FPSC CSS Screening Test (MPT) General Abilities Syllabus

How does the average change if every number in a set is increased by the same amount k?

The average also increases by k. Adding (or subtracting) a constant to every value shifts the mean by that same constant.

State the fundamental Time and Work formula relating work, rate, and time.

Work = Rate x Time. If a person completes a job in n days, their rate (work per day) is 1/n.

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

Combined time = (x*y)/(x+y) days, because their combined rate is 1/x + 1/y per day.

State the relationship between speed, distance, and time.

Speed = Distance / Time; Distance = Speed x Time; Time = Distance / Speed.

How do you convert km/h to m/s and m/s to km/h?

km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.

What is the average speed for a journey covering equal distances at speeds u and v?

Average speed = 2uv / (u + v), the harmonic mean. It is NOT the simple average (u+v)/2 when distances are equal.

How do relative speeds combine for objects moving toward each other versus in the same direction?

Opposite directions (toward each other or apart): relative speed = sum of speeds. Same direction: relative speed = difference of speeds.

What is the general form of a linear equation in one variable and how many solutions does it have?

General form: ax + b = 0 (a not equal to 0). It has exactly one solution: x = -b/a.

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

Standard form: ax^2 + bx + c = 0 (a not equal to 0). Quadratic formula: x = [-b plus or minus sqrt(b^2 - 4ac)] / (2a).

What is the discriminant of a quadratic equation and what does it indicate?

Discriminant D = b^2 - 4ac. If D > 0: two distinct real roots; D = 0: one repeated real root; D < 0: no real roots (two complex roots).

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

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

State the expansion of (a + b)^2, (a - b)^2, and (a + b)(a - b).

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

State the factorization formulas 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 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 is the difference between a monomial, binomial, and trinomial?

A monomial has one term (5x), a binomial has two terms (x + 3), and a trinomial has three terms (x^2 + 2x + 1).

State the product rule and quotient rule for exponents with the same base.

Product rule: a^m x a^n = a^(m+n). Quotient rule: a^m / a^n = a^(m-n).

State the power rule for exponents: (a^m)^n.

(a^m)^n = a^(mn). Multiply the exponents when raising a power to another power.

What do a^0 and a^(-n) equal (for a not equal to 0)?

a^0 = 1 for any nonzero a. a^(-n) = 1/a^n (a negative exponent means reciprocal).

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

a^(m/n) = nth root of (a^m) = (nth root of a)^m. The denominator is the root and the numerator is the power.

What is an arithmetic progression (AP) and its nth term formula?

An AP has a constant common difference d between consecutive terms. The nth term is a_n = a + (n - 1)d, where a is the first term.

What is the sum of the first n terms of an arithmetic progression?

S_n = n/2 [2a + (n - 1)d] or equivalently S_n = n/2 (first term + last term).

What is a geometric progression (GP) and its nth term formula?

A GP has a constant common ratio r between consecutive terms. The nth term is a_n = a r^(n-1), where a is the first term.

What is the sum of the first n terms of a geometric progression (r not equal to 1)?

S_n = a(r^n - 1)/(r - 1) when r > 1, or a(1 - r^n)/(1 - r) when r < 1.

What is the sum of the first n natural numbers, and the sum of their squares?

Sum of first n natural numbers = n(n+1)/2. Sum of their squares = n(n+1)(2n+1)/6.

What are complementary and supplementary angles?

Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees.

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

Vertically opposite angles are formed when two lines intersect, lying opposite each other at the intersection. They are always equal.

When a transversal crosses two parallel lines, what is the relationship between alternate interior angles and corresponding angles?

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

What is the angle sum property of a triangle, and the sum of interior angles of a polygon with n sides?

The interior angles of a triangle sum to 180 degrees. For an n-sided polygon, the interior angles sum to (n - 2) x 180 degrees.

Classify triangles by their sides: scalene, isosceles, and equilateral.

Scalene: all three sides (and angles) different. Isosceles: two equal sides and two equal base angles. Equilateral: all three sides equal and each angle is 60 degrees.

State the Pythagorean theorem and the triangle inequality.

Pythagorean theorem: in a right triangle, hypotenuse^2 = base^2 + height^2. Triangle inequality: the sum of any two sides must be greater than the third side.