🚀 Basic Maths | Complete Chapter in ONE SHOT | MDCAT 2026 Physics | Manzil Series | PMDC MDCAT

Summary

The lecture is a foundational mathematics revision for physics. The instructor emphasizes that calculation skills—especially squaring, square roots, proportionality, and trigonometry—will make later physics topics easier. Students are encouraged to watch attentively, take notes, attempt worked questions before looking at solutions, and complete the assigned practice rather than relying on memorization alone.

1. Squaring Numbers

For a two-digit number, the instructor demonstrates a shortcut that combines the squares of the digits with twice their product.

Method for a number (10a+b):

1. Find (a^2) and (b^2).
2. Find (2ab).
3. Combine these using place value, carrying digits where needed.
4. Pay attention to the rightmost digit: write its square using two digits when necessary (for example, (3^2=09)).

Examples include (58^2=3364), (56^2=3136), and (74^2=5476). The same idea is extended to numbers such as 104 by treating the leading portion as a group.

For numbers ending in 5, the instructor gives a faster shortcut:

1. Drop the final 5.
2. Multiply the remaining number by the next integer.
3. Append 25.

For example, (25^2) is (2\times3) followed by 25, giving 625. Similarly, (45^2) is (4\times5) followed by 25, giving 2025.

The instructor recommends learning multiplication tables up to 20 and memorizing squares at least through 15. Students are assigned to calculate squares from 1 to 99, with further examples involving three-digit numbers.

2. Estimating Square Roots

Perfect squares, such as 81 and 100, have straightforward roots. For numbers between perfect squares, the instructor teaches an approximation method based on averaging.

Approximation method:

1. Let (y) be the number whose square root is required.
2. Choose (x), the root of a nearby perfect square below (y).
3. Calculate (z=y/x).
4. Estimate (\sqrt{y}) by averaging: ((x+z)/2).
5. For a closer estimate, repeat the process using the new estimate.

Examples shown include (\sqrt{42}\approx6.5), (\sqrt{90}\approx9.5), and (\sqrt{95}\approx9.77). The instructor describes these as approximations and recommends checking calculations with a calculator after practising the method manually.

3. Standard Form and Decimal Calculations

Standard form (scientific notation)

• Move the decimal point until exactly one non-zero digit remains to its left.
• Count how many places the decimal moved.
• If it moved left, the power of 10 is positive; if it moved right, the power is negative.
• Reverse the process to convert back to ordinary decimal notation.

The instructor also shows how decimals can be rewritten as fractions with powers of 10. For example, (4.2=42/10) and (0.302=302/1000). This is useful when squaring decimals.

Squaring decimals

1. Rewrite the decimal as a fraction over a power of 10.
2. Square the numerator and denominator separately.
3. Convert the result back to decimal form by placing the decimal point appropriately.

The assigned exercise is to calculate the squares of decimal numbers from 2.1 through 9.9.

Adding and subtracting decimals

• Align decimal points vertically.
• Add or subtract each place value.
• Keep the decimal point aligned in the result.
• In subtraction, borrow as needed and use the sign implied by the relative sizes of the numbers.

Multiplying decimals

• Multiply as if the numbers were whole numbers.
• Count the total number of decimal places in the factors.
• Place the decimal point that many places from the right in the product.

Dividing decimals

• Rewrite each decimal using a power-of-10 denominator.
• Convert the division into multiplication by the reciprocal where helpful.
• Simplify the resulting fractions and powers of 10.

4. Fractions and Powers of 10

Fractions

• To add or subtract fractions, first make their denominators the same, using an LCM or equivalent multiplication and division.
• For multiplication, multiply the numerators together and the denominators together.
• For division, multiply by the reciprocal of the second fraction.
• In nested division, rewrite the expression carefully before simplifying. Do not cancel terms unless they are factors in the same numerator and denominator.

Powers of 10

• When multiplying powers with the same base, add exponents: (10^a\times10^b=10^{a+b}).
• When dividing powers with the same base, subtract exponents: (10^a/10^b=10^{a-b}).
• For addition or subtraction, express the terms with matching powers of 10, then factor out the common power before combining coefficients.
• When a power is raised to another power, multiply the exponents.

5. Trigonometric Ratios and Standard Angles

The lecture introduces right-triangle terminology:

• The hypotenuse is opposite the right angle.
• The base and perpendicular depend on which acute angle is being used as the reference angle.

The basic ratios are:

• (\sin\theta=\text{perpendicular}/\text{hypotenuse})
• (\cos\theta=\text{base}/\text{hypotenuse})
• (\tan\theta=\text{perpendicular}/\text{base}=\sin\theta/\cos\theta)

The instructor also reviews Pythagoras’ theorem:

[ \text{hypotenuse}^2=\text{base}^2+\text{perpendicular}^2 ]

For the standard angles (0^\circ,30^\circ,45^\circ,60^\circ,90^\circ), the lecture builds the sine table from square roots divided by 2. The cosine values are the sine values in reverse order, and tangent values are found by dividing sine by cosine. Students are encouraged to practise constructing the table rather than only memorizing it. The lecture also explains how to use inverse trigonometric functions to work backward from a ratio to an angle.

6. Trigonometric Functions in Other Quadrants

Angles are located in one of four quadrants, and the sign of each trigonometric function depends on the quadrant. The mnemonic given is “All Silly Tom Cats”:

• Quadrant I: all trigonometric functions are positive.
• Quadrant II: sine is positive.
• Quadrant III: tangent is positive.
• Quadrant IV: cosine is positive.

To evaluate an angle outside the first quadrant:

1. Identify its quadrant.
2. Find its reference angle using the relevant relationship:
   • Quadrant II: (180^\circ-\theta)
   • Quadrant III: (\theta-180^\circ)
   • Quadrant IV: (360^\circ-\theta)
3. Evaluate the trigonometric function at the reference angle.
4. Apply the sign for the original quadrant.

Examples include (\cos120^\circ=-\frac12), (\sin150^\circ=\frac12), and (\tan240^\circ=\sqrt3).

Students are assigned to calculate sine, cosine, and tangent for a list of standard angles in the other quadrants without a calculator.

7. Proportionality and Linearity

The instructor distinguishes two ideas:

• Direct proportionality: when one variable increases, the other increases; when one decreases, the other decreases.
• Inverse proportionality: when one variable increases, the other decreases.

Linearity concerns how much one quantity changes relative to another:

• A relation with matching powers, such as (A\propto B), is linear.
• A relation such as (A\propto B^2) is non-linear: multiplying (B) by a factor changes (A) by the square of that factor.
• Inverse-square relations, such as (F\propto1/r^2), are inverse and non-linear.

The instructor connects these ideas to examples from physics, including Ohm’s law, Boyle’s law, Coulomb’s law, gravitation, and the motion equation (s=\frac12at^2). The main lesson is to identify both the direction of the relationship and the power governing the change.

8. Angle Measurement: Degrees and Radians

Degrees are commonly used, but radians are the SI unit for measuring angles.

Key conversions:

• (\pi) radians (=180^\circ)
• Degrees to radians: multiply by (\pi/180).
• Radians to degrees: multiply by (180/\pi).

Examples include converting (90^\circ) to (\pi/2) radians and (45^\circ) to (\pi/4) radians. Students are also asked to convert the assigned standard angles into radians.

9. Graphs: Slope, Area, and Straight-Line Equations

Slope

• Slope can be calculated as (m=\Delta y/\Delta x), or as (\tan\theta) when the angle to the positive (x)-axis is known.
• Given two points, (m=(y_2-y_1)/(x_2-x_1)).
• A line rising as (x) increases has positive slope; a line falling as (x) increases has negative slope.
• The angle for interpreting slope is measured from the positive (x)-axis.

Area under a curve

The area under a graph is the region between the graph and the positive (x)-axis over the interval considered. The calculation depends on the shape formed; for example, a triangular region uses the triangle-area formula. The instructor notes that not every graph region is a triangle.

Equation of a straight line

The form introduced is (y=mx+c):

• (m) is the slope or gradient.
• (c) is the (y)-intercept—the value where the line crosses the (y)-axis.
• The (y)-intercept can be positive or negative.

10. Percentages and Trigonometric Formulae

To express a number as a percentage, multiply it by 100 and attach the percent sign. To convert a percentage to a decimal, divide by 100.

The lecture concludes with angle-sum and angle-difference identities:

• (\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta)
• (\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta)
• (\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta)
• (\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta)

The instructor says these identities will be useful in later physics work, particularly alternating-current problems.

Speakers and Sources Featured

• Instructor/lecturer: Teaches the lesson and gives the examples, explanations, and homework. The video context identifies the channel as Physics Ka Manjan – Bilal Zia, though the subtitles do not clearly state the lecturer’s name.
• Muzaffar Bhai: Addressed by the instructor as managing or interrupting the chat.
• Students/viewers: Their questions and answers appear through the live chat and are addressed by the instructor; no individual student is clearly identified.