Question

# Is the given conjecture correct? Provide arguments to support your answer.

The product of any two consecutive odd numbers is 1 less than a perfect square.

Hint:

### Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.

Deductive reasoning is the process by which a person makes conclusions based on previously known facts.

## The correct answer is: odd numbers.

### Let the odd number be 2n+1. Its consecutive even number will be 2n+3

Finding the product of (2n+1) and (2n+3)

(2n+1)(2n+3) = 2n(2n+3) + 1(2n+3)

(2n+1)(2n+3) = 2n(2n) + 2n(3) + 1(2n) + 1(3)

(2n+1)(2n+3) = 4n^{2} + 6n + 2n + 3

(2n+1)(2n+3) = 4n^{2} + 8n + 3

The given conjecture is “The product of any two consecutive odd numbers is 1 less than a perfect square”

So, add 1 to 4n^{2} + 8n + 3

4n^{2} + 8n + 4 = (2n)^{2} + 2(2n)(2) + 2^{2}

= (2n + 2)^{2}

So, we can see that the perfect square of 2n+2 is 4n^{2} + 8n + 4 and 1 less than 4n^{2} + 8n + 4 is a product of any two consecutive odd numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

^{2}+ 6n + 2n + 3

^{2}+ 8n + 3

The given conjecture is “The product of any two consecutive odd numbers is 1 less than a perfect square”

So, add 1 to 4n

^{2}+ 8n + 3

^{2}+ 8n + 4 = (2n)

^{2}+ 2(2n)(2) + 2

^{2}

^{2}

So, we can see that the perfect square of 2n+2 is 4n

^{2}+ 8n + 4 and 1 less than 4n

^{2}+ 8n + 4 is a product of any two consecutive odd numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

### Related Questions to study

### Name the polynomial based on its degree and number of terms.

- We have been given a function in the question.
- We will have to name the polynomial based on its degree and number of terms.

We have given a polynomial .

The degree of is 3.

And degree of 3 is zero.

So, The degree of the polynomial is 3, So the given polynomial is cubic.

And the number of terms is 2.

### Name the polynomial based on its degree and number of terms.

- We have been given a function in the question.
- We will have to name the polynomial based on its degree and number of terms.

We have given a polynomial .

The degree of is 3.

And degree of 3 is zero.

So, The degree of the polynomial is 3, So the given polynomial is cubic.

And the number of terms is 2.

### In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

ii)How many answers did each school get correct in the final round?

- Step by step explanation:

○ Solve equation: 165 + 12x - 5x = 65 + 12x

165 + 12x - 5x = 65 + 12x

165 + 7x = 65 + 12x

165 - 65 = 12x - 7x

100 = 5x

= x

20 = x

- Final Answer:

### In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

ii)How many answers did each school get correct in the final round?

- Step by step explanation:

○ Solve equation: 165 + 12x - 5x = 65 + 12x

165 + 12x - 5x = 65 + 12x

165 + 7x = 65 + 12x

165 - 65 = 12x - 7x

100 = 5x

= x

20 = x

- Final Answer:

### Determine the equation of the line that passes through

We are given two points and we need to find the equation of the line passing through them. The equation of a line passing through two points (a, b) and (c, d) is

Step by step solution:

Let the given points be denoted by

(a, b) = (-3, -6)

(c, d) = (2, 34)

The equation of a line passing through two points and is

Using the above points, we have

Simplifying the above equation, we have

Cross multiplying, we get

5(y - 34) = 40(x - 2)

Expanding the factors, we have

5y - 170 = 40 x -80

Taking all the terms in the left hand side, we have

-40x + 5y - 170 + 80 = 0

Finally, the equation of the line is

-40x + 5y - 90=0

Dividing the equation throughout by(- 5), we get

This is the required equation.

Note:

We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by a + by + c=0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.

### Determine the equation of the line that passes through

We are given two points and we need to find the equation of the line passing through them. The equation of a line passing through two points (a, b) and (c, d) is

Step by step solution:

Let the given points be denoted by

(a, b) = (-3, -6)

(c, d) = (2, 34)

The equation of a line passing through two points and is

Using the above points, we have

Simplifying the above equation, we have

Cross multiplying, we get

5(y - 34) = 40(x - 2)

Expanding the factors, we have

5y - 170 = 40 x -80

Taking all the terms in the left hand side, we have

-40x + 5y - 170 + 80 = 0

Finally, the equation of the line is

-40x + 5y - 90=0

Dividing the equation throughout by(- 5), we get

This is the required equation.

Note:

We can simplify the equation in any other way and we would still reach the same equation. The general form of an equation in two variables is given by a + by + c=0, where a, b, c are real numbers. The student is advised to remember all the different forms of a line, like, slope-intercept form, axis-intercept form, etc.

### Is the given conjecture correct? Provide arguments to support your answer.

The product of any two consecutive even numbers is 1 less than a perfect square

Finding the product of 2n and (2n+2)

2n(2n+2) = 2n(2n) + 2n(2)

2n(2n+2) = 4n^{2} + 4n

The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

So, add 1 to 4n^{2} + 4n

4n^{2} + 4n +1 = (2n)^{2} + 2(2n)(1) + 1^{2}

= (2n + 1)^{2}

So, we can see that the perfect square of 2n+1 is 4n^{2} + 4n +1 and 1 less than 4n^{2} + 4n +1 is a product of any two consecutive even numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

### Is the given conjecture correct? Provide arguments to support your answer.

The product of any two consecutive even numbers is 1 less than a perfect square

Finding the product of 2n and (2n+2)

2n(2n+2) = 2n(2n) + 2n(2)

2n(2n+2) = 4n^{2} + 4n

The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

So, add 1 to 4n^{2} + 4n

4n^{2} + 4n +1 = (2n)^{2} + 2(2n)(1) + 1^{2}

= (2n + 1)^{2}

So, we can see that the perfect square of 2n+1 is 4n^{2} + 4n +1 and 1 less than 4n^{2} + 4n +1 is a product of any two consecutive even numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

### Identify a linear polynomial.

We have to identify the linear polynomial from the given four options in the question

Step 1 of 1:

A linear polynomial is defined as any polynomial expressed in the form of an equation of .

From the option We can see that option B is in the form of

So, Option B is correct.

### Identify a linear polynomial.

We have to identify the linear polynomial from the given four options in the question

Step 1 of 1:

A linear polynomial is defined as any polynomial expressed in the form of an equation of .

From the option We can see that option B is in the form of

So, Option B is correct.

### Find the value of x. Identify the theorem used to find the answer.

- Hint:
- Mid-point theorem:
- According to mid-point theorem, in triangle, the line segment which joins the midpoint of two sides is parallel to third side and is equal to half of third side,

- Step by step explanation:
- Given:

BM = MC,

hence M is midpoint of BC.

BN = NA,

hence N is midpoint of AB.

MN = x

AC = 24

- Step 1:

The line segment MN joins midpoint of AB and BC

So,

According to midpoint theorem,

MN is parallel to AC and

MN =

MN =

MN = 12

∴ x = 12.

Final Answer:

x = 12.

### Find the value of x. Identify the theorem used to find the answer.

- Hint:
- Mid-point theorem:
- According to mid-point theorem, in triangle, the line segment which joins the midpoint of two sides is parallel to third side and is equal to half of third side,

- Step by step explanation:
- Given:

BM = MC,

hence M is midpoint of BC.

BN = NA,

hence N is midpoint of AB.

MN = x

AC = 24

- Step 1:

The line segment MN joins midpoint of AB and BC

So,

According to midpoint theorem,

MN is parallel to AC and

MN =

MN =

MN = 12

∴ x = 12.

Final Answer:

x = 12.

### identify a cubic polynomial.

- We have to identify the cubic polynomial from the given four options.

We know that a cubic polynomial is a polynomial with degree 3.

Option A:

x + 3

The degree of this polynomial 1.

Option B:

The degree of the polynomial is 4.

Option C:

The degree of this polynomial is 3.

Option D:

The degree of this polynomial 2.

### identify a cubic polynomial.

- We have to identify the cubic polynomial from the given four options.

We know that a cubic polynomial is a polynomial with degree 3.

Option A:

x + 3

The degree of this polynomial 1.

Option B:

The degree of the polynomial is 4.

Option C:

The degree of this polynomial is 3.

Option D:

The degree of this polynomial 2.

### Find the value of 𝑚 & 𝑛 to make a true statement.

(𝑚𝑥 + 𝑛𝑦)^{2} = 4𝑥^{2} + 12𝑥𝑦 + 9𝑦^{2}

^{2}can be written as (mx + ny)(mx + ny)

(mx + ny)(mx + ny) = mx(mx + ny) + ny(mx + ny)

= mx(mx) + mx(ny) + ny(mx) + ny(ny)

= m

^{2}x

^{2}+ mnxy + mnxy + n

^{2}y

^{2}

= m

^{2}x

^{2}+ 2mnxy + n

^{2}y

^{2}

Now, m

^{2}x

^{2}+ 2mnxy + n

^{2}y

^{2}= 4𝑥

^{2}+ 12𝑥𝑦 + 9𝑦

^{2}

Comparing both sides, we get

m

^{2}= 4, n = 9, 2mn = 12

So, m = +2 or -2 , n = +3 or -3

Considering 2mn = 12, there are two combinations possible

- m = +2 and n = +2
- m = -2 and n = -2

Hence, the values of (m, n) are (2,2) and (-2,-2).

### Find the value of 𝑚 & 𝑛 to make a true statement.

(𝑚𝑥 + 𝑛𝑦)^{2} = 4𝑥^{2} + 12𝑥𝑦 + 9𝑦^{2}

^{2}can be written as (mx + ny)(mx + ny)

(mx + ny)(mx + ny) = mx(mx + ny) + ny(mx + ny)

= mx(mx) + mx(ny) + ny(mx) + ny(ny)

= m

^{2}x

^{2}+ mnxy + mnxy + n

^{2}y

^{2}

= m

^{2}x

^{2}+ 2mnxy + n

^{2}y

^{2}

Now, m

^{2}x

^{2}+ 2mnxy + n

^{2}y

^{2}= 4𝑥

^{2}+ 12𝑥𝑦 + 9𝑦

^{2}

Comparing both sides, we get

m

^{2}= 4, n = 9, 2mn = 12

So, m = +2 or -2 , n = +3 or -3

Considering 2mn = 12, there are two combinations possible

- m = +2 and n = +2
- m = -2 and n = -2

Hence, the values of (m, n) are (2,2) and (-2,-2).

### 37. In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

i)Which equation models the scoring in the final round and the outcome of the contest

- Hint:

○ Take the variable value as x or any alphabet.

- Step by step explanation:

correct answer = 12 points

incorrect answers = -5 points

School A starts with 165 points and gives the same number of correct and incorrect answers.

School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A

○ Step 1:

○ Let the number of correct answers given by school A be x.

So, the number of incorrect answers is also x.

At school A starts with 165 points. After giving x correct and a incorrect answers points will be

165 + 12x - 5x

School B starts with 65 points. Schools are given the same number of correct answers as school A and no incorrect answers. So, their points will be

65 + 12x

○ Step 2:

○ As both schools tied

∴ 165 + 12x - 5x = 65 + 12x

- Final Answer:

Option B. 165 + 12x - 5x = 65 + 12x

### 37. In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

i)Which equation models the scoring in the final round and the outcome of the contest

- Hint:

○ Take the variable value as x or any alphabet.

- Step by step explanation:

correct answer = 12 points

incorrect answers = -5 points

School A starts with 165 points and gives the same number of correct and incorrect answers.

School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A

○ Step 1:

○ Let the number of correct answers given by school A be x.

So, the number of incorrect answers is also x.

At school A starts with 165 points. After giving x correct and a incorrect answers points will be

165 + 12x - 5x

School B starts with 65 points. Schools are given the same number of correct answers as school A and no incorrect answers. So, their points will be

65 + 12x

○ Step 2:

○ As both schools tied

∴ 165 + 12x - 5x = 65 + 12x

- Final Answer:

Option B. 165 + 12x - 5x = 65 + 12x

### Find the value of x. Identify the theorem used to find the answer.

- Hints:
- Perpendicular bisector theorem
- According to perpendicular bisector theorem, in the triangle, any point on perpendicular bisector is at equal distance from both end points of the line segment on which it is drawn.

- Step by step explanation:
- Given:

AC = 4x – 4

AD is perpendicular bisector at BC.

- Step 1:
- In

A is point on AD

A is equidistant from B and C.

So,

AB = AC

2x = 4x – 4

4 = 4x – 2x

4 = 2x

x = 2

- Final Answer:

Perpendicular bisector theorem is used.

### Find the value of x. Identify the theorem used to find the answer.

- Hints:
- Perpendicular bisector theorem
- According to perpendicular bisector theorem, in the triangle, any point on perpendicular bisector is at equal distance from both end points of the line segment on which it is drawn.

- Step by step explanation:
- Given:

AC = 4x – 4

AD is perpendicular bisector at BC.

- Step 1:
- In

A is point on AD

A is equidistant from B and C.

So,

AB = AC

2x = 4x – 4

4 = 4x – 2x

4 = 2x

x = 2

- Final Answer:

Perpendicular bisector theorem is used.

### Where is the circumcentre located in any right triangle? Write a coordinate proof of this result.

- Hints:

- Distance between two points having coordinates (x
_{1}, y_{1}) and (x_{2}, y_{2}) is given by formula:

- Distance =

- Step by step explanation:
- Step 1:
- Let triangle ABO,

O = (0, 0)

A = (2a, 0)

B = (0, 2b).

- Step 1:
- Let triangle ABO, where:

The midpoint of BC is given by,

So, the perpendicular bisector will intersect BC at M (a, b).

Equation of line BC is

And point M (a, b) satisfy above equation

b = b

Hence, point M (a, b) lies on BC.

- Final Answer:

### Where is the circumcentre located in any right triangle? Write a coordinate proof of this result.

- Hints:

- Distance between two points having coordinates (x
_{1}, y_{1}) and (x_{2}, y_{2}) is given by formula:

- Distance =

- Step by step explanation:
- Step 1:
- Let triangle ABO,

O = (0, 0)

A = (2a, 0)

B = (0, 2b).

- Step 1:
- Let triangle ABO, where:

The midpoint of BC is given by,

So, the perpendicular bisector will intersect BC at M (a, b).

Equation of line BC is

And point M (a, b) satisfy above equation

b = b

Hence, point M (a, b) lies on BC.

- Final Answer:

### Ayush is choosing between two health clubs. Health club 1: Membership R s 22 and

Monthly fee R s 24.50. Health club 2: Membership R s 47.00 , monthly fee R s 18.25

. After how many months will the total cost for each health club be the same ?

- Hint:

- Step by step explanation:

Health club 1: Membership R s 22 and Monthly fee R s 24.50

Health club 2: Membership R s 47.00 , monthly fee R s 18.25.

○ Step 1:

○ Let the number of months after which the total cost is equal be x.

So, for Health club 1:

After x months, total cost will be =

Membership Rs 22 + fee for x months

22 + 24.50x

Health club 2:

After x months, total cost will be =

Membership R s 47 + fee for x months

47 + 18.25x

○ Step 2:

○ Equalize both costs to get number of months

22 + 24.50x = 47 + 18.25x

24.50x - 18.25x= 47 - 22

6.25x = 25

x =

x = 4

- Final Answer:

### Ayush is choosing between two health clubs. Health club 1: Membership R s 22 and

Monthly fee R s 24.50. Health club 2: Membership R s 47.00 , monthly fee R s 18.25

. After how many months will the total cost for each health club be the same ?

- Hint:

- Step by step explanation:

Health club 1: Membership R s 22 and Monthly fee R s 24.50

Health club 2: Membership R s 47.00 , monthly fee R s 18.25.

○ Step 1:

○ Let the number of months after which the total cost is equal be x.

So, for Health club 1:

After x months, total cost will be =

Membership Rs 22 + fee for x months

22 + 24.50x

Health club 2:

After x months, total cost will be =

Membership R s 47 + fee for x months

47 + 18.25x

○ Step 2:

○ Equalize both costs to get number of months

22 + 24.50x = 47 + 18.25x

24.50x - 18.25x= 47 - 22

6.25x = 25

x =

x = 4

- Final Answer:

### Find the gradient and y- Intercept of the line

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

x + 2y = 14

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c

Rewriting the given equation, we have

2y = 14 - x

Dividing by 2, we get

Simplifying, we get

Comparing the above equation with , we get

Thus, we get

Gradient =

y-intercept = 7

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula.

### Find the gradient and y- Intercept of the line

Gradient is also called the slope of the line. The slope intercept form of the equation of the line is y = mx + c, where m is the slope of the line and c is the y-intercept. First we convert the given equation in this form. Further, compare the equation with the standard form to get the slope and the y-intercept.

Step by step solution:

The given equation of the line is

x + 2y = 14

We need to convert this equation in the slope-intercept form of the line, which is

y = mx + c

Rewriting the given equation, we have

2y = 14 - x

Dividing by 2, we get

Simplifying, we get

Comparing the above equation with , we get

Thus, we get

Gradient =

y-intercept = 7

Note:

We can find the slope and y-intercept directly from the general form of the equation too; slope = and y-intercept = , where the general form of equation of a line is ax + by + c = 0. Using this method, be careful to check that the equation is in general form before applying the formula.

### We can express any constant in the variable form without changing its value as

- We have given a constant
- We have to find how we can express any constant 𝑘 in the variable form without changing its value.

We have given a constant

*k*

We have to find how we can express any constant 𝑘 in the variable form without changing its value

We know that x

^{0}= 1

So After multiplying it with any constant, it will not change its value.

So,kx

^{0}is the answer

Hence, Option C is correct.

### We can express any constant in the variable form without changing its value as

- We have given a constant
- We have to find how we can express any constant 𝑘 in the variable form without changing its value.

We have given a constant

*k*

We have to find how we can express any constant 𝑘 in the variable form without changing its value

We know that x

^{0}= 1

So After multiplying it with any constant, it will not change its value.

So,kx

^{0}is the answer

Hence, Option C is correct.