COMSATS Admission Test Quantitative Reasoning Syllabus

State the basic relationship between distance, speed, and time.

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

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

Multiply by 5/18 (since 1 km/h = 1000m/3600s = 5/18 m/s).

Two trains move toward each other at speeds u and v. What is their relative speed?

The sum, u + v (relative speed for opposite directions).

Two objects move in the same direction at speeds u and v. What is their relative speed?

The difference, |u - v| (relative speed for same direction).

A boat's speed in still water is b and stream speed is s. What are its downstream and upstream speeds?

Downstream = b + s; Upstream = b - s.

If A can do a job in n days, what fraction of the work does A complete in one day?

1/n of the work per day.

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

Together time = xy/(x + y) days (since combined rate = 1/x + 1/y).

What is the relationship between number of workers and time taken (for a fixed job)?

They are inversely proportional: more workers means less time (Men x Days = constant work).

A and B together finish a job in 6 days; A alone in 10 days. How long does B take alone?

15 days. B's rate = 1/6 - 1/10 = 1/15, so B takes 15 days.

What is the standard form of a linear equation in two variables?

ax + by + c = 0 (or ax + by = c), where a and b are not both zero.

What does the solution of a system of two linear equations represent graphically?

The point of intersection of the two straight lines.

When does a system of two linear equations have no solution (parallel lines)?

When a1/a2 = b1/b2 is not equal to c1/c2 (consistent ratios of coefficients but different constant ratio).

When does a system of two linear equations have infinitely many solutions?

When a1/a2 = b1/b2 = c1/c2 (the two lines coincide / are dependent).

What is the standard form of a quadratic equation?

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

State the quadratic formula for the roots of ax^2 + bx + c = 0.

x = (-b +/- sqrt(b^2 - 4ac)) / (2a).

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

D = b^2 - 4ac; it determines the nature of the roots (real/equal/complex).

For ax^2 + bx + c = 0, what do the discriminant values D>0, D=0, and D<0 indicate?

D>0: two distinct real roots; D=0: two equal real roots; D<0: two complex (no real) roots.

For 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.

Expand the algebraic identity (a + b)^2.

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

Expand the algebraic identity (a - b)^2.

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

Factorize the difference of two squares a^2 - b^2.

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

Give 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).

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

a^m x a^n = a^(m+n); (a^m)^n = a^(mn).

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

a^0 = 1; a^(-n) = 1/a^n.

Express the fractional exponent a^(m/n) in radical (surd) form.

a^(m/n) = the n-th root of a^m, i.e., (n-th root of a)^m.

How do you rationalize the denominator of a fraction like 1/(a + sqrt(b))?

Multiply numerator and denominator by the conjugate (a - sqrt(b)), giving (a - sqrt(b))/(a^2 - b).

What is the measure of angles in a linear pair, and what are complementary and supplementary angles?

A linear pair sums to 180 degrees; complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees.

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.

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

Triangle = 180 degrees; n-sided polygon = (n - 2) x 180 degrees.

State the Pythagorean theorem and the exterior angle theorem for triangles.

Pythagoras: in a right triangle, hypotenuse^2 = sum of squares of the other two sides; Exterior angle = sum of the two remote (opposite) interior angles.