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

Explanation:

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

Step 1 of 1:
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.

(mx + ny)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)
= m²x² + mnxy + mnxy + n²y²
= m²x² + 2mnxy + n²y²
Now, m²x² + 2mnxy + n²y² = 4𝑥² + 12𝑥𝑦 + 9𝑦²
Comparing both sides, we get
m² = 4, n = 9, 2mn = 12
So, m = +2 or -2 , n = +3 or -3
Considering 2mn = 12, there are two combinations possible

1. m = +2 and n = +2
2. m = -2 and n = -2

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

Answer:

• Step by step explanation:

○    Given:
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:

Correct option:
Option B. 165 + 12x - 5x = 65 + 12x

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:

AB = 2x.
AC = 4x – 4
AD is perpendicular bisector at BC.

• Step 1:
• In  

AD is perpendicular bisector.
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: 

             Hence, x = 2.
Perpendicular bisector theorem is used.

Answer:

• Hints:

• Distance between two points having coordinates (x₁, y₁) and (x₂, y₂) is given by formula:

• Distance =

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

where,
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: 

Hence, circumcentre of right angle triangle lie on midpoint of hypotenuse.

Answer:

• Hint:

○    The concept used in the question is of the quadratic equation.

• Step by step explanation:

○    Given:
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:

Hence, after 4 months the total cost of each health club will be equal.

Hint:
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.

Explanation:

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

Step 1 of 1:
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 x⁰ = 1
So After multiplying it with any constant, it will not change its value.
So,kx⁰ is the answer
Hence, Option C is correct.

Answer:

• To prove: 
• Perpendicular Bisector Theorem:

• Proof: 
• Statement:
• 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 1:

Let consider given figure,

Let arbitrary point C on perpendicular bisector.
In which,
CD is perpendicular bisector on AB.
Hence,
AD = DB
CD = CD (common)

So, according to SAS rule

Hence,
CA = CB
So, any point on perpendicular bisector is at equal distance from end points of line segment.
Hence proved.

Explanation:

• We have to define monomial by giving an example.

Step 1 of 1:
A monomial is an expression with only one term.
The examples are: 3x, 6y

Hint:
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, 1)
(c, d) = (2, -14)
The equation of a line passing through two points (a, b) and (c, d) is

Using the above points, we have

Simplifying the above equation, we have


Cross multiplying, we get
5(y + 14) = -15(x - 2)
Expanding the factors, we have
5y + 70 = -15x + 30
Taking all the terms in the left hand side, we have
15x + 5y + 70 - 30 = 0
Finally, the equation of the line is
15x + 5y + 40 = 0
Dividing the equation throughout by 5, we get
3x + y + 8 = 0
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 ax + 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.

Explanation:

• We have been given a polynomial expression in the question for which we have to find its degree.

Step 1 of 1:
We have given an expression
We know that degree is highest power of variable present in polynomial.
So,
The degree of is
Degree of  is
Degree of is
So, Total degree is

Hence, Option B is correct.

Answer:

• Hint:

○   To solve the equation, group terms with the same coefficient on one side and numbers on                      the other side.

• Step by step explanation:

○    Given:
10.25x + 680 = 12.5x + 480
○    Step 1:
○    Solve the equation.
○    Group the like terms
10.25x + 680 = 12.5x + 480
680 - 480 = 12.5x - 10.25x
200 = 2.25x
○    Step 2:
○    Divide both side with 2.25

88.88 = x

• Final Answer:

Hence, the x = 88.88.

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: 
• Step 1:

Let consider given figure,

Let arbitrary point C on perpendicular bisector.
In which,
CD is perpendicular bisector on AB.
Hence,
AD = DB
CD = CD (common)
ADC = BDC = 90^o
So, according to SAS rule
ACD  BCD
Hence,
CA = CB
So, triangle is always isosceles.
Hence proved.

• Final Answer: 

Correct option A always.