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Concept used:

The Sum of the angle of a triangle is 180° 

When a transversal intersects two parallel lines then alternate interior angles are equal 

Calculations:

ΔQSR 

∠SQR + ∠QRS + ∠RSQ = 180° 

⇒ 2x + 3x + 5x = 180°

⇒ 10x = 180° 

⇒ x = 18° 

⇒ ∠SQR = 2 × 18° = 36° 

⇒ ∠QSR = 3 × 18° = 54° 

⇒ ∠SRQ = 5 × 18° = 90° 

Now, PS II QR and SQ is transversal

⇒ ∠RSQ = ∠SQP = 54°  --(as PQ||SR)

⇒ ∠RQS = ∠QSP = 36°  --(as PS||QR)

In ΔPQS

∠PQS + ∠PSQ + ∠QPS = 180° 

⇒ 54° + 36° + ∠QPS = 180° 

⇒ ∠QPS = 180° - (54° + 36°)

⇒ ∠QPS = 180° - 90° 

∴ ∠QPS = 90°

Given:

A Pentagon 

Formula Used:

Sum of Interior angle of a polygon: (n - 2) × 180°, where n is the number of sides of a polygon. 

Calculations:

Number of Sides of Pentagon: 5

Sum of Interior angle of a Pentagon = (5 - 2)× 180° 

Sum of Interior angle of a Pentagon = 3 × 180° 

∴ Sum of Interior angle of a Pentagon = 540°

Given:

The radius of a circle is 6 cm and P is any point which is 10 cm away from the centre.

Concept used:

In a Right Angled Triangle 

Hypotenuse² = Perpendicular² + Base²

Calculations:

Hypotenuse = The distance of the point P to the Centre of the Circle = 10cm 

Base = Radius of the circle = 6cm

The length of the Tangent drawn from point P to the Circle = Perpendicular

⇒ Hypotenuse2 = Perpendicular2 + Base2

⇒ 10² = 6² + Base²

⇒ 100 - 36 = Base²

⇒ 64 = Base²

⇒ Base = 8cm

∴ The length of the tangent drawn from P to the circle will be 8cm

Given:

In a right-angled triangle, if its sides are 15 cm and 8 cm perpendicular to each other

Concept used:

In a Right angled triangle

(Hypotenuse)² = (Perpendicular)² + (Base)²

The perimeter of the Triangle: Sum of all the sides of the triangle.

Calculations:

In a right-angled triangle, if its sides are 15 cm and 8 cm perpendicular to each other

⇒ Perpendicular is 15cm and Base is 8cm

⇒ (Hypotenuese)² = (Perpendicular)² + (Base)²

⇒ (Hypotenuese)2 = (15)2 + (8)2

⇒ (Hypotenuese)2 = 225 + 64

⇒ (Hypotenuese)2 = 289 

⇒ Hypotenuse = 17 cm 

⇒ The sides of the Right Angled Triangle = 17, 15, 8

⇒ Perimeter = 17 + 15 + 8 

⇒ Perimeter of the Triangle = 40cm 

∴ The Perimeter of the Right Angled Triangle = 40 cm

Formula Used:

Mean = Sum of all observations/ Number of observation

Sum of 'n' natural number = \(\frac{n(n + 1)}{2}\)

Calculation:

The number of observation = n

⇒ Mean = \(\frac{n(n + 1)}{2n}\)

⇒ Mean = \(\frac{(n+1)}{2}\)

∴ The mean is \(\frac{(n + 1)}{2}\)

Calculation:

Class size = Upper limit - Lower limit

∴ The difference between the upper limit and the lower limit of a class is called class size.

Additional Information

Frequency: The number of times something happens within a particular period, or the fact of something happening often or a large number of times is called frequency.

Mode: The number or value that appears the most often in a particular set of numbers or values is called mode.

Class size: The difference between the upper limit and the lower limit of a class is called class size.

Calculation:

The coordinates of the origin are (0,0)

Additional Information 

The coordinates of a point on a map or graph are the two sets of numbers or letters that you need in order to find that point.

X and Y coordinates are addresses that help us to locate a point in two-dimensional space having two axes. Any point in the coordinate plane (position plane) is referred to by a point (x, y), where the x value is the position of the point with reference to the x-axis, and the y-axis value is the position of the point with reference to the y.

Formula Used:

Probability = \(\frac{Number\space of\space favourable\space outcome}{Total\space outcomes}\)

Calculation:

Total outcomes (1, 2, 3, 4, 5, 6) = 6

Number of favorable outcomes = 0 

Dice have number 6 and less than 6

⇒ Probability = 0/6 = 0

∴ The probability of getting a number greater than 6 is 0.

Mathematics is a subject that finds application in every walk of our life. Knowingly or unknowingly, people use concepts of Mathematics in their daily life.

Key Points The Nature of Mathematics is - 

• Mathematics is the study of numbers, places, measurements, etc.
• Mathematics has its own language terms, concepts, formulas, etc.
• The knowledge of Mathematics is the same in the whole world.
• The knowledge of Mathematics is logical.
• Mathematical ideas grow from concrete to abstract.
• Mathematics follows the pattern of particular to the general.
• Mathematical knowledge is conceptual as well procedural.
• Learning Mathematics is concerned with both comprehending the facts, theories, rules, and laws, and with the aim of learning mathematics which is much broader.

​Hence, it is concluded that the knowledge of Mathematics is not logical is not the nature of mathematics.

Additional Information

• Mathematics is a diverse discipline that deals with data analysis, integration of various fields of knowledge, involves proofs, deductive and inductive reasoning, and generalizations to give an explanation of natural phenomena and human behavior.
• Mathematics also helps to understand the world around us by exploring the hidden patterns in a systematic and organized manner and it has universal applicability.
• Mathematics is considered a formally organized subject of study, and hence, it is treated as a discipline.
• Mathematics deal with various branches such as arithmetic, algebra, geometry, trigonometry, etc. The branches of Mathematics find application in our daily life.

Mathematics is a subject that finds application in every walk of our life. Knowingly or unknowingly, people use concepts of Mathematics in their daily life.

 Key Points

• "Mathematics is the language in which God has written the universe. It is quoted by Galileo.
• He was an Italian professor of mathematics.
• He believed that we can not understand the universe until and unless we become familiar with the characters in which it is written.
• He believed that the universe is written in mathematical language and the alphabets are circle, triangle, rectangle, and geometrical figures.

Thus, it is concluded that “Mathematics is the language in which God has written the Universe.” This statement is told by Galileo.