NTS GAT-General Quantitative Reasoning Syllabus

What is the formula for the average (arithmetic mean) of a set of numbers?

Average = (sum of all observations) / (number of observations).

State the relationship between speed, distance, and time.

Speed = Distance / Time; Distance = Speed × Time; Time = Distance / Speed.

What is the formula for average speed over a whole journey?

Average speed = Total distance travelled / Total time taken (not the simple average of the speeds).

How do you convert a speed from km/h to m/s and vice versa?

To convert km/h to m/s multiply by 5/18; to convert m/s to km/h multiply by 18/5.

In time and work problems, if a person completes a job in n days, what fraction is done in one day?

They complete 1/n of the work in one day; this one-day work rate is the basis for combining workers.

If A can do a job in x days and B in y days, how long do they take working together?

Together they take xy/(x+y) days, because their combined one-day work is 1/x + 1/y.

What is a monomial, binomial, and trinomial?

A monomial has one term, a binomial has two terms, and a trinomial has three terms in an algebraic expression.

State the identity for (a + b)² and (a − b)².

(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².

State the identity for a² − b² and for (a + b)(a − b).

a² − b² = (a + b)(a − b); equivalently (a + b)(a − b) = a² − b².

State the identity for (a + b)³ and a³ + b³.

(a + b)³ = a³ + 3a²b + 3ab² + b³; a³ + b³ = (a + b)(a² − ab + b²).

What is the standard form and solution of a linear equation in one variable, ax + b = 0?

A linear equation in one variable has the form ax + b = 0 (a ≠ 0), with solution x = −b/a.

What does it mean for a system of two linear equations to have no solution or infinitely many solutions?

For a₁x+b₁y=c₁ and a₂x+b₂y=c₂: no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines); infinitely many if a₁/a₂ = b₁/b₂ = c₁/c₂ (same line); unique solution if a₁/a₂ ≠ b₁/b₂.

What is the standard form of a quadratic equation and the quadratic formula?

Standard form: ax² + bx + c = 0 (a ≠ 0). Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).

What does the discriminant of a quadratic equation tell you?

For ax²+bx+c=0, discriminant D = b² − 4ac. If D > 0 there are two distinct real roots; if D = 0 two equal real roots; if D < 0 no real roots (complex roots).

State the sum and product of the roots of ax² + bx + c = 0.

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

State the laws of exponents for a^m × a^n and (a^m)^n.

a^m × a^n = a^(m+n); (a^m)^n = a^(m·n).

What do a^0 and a^(−n) equal?

a^0 = 1 (for a ≠ 0); a^(−n) = 1/a^n.

How is a fractional exponent a^(m/n) expressed as a radical?

a^(m/n) = the n-th root of a^m = (ⁿ√a)^m = ⁿ√(a^m).

What is the formula for the nth term of an arithmetic progression (AP)?

aₙ = a + (n − 1)d, where a is the first term, d is the common difference, and n is the term number.

What is the formula for the sum of the first n terms of an AP?

Sₙ = n/2 × [2a + (n − 1)d], or equivalently Sₙ = n/2 × (first term + last term).

What is the formula for the nth term of a geometric progression (GP)?

aₙ = a·r^(n−1), where a is the first term and r is the common ratio.

What is the formula for the sum of the first n terms of a GP?

Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; the sum to infinity (when |r| < 1) is S∞ = a/(1 − r).

What is a function, and what are its domain and range?

A function is a relation that assigns each input exactly one output. The domain is the set of all valid inputs (x-values); the range is the set of all resulting outputs (y-values).

In the line y = mx + c, what do m and c represent?

m is the slope (gradient) of the line and c is the y-intercept (the y-value where the line crosses the y-axis).

What is the formula for the slope of a line through points (x₁, y₁) and (x₂, y₂)?

Slope m = (y₂ − y₁) / (x₂ − x₁).

What is the order of a matrix, and when can two matrices be multiplied?

The order is rows × columns (m × n). Two matrices can be multiplied only if the number of columns of the first equals the number of rows of the second.

How do you find the determinant of a 2×2 matrix [[a, b], [c, d]]?

Determinant = ad − bc.

What are complementary and supplementary angles?

Complementary angles sum to 90°; supplementary angles sum to 180°.

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

The three interior angles of a triangle sum to 180°. The exterior angle theorem states an exterior angle equals the sum of the two non-adjacent (remote) interior angles.

State the formula for the sum of interior angles of a polygon with n sides.

Sum of interior angles = (n − 2) × 180°.

What is the relationship between the inscribed angle and the central angle subtending the same arc of a circle?

The angle subtended at the centre is twice the angle subtended at any point on the remaining circumference by the same arc (inscribed angle theorem).