Question

# If F(x) = ax + b, where a and b are integers, f(2) = 6 and F(3) = 7 then find the values of a and b.

Hint:

### Put the given values of x and functions and find the required value.

## The correct answer is: a = 1

### We have given the values of f(x)

F(x) = ax + b

F(2) = 6

F(3) = 7

When x = 2

a(2) + b =6

2a + b = 6 ------(i)

when x = 3

a(3) + b = 7

3a + b = 7 -------(ii)

Subtract equation (ii) from (i)

3a – 2a + b – b = 7 – 6

a = 1

Put this value in equation (i)

2(1) + b = 6

b = 6 – 2

b = 4

Therefore the value of a = 1

Subtract equation (ii) from (i)

Put this value in equation (i)

Therefore the value of a = 1

### Related Questions to study

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials

We have given two monomials .

As we can see there is only one common factor in this two monomials

So, The highest factor in these two monomial is 1.

So, The GCF 4x

^{3 }, 9y

^{5}is 1.

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials

We have given two monomials .

As we can see there is only one common factor in this two monomials

So, The highest factor in these two monomial is 1.

So, The GCF 4x

^{3 }, 9y

^{5}is 1.

### If the relation R: A → B, where A = {2, 3, 4} and B = {3, 5} is defined by R = {(x, y):x < y, x ϵ A, y ϵ B}, then find R^{-1}

A = {2, 3, 4}

B = {3, 5}

We have given the relation in Set- builder form ,

R = {(x, y):x < y, x ϵ A, y ϵ B}

We will first find the Cartesian product of set A and B

A X B = {(2,3),(2,5),(3,3),(3,5)(4,3),(4,5)}

R = {(2,3),(2,5),(3,5),(4,5)}

Therefore, R^{-1} = (A X B) – R

R^{-1} = {(3,3),(4,3)}

R^{-1} = {(x, y):x >= y, x ϵ A, y ϵ B}

### If the relation R: A → B, where A = {2, 3, 4} and B = {3, 5} is defined by R = {(x, y):x < y, x ϵ A, y ϵ B}, then find R^{-1}

A = {2, 3, 4}

B = {3, 5}

We have given the relation in Set- builder form ,

R = {(x, y):x < y, x ϵ A, y ϵ B}

We will first find the Cartesian product of set A and B

A X B = {(2,3),(2,5),(3,3),(3,5)(4,3),(4,5)}

R = {(2,3),(2,5),(3,5),(4,5)}

Therefore, R^{-1} = (A X B) – R

R^{-1} = {(3,3),(4,3)}

R^{-1} = {(x, y):x >= y, x ϵ A, y ϵ B}

### The shape of the window shown is a regular polygon. The window has a boundary made of silver wire. How many meters of silver wire are needed for this border if two sides are given to be

Hint:

- A regular polygon is a polygon with congruent sides and equal angles and are symmetrically placed about a common center.

- We have been given a figure of window in the question which is in the shape of a regular polygon, the boundary of the window is made up of silver wire.
- We have also been given the two sides of it that is - (𝑥 + 7) 𝑎𝑛𝑑 (2𝑥 − 3)
- We have to find out how many meters of sliver wire are needed for the border.

We have given a window is of shape of regular polygon

Two of its side is represented by x + 7; 2x - 3

Since, It is regular polygon, all the sides length will be equal.

So,

X + 7 = 2x - 3

x = 7 + 3

x = 10

So,

The side length is

= x + 7

= 7 + 10

= 17

Now the perimeter of octagon will be

= 8 × side

= 8 × 17

= 136

### The shape of the window shown is a regular polygon. The window has a boundary made of silver wire. How many meters of silver wire are needed for this border if two sides are given to be

Hint:

- A regular polygon is a polygon with congruent sides and equal angles and are symmetrically placed about a common center.

- We have been given a figure of window in the question which is in the shape of a regular polygon, the boundary of the window is made up of silver wire.
- We have also been given the two sides of it that is - (𝑥 + 7) 𝑎𝑛𝑑 (2𝑥 − 3)
- We have to find out how many meters of sliver wire are needed for the border.

We have given a window is of shape of regular polygon

Two of its side is represented by x + 7; 2x - 3

Since, It is regular polygon, all the sides length will be equal.

So,

X + 7 = 2x - 3

x = 7 + 3

x = 10

So,

The side length is

= x + 7

= 7 + 10

= 17

Now the perimeter of octagon will be

= 8 × side

= 8 × 17

= 136

### If f(x) = (25 – x2)1/2 then f( ) =…..

f(x) = (25 – x^{2})^{1/2}

We have to find out the value of f()

f() = (25 – ()^{2})^{1/2}

= (25 – 5)^{ ½}

= 20 ^{1/2}

= 2

### If f(x) = (25 – x2)1/2 then f( ) =…..

f(x) = (25 – x^{2})^{1/2}

We have to find out the value of f()

f() = (25 – ()^{2})^{1/2}

= (25 – 5)^{ ½}

= 20 ^{1/2}

= 2

### Given x = {(2, 7), (3, 9), (5, 13), (0, 3)} be a function from Z to Z defined by f(x) = ax + b for some integral a and b. What are the values of a and b?

Given x = {(2, 7), (3, 9), (5, 13), (0, 3)}

And also we have given that

f(x) = ax + b

We have to find the value of a and b .

First of all if the f is a function then its points will satisfy f(x) = ax + b

f(2) = 7

f(3) = 9

f(5) = 13

f(0) = 3

i) (2,7)

f(2) = a (2) + b

7 = 2a + b

ii) (3,9)

f(3) = a(3) + b

9 = 3a + b

Subtract equation (i) from (ii)

3a – 2a + b – b = 9 – 7

a = 2

Putting this value in equation (i)

7 = 2(2) + b

b = 7 – 4

b = 3

Therefore, value of a = 2 and b = 3.

### Given x = {(2, 7), (3, 9), (5, 13), (0, 3)} be a function from Z to Z defined by f(x) = ax + b for some integral a and b. What are the values of a and b?

Given x = {(2, 7), (3, 9), (5, 13), (0, 3)}

And also we have given that

f(x) = ax + b

We have to find the value of a and b .

First of all if the f is a function then its points will satisfy f(x) = ax + b

f(2) = 7

f(3) = 9

f(5) = 13

f(0) = 3

i) (2,7)

f(2) = a (2) + b

7 = 2a + b

ii) (3,9)

f(3) = a(3) + b

9 = 3a + b

Subtract equation (i) from (ii)

3a – 2a + b – b = 9 – 7

a = 2

Putting this value in equation (i)

7 = 2(2) + b

b = 7 – 4

b = 3

Therefore, value of a = 2 and b = 3.

### Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.

And n( A)= 3

n(B) = 2

Therefore, n(AXB) = 3X2 = 6

Since, {(x,1),(y, 2),(z,1)} are the elements of A×B.

It follows that the elements of

set A={x, y, z}

and B={1,2}

Hence,

A×B={(x,1),(x,2),(y,1),(y,2),(z,1),(z,2)}

### Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.

And n( A)= 3

n(B) = 2

Therefore, n(AXB) = 3X2 = 6

Since, {(x,1),(y, 2),(z,1)} are the elements of A×B.

It follows that the elements of

set A={x, y, z}

and B={1,2}

Hence,

A×B={(x,1),(x,2),(y,1),(y,2),(z,1),(z,2)}

### Fundamental Algebra:

1. Add:

(ii) –7y^{2} + 8y + 2 ; 3y^{2} – 7y + 1

- Hint:

○ Arrange the term in the decreasing order of their power.

○ To add polynomials simply add any like terms together.

○ Like terms are terms whose variables and exponents are the same.

- Step by step explanation:

Expression: –7y

^{2}+ 8y + 2 ;

3y

^{2}– 7y + 1 ;

○ Step 1:

○ Simplify the expression.

(–7y

^{2}+ 8y + 2 ) + ( 3y

^{2}– 7y + 1 )

(–7y

^{2}+ 8y + 2 ) + ( 3y

^{2}– 7y + 1 )

○ Step 2:

○ Group the like terms.

( –7y

^{2.}+ 3y

^{2}) + ( 8y - 7y ) + ( 2 + 1 )

○ Step 3:

○ Add like terms,

– 4y

^{2}+ y + 3

- Final Answer:

^{2}+ y + 3

### Fundamental Algebra:

1. Add:

(ii) –7y^{2} + 8y + 2 ; 3y^{2} – 7y + 1

- Hint:

○ Arrange the term in the decreasing order of their power.

○ To add polynomials simply add any like terms together.

○ Like terms are terms whose variables and exponents are the same.

- Step by step explanation:

Expression: –7y

^{2}+ 8y + 2 ;

3y

^{2}– 7y + 1 ;

○ Step 1:

○ Simplify the expression.

(–7y

^{2}+ 8y + 2 ) + ( 3y

^{2}– 7y + 1 )

(–7y

^{2}+ 8y + 2 ) + ( 3y

^{2}– 7y + 1 )

○ Step 2:

○ Group the like terms.

( –7y

^{2.}+ 3y

^{2}) + ( 8y - 7y ) + ( 2 + 1 )

○ Step 3:

○ Add like terms,

– 4y

^{2}+ y + 3

- Final Answer:

^{2}+ y + 3

### A regular polygon has 9 diagonals. Find the number of sides and classify it based on the number of sides.

Hint:

- A regular polygon is a polygon with congruent sides and equal angles and are symmetrically placed about a common center

- We have been given in the question information about a regular polygon that has 9 diagonals which means it is a hexagon.
- We have to find the number of sides and classify it based on the number of sides.

We know that the number of diagonal in n - side polygon

Here, the number of diagonals is 9

So,

(n - 6) (n + 3) = 0

n = 6

The number of sides of the given polygon are 6

And it is Hexagon.

### A regular polygon has 9 diagonals. Find the number of sides and classify it based on the number of sides.

Hint:

- A regular polygon is a polygon with congruent sides and equal angles and are symmetrically placed about a common center

- We have been given in the question information about a regular polygon that has 9 diagonals which means it is a hexagon.
- We have to find the number of sides and classify it based on the number of sides.

We know that the number of diagonal in n - side polygon

Here, the number of diagonals is 9

So,

(n - 6) (n + 3) = 0

n = 6

The number of sides of the given polygon are 6

And it is Hexagon.

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials

We have given two monomials .

The highest factor in these two monomial is a × a

Ie, a

^{2}.

So, The GCF 8a

^{2}, 28a

^{5}is a

^{2}.

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials

We have given two monomials .

The highest factor in these two monomial is a × a

Ie, a

^{2}.

So, The GCF 8a

^{2}, 28a

^{5}is a

^{2}.

### A regular polygon has 20 diagonals. Find the number of sides and classify it based on the number of sides.

Hint:

- A regular polygon is a polygon with congruent sides and equal angles and are symmetrically placed about a common center.

- We have been given in the question information about a regular polygon that has 20 diagonals which means it is an octagon.
- We have to find the number of sides and classify it based on the number of sides.

We know that the number of diagonal in n - side polygon

Here, the number of diagonals is 20

So,

(n - 8) (n + 5) = 0

n = 8

So, The number of sides are 8.

And It is Octagon.

### A regular polygon has 20 diagonals. Find the number of sides and classify it based on the number of sides.

Hint:

- We have been given in the question information about a regular polygon that has 20 diagonals which means it is an octagon.
- We have to find the number of sides and classify it based on the number of sides.

We know that the number of diagonal in n - side polygon

Here, the number of diagonals is 20

So,

(n - 8) (n + 5) = 0

n = 8

So, The number of sides are 8.

And It is Octagon.

### A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.

A = {1, 2, 3, 5} and B = {4, 6, 9}

And

R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}

First of all we will find the cartesian product of set A and set B

A X B = {(1,4),(1,6),(1,9),(2,4),(2,6),(2,9),(3,4),(3,6),(3,9),(5,4),(5,6),(5,9)}

We have relation R , with condition difference between x and y is odd, in this cartesian product we will find out the points satisfying the condition.

So, R in roaster form is

R = {(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}

### A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.

A = {1, 2, 3, 5} and B = {4, 6, 9}

And

R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}

First of all we will find the cartesian product of set A and set B

A X B = {(1,4),(1,6),(1,9),(2,4),(2,6),(2,9),(3,4),(3,6),(3,9),(5,4),(5,6),(5,9)}

We have relation R , with condition difference between x and y is odd, in this cartesian product we will find out the points satisfying the condition.

So, R in roaster form is

R = {(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}

### Which of the following relations are functions? Give reasons.

(i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

(ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}

As {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

Domain:- { 2 , 5, 8, 11, 14, 17}

Co- Domain:- {1}

From the given data we can analyse that each element in the domain input set has exactly one output.

Therefore, the given data is a function.

### Which of the following relations are functions? Give reasons.

(i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

(ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}

As {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

Domain:- { 2 , 5, 8, 11, 14, 17}

Co- Domain:- {1}

From the given data we can analyse that each element in the domain input set has exactly one output.

Therefore, the given data is a function.

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials.

We have given two monomials .

The highest factor in these two monomial is x × x × x × y

Ie, x

^{3}y.

So, The GCF x

^{3}y

^{2}, x

^{5}y is x

^{3}y.

### Find the GCF (GCD) of the given pair of monomials.

Hint:

- The highest number that divides exactly into two or more numbers is known as GCF.
- An algebraic expression consisting only one term is called monomial.

- We have been given a monomial in the question
- We have to find the GCF of the given pair of monomials.

We have given two monomials .

The highest factor in these two monomial is x × x × x × y

Ie, x

^{3}y.

So, The GCF x

^{3}y

^{2}, x

^{5}y is x

^{3}y.

### A square has lines of symmetry

Hint:

- A square is a regular quadrilateral which has four equal sides and four equal angles or four right angles.

- We have been given a statement in the question for which we have to fill the blank from the given four options.

We know that in regular polygon, the number of line of symmetry is equal to number of sides of polygon.

A square is regular polygon with 4 sides.

So, The number of line of symmetry will be 4.

### A square has lines of symmetry

Hint:

- A square is a regular quadrilateral which has four equal sides and four equal angles or four right angles.

- We have been given a statement in the question for which we have to fill the blank from the given four options.

We know that in regular polygon, the number of line of symmetry is equal to number of sides of polygon.

A square is regular polygon with 4 sides.

So, The number of line of symmetry will be 4.

### Let R be a relation from N-N defined by R= { (a,b) : a, b belong to N and a= b2 }.Are the following true ?

a) (a,a) ∈R , for all a ∈N

b) (a,b) ∈R implies (b,a) ∈R

c) (a,b) ∈R , (b,c) ∈R implies (a,c) ∈R

Justify your answer in each case.

^{2}}.

We will consider all the given cases

i) It is not true because a = a

^{2}

a^{2}-a = 0

a(a-1) = 0

a = 0 ,1

Hence the given condition is only true for a = 1

Not for all natural numbers.

ii) It is not true because if a = b^{2} , then b = a^{2} is not possible

Eg. 3^{2} = 9 but 9^{2} > 3

iii) It is not true because

If then

It means a = b^{2} and b = c^{2} , a = c^{2}

This means a = (c^{2})^{2} = c^{4}

But a = c^{2}

So the given statement is wrong.

### Let R be a relation from N-N defined by R= { (a,b) : a, b belong to N and a= b2 }.Are the following true ?

a) (a,a) ∈R , for all a ∈N

b) (a,b) ∈R implies (b,a) ∈R

c) (a,b) ∈R , (b,c) ∈R implies (a,c) ∈R

Justify your answer in each case.

^{2}}.

We will consider all the given cases

i) It is not true because a = a

^{2}

a^{2}-a = 0

a(a-1) = 0

a = 0 ,1

Hence the given condition is only true for a = 1

Not for all natural numbers.

ii) It is not true because if a = b^{2} , then b = a^{2} is not possible

Eg. 3^{2} = 9 but 9^{2} > 3

iii) It is not true because

If then

It means a = b^{2} and b = c^{2} , a = c^{2}

This means a = (c^{2})^{2} = c^{4}

But a = c^{2}

So the given statement is wrong.