IPMAT Quantitative Ability Syllabus

For 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 happens to an inequality when you multiply or divide both sides by a negative number?

The direction of the inequality sign reverses (e.g. < becomes >).

How do you represent the solution set of an inequality such as x > 3 on a number line?

Draw an open circle at 3 (value not included) and shade the line to the right toward larger values; use a closed/filled circle if the inequality is >= (value included).

What is a polynomial, and what is its degree?

A polynomial is an expression of the form a_n x^n + ... + a_1 x + a_0 with non-negative integer exponents and real coefficients. Its degree is the highest power of the variable with a non-zero coefficient.

Name polynomials of degree 0, 1, 2, and 3.

Degree 0 = constant; degree 1 = linear; degree 2 = quadratic; degree 3 = cubic.

State the Remainder Theorem.

When a polynomial p(x) is divided by (x - a), the remainder equals p(a).

State the Factor Theorem.

(x - a) is a factor of polynomial p(x) if and only if p(a) = 0.

What are complementary, supplementary, and vertically opposite angles?

Complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees; vertically opposite angles are formed by two intersecting lines and are equal.

When a transversal cuts two parallel lines, which angle pairs are equal and which are supplementary?

Corresponding angles and alternate (interior/exterior) angles are equal; co-interior (allied) angles on the same side of the transversal are supplementary (sum 180 degrees).

What is the sum of the interior angles of a triangle, and the exterior angle property?

The interior angles of a triangle sum to 180 degrees. An exterior angle equals the sum of the two opposite (remote) interior angles.

Classify triangles by their sides.

Equilateral (all three sides equal), isosceles (two sides equal), and scalene (all sides different).

State the Pythagoras theorem.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: hypotenuse^2 = base^2 + height^2.

List the standard criteria for congruence of two triangles.

SSS, SAS, ASA, AAS, and RHS (right angle-hypotenuse-side).

What is the condition for two triangles to be similar (and the relation of their areas)?

Triangles are similar if corresponding angles are equal and corresponding sides are in proportion (AA, SSS, or SAS similarity). The ratio of their areas equals the square of the ratio of corresponding sides.

Define the radius, diameter, chord, and tangent of a circle.

Radius: segment from centre to the circle. Diameter: chord through the centre (= 2 x radius). Chord: segment joining two points on the circle. Tangent: a line touching the circle at exactly one point.

What is the relationship between a tangent to a circle and the radius at the point of contact?

The tangent is perpendicular to the radius drawn to the point of contact (angle of 90 degrees).

State the angle-at-centre theorem and the angle-in-a-semicircle property.

The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining circle. The angle in a semicircle (subtended by a diameter) is 90 degrees.

What is the distance formula between two points (x1, y1) and (x2, y2)?

Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2].

State the midpoint formula and the section formula (internal division in ratio m:n).

Midpoint = ((x1+x2)/2, (y1+y2)/2). Section point = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)).

What is the slope of a line through (x1,y1) and (x2,y2), and the slope-intercept form?

Slope m = (y2 - y1)/(x2 - x1). Slope-intercept form: y = mx + c, where c is the y-intercept.

What are the slope conditions for two lines to be parallel or perpendicular?

Parallel lines have equal slopes (m1 = m2). Perpendicular lines have slopes whose product is -1 (m1 x m2 = -1).

Give the area and perimeter formulas for a rectangle and a square.

Rectangle: area = length x breadth, perimeter = 2(length + breadth). Square: area = side^2, perimeter = 4 x side.

State the area of a triangle (base-height) and the area and circumference of a circle.

Triangle area = (1/2) x base x height. Circle area = pi x r^2; circumference = 2 x pi x r.

State Heron's formula for the area of a triangle with sides a, b, c.

Area = sqrt[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter.

Give the volume and total surface area of a cuboid with dimensions l, b, h.

Volume = l x b x h. Total surface area = 2(lb + bh + hl).

State the volume and curved surface area of a cylinder and a sphere.

Cylinder: volume = pi x r^2 x h, curved surface area = 2 x pi x r x h. Sphere: volume = (4/3) x pi x r^3, surface area = 4 x pi x r^2.

State the volume of a cone and of a sphere in relation to a cylinder, and the cone's slant height.

Cone volume = (1/3) x pi x r^2 x h (one-third of a cylinder with the same base and height). Slant height l = sqrt(r^2 + h^2). Sphere volume = (4/3) x pi x r^3.

Define the mean, median, and mode of a data set.

Mean = sum of all values divided by the number of values. Median = the middle value when data are arranged in order. Mode = the value that occurs most frequently.

State the empirical relationship between mean, median, and mode.

Mode = 3 x Median - 2 x Mean.

What does standard deviation measure, and what is its formula for a population?

It measures the spread/dispersion of data about the mean. Population SD = sqrt[ (Σ(x - mean)^2) / N ], the square root of the variance.

What is the basic (classical) definition of the probability of an event?

P(event) = (number of favourable outcomes) / (total number of equally likely outcomes). Its value ranges from 0 (impossible) to 1 (certain).

What is the probability of the complement of an event, and the addition rule for mutually exclusive events?

P(not A) = 1 - P(A). For mutually exclusive events, P(A or B) = P(A) + P(B).