By angle sum property we know that the sum of all the angles of a triangle is 180⁰

We know that all the sides of a triangle are equal.

i.e all the angles of an equilateral triangle are equal.

Let each angle be x.

x + x + x = 180⁰

3x = 180⁰

⇒x = 60⁰

Hence the answe

WAVY CURVE METHOD

inequalities

If p, q and r are real numbers, then 

• 

• 

• and 

• 

• 

•  and equality holds for a = 1

•  and equality holds for a = –1

Inequation Involving Exponential Expression:

• If k > 0, then kx > 0 for all real x.

• If k > 1, then kx > 1, when x > 0

• If 0 < k < 1, then kx < 1, when x > 0 and kx > 1, when x < 0.

How to solve inequalities by wavy curve method

In order to solve the inequalities of the form 

where n₁, n₂, ……. , n k , m1, m₂, ……. , mp are real numbers and a₁, a₂, ……. , a_k, b₁, b₂, ……., bp are any real number such that ai ≠ bj where i = 1, 2, 3, ….k and j = 1, 2, 3, ….p.

Method:

Step - 1→    First arrange all values of x at which either numerator or denominator is becomes zero, that means a1, a2,….., ak, b1, b2, ….bp in increasing order say c1, c2, c3,……. cp + k. Plot them on real line

Step -2 →     Value of x at which numerator becomes zero should be marked with dark circles.

Step - 3 →    All pints of discontinuities (x at which denominator becomes zero) should be marked on number line with empty circles. Check the value of ƒ(x) for any real number greater than the right most marked number on the number line.

Step - 4 →    From right to left draw a wavy curve (beginnings above the number line in case of value of ƒ(x) is positive in step–3 otherwise from below the number line), passing thoroughly all the marked points. So that when passes through a point (exponent whose corresponds factor is odd) intersects the number line, and when passing thoroughly a point (exponent whose corresponds factor is even) the curve doesn’t intersect the real line and remain on the same side of real line.

Step - 5 →    The appropriate intervals are chosen in accordance with the sign of inequality (the function ƒ(x) is positive wherever the curve is above the number line, it is negative if the curve is found below the number line). Their union represents the Detail Explanation  of inequality

Example :   Let 

Detail Explanation : 

Step - 1 →     make on real line all x at which numerator becomes zero with dark circles.

Step - 2 →     mark point of discontinuity (value of x at which denominator becomes zero) with empty circles

Step - 3 →     Check ƒ(x) for x > 7, ƒ(8) > 0

Exponents of factors of –1, 3, 4 is even, hence wave will not change the direction at these points.

 Hence 

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

Solution:

True

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = ‎π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, 0.102…

Every irrational number is a real number, however, every real numbers are not irrational numbers.

For more click on the given link:

<a href="https://mycbseguide.com/blog/ncert-solutions-class-9-maths-exercise-1-2/" ping="/url?sa=t&source=web&rct=j&url=https://mycbseguide.com/blog/ncert-solutions-class-9-maths-exercise-1-2/&ved=2ahUKEwjL18TnpPLsAhWswjgGHWopB_YQFjADegQIBBAC" rel="noopener" target="_blank">NCERT Solutions for Class 9 Maths Exercise 1.2 ...</a>

HRD Minister Ramesh Nishank announced a major CBSE syllabus reduction for the new academic year 2020-21 on July 7 which was soon followed by an official notification by CBSE on the same.

Considering the loss of classroom teaching time due to the Covid-19 pandemic and lockdown, CBSE reduced the syllabus of classes 9 to 12 with the help of suggestions from NCERT.

Click on the given links:

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The Range is the difference between the lowest and highest values. Example: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9. So the range is 9 − 3 = 6. It is that simple!

In the given sum

Highest value = 67

Lowest value = 17

Range = Highest - Lowest value  67 - 17

Therefore rANGE =  50

A n s w e r

Area of an Equilateral Triangle Formula

The formula for the area of an equilateral triangle is given as:

Where a = length of sides

Consider a polynomial f(x) which is divided by (x-c), then f(c)=0.

Using remainder theorem,

f(x)= (x-c)q(x)+f(c)

Where f(x) is the target polynomial and q(x) is the quotient polynomial.

Since, f(c) = 0, hence,

f(x)= (x-c)q(x)+f(c)

f(x) = (x-c)q(x)+0

f(x) = (x-c)q(x)

Therefore, (x-c) is a factor of the polynomial f(x).

Another Method

By <a href="https://byjus.com/maths/remainder-theorem/">remainder theorem</a>,

f(x)= (x-c)q(x)+f(c)

If (x-c) is a factor of f(x), then the remainder must be zero.

(x-c) exactly divides f(x)

Therefore, f(c)=0.

The following statements are equivalent for any polynomial f(x)

• The remainder is zero when f(x) is exactly divided by (x-c)
• (x-c) is a factor of f(x)
• c is the solution to f(x)
• c is a zero of the function f(x), or f(c) =0

A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?

Let the height of the cuboidal vessel be h m. l = 10 m b = 8 m
Capacity of the cuboidal vessel = 380 m³

                 lbh = 380
         (10)(8)h = 380

Hence, the cuboidal vessel must be made 4.75 m high.

(1,-3) = (x,y)

x= 1

y=-3

now we will put these values in equation;x - 2y =k

1 - 2 X (-3) = k    

1 + 6 = k ( because - and - will change in + )

and

then

k= 7.

Yes zero is a rational number.

We know that the integer 0 can be written in any one of the following forms.

For example, 0/1, 0/-1, 0/2, 0/-2, 0/3, 0/-3, 0/4, 0/-4 and so on …..
In other words, 0 = 0/b, where b is any non-zero integer

Thus, 0 can be written as, where a/b = 0, where a = 0 and b is any non-zero integer.

Hence, 0 is a rational number.

A n s w e r:
2) option is correct.

y = 2, means point is represted on y-axis.

If we draw a line through it then it will parallel to the x-axis.

Remainder Theorem Proof

Theorem functions on an actual case that a polynomial is comprehensively dividable, at least one time by its factor in order to get a smaller polynomial and ‘a’ remainder of zero. This acts as one of the simplest ways to determine whether the value ‘a’ is a root of the polynomial P(x).

That is when we divide p(x) by x-a we obtain

p(x) = (x-a)·q(x) + r(x),

as we know that Dividend = (Divisor × Quotient) + Remainder

But if r(x) is simply the constant r (remember when we divide by (x-a) the remainder is a constant)…. so we obtain the following solution, i.e

p(x) = (x-a)·q(x) + r

Observe what happens when we have x equal to a:

p(a) = (a-a)·q(a) + r

p(a) = (0)·q(a) + r

p(a) = r

Hence, proved.

The vertical line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

The horizontal line in the Cartesian plane which determines the position of a point is called THE X-AXIS

The vertical line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

The horizontal line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

we know ,

a³ + b³ + c³ -3abc = (a + b + c )(a² + b² + c² -ab -bc-ca)

now ,

a + b + c = 5
ab + bc + ca = 10

(a + b + c)² = a² + b² + c² +2(ab + bc+ca)
(5)² -2×10 = a² + b² + c²
a² + b² + c² =5

hence ,

a³ + b³ +c³ -3abc = ( a + b + c )(a² + b² + c² -ab- bc-ca)

=( 5)( 5 - 10) = 5 × (-5) = -25

hence proved