Master the basics - and everything after gets easier

What it is

Euclid's algorithm finds the HCF by repeated division until the remainder becomes zero.

Answer

Apply repeated division: 404 = 96 x 4 + 20; 96 = 20 x 4 + 16; 20 = 16 x 1 + 4; 16 = 4 x 4 + 0. The last non-zero remainder is 4, so HCF(96, 404) = 4.

What it is

The zeroes of a quadratic are where it equals zero; their sum is -b/a and product is c/a.

Answer

Factorise: x^2 - 7x + 10 = (x - 2)(x - 5), so the zeroes are 2 and 5. Sum of zeroes = 2 + 5 = 7 = -(-7)/1 = -b/a. Product of zeroes = 2 x 5 = 10 = 10/1 = c/a. Both relations hold.

What it is

Two linear equations meet at one point - the (x, y) that satisfies both at once.

Answer

From the second equation, x = y - 1. Substitute into the first: 2(y - 1) + 3y = 13, so 5y - 2 = 13, giving 5y = 15 and y = 3. Then x = y - 1 = 2. Solution: x = 2, y = 3.

What it is

The discriminant D = b^2 - 4ac reveals the nature of a quadratic's roots before you solve it.

Answer

Factorise: (x - 2)(x - 3) = 0, so the roots are x = 2 and x = 3. The discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0 and is a perfect square, the roots are real, distinct and rational.

What it is

In an AP each term rises by a fixed step d; the nth term is a + (n - 1)d.

Answer

The nth term of an AP is a_n = a + (n - 1)d. Here a = 3, d = 5, n = 10, so a_10 = 3 + (10 - 1) x 5 = 3 + 45 = 48.

What it is

The sum of an AP pairs the first and last terms: S = n/2 [2a + (n - 1)d].

Answer

Here a = 2 and d = 3. The sum of n terms is S_n = n/2 [2a + (n - 1)d]. So S_20 = 20/2 [2(2) + 19(3)] = 10 [4 + 57] = 10 x 61 = 610.

What it is

A line parallel to one side of a triangle cuts the other two sides in the same ratio.

Answer

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides those two sides in the same ratio. In triangle ABC, if DE is parallel to BC and meets AB at D and AC at E, then AD/DB = AE/EC.

What it is

The distance between two points is the hypotenuse of the right triangle of their coordinate gaps.

Answer

The distance formula is d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Here d = sqrt[(5 - 2)^2 + (7 - 3)^2] = sqrt[9 + 16] = sqrt[25] = 5 units.

What it is

The identity sin^2 + cos^2 = 1 lets you find one trig ratio from another for the same angle.

Answer

Using the identity sin^2(theta) + cos^2(theta) = 1, cos^2(theta) = 1 - (3/5)^2 = 1 - 9/25 = 16/25, so cos(theta) = 4/5 (positive since theta is acute). Then tan(theta) = sin/cos = (3/5) / (4/5) = 3/4.

What it is

An angle of elevation turns a triangle into a height problem using tan = opposite / adjacent.

Answer

Let the height be h. tan(30 degrees) = h / 30, and tan(30 degrees) = 1/sqrt(3). So h = 30 x (1/sqrt(3)) = 30/sqrt(3) = 10 sqrt(3) m, which is about 17.3 m.

What it is

A tangent touches a circle at one point and is perpendicular to the radius at that point.

Answer

The tangent is perpendicular to the radius at the point of contact, so the radius, tangent and line OP form a right-angled triangle with OP as the hypotenuse. Length of tangent = sqrt[OP^2 - r^2] = sqrt[13^2 - 5^2] = sqrt[169 - 25] = sqrt[144] = 12 cm.

What it is

A cone's volume is exactly one-third of a cylinder with the same base and height.

Answer

Volume of a cone = (1/3) x pi x r^2 x h = (1/3) x (22/7) x 3^2 x 4 = (1/3) x (22/7) x 36 = (22 x 12) / 7 = 264/7, which is about 37.7 cubic cm.

What it is

For a moderately skewed data set, the three averages link as Mode = 3 Median - 2 Mean.

Answer

The empirical relationship between the three measures of central tendency is Mode = 3 Median - 2 Mean. Substituting, Mode = 3(30) - 2(27) = 90 - 54 = 36.

What it is

Probability = favourable outcomes / total outcomes, always a number between 0 and 1.

Answer

There are 4 kings in a deck of 52 cards. Probability = number of favourable outcomes / total outcomes = 4/52 = 1/13.

What it is

The standard angles 0, 30, 45, 60 and 90 degrees have fixed trig values worth memorising.

Answer

Use standard values: tan(45 deg) = 1, cos(30 deg) = sqrt(3)/2, sin(60 deg) = sqrt(3)/2. So the expression = 2(1)^2 + (sqrt(3)/2)^2 - (sqrt(3)/2)^2 = 2 + 3/4 - 3/4 = 2.

What it is

Probability measures how likely an event is - favourable outcomes out of all equally likely outcomes.

Answer

Probability = favourable outcomes / total outcomes = 1 / 6 = 0.167 (about 17%). A fair die has 6 equally likely faces and only one of them is a 4. Every probability lies between 0 (impossible) and 1 (certain).

What it is

The volume of a cylinder is the area of its circular base times its height.

Answer

Volume = pi x r^2 x h = 3.14159 x 3^2 x 7 = 3.14159 x 9 x 7 = 197.9 cubic units. The base is a circle of area pi x r^2, and stacking it to a height h gives the volume.

What it is

The surface area of a sphere is four times pi times the square of its radius.

Answer

Surface area = 4 x pi x r^2 = 4 x 3.14159 x 7^2 = 4 x 3.14159 x 49 = 615.75 square units. Remarkably, a sphere's surface is exactly four times the area of its flat circular cross-section (pi x r^2).

What it is

In an arithmetic progression, each term is found by adding the common difference to the first term repeatedly.

Answer

nth term = a + (n - 1) d = 2 + (5 - 1) x 3 = 2 + 12 = 14. We add the common difference (n - 1) times because the first term already counts as term 1.

What it is

The sum of the first n terms of an arithmetic progression is n/2 times the sum of twice the first term and (n - 1) times the common difference.

Answer

Sum = n/2 x (2a + (n - 1) d) = 5/2 x (2x2 + (5 - 1) x 3) = 5/2 x (4 + 12) = 5/2 x 16 = 40. This equals the number of terms times the average of the first and last term.

What it is

The slope of a line is how much it rises for each unit it runs across.

Answer

Slope = rise / run = 6 / 3 = 2. A slope of 2 means the line climbs 2 units for every 1 unit across. A negative slope would mean the line falls as you move right.

What it is

The area of a sector is the fraction of the circle its angle covers, times the circle's area.

Answer

Sector area = (angle / 360) x pi x r^2 = (90 / 360) x 3.14159 x 7^2 = 0.25 x 3.14159 x 49 = 38.48 square units. A 90 degree sector is one quarter of the circle, so it has a quarter of the circle's area.

What it is

The length of an arc is the fraction of the circumference that its angle subtends.

Answer

Arc length = (angle / 360) x 2 x pi x r = (90 / 360) x 2 x 3.14159 x 7 = 0.25 x 43.98 = 11.0 units. A 90 degree arc is one quarter of the full circumference of the circle.

What it is

The discriminant of a quadratic equation tells how many real roots it has.

Answer

Discriminant D = b^2 - 4ac = 5^2 - 4 x 1 x 6 = 25 - 24 = 1. Since D is positive, the equation has two distinct real roots. If D were 0 there would be one repeated root, and if D were negative there would be no real roots.

What it is

The sum of the roots of a quadratic equation is minus b divided by a.

Answer

Sum of roots = -b / a = -(-5) / 1 = 5. Indeed the roots of x^2 - 5x + 6 = 0 are 2 and 3, which add to 5. The product of the roots is c / a = 6, matching 2 x 3.

What it is

In a right triangle, the sine of an angle is the opposite side divided by the hypotenuse.

Answer

sin(theta) = opposite / hypotenuse = 3 / 5 = 0.6. The sine of an acute angle is always between 0 and 1, because the opposite side is never longer than the hypotenuse.

What it is

In a right triangle, the cosine of an angle is the adjacent side divided by the hypotenuse.

Answer

cos(theta) = adjacent / hypotenuse = 4 / 5 = 0.8. Like sine, it lies between 0 and 1, and the two obey the identity sin^2(theta) + cos^2(theta) = 1.

What it is

In a right triangle, the tangent of an angle is the opposite side divided by the adjacent side.

Answer

tan(theta) = opposite / adjacent = 3 / 4 = 0.75. It also equals sin(theta) / cos(theta), and it grows without limit as the angle approaches 90 degrees.

What it is

The distance between two points is the square root of the sum of the squares of the differences in their coordinates.

Answer

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((3 - 0)^2 + (4 - 0)^2) = sqrt(9 + 16) = sqrt(25) = 5 units. It is simply the Pythagoras theorem applied to the horizontal and vertical gaps between the points.

What it is

The sum of the first n natural numbers is n times n plus one, divided by two.

Answer

Sum = n(n + 1) / 2 = 10 x 11 / 2 = 55. This is an arithmetic progression with first term 1 and common difference 1; pairing the first and last terms gives the neat n(n + 1) / 2 formula.

What it is

The perimeter of a sector is its two straight radii plus the curved arc.

Answer

Perimeter = 2r + arc = 2 x 7 + (90 / 360) x 2 x 3.14159 x 7 = 14 + 11.0 = 25.0 units. It is the two radii bounding the sector plus the curved arc along the circle.

What it is

The probability that an event does not happen is one minus the probability that it does.

Answer

P(not E) = 1 - P(E) = 1 - 1/6 = 5/6 (about 0.833). The event and its complement together cover every outcome, so their probabilities always add up to 1.

What it is

The product of the roots of a quadratic equation is c divided by a.

Answer

Product of roots = c / a = 6 / 1 = 6. The roots of x^2 - 5x + 6 = 0 are 2 and 3, and indeed 2 x 3 = 6. Their sum is -b / a = 5, matching 2 + 3.

What it is

Two angles are complementary when they add up to 90 degrees.

Answer

Complement = 90 - angle = 90 - 30 = 60 degrees. Complementary angles add to a right angle, which is why the sine of an angle equals the cosine of its complement: sin(30) = cos(60).

What it is

The perimeter of a semicircle is the curved half of the circumference plus the straight diameter.

Answer

Perimeter = pi x r + 2r = 3.14159 x 7 + 2 x 7 = 21.99 + 14 = 35.99 units. The curved part is half the circumference (pi x r), and we must add the straight diameter (2r) that closes the shape.

What it is

The area of a semicircle is half the area of the full circle.

Answer

Area = 1/2 x pi x r^2 = 1/2 x 3.14159 x 7^2 = 1/2 x 3.14159 x 49 = 76.97 square units. A semicircle is exactly half a circle, so we take half of pi x r^2.

What it is

The cosecant of an angle is the hypotenuse divided by the opposite side, the reciprocal of sine.

Answer

cosec(theta) = hypotenuse / opposite = 5 / 3 = 1.667. It is the reciprocal of the sine ratio, so cosec(theta) = 1 / sin(theta).

What it is

The secant of an angle is the hypotenuse divided by the adjacent side, the reciprocal of cosine.

Answer

sec(theta) = hypotenuse / adjacent = 5 / 4 = 1.25. It is the reciprocal of the cosine ratio, so sec(theta) = 1 / cos(theta).

What it is

The cotangent of an angle is the adjacent side divided by the opposite side, the reciprocal of tangent.

Answer

cot(theta) = adjacent / opposite = 4 / 3 = 1.333. It is the reciprocal of the tangent ratio, so cot(theta) = 1 / tan(theta).

What it is

The height of an object is the horizontal distance to it times the tangent of the angle of elevation.

Answer

Height = distance x tan(theta) = 50 x tan(30) = 50 x 0.577 = 28.87 m. The horizontal distance is the adjacent side and the height is the opposite side, so tan links them.

What it is

A quadrant is a quarter of a circle, so its area is a quarter of the circle's area.

Answer

Area = 1/4 x pi x r^2 = 1/4 x 3.14159 x 7^2 = 1/4 x 3.14159 x 49 = 38.48 square units. A quadrant is a 90 degree sector, which is one quarter of the whole circle.

What it is

The sum of the first n odd numbers is always n squared.

Answer

The sum of the first n odd numbers = n^2. For n = 5 that is 1 + 3 + 5 + 7 + 9 = 25 = 5^2. It is an arithmetic progression with first term 1 and common difference 2.

What it is

The sum of the first n even numbers is n times n plus one.

Answer

The sum of the first n even numbers = n(n + 1). For n = 5 that is 2 + 4 + 6 + 8 + 10 = 30 = 5 x 6. It is an arithmetic progression with first term 2 and common difference 2.

What it is

The quadratic formula gives the roots of any quadratic equation from its coefficients.

Answer

x = (-b + sqrt(b^2 - 4ac)) / 2a = (5 + sqrt(25 - 24)) / 2 = (5 + 1) / 2 = 3. Using the minus sign gives the other root, 2. If b^2 - 4ac is negative there are no real roots.

What it is

For two numbers, the product of the HCF and LCM equals the product of the numbers.

Answer

Since HCF x LCM = a x b, LCM = (a x b) / HCF = (12 x 18) / 6 = 216 / 6 = 36. This identity holds for any pair of positive integers.

What it is

The area of a ring is the area of the outer circle minus the area of the inner circle.

Answer

Ring area = pi x (R^2 - r^2) = 3.14159 x (10^2 - 6^2) = 3.14159 x (100 - 36) = 3.14159 x 64 = 201.06 square units. We take the big circle's area and subtract the hole in the middle.

What it is

The radius of a circle is the square root of its area divided by pi.

Answer

Radius = sqrt(area / pi) = sqrt(154 / 3.14159) = sqrt(49.02) = 7.0 units. This reverses the area formula area = pi x r^2 to make the radius the subject.

What it is

The mean proportional between two numbers is the square root of their product.

Answer

Mean proportional = sqrt(a x b) = sqrt(4 x 9) = sqrt(36) = 6. It is the value x for which a : x = x : b, so x^2 = a x b.

What it is

The average of all terms of an arithmetic progression equals the average of its first and last term.

Answer

Average term = (first + last) / 2 = (2 + 14) / 2 = 16 / 2 = 8. Because the terms are evenly spaced, the middle value is exactly the average of the two ends.

What it is

The number of terms in an arithmetic progression is found from the first term, last term and common difference.

Answer

Number of terms n = (last - first) / d + 1 = (14 - 2) / 3 + 1 = 12 / 3 + 1 = 4 + 1 = 5. We count the steps of size d from the first to the last term, then add one for the starting term.

What it is

The midpoint of two points is found by averaging their x-coordinates and their y-coordinates.

Answer

Midpoint x = (x1 + x2) / 2 = (2 + 8) / 2 = 10 / 2 = 5. The full midpoint formula averages both coordinates: ((x1 + x2)/2, (y1 + y2)/2).

What it is

An empirical relation estimates the mode from the mean and the median.

Answer

Mode = 3 x Median - 2 x Mean = 3 x 40 - 2 x 38 = 120 - 76 = 44. This empirical formula connects the mean, median and mode for moderately skewed data.

What it is

The length of a shadow is the object's height divided by the tangent of the sun's angle of elevation.

Answer

Shadow = height / tan(angle) = 50 / tan(45) = 50 / 1 = 50 m. As the sun gets lower, the angle shrinks, tan gets smaller, and the shadow grows much longer.

What it is

For two numbers, the product of their HCF and LCM equals the product of the numbers.

Answer

For any two numbers, HCF x LCM = product of the two numbers. So LCM = product / HCF = 216 / 6 = 36. Therefore the LCM of the two numbers is 36.

What it is

Every composite number can be expressed as a unique product of prime numbers.

Answer

The fundamental theorem of arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique except for the order of the factors. Prime factorising: 12 = 2^2 x 3 and 18 = 2 x 3^2. The HCF is the product of the smallest powers of common primes = 2^1 x 3^1 = 6. The LCM is the product of the greatest powers of all primes = 2^2 x 3^2 = 36.

What it is

A number that cannot be written as p/q (with q not zero) is irrational, and can be proved so by contradiction.

Answer

We prove sqrt(2) is irrational by contradiction. Assume that sqrt(2) is rational, so sqrt(2) = p/q where p and q are co-prime integers (no common factor) and q is not zero. Squaring gives 2 = p^2/q^2, so p^2 = 2q^2, which means p^2 is even, and hence p is even. Let p = 2m; then 2q^2 = 4m^2, so q^2 = 2m^2, meaning q^2 is even and q is even too. But then p and q have a common factor 2, contradicting our assumption that they are co-prime. Hence our assumption is wrong, and sqrt(2) is irrational.

What it is

For a quadratic ax^2 + bx + c, the sum of zeroes is -b/a and the product is c/a.

Answer

For a quadratic polynomial ax^2 + bx + c, the sum of the zeroes is -b/a and the product of the zeroes is c/a. Here a = 1, b = -5, c = 6. Sum of zeroes = -b/a = -(-5)/1 = 5. Product of zeroes = c/a = 6/1 = 6. (Indeed the zeroes are 2 and 3, whose sum is 5 and product is 6.)

What it is

A pair of linear equations can be solved by expressing one variable in terms of the other and substituting.

Answer

From the first equation, x = 10 - y. Substitute this into the second equation: (10 - y) - y = 4, so 10 - 2y = 4, which gives 2y = 6 and y = 3. Then x = 10 - y = 10 - 3 = 7. So the solution is x = 7 and y = 3. Check: 7 + 3 = 10 and 7 - 3 = 4, both correct.

What it is

The ratios of coefficients decide whether a pair of linear equations has one solution, no solution or infinitely many.

Answer

For a pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0: if a1/a2 is not equal to b1/b2, the lines intersect at one point and there is a unique solution (consistent). If a1/a2 = b1/b2 = c1/c2, the lines are coincident and there are infinitely many solutions (consistent and dependent). If a1/a2 = b1/b2 but not equal to c1/c2, the lines are parallel and there is no solution (inconsistent).

What it is

A quadratic equation can be solved by splitting the middle term and factorising into two brackets.

Answer

To factorise x^2 - 5x + 6, split the middle term -5x into two terms whose product is 1 x 6 = 6 and whose sum is -5; these are -2 and -3. So x^2 - 2x - 3x + 6 = 0, giving x(x - 2) - 3(x - 2) = 0, that is (x - 2)(x - 3) = 0. Setting each factor to zero, x = 2 or x = 3. So the roots are 2 and 3.

What it is

Any quadratic ax^2 + bx + c = 0 can be solved using the quadratic formula.

Answer

Here a = 1, b = -4, c = 1. The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The discriminant b^2 - 4ac = (-4)^2 - 4(1)(1) = 16 - 4 = 12. So x = (4 +/- sqrt(12)) / 2 = (4 +/- 2sqrt(3)) / 2 = 2 +/- sqrt(3). So the roots are 2 + sqrt(3) and 2 - sqrt(3).

What it is

The discriminant D = b^2 - 4ac tells the nature of the roots of a quadratic equation.

Answer

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac. If D > 0 the roots are real and distinct; if D = 0 the roots are real and equal; and if D < 0 the equation has no real roots. For x^2 + 4x + 4 = 0, a = 1, b = 4, c = 4, so D = 4^2 - 4(1)(4) = 16 - 16 = 0. Since D = 0, the roots are real and equal (both equal to -2).

What it is

The nth term of an arithmetic progression is a + (n - 1)d, where a is the first term and d the common difference.

Answer

In this AP, the first term a = 3 and the common difference d = 7 - 3 = 4. The nth term is given by an = a + (n - 1)d. For the 15th term, a15 = 3 + (15 - 1) x 4 = 3 + 14 x 4 = 3 + 56 = 59. So the 15th term is 59.

What it is

The sum of the first n terms of an AP is (n/2)[2a + (n - 1)d].

Answer

Here a = 2 and d = 5 - 2 = 3, and n = 20. The sum is Sn = (n/2)[2a + (n - 1)d] = (20/2)[2 x 2 + (20 - 1) x 3] = 10[4 + 57] = 10 x 61 = 610. So the sum of the first 20 terms is 610.

What it is

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.

Answer

Two triangles are similar if they have the same shape, that is, their corresponding angles are equal and their corresponding sides are in the same ratio. The criteria for similarity are: AA (or AAA), when two angles of one triangle equal two angles of the other; SSS, when the three pairs of corresponding sides are in the same ratio; and SAS, when one angle equals the corresponding angle and the two sides including these angles are in the same ratio. Similar triangles are written with the symbol for similarity.

What it is

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

Answer

The basic proportionality theorem, also called Thales theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then it divides those two sides in the same ratio. For a triangle ABC with a line DE parallel to BC cutting AB at D and AC at E, this means AD/DB = AE/EC. The converse is also true: if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.

What it is

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

Answer

By the Pythagoras theorem, in a right-angled triangle, hypotenuse^2 = base^2 + height^2. Here hypotenuse^2 = 6^2 + 8^2 = 36 + 64 = 100, so the hypotenuse = sqrt(100) = 10 cm. Therefore the hypotenuse is 10 centimetres.

What it is

The distance between two points is found from the differences of their coordinates using the Pythagoras theorem.

Answer

The distance between two points (x1, y1) and (x2, y2) is given by sqrt((x2 - x1)^2 + (y2 - y1)^2). Here = sqrt((4 - 1)^2 + (6 - 2)^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. So the distance between the two points is 5 units.

What it is

The section formula gives the coordinates of a point dividing a line segment in a given ratio.

Answer

The section formula for a point dividing the join of (x1, y1) and (x2, y2) in the ratio m:n is ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n)). For the ratio 1:1 this becomes the midpoint formula ((x1 + x2)/2, (y1 + y2)/2). Here midpoint = ((2 + 6)/2, (3 + 7)/2) = (8/2, 10/2) = (4, 5). So the midpoint is (4, 5).

What it is

In a right triangle, the trigonometric ratios sine, cosine and tangent relate an angle to pairs of its sides.

Answer

In a right-angled triangle, for an acute angle A: the sine of A is the ratio of the side opposite to A to the hypotenuse (sin A = opposite/hypotenuse); the cosine of A is the ratio of the side adjacent to A to the hypotenuse (cos A = adjacent/hypotenuse); and the tangent of A is the ratio of the side opposite to A to the side adjacent to A (tan A = opposite/adjacent). Also, tan A = sin A / cos A. The reciprocals are cosec A, sec A and cot A.

What it is

The trig ratios of 0, 30, 45, 60 and 90 degrees have fixed standard values.

Answer

Using the standard values: sin 30 = 1/2 and cos 60 = 1/2, so sin 30 + cos 60 = 1/2 + 1/2 = 1. Also tan 45 = 1. So sin 30 + cos 60 = 1 and tan 45 = 1.

What it is

The fundamental trigonometric identity is sin^2 A + cos^2 A = 1.

Answer

The three fundamental trigonometric identities are: sin^2 A + cos^2 A = 1; 1 + tan^2 A = sec^2 A; and 1 + cot^2 A = cosec^2 A. Given sin A = 3/5, use sin^2 A + cos^2 A = 1: cos^2 A = 1 - sin^2 A = 1 - (3/5)^2 = 1 - 9/25 = 16/25. So cos A = sqrt(16/25) = 4/5 (taking the positive value for an acute angle).

What it is

Trigonometry finds unknown heights and distances using an angle of elevation or depression.

Answer

Let the height of the tower be h. The point is 30 m from the base, and the angle of elevation of the top is 45 degrees. Using tan(angle) = opposite/adjacent = height/distance: tan 45 = h/30. Since tan 45 = 1, we get 1 = h/30, so h = 30 m. Therefore the height of the tower is 30 metres.

What it is

A tangent to a circle touches it at exactly one point and is perpendicular to the radius at that point.

Answer

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact. A key theorem states that the tangent at any point of a circle is perpendicular to the radius drawn to the point of contact; that is, the radius and the tangent make a right angle (90 degrees) at the point of contact. A line that cuts the circle at two points is called a secant, not a tangent.

What it is

The lengths of the two tangents drawn from an external point to a circle are equal.

Answer

From a point lying outside a circle, exactly two tangents can be drawn to the circle. A key theorem states that the lengths of these two tangents drawn from an external point to a circle are equal. From a point on the circle, only one tangent can be drawn, and from a point inside the circle, no tangent can be drawn. These equal-tangent lengths are used in many geometry proofs and constructions.

What it is

The area of a sector of a circle is a fraction of the whole circle's area, proportional to its central angle.

Answer

The area of a sector = (central angle / 360) x pi r^2 = (90/360) x (22/7) x 7^2 = (1/4) x (22/7) x 49 = (1/4) x 22 x 7 = (1/4) x 154 = 38.5 cm^2. So the area of the sector is 38.5 square centimetres.

What it is

The mean of grouped data is found using the class marks and their frequencies.

Answer

For grouped data, the mean = sum of (frequency x class mark) / sum of frequencies. Compute the products: 2 x 10 = 20, 3 x 20 = 60, 5 x 30 = 150. Sum of products = 20 + 60 + 150 = 230. Sum of frequencies = 2 + 3 + 5 = 10. Mean = 230 / 10 = 23. So the mean of the data is 23.

What it is

Theoretical probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes.

Answer

When a die is rolled, the total number of equally likely outcomes is 6 (the numbers 1 to 6). The even numbers are 2, 4 and 6, so the number of favourable outcomes is 3. Therefore the probability of getting an even number = favourable outcomes / total outcomes = 3/6 = 1/2. So the probability is 1/2.