eamcet maths formulae with examples

eamcet maths formulae with examples

EAMCET (Engineering Agriculture and Medical Common Entrance Test) is a significant examination conducted in various states of India, especially Andhra Pradesh and Telangana. It primarily assesses candidates' proficiency in physics, chemistry, and mathematics. A strong grasp of mathematical formulae is crucial for securing a good score in the mathematics section of the exam. This article provides a comprehensive overview of essential EAMCET maths formulae, accompanied by relevant examples to facilitate better understanding and application.

Essential Mathematics Formulae for EAMCET

Understanding core formulae is vital for solving problems efficiently. Below, we categorize the key formulae into different topics commonly encountered in the EAMCET mathematics syllabus.

1. Number Systems and Polynomials

• Laws of Exponents:
• \(a^m \times a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\) (for \(a \neq 0\))
• \((a^m)^n = a^{mn}\)

• Quadratic Equations:
• Standard form: \(ax^2 + bx + c = 0\)
• Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
• Product of roots: \(\alpha \beta = \frac{c}{a}\)

Example:

Find the roots of \(2x^2 - 4x - 6 = 0\).

Solution:

Sum of roots = \(\frac{4}{2} = 2\)

Product of roots = \(\frac{-6}{2} = -3\)

Roots are \(\alpha, \beta\) satisfying:

\(\alpha + \beta = 2\) and \(\alpha \beta = -3\)

2. Arithmetic Progression (AP)

• \(a_n = a + (n-1)d\) — nth term
• Sum of first n terms: \(S_n = \frac{n}{2} [2a + (n-1)d]\)
• Common difference: \(d = a_{n} - a_{n-1}\)

Example:

Find the sum of the first 10 terms of the AP: 3, 7, 11, ...

Solution:

First term \(a = 3\)

Common difference \(d = 4\)

Sum \(S_{10} = \frac{10}{2} [2 \times 3 + (10-1) \times 4] = 5 [6 + 36] = 5 \times 42 = 210\)

3. Geometric Progression (GP)

• \(a_n = ar^{n-1}\) — nth term
• Sum of first n terms: \(S_n = a \frac{r^n - 1}{r - 1}\), \(r \neq 1\)
• Infinite sum (if \(|r| < 1\)): \(S_\infty = \frac{a}{1 - r}\)

Example:

Find the sum of the first 5 terms of GP: 2, 6, 18, ...

Solution:

\(a = 2\), \(r = 3\)

\(S_5 = 2 \frac{3^5 - 1}{3 - 1} = 2 \times \frac{243 - 1}{2} = 2 \times 121 = 242\)

4. Coordinate Geometry

• Distance between points \((x_1, y_1)\) and \((x_2, y_2)\):

\[

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

\]

• Midpoint:

\[

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

\]

• Slope of the line through two points:

\[

m = \frac{y_2 - y_1}{x_2 - x_1}

\]

Example:

Calculate the distance between points \((1, 2)\) and \((4, 6)\).

Solution:

\[

d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

\]

5. Trigonometry

• Basic identities:
• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

• Sum and difference formulas:
• \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
• \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)

• Double angle formulas:
• \(\sin 2A = 2 \sin A \cos A\)
• \(\cos 2A = \cos^2 A - \sin^2 A\)

Example:

Evaluate \(\sin 75^\circ\) using known angle formulas.

Solution:

\(\sin 75^\circ = \sin (45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

6. Mensuration

• Surface Area and Volume of Basic Solids:

Cube:

• Surface Area: \(6a^2\)
• Volume: \(a^3\)

Cuboid:

• Surface Area: \(2(lb + bh + hl)\)
• Volume: \(l \times b \times h\)

Cylinder:

• Surface Area: \(2\pi r(h + r)\)
• Volume: \(\pi r^2 h\)

Sphere:

• Surface Area: \(4 \pi r^2\)
• Volume: \(\frac{4}{3} \pi r^3\)

Cone:

• Surface Area: \(\pi r (l + r)\)
• Volume: \(\frac{1}{3} \pi r^2 h\)

Example:

Calculate the volume of a sphere with radius 7 cm.

Solution:

\[

V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times \frac{22}{7} \times 7^3 = \frac{4}{3} \times \frac{22}{7} \times 343 = \frac{4 \times 22 \times 343}{3 \times 7} = \frac{4 \times 22 \times 343}{21}

\]

Simplify:

\[

= \frac{4 \times 22 \times 343}{21} = \frac{4 \times 22 \times 343}{21}

\]

Calculate numerator: \(4 \times 22 = 88\),

then \(88 \times 343 = 88 \times 343 = 30,184\)

Divide by 21: \(30,184 / 21 \approx 1437.33 \text{ cm}^3\)

7. Sets and Probability

• Union of two sets:

\[

A \cup B = A + B - (A \cap B)

\]

• Probability of an event:

\[

P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}

\]

Example:

In a deck of 52 cards, what is the probability of drawing a king or a queen?

Solution:

Number of kings = 4, number of queens = 4,

Total favorable outcomes = 4 + 4 = 8

Probability = \(8/52 = 2/13\)

Additional Tips for Using Formulae Effectively

• Memorize key formulae and practice applying them in various contexts.
• Learn to derive lesser-known formulae quickly to save time during exams.
• Use diagrams wherever possible for clarity and to identify the relevant formula.
• Practice solving previous years' question papers to familiarize yourself with common

EAMCET Maths Formulae with Examples: A Comprehensive Guide for Aspirants

In the realm of engineering and medical entrance examinations, EAMCET (Engineering Agricultural and Medical Common Entrance Test) holds a pivotal position for students aspiring to secure admission in various colleges across Andhra Pradesh and Telangana. Among the subjects tested, Mathematics often presents a significant challenge due to its diverse range of concepts and the extensive formulae involved. Mastery over these formulae is not merely about rote memorization but about understanding their applications and nuances, enabling aspirants to solve problems efficiently and accurately. This article aims to serve as a detailed, analytical resource that consolidates essential EAMCET Maths formulae, supplemented with illustrative examples, to empower students in their preparation journey.

Understanding the Importance of Maths Formulae in EAMCET

Mathematics in EAMCET is designed to assess logical reasoning, problem-solving skills, and conceptual clarity. The formulae act as the foundational tools that simplify complex problems into manageable calculations. A solid grasp of these formulae enhances speed and accuracy, which are crucial given the exam's time constraints. Moreover, recognizing the underlying patterns and relationships facilitated by formulas boosts confidence and minimizes errors.

Core Areas Covered in EAMCET Maths Formulae

EAMCET Maths syllabus broadly encompasses several topics, each with specific formulae. These include:

• Algebra
• Trigonometry
• Coordinate Geometry
• Geometry (Triangles, Circles)
• Mensuration
• Probability and Statistics
• Sequences and Series

Each section has unique formulae, which we will explore comprehensively below.

Algebra: Essential Formulae and Examples

Quadratic Equations

• Standard form: ax² + bx + c = 0
• Sum of roots (α + β) = -b/a
• Product of roots (αβ) = c/a
• Discriminant (D) = b² - 4ac

Example:

Solve x² - 5x + 6 = 0

• Sum of roots = 5
• Product of roots = 6
• Roots: x = (5 ± √(25 - 24))/2 = (5 ± 1)/2
• Roots: x = 3, 2

Arithmetic Progression (AP)

• nth term: aₙ = a + (n - 1)d
• Sum of first n terms: Sₙ = n/2 [2a + (n - 1)d]

Example:

Find the 10th term of AP: 3, 7, 11, ...

• a = 3, d = 4
• a₁₀ = 3 + (10 - 1)×4 = 3 + 36 = 39

Geometric Progression (GP)

• nth term: aₙ = a × r^(n-1)
• Sum of first n terms: Sₙ = a( r^n - 1 ) / (r - 1), r ≠ 1

Example:

Find the sum of first 5 terms of GP: 2, 6, 18, ...

• a = 2, r = 3
• S₅ = 2(3^5 - 1)/(3 - 1) = 2(243 - 1)/2 = 2×242/2 = 242

Trigonometry: Key Formulae and Applications

Basic Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Sum and Difference Formulas

• sin(A ± B) = sinA cosB ± cosA sinB
• cos(A ± B) = cosA cosB ∓ sinA sinB
• tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

Example:

Calculate sin(75°) using sum formula:

• sin(75°) = sin(45° + 30°)
• = sin45° cos30° + cos45° sin30°
• = (√2/2) × (√3/2) + (√2/2) × (1/2)
• = (√6/4) + (√2/4) = (√6 + √2)/4

Double Angle and Half Angle Formulae

• sin2θ = 2 sinθ cosθ
• cos2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ
• tan2θ = 2 tanθ / (1 - tan²θ)

Example:

Find cos(2θ) given sinθ = 3/5 and θ in the first quadrant.

• cosθ = √(1 - sin²θ) = √(1 - 9/25) = √(16/25) = 4/5
• cos(2θ) = cos²θ - sin²θ = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25

Coordinate Geometry: Formulae and Applications

Distance Formula

• Distance between points (x₁, y₁) and (x₂, y₂):

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Example:

Distance between (1, 2) and (4, 6):

d = √[(4 - 1)² + (6 - 2)²] = √(3² + 4²) = √(9 + 16) = √25 = 5

Midpoint Formula

• Midpoint of segment joining (x₁, y₁) and (x₂, y₂):

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example:

Midpoint of (2, 3) and (4, 7):

M = ((2 + 4)/2, (3 + 7)/2) = (3, 5)

Slope Formula

• Slope of line through (x₁, y₁) and (x₂, y₂):

m = (y₂ - y₁) / (x₂ - x₁)

Example:

Slope between (1, 2) and (3, 8):

m = (8 - 2) / (3 - 1) = 6/2 = 3

Geometry: Circles, Triangles, and Their Formulae

Circle Formulae

• Radius (r) from area: r = √(A/π)
• Circumference: C = 2πr
• Area: A = πr²
• Length of chord at a distance d from center:

l = 2√(r² - d²)

Example:

Find the length of a chord 4 units from the center of a circle with radius 5.

l = 2√(25 - 16) = 2√9 = 2×3 = 6

Triangle Formulae

• Area (Heron’s formula):

A = √[s(s - a)(s - b)(s - c)]

where s = (a + b + c)/2 (semi-perimeter)

• Pythagoras theorem (for right-angled triangles):

c² = a² + b²

Example:

Sides of a triangle: 7, 24, 25

• Check if right-angled: 25² = 7² + 24²?
• 625 = 49 + 576 = 625 → Yes, right-angled at the corner with sides 7 and 24.

Mensuration: Volume and Surface Area Formulae

Cube

• Volume: V = a³
• Surface Area: SA = 6a²

Cuboid

• Volume: V = l × b × h
• Surface Area: SA = 2(lb + bh + hl)

Sphere

• Volume: V = (4/3)πr³
• Surface Area: SA = 4πr²

Cylinder

• Volume: V = πr²h
• Surface Area: SA = 2πr(h + r)

Cone

• Volume: V = (1/3)πr²h
• Surface Area: SA = πr(l + r), where l is the slant height

Example:

Calculate the volume of a sphere with radius 3 units.

V = (4/3)π(3)³ = (4

Question	Answer
What are the key Maths formulae to remember for EAMCET exams?	Some essential Maths formulae include the quadratic formula, distance formula, midpoint formula, area and volume formulas for various shapes, and trigonometric identities like sin²θ + cos²θ = 1. Memorizing these helps solve problems efficiently during the exam.
How is the quadratic formula derived and how can I use it with an example?	The quadratic formula x = (-b ± √(b² - 4ac)) / 2a is derived by completing the square for ax² + bx + c = 0. For example, for 2x² + 3x - 2 = 0, a=2, b=3, c=-2. Plugging in: x = (-3 ± √(9 - 42(-2))) / 4 = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4. So, x = (-3 ± 5)/4, giving x= (2)/4=0.5 or x= (-8)/4=-2.
What is the formula for the area of a triangle using Heron’s formula?	Heron’s formula states that the area of a triangle with sides a, b, and c is √[s(s - a)(s - b)(s - c)], where s = (a + b + c)/2 is the semi-perimeter. For example, if a=5, b=6, c=7, then s=9. Area=√[9(9-5)(9-6)(9-7)] = √[9432] = √[216] ≈ 14.7.
How do I apply the distance formula with an example?	The distance formula between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²]. For example, between (1, 2) and (4, 6): √[(4 - 1)² + (6 - 2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
What is the formula for the sum of an arithmetic series?	The sum of the first n terms of an arithmetic series is Sₙ = n/2 (2a + (n - 1)d), where a is the first term and d is the common difference. For example, sum of first 5 terms with a=2 and d=3: S₅= 5/2 [22 + (5-1)3]= 2.5 [4 + 12]= 2.516=40.
How do I use the sine rule with an example?	The sine rule states that a/sin A = b/sin B = c/sin C. For example, in a triangle with sides a=7, b=9, and angle A=30°, to find angle B: b/sin B = a/sin A → 9/ sin B = 7/ sin 30° → 9/ sin B = 7/0.5 → 9/ sin B = 14. Solving for sin B: sin B= 9/14 ≈ 0.643. Then, B ≈ sin⁻¹(0.643) ≈ 40°.
What is the volume formula for a cylinder, and can you give an example?	The volume of a cylinder is V = πr²h, where r is radius and h is height. For example, if r=3 cm and h=10 cm, V= π(3)²10= π910= 90π ≈ 282.74 cubic centimeters.
How can I remember the basic trigonometric identities for exams?	Key identities include sin²θ + cos²θ=1, 1 + tan²θ= sec²θ, and 1 + cot²θ= csc²θ. Remembering these helps simplify problems involving trigonometric functions. For example, if sin θ=3/5, then cos θ=4/5, using the Pythagorean identity.

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