y′)² + x² = 0

Hence, this is the required differential equation.

Q. No. 7: Find the general solution of the differential equation dy/dx =1+y²/1+x².

Solution:

Given differential equation is dy/dx =1+y²/1+x²

Since 1 + y² ≠ 0, therefore by separating the variables, the given differential equation can be written as:

dy/1+y² = dx/1+x² …….(i)

Integrating equation (i) on both sides,

tan^-1y = tan^-1x + C

This is the general solution of the given differential equation.

Q. No. 8: For each of the given differential equation, find a particular solution satisfying the given condition:

dy/dx = y tan x ; y = 1 when x = 0

Solution:

dy/dx = y tan x

dy/y = tan x dx

Integrating on both sides,

log y = log (sec x) +C

log y = log (C sec x)

⇒ y = C sec x ……..(i)

Now consider y = 1 when x= 0.

1 = C sec 0

1 = C (1)

C = 1

Substituting C = 1 in (i)

y = sec x

Hence, this is the required particular solution of the given differential equation.

Q. No. 9: Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos²(y/x).

Solution:

According to the given condition,

dy/dx = y/x – cos²(y/x) ………….(i)

This is a homogeneous differential equation.

Substituting y = vx in (i),

v + (x) dv/dx = v – cos²v

⇒ (x)dv/dx = – cos²v

⇒ sec²v dv = – dx/x

By integrating on both the sides,

⇒ ∫sec²v dv = – ∫dx/x

⇒ tan v = – log x + c

⇒ tan (y/x) + log x = c ……….(ii)

Substituting x = 1 and y = π/4,

⇒ tan (π/4) + log 1 = c

⇒ 1 + 0 = c

⇒ c = 1

Substituting c = 1 in (ii),

tan (y/x) + log x = 1

Q. No. 10: Integrating factor of the differential equation (1 – x²)dy/dx – xy = 1 is

(A) -x

(B) x/ (1 + x²)

(C) √(1- x²)

(D) ½ log (1 – x²)

Solution:

Given differential equation is (1 – x²)dy/dx – xy = 1

(1 – x²)dy/dx = 1 + xy

dy/dx = (1/1 – x²) + (x/1 – x²)y

dy/dx – (x/1 – x²)y = 1/1-x²

This is of the form dy/dx + Py = Q

We can get the integrating factor as below:

Let 1 – x² = t

Differentiating with respect to x

-2x dx = dt

-x dx = dt/2

Now,

I.F = √t = √(1- x²)

Hence, option C is the correct answer.

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Indian Airforce X Group Mathematics Model Paper PDF- Download PDF

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Indian Air Force X Group Mathematics Model Paper PDF 2022

Indian Air Force X Group Math Model Papers PDF files

AirForce X&Y Group Syllabus

Mathematics:-

Other Competitive Exams PDF

Maths Question and Answer

Q.4. If sin 0 – cos 0 = 0, and O < 0 < 90° , then 0 = 1. 30° 2. 45° 3. 60° 4. 90° Answer Correct Option – 2

Q. No.1: Determine order and degree (if defined) of differential equation (y′′′)2 + (y″)3 + (y′)4 + y5 = 0

Q. No. 4: Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.

Q. No. 6: Form the differential equation of the family of circles having a centre on y-axis and radius 3 units.

Q. No. 9: Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos2(y/x).

Download notes and pdf

Download Important Notes and PDF Files-

Indian Airforce X Group Mathematics Model Paper PDF- Download PDF

Related PDF

Please Support us By Joining the Below Groups And Like Our Pages We Will be very thankful to you.

Q.4. If sin 0 – cos 0 = 0, and O < 0 < 90°
, then 0 =
1. 30°
2. 45°
3. 60°
4. 90°
Answer
Correct Option – 2

Q. No.1: Determine order and degree (if defined) of differential equation (y′′′)² + (y″)³ + (y′)⁴ + y⁵ = 0

Q. No. 9: Find the equation of a curve passing through (1, π/4) if the slope of the tangent to the curve at any point P (x, y) is y/x – cos²(y/x).