PMA Long Course Mathematics Syllabus

What is the LCM (Least Common Multiple) of two or more numbers?

The LCM is the smallest number that is exactly divisible by each of the given numbers.

What is the relationship between HCF, LCM, and the product of two numbers?

For any two numbers, HCF x LCM = Product of the two numbers.

How do you find the HCF of two numbers using prime factorization?

Express each number as a product of prime factors, then multiply the common prime factors taken with their lowest powers.

How do you find the LCM of numbers using prime factorization?

Express each number as a product of prime factors, then multiply all distinct prime factors taken with their highest powers.

What is HCF of fractions and LCM of fractions?

HCF of fractions = HCF of numerators / LCM of denominators. LCM of fractions = LCM of numerators / HCF of denominators.

What is the difference between a factor and a multiple of a number?

A factor of N divides N exactly (e.g., 3 is a factor of 12). A multiple of N is obtained by multiplying N by an integer (e.g., 12 is a multiple of 3).

What is a prime number and what is a composite number?

A prime number has exactly two distinct factors: 1 and itself (e.g., 2, 3, 5, 7). A composite number has more than two factors (e.g., 4, 6, 9). Note: 1 is neither prime nor composite.

What is the divisibility rule for 3 and for 9?

A number is divisible by 3 if the sum of its digits is divisible by 3. It is divisible by 9 if the sum of its digits is divisible by 9.

What is the divisibility rule for 4 and for 8?

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if its last three digits form a number divisible by 8.

What is the divisibility rule for 11?

A number is divisible by 11 if the difference between the sum of digits at odd positions and the sum of digits at even positions is 0 or a multiple of 11.

What are proper, improper, and mixed fractions?

A proper fraction has numerator < denominator (e.g., 3/5). An improper fraction has numerator >= denominator (e.g., 7/4). A mixed fraction combines a whole number and a proper fraction (e.g., 1 3/4).

How do you add or subtract two fractions with different denominators?

Convert both to equivalent fractions with a common denominator (the LCM of the denominators), then add or subtract the numerators while keeping the common denominator.

How do you divide one fraction by another?

Multiply the first fraction by the reciprocal (inverse) of the second: (a/b) / (c/d) = (a/b) x (d/c).

How do you convert a fraction into a decimal, and a terminating decimal back into a fraction?

Divide the numerator by the denominator to get the decimal. To convert a terminating decimal to a fraction, write the digits over the appropriate power of 10 and simplify (e.g., 0.25 = 25/100 = 1/4).

What is a linear equation in one variable, and what is its general form?

A linear equation in one variable has the highest power of the variable equal to 1. General form: ax + b = 0, where a is not 0; its solution is x = -b/a.

What is the general form of a linear equation in two variables, and how many solutions does it have?

General form: ax + by + c = 0. A single such equation has infinitely many solutions, represented graphically by a straight line.

What are the two common algebraic methods to solve a pair of simultaneous linear equations?

The substitution method (solve one equation for one variable and substitute into the other) and the elimination method (add/subtract equations to eliminate one variable).

What is the standard form of a quadratic equation?

ax^2 + bx + c = 0, where a, b, c are constants and a is not equal to 0.

What is the quadratic formula for solving ax^2 + bx + c = 0?

x = [-b plus or minus sqrt(b^2 - 4ac)] / (2a).

What is the discriminant of a quadratic equation, and what does it tell us?

The discriminant D = b^2 - 4ac. If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are imaginary (no real roots).

For the roots of ax^2 + bx + c = 0, what are the sum and product of the roots?

Sum of roots = -b/a; Product of roots = c/a.

What are the identities for (a + b)^2 and (a - b)^2?

(a + b)^2 = a^2 + 2ab + b^2. (a - b)^2 = a^2 - 2ab + b^2.

What is the factorization identity for a^2 - b^2 (difference of two squares)?

a^2 - b^2 = (a + b)(a - b).

What is the expansion of (a + b)^3 and (a - b)^3?

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.

What are the factorization identities for a^3 + b^3 and a^3 - b^3?

a^3 + b^3 = (a + b)(a^2 - ab + b^2). a^3 - b^3 = (a - b)(a^2 + ab + b^2).

What is the identity for (x + a)(x + b)?

(x + a)(x + b) = x^2 + (a + b)x + ab.

What are complementary angles and supplementary angles?

Two angles are complementary if their sum is 90 degrees. Two angles are supplementary if their sum is 180 degrees.

What are vertically opposite angles, and what is their property?

When two lines intersect, the angles opposite each other are vertically opposite angles. They are always equal.

When a transversal cuts two parallel lines, what is true of alternate interior angles and corresponding angles?

Alternate interior angles are equal, and corresponding angles are equal. Co-interior (allied) angles are supplementary (sum to 180 degrees).

What is the angle sum property of a triangle, and what is the exterior angle theorem?

The sum of the three interior angles of a triangle is 180 degrees. The exterior angle theorem states that an exterior angle equals the sum of the two opposite (non-adjacent) interior angles.

Classify triangles by their sides, and state the Pythagoras theorem for a right-angled triangle.

By sides: equilateral (all sides equal), isosceles (two sides equal), scalene (all sides different). Pythagoras theorem: in a right triangle, (hypotenuse)^2 = (base)^2 + (perpendicular)^2.