MAT Mathematical Skills Syllabus

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 are relative speeds for objects moving in the same vs opposite directions?

Opposite directions: relative speed = sum of speeds. Same direction: relative speed = difference of speeds.

How long does a train of length L take to cross a platform of length P at speed S?

Time = (L + P) / S. To cross a pole or stationary point, time = L / S (platform length is zero).

In boats and streams, what are the downstream and upstream speeds?

Downstream speed = boat speed + stream speed (b + s); Upstream speed = boat speed - stream speed (b - s). Boat speed = (down + up)/2; stream speed = (down - up)/2.

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

Average = (Sum of all observations) / (Number of observations). Conversely, Sum = Average x Number of observations.

What is the average of the first n natural numbers?

Average = (n + 1) / 2. The sum of the first n natural numbers is n(n+1)/2.

How does an average change when each observation is increased by a constant k?

The average also increases by k. Adding/subtracting a constant to every term shifts the mean by that constant; multiplying every term by k multiplies the mean by k.

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

Standard form: ax^2 + bx + c = 0 (a not 0). Roots x = (-b ± sqrt(b^2 - 4ac)) / 2a.

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 does the discriminant of a quadratic tell you about its roots?

D = b^2 - 4ac. If D > 0: two distinct real roots; D = 0: two equal real roots; D < 0: no real roots (complex conjugates).

What is the condition for a system of two linear equations to have a unique solution, no solution, or infinitely many solutions?

For a1x+b1y=c1 and a2x+b2y=c2: unique if a1/a2 != b1/b2; no solution if a1/a2 = b1/b2 != c1/c2; infinite if a1/a2 = b1/b2 = c1/c2.

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. if x > y and you multiply by -1, then -x < -y).

What is the formula for the nth term and sum of an Arithmetic Progression (AP)?

nth term: a_n = a + (n-1)d. Sum of n terms: S_n = n/2 [2a + (n-1)d] = n/2 (first term + last term).

What is the formula for the nth term and sum of a Geometric Progression (GP)?

nth term: a_n = a x r^(n-1). Sum of n terms: S_n = a(r^n - 1)/(r - 1) for r != 1. Sum to infinity (|r| < 1): a/(1 - r).

What is a Harmonic Progression (HP) and how is its nth term found?

An HP is a sequence whose reciprocals form an AP. To find the nth term, take the reciprocal of the nth term of the corresponding AP.

What is the relationship between Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) of two positive numbers?

GM^2 = AM x HM, and AM >= GM >= HM (equality only when the numbers are equal).

What is the domain and range of a function?

The domain is the set of all permissible input values (x); the range is the set of all resulting output values (f(x)).

State the key logarithm laws for product, quotient, and power.

log(mn) = log m + log n; log(m/n) = log m - log n; log(m^k) = k log m. Also log_b(b) = 1 and log_b(1) = 0.

What is the change of base formula for logarithms?

log_b(x) = log_c(x) / log_c(b), for any valid base c. Commonly used to convert to base 10 or base e.

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

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

State the Pythagorean theorem and its converse.

In a right triangle, hypotenuse^2 = base^2 + height^2 (c^2 = a^2 + b^2). Converse: if c^2 = a^2 + b^2, the triangle is right-angled at the vertex opposite c.

What are the conditions for two triangles to be congruent?

Congruence criteria: SSS, SAS, ASA, AAS, and RHS (right angle-hypotenuse-side). Congruent triangles are identical in shape and size.

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

Sum of interior angles = (n - 2) x 180 degrees. Sum of exterior angles = 360 degrees (for any convex polygon).

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

The central angle is twice the inscribed angle subtending the same arc. Angles in the same segment are equal, and an angle in a semicircle is 90 degrees.

What are the formulas for the circumference and area of a circle?

Circumference = 2(pi)r; Area = (pi)r^2, where r is the radius. Arc length = (theta/360) x 2(pi)r; Sector area = (theta/360) x (pi)r^2.

What is the distance formula and the midpoint formula in coordinate geometry?

Distance between (x1,y1) and (x2,y2) = sqrt((x2-x1)^2 + (y2-y1)^2). Midpoint = ((x1+x2)/2, (y1+y2)/2).

What is the slope of a line and the condition for two lines to be parallel or perpendicular?

Slope m = (y2 - y1)/(x2 - x1). Parallel lines have equal slopes (m1 = m2); perpendicular lines have m1 x m2 = -1.

What are the volume and total surface area formulas for a cylinder and a sphere?

Cylinder: Volume = (pi)r^2h, TSA = 2(pi)r(r + h). Sphere: Volume = (4/3)(pi)r^3, Surface area = 4(pi)r^2.

What is the difference between permutation and combination, and their formulas?

Permutation (order matters): nPr = n!/(n-r)!. Combination (order doesn't matter): nCr = n!/[r!(n-r)!].

What is the classical (theoretical) probability of an event, and the range of its value?

P(event) = (Number of favourable outcomes) / (Total number of equally likely outcomes). The value lies between 0 and 1 inclusive; P(not E) = 1 - P(E).

State the addition formula for the union of two sets (inclusion-exclusion for 2 sets).

n(A union B) = n(A) + n(B) - n(A intersection B). For mutually exclusive sets the intersection term is 0.