Master the basics - and everything after gets easier

What it is

Rationalising clears a surd (root) from the denominator by multiplying top and bottom by it.

Answer

Multiply the numerator and denominator by sqrt(2): (1/sqrt(2)) x (sqrt(2)/sqrt(2)) = sqrt(2) / 2. So 1/sqrt(2) = sqrt(2)/2, which is about 0.707. Rationalising removes the irrational number (surd) from the denominator.

What it is

The remainder theorem says the remainder of p(x) divided by (x - a) is simply p(a).

Answer

By the remainder theorem, the remainder when p(x) is divided by (x + 1) is p(-1). p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. Since the remainder is 0, (x + 1) is a factor of p(x).

What it is

A straight line meets the axes at its intercepts - put y = 0 for the x-axis, x = 0 for the y-axis.

Answer

On the x-axis, y = 0, so x + 0 = 5 gives x = 5; the line meets the x-axis at (5, 0). On the y-axis, x = 0, so 0 + y = 5 gives y = 5; it meets the y-axis at (0, 5). These two intercept points are enough to draw the straight line.

What it is

Heron's formula finds a triangle's area from just its three sides, using the semi-perimeter.

Answer

First find the semi-perimeter s = (3 + 4 + 5)/2 = 6 cm. Heron's formula gives Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[6(6 - 3)(6 - 4)(6 - 5)] = sqrt[6 x 3 x 2 x 1] = sqrt[36] = 6 square cm.

What it is

A sphere's volume grows with the cube of its radius: V = (4/3) pi r^3.

Answer

Volume of a sphere = (4/3) x pi x r^3 = (4/3) x (22/7) x 7^3 = (4/3) x (22/7) x 343 = (4 x 22 x 49)/3 = 4312/3, which is about 1437.3 cubic cm.

What it is

The mean (average) is the total of all values divided by how many values there are.

Answer

Mean = (sum of all observations) / (number of observations) = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20.

What it is

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Answer

By the Pythagoras theorem, c^2 = a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = sqrt(25) = 5. The hypotenuse is always the longest side and lies opposite the right angle.

What it is

The mean of a set of values is their sum divided by how many values there are.

Answer

Mean = sum of values / number of values = 250 / 5 = 50. The mean is the value each item would have if the total were shared out equally among them.

What it is

The surface area of a cube is six times the area of one square face.

Answer

Surface area = 6 x side^2 = 6 x 4^2 = 6 x 16 = 96 square units. A cube has six identical square faces, so we find the area of one face and multiply by six.

What it is

The volume of a cone is one third of pi times the square of the radius times the height.

Answer

Volume = 1/3 x pi x r^2 x h = 1/3 x 3.14159 x 3^2 x 7 = 1/3 x 3.14159 x 9 x 7 = 65.97 cubic units. A cone holds exactly one third of the cylinder with the same base and height.

What it is

The total surface area of a cylinder is the curved side plus its two circular ends.

Answer

Total surface area = 2 pi r (r + h) = 2 x 3.14159 x 3 x (3 + 7) = 2 x 3.14159 x 3 x 10 = 188.5 square units. This is the curved surface 2 pi r h plus the two circular ends 2 pi r^2.

What it is

The surface area of a cuboid is twice the sum of the areas of its three different faces.

Answer

Surface area = 2(lb + bh + hl) = 2(3x4 + 4x5 + 5x3) = 2(12 + 20 + 15) = 2 x 47 = 94 square units. A cuboid has three pairs of identical faces, so we add the three distinct faces and double the total.

What it is

The volume of a hemisphere is two thirds of pi times the cube of the radius.

Answer

Volume = 2/3 x pi x r^3 = 2/3 x 3.14159 x 6^3 = 2/3 x 3.14159 x 216 = 452.39 cubic units. A hemisphere is half a sphere, so its volume is half of 4/3 x pi x r^3, which is 2/3 x pi x r^3.

What it is

The slant height of a cone is found from its radius and vertical height using the Pythagoras theorem.

Answer

Slant height l = sqrt(r^2 + h^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The radius, the vertical height and the slant height form a right triangle, so Pythagoras gives the slant.

What it is

The curved surface area of a cone is pi times the radius times the slant height.

Answer

Curved surface area = pi x r x l = 3.14159 x 3 x 5 = 47.12 square units. Note that l is the slant height along the sloping side, not the vertical height of the cone.

What it is

The total surface area of a cone is its curved surface plus its circular base.

Answer

Total surface area = pi x r x (l + r) = 3.14159 x 3 x (5 + 3) = 3.14159 x 3 x 8 = 75.40 square units. This is the curved surface (pi x r x l) plus the base circle (pi x r^2).

What it is

The curved surface area of a hemisphere is two pi times the square of the radius.

Answer

Curved surface area = 2 x pi x r^2 = 2 x 3.14159 x 6^2 = 2 x 3.14159 x 36 = 226.19 square units. A full sphere has surface 4 x pi x r^2, so the curved part of a hemisphere is exactly half of that.

What it is

The diagonal of a square is its side length times the square root of two.

Answer

Diagonal = side x sqrt(2) = 10 x 1.414 = 14.14 units. This comes from the Pythagoras theorem on the two equal sides of the square: diagonal^2 = side^2 + side^2 = 2 x side^2.

What it is

The diagonal of a rectangle is the square root of the sum of the squares of its length and breadth.

Answer

Diagonal = sqrt(l^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The length, breadth and diagonal form a right triangle, so the Pythagoras theorem gives the diagonal.

What it is

The longest diagonal of a cuboid is the square root of the sum of the squares of its three dimensions.

Answer

Space diagonal = sqrt(l^2 + b^2 + h^2) = sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13 units. It extends the Pythagoras theorem into three dimensions.

What it is

Heron's formula gives the area of a triangle from the lengths of its three sides.

Answer

First the semi-perimeter s = (a+b+c)/2 = (3+4+5)/2 = 6. Then area = sqrt(s(s-a)(s-b)(s-c)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units. (This is a right triangle, so 1/2 x 3 x 4 = 6 agrees.)

What it is

The area of an equilateral triangle is root three over four times the square of its side.

Answer

Area = (sqrt(3) / 4) x side^2 = (1.732 / 4) x 6^2 = 0.433 x 36 = 15.59 square units. This is a special case of Heron's formula when all three sides are equal.

What it is

The curved surface area of a cylinder is two pi times the radius times the height.

Answer

Curved surface area = 2 x pi x r x h = 2 x 3.14159 x 3 x 7 = 131.95 square units. It is the rectangle you would get by unrolling the curved side, with width equal to the circumference and height equal to the cylinder's height.

What it is

The longest diagonal of a cube is its side length times the square root of three.

Answer

Space diagonal = side x sqrt(3) = 5 x 1.732 = 8.66 units. It comes from applying Pythagoras twice - once across a face, then through the solid - giving sqrt(side^2 + side^2 + side^2) = side x sqrt(3).

What it is

The total surface area of a solid hemisphere is three pi times the square of the radius.

Answer

Total surface area = 3 x pi x r^2 = 3 x 3.14159 x 6^2 = 3 x 3.14159 x 36 = 339.29 square units. It is the curved dome (2 pi r^2) plus the flat circular base (pi r^2), which together make 3 pi r^2.

What it is

The range of a set of data is the difference between its highest and lowest values.

Answer

Range = highest value - lowest value = 80 - 20 = 60. A larger range means the data is more spread out; a small range means the values are close together.

What it is

The class mark is the midpoint of a class interval, the average of its lower and upper limits.

Answer

Class mark = (lower limit + upper limit) / 2 = (10 + 20) / 2 = 15. It is the midpoint of the interval and stands in for every value in that class when we calculate a grouped mean.

What it is

An exterior angle of a triangle equals the sum of the two interior angles opposite to it.

Answer

Exterior angle = angle 1 + angle 2 = 60 + 70 = 130 degrees. This works because the exterior angle and the third interior angle sit on a straight line (180 degrees), and the three interior angles also add to 180 degrees.

What it is

Two angles are supplementary when they add up to 180 degrees.

Answer

Supplement = 180 - angle = 180 - 60 = 120 degrees. Supplementary angles add to a straight angle (180 degrees), so the angles on one side of a straight line always sum to 180.

What it is

The edge of a cube is the cube root of its volume.

Answer

Edge = cube root of volume = cube root of 64 = 4 units, because 4 x 4 x 4 = 64. It simply reverses the volume formula V = side^3.

What it is

The height of a triangle is twice its area divided by the base.

Answer

Height = 2 x area / base = 2 x 40 / 10 = 80 / 10 = 8 units. This is the area formula area = 1/2 x base x height rearranged to make the height the subject.

What it is

A straight line is described by y = mx + c, where m is the slope and c is the y-intercept.

Answer

y = m x + c = 2 x 3 + 1 = 6 + 1 = 7. The slope m tells how steeply y changes with x, and c is where the line crosses the y-axis (its value when x = 0).

What it is

A rational number can be written as p/q with integers p and q (q not 0); an irrational number cannot.

Answer

A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not zero; its decimal expansion is either terminating or non-terminating but recurring, for example 3/4 = 0.75 or 1/3 = 0.333... An irrational number cannot be written as p/q, and its decimal expansion is non-terminating and non-recurring, for example sqrt(2) = 1.41421... or pi.

What it is

Rationalising removes a surd from the denominator by multiplying by a suitable factor.

Answer

To rationalise 1/sqrt(3), multiply both the numerator and the denominator by sqrt(3): (1 x sqrt(3)) / (sqrt(3) x sqrt(3)) = sqrt(3) / 3. So 1/sqrt(3) = sqrt(3)/3, which now has a rational denominator.

What it is

The laws of exponents also apply to rational powers (roots) of positive real numbers.

Answer

Using a^m x a^n = a^(m+n): 2^(1/2) x 2^(1/2) = 2^(1/2 + 1/2) = 2^1 = 2. For (8)^(2/3), write 8 = 2^3, so (2^3)^(2/3) = 2^(3 x 2/3) = 2^2 = 4. So the answers are 2 and 4.

What it is

The degree of a polynomial is the highest power of the variable; polynomials are named by their degree.

Answer

The degree of a polynomial is the highest power of the variable present in it. A polynomial of degree 1, such as 2x + 3, is called a linear polynomial; a polynomial of degree 2, such as x^2 + 5x + 6, is called a quadratic polynomial; and a polynomial of degree 3, such as x^3 - 2x + 1, is called a cubic polynomial. A constant like 7 has degree 0.

What it is

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

Answer

By the remainder theorem, the remainder when p(x) is divided by (x - a) is p(a). Here the divisor is (x + 1) = (x - (-1)), so a = -1. Substitute: p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. So the remainder is 0, which means (x + 1) is a factor.

What it is

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

Answer

The factor theorem states that (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0. To test (x - 2), substitute a = 2: p(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0. Since p(2) = 0, (x - 2) is a factor of x^2 - 5x + 6.

What it is

The identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca expands the square of a trinomial.

Answer

Using (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca with a = x, b = 2y and c = 3z: = x^2 + (2y)^2 + (3z)^2 + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x) = x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6zx.

What it is

The Cartesian plane is formed by two perpendicular axes and is divided into four quadrants.

Answer

On the Cartesian plane, the x-axis and y-axis meet at the origin and divide the plane into four quadrants. In a point (x, y): the first quadrant has both coordinates positive, the second has x negative and y positive, the third has both negative, and the fourth has x positive and y negative. The point (3, -4) has x positive and y negative, so it lies in the fourth quadrant. The point (-2, 5) has x negative and y positive, so it lies in the second quadrant.

What it is

A linear equation in two variables has infinitely many solutions, which lie on a straight line when graphed.

Answer

A linear equation in two variables of the form ax + by + c = 0 has infinitely many solutions, and all of them lie on a straight line when plotted on a graph. For x + y = 5, we can choose values of x and find y: if x = 0, then y = 5, giving (0, 5); if x = 2, then y = 3, giving (2, 3). So (0, 5) and (2, 3) are two of its infinitely many solutions.

What it is

When lines meet, they form angle pairs such as linear pairs (sum 180) and vertically opposite angles (equal).

Answer

A linear pair is a pair of adjacent angles formed when two lines intersect (or a ray stands on a line) such that their non-common arms form a straight line; the sum of a linear pair is 180 degrees. Vertically opposite angles are the pairs of opposite angles formed when two lines intersect; vertically opposite angles are always equal. For example, if two lines cross and one angle is 70 degrees, the angle opposite it is also 70 degrees, and its linear pair is 110 degrees.

What it is

A transversal cutting two parallel lines makes equal corresponding angles, equal alternate angles and co-interior angles that sum to 180.

Answer

When a transversal intersects two parallel lines, the following angle relationships hold: corresponding angles are equal; alternate interior angles are equal; alternate exterior angles are equal; and co-interior (allied or same-side interior) angles are supplementary, that is, they add up to 180 degrees. These properties can be used both to find unknown angles and to prove that two lines are parallel.

What it is

The interior angles of a triangle sum to 180 degrees, and an exterior angle equals the sum of the two opposite interior angles.

Answer

By the angle sum property, the three interior angles add to 180 degrees, so the third angle = 180 - (50 + 60) = 180 - 110 = 70 degrees. By the exterior angle property, the exterior angle at the third vertex equals the sum of the two opposite interior angles = 50 + 60 = 110 degrees (which is also 180 - 70).

What it is

Two triangles are congruent if they satisfy SSS, SAS, ASA, AAS or RHS.

Answer

Two triangles are congruent (equal in all respects) if any one of the following holds: SSS, when the three sides of one equal the three sides of the other; SAS, when two sides and the included angle are equal; ASA, when two angles and the included side are equal; AAS, when two angles and a non-included side are equal; and RHS, when in two right triangles the hypotenuse and one side are equal. Note that there is no SSA/ASS congruence rule.

What it is

In an isosceles triangle the angles opposite the equal sides are equal, and vice versa.

Answer

An isosceles triangle has two sides equal. Its key properties are: the angles opposite the two equal sides are equal (the base angles are equal); and conversely, if two angles of a triangle are equal, then the sides opposite them are equal, so the triangle is isosceles. Also, the perpendicular drawn from the vertex angle to the base bisects the base and the vertex angle.

What it is

In a parallelogram opposite sides and opposite angles are equal, and the diagonals bisect each other.

Answer

Three properties of a parallelogram are: opposite sides are equal and parallel; opposite angles are equal; and the diagonals bisect each other. A further property is that each diagonal divides the parallelogram into two congruent triangles, and any pair of adjacent angles is supplementary (sums to 180 degrees).

What it is

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Answer

The midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half of it. If the third side is 10 cm, then the segment joining the midpoints of the other two sides is parallel to it and its length is 10 / 2 = 5 cm.

What it is

The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

Answer

Two important chord properties are: the perpendicular drawn from the centre of a circle to a chord bisects the chord (and conversely, the line from the centre to the midpoint of a chord is perpendicular to it); and equal chords of a circle are equidistant from the centre (and chords that are equidistant from the centre are equal). These follow from the fact that all radii of a circle are equal.

What it is

The angle at the centre is twice the angle at the circumference on the same arc, and opposite angles of a cyclic quadrilateral sum to 180 degrees.

Answer

The angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference. A consequence is that angles in the same segment are equal, and the angle in a semicircle is a right angle. In a cyclic quadrilateral (one whose four vertices lie on a circle), the sum of each pair of opposite angles is 180 degrees, that is, the opposite angles are supplementary.

What it is

Heron's formula finds the area of a triangle from the lengths of its three sides.

Answer

First find the semi-perimeter s = (a + b + c)/2 = (3 + 4 + 5)/2 = 12/2 = 6 cm. Then area = sqrt(s(s - a)(s - b)(s - c)) = sqrt(6(6 - 3)(6 - 4)(6 - 5)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 cm^2. So the area of the triangle is 6 square centimetres.

What it is

A cone has curved surface area pi r l and volume (1/3) pi r^2 h.

Answer

Volume of a cone = (1/3) pi r^2 h = (1/3) x (22/7) x 3^2 x 7 = (1/3) x (22/7) x 9 x 7 = (1/3) x 22 x 9 = (1/3) x 198 = 66 cm^3. So the volume of the cone is 66 cubic centimetres.

What it is

A sphere has surface area 4 pi r^2 and volume (4/3) pi r^3.

Answer

Surface area of a sphere = 4 pi r^2 = 4 x (22/7) x 7^2 = 4 x (22/7) x 49 = 4 x 22 x 7 = 616 cm^2. So the surface area of the sphere is 616 square centimetres.

What it is

A cylinder has curved surface area 2 pi r h and total surface area 2 pi r (r + h).

Answer

Total surface area of a closed cylinder = 2 pi r (r + h) = 2 x (22/7) x 7 x (7 + 10) = 2 x 22 x 17 = 748 cm^2. So the total surface area of the cylinder is 748 square centimetres.

What it is

The mean (average) is the sum of all observations divided by the number of observations.

Answer

The mean is the sum of all observations divided by the number of observations. Sum = 10 + 12 + 14 + 16 + 18 = 70. Number of observations = 5. Mean = 70 / 5 = 14. So the mean of the data is 14.

What it is

The median is the middle value of ordered data; the mode is the most frequently occurring value.

Answer

The data is already arranged in ascending order: 4, 6, 6, 8, 10. There are 5 observations (odd number), so the median is the middle (3rd) value, which is 6. The mode is the value that occurs most often; here 6 occurs twice while all others occur once, so the mode is 6. Thus both the median and the mode are 6.

What it is

Empirical (experimental) probability is the ratio of the number of times an event happens to the total number of trials.

Answer

Empirical probability of an event = (number of times the event occurs) / (total number of trials). Here a head appeared 60 times out of 100 tosses, so P(head) = 60/100 = 0.6. The probability of any event lies between 0 and 1, and the probabilities of all outcomes add up to 1.