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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

The points P(1, −2) and Q(7, 1) lie on the circumference of a circle.Show that the centre of the circle lies on the line 4x + 2y = 15.
Three points have coordinates A(7, 4), B(19, 8) and C(k, 2k).Find the value of the constant k for which:a. C lies on the line that passes through the points A and Bb. angle CAB is 90°.
The perpendicular bisector of the line joining A(−10, 5) and B(−2, −1) intersects the x-axis at P and the y-axis at Q.a. Find the equation of the line PQ.b. Find the coordinates of P and Q.c.
Triangle ABC has vertices at A(−6, 3), B(3, 5) and C(1, −4). Show that triangle ABC is isosceles and find the area of this triangle.
The line x + ky + k2 = 0, where k is a constant, is a tangent to the curve y2 = 4x at the point P.Find, in terms of k, the coordinates of P.
A circle passes through the points (3, 2) and (7, 2) and has radius 2√2.Find the two possible equations for this circle.
The line y = mx + 1 intersects the circle x2 + y2 − 19x − 51 = 0 at the point P(5, 11).a. Find the coordinates of the point Q where the line meets the curve again.b. Find the equation of the
The line l1 has equation 2x + 5y = 10.The line l2 passes through the point A(−9, −6) and is perpendicular to the line l1.a. Find the equation of the line l2.b. Given that the lines l1 and l2
Triangle ABC has vertices at A(−7, 8), B(3, k) and C(8, 5). Given that AB = 2BC, find the value of k.
A is the point (4, −6) and B is the point (12, 10). The perpendicular bisector of AB intersects the x-axis at C and the y-axis at D. Find the length of CD.
A circle passes through the points O(0, 0), A(8, 4) and B(6, 6).Show that OA is a diameter of the circle and find the equation of this circle.
The diagram shows a triangle ABC in which A is (3, −2) and B is (15, 22). The gradients of AB, AC and BC are 2m, −2m and m respectively, where m is a positive constant.i. Find the gradient of AB
The diagram shows the points E, F and G lying on the line x + 2y = 16. The point G lies on the x-axis and EF = FG. The line FH is perpendicular to EG.Find the coordinates of E and F. E F H (5,-7) G X
P is the point (a, a − 2) and Q is the point (4 − 3a, −a).a. Find the gradient of the line PQ.b. Find the gradient of a line perpendicular to PQ.c. Given that the distance PQ is 10√5 , find
a. The design shown is made from four green circles and one orange circle.i. The radius of each green circle is 1 unit.Find the radius of the orange circle.ii. Use graphing software to draw the
The points A, B and C have coordinates A(2, 8), B(9, 7) and C(k, k − 2).a. Given that AB = BC, show that a possible value of k is 4 and find the other possible value of k.b. For the case where k =
Show that x2 + y2 − 6x + 2y = 6 can be written in the form (x − a)2 + (y − b)2 = r2, where a, b and r are constants to be found. Hence, write down the coordinates of the centre of the circle
The function f is such that f(x) = 3x − 2 for x ≥ 0.The function g is such that g(x) = 2x2 − 8 for x ≤ k, where k is a constant.a. Find the greatest value of k for which the composite
The line L1 passes through the points A(−6, 10) and B(6, 2). The line L2 is perpendicular to L1 and passes through the point C(−7, 2).a. Find the equation of the line L2.b. Find the coordinates
The diagram shows a rhombus ABCD. M is the midpoint of BD.a. Find the coordinates of M.b. Find the value of a, the value of b and the value of c.c. Find the perimeter of the rhombus.d. Find the area
The point A has coordinates (−1, 6) and the point B has coordinates (7, 2).i. Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx + c.ii. A point C on the
The coordinates of three points are A(−4, −1), B(8, −9) and C(k, 7).M is the midpoint of AB and MC is perpendicular to AB. Find the value of k.
The line y = x − 3 meets the curve y2 = 4x at the points A and B.a. Find the coordinates of the midpoint of AB.b. Find the length of the line segment AB.
A curve has equation xy = 12 + x and a line has equation y = kx − 9, where k is a constant.a. In the case where k = 2, find the coordinates of the points of intersection of the curve and the
ABCD is a trapezium with AB parallel to DC and angle BAD = 90°.a. Calculate the coordinates of D.b. Calculate the area of trapezium ABCD. B (3, 2) A (13, 17) D C(13,4) X
The equation of a circle is (x − 3)2 + (y + 2)2 = 25. Show that the point A(6, −6) lies on the circle and find the equation of the tangent to the circle at the point A.
The coordinates of A are (−3, 2) and the coordinates of C are (5, 6).The mid-point of AC is M and the perpendicular bisector of AC cuts the x-axis at B.i. Find the equation of MB and the
The point P is the reflection of the point (−2, 10) in the line 4x − 3y = 12.Find the coordinates of P.
In triangle ABC, the midpoints of the sides AB, BC and AC are (1, 4), (2, 0) and (−4, 1), respectively. Find the coordinates of points A, B and C.
The function f is such that f(x) = 2x − 3 for x ≥ k, where k is a constant. The function g is such that g(x) = x2 − 4 for x ≥ −4.a. Find the smallest value of k for which the composite
The line 2x + 5y = 20 cuts the x-axis at A and the y-axis at B. The point C is the midpoint of the line AB. Find the equation of the circle that has centre C and that passes through the points A and
The points A(1, −2) and B(5, 4) lie on a circle with centre C(6, p).a. Find the equation of the perpendicular bisector of the line segment AB.b. Use your answer to part a to find the value of p.c.
The coordinates of triangle ABC are A(−7, 3), B(3, −7) and C(8, 8). P is the foot of the perpendicular from B to AC.a. Find the equation of the line BP.b. Find the coordinates of P.c. Find the
The points P(−5, 6), Q(−3, 8) and R(3, 2) are joined to form a triangle.a. Show that angle PQR is a right angle.b. Find the equation of the circle that passes through the points P, Q and R.
The coordinates of triangle PQR are P(1, 1), Q(1, 8) and R(6, 6).a. Find the equation of the perpendicular bisectors of:i. PQ ii. PRb. Find the coordinates of the point that is equidistant from
The equation x2 + bx + c = 0 has roots −2 and 7.a. Find the value of b and the value of c.b. Using these values of b and c, find:i. The coordinates of the vertex of the curve y = x2 + bx + cii. The
Find the equation of the circle that passes through the points (7, 3) and (11, −1) and has its centre lying on the line 2x + y = 7.
The equation of a curve is xy = 12 and the equation of a line is 3x + y = k, where k is a constant.a. In the case where k = 20, the line intersects the curve at the points A and B.Find the midpoint
The equations of two of the sides of triangle ABC are x + 2y = 8 and 2x + y = 1. Given that A is the point (2, −3) and that angle ABC = 90°, find:a. The equation of the third sideb. The
The function f is such that f(x) = 3x − 7 for x ∈ ℝ.The function g is such thata. Find the value of x for which fg(x) = 5.b. Find f−1(x) and g−1(x).c. Show that the equation f−1(x) =
A circle passes through the points O(0, 0), P(3, 9) and Q(11, 11).Find the equation of the circle.
A circle has radius 10 units and passes through the point (5, −16). The x-axis is a tangent to the circle. Find the possible equations of the circle.
A is the point (−3, 6) and B is the point (9, −10).a. Find the equation of the line through A and B.b. Show that the perpendicular bisector of the line AB is 3x − 4y = 17.c. A circle
Find two straight lines whose x-intercepts differ by 7, whose y-intercepts differ by 5 and whose gradients differ by 2. Is your solution unique? Investigate further.
A curve has equation y = 12x − x2.a. Express 12x − x2 in the form a − (x + b)2, where a and b are constants to be determined.b. State the maximum value of 12x − x2.The function g is defined
a. Express 3x2 + 12x − 1 in the form a(x + b)2 + c, where a, b and c are constants.b. Write down the coordinates of the vertex of the curve y = 3x2 + 12x − 1.c. Find the set of values of k for
The equation of a circle is x2 + y2 − 8x + 4y + 4 = 0.a. Find the radius of the circle and the coordinates of its centre.b. Find the x-coordinates of the points where the circle crosses the x-axis,
The function f is such that f(x) = 2x + 1 for x ∈ ℝ.The function g is such that g(x) = 8 − ax − bx2 for x ≥ k, where a, b and k are constants.The function fg is such that fg(x) = 17 − 24x
A circle has centre (8, 3) and passes through the point P(13, 5).a. Find the equation of the circle.b. Find the equation of the tangent to the circle at the point P.Give your answer in the form ax +
A curve has equation y = 2 − 3x − x2.a. Express 2 − 3x − x2 in the form a −(x + b)2, where a and b are constants.b. Write down the coordinates of the maximum point on the curve.c. Find the
A circle, C, has equation x2 + y2 − 16x − 36 = 0.a. Find the coordinates of the centre of the circle.b. Find the radius of the circle.c. Find the coordinates of the points where the circle meets
A curve has equation xy = 20 and a line has equation x + 2y = k, where k is a constant.a. In the case where k = 14, the line intersects the curve at the points A and B.Find:i. The coordinates of the
The distributions of the heights of 1000 women and of 1000 men both produce normal curves, as shown. The mean height of the women is 160 cm and the mean height of the men is 180 cm.The heights of
Two normally distributed continuous random variables are X and Y. It is given that X ~N(1.5, 0.22 ) and that Y ~N(2.0, 0.52 ). On the same diagram, sketch graphs showing the probability density
The masses of all the different pies sold at a market are normally distributed with mean 400 g and standard deviation 61g. Find the probability that:a. The mass of a randomly selected pie is less
A fair coin is tossed 400 times. Given that it shows a head on more than 205 occasions, find an approximate value for the probability that it shows a head on fewer than 215 occasions.
A law firm has found that their assistants make, on average, one error on every 36 pages that they type. A random sample of 90 typed documents, with a mean of 62 pages per document, is selected.
The height of a female university student is normally distributed with mean 1.74m and standard deviation 12.3cm. Find the probability that:a. A randomly selected female student is between 1.71 and
An ordinary fair die is rolled 450 times. Given that a 6 is rolled on fewer than 80 occasions, find approximately the probability that a 6 is rolled on at least 70 occasions.
a. The following table shows the probability distribution for the random variable X.i. Show that k = 2.ii. Calculate E(X ) and Var(X ).iii. Find the probability that two independent observations of X
The random variable X has a geometric distribution such that P(X = 2)/P(X = 5) = 3 3/8. Find P(X ≤ 3).
The variable X has a normal distribution with mean μ and standard deviation σ. Given that P(X < 32.83) = 0.834 and that P(X ≥ 27.45) = 0.409, find the value of μ and of σ.
The length of time, in seconds, that it takes to transfer a photograph from a camera to a computer can be modelled by a normal distribution with mean 4.7 and variance 0.7225. Find the probability
The mass of a berry from a particular type of bush is normally distributed with mean 7.08 grams and standard deviation σ . It is known that 5% of the berries have a mass of exactly 12 grams or
The time taken, in minutes, to fit a new windscreen to a car is normally distributed with mean μ and standard deviation 16.32. Given that three-quarters of all windscreens are fitted in less than 45
The mid-day wind speed, in knots, at a coastal resort is normally distributed with mean 12.8 and standard deviation σ.a. Given that 15% of the recorded wind speeds are less than 10 knots, find the
A technical manual contains 10 pages of text, 7 pages of diagrams and 3 pages of colour illustrations. Four different pages are selected at random from the manual. Let X be the number of pages of
A fair six-sided die is numbered 1, 1, 2, 3, 5 and 8. The die is rolled twice and the two numbers obtained are added together to give the score, X.a. Find E(X).b. Given that the first number rolled
The following table shows the probability distribution table for the random variable Q.a. Find the value of x.b. Evaluate Var(Q). Camb P(Q= 3 x-2 x+1 18 x- 3 2.
Research shows that 17% of children are absent from school on at least five days during winter because of ill health. A random sample of 55 children is taken.a. Find the probability that exactly 10
The ratio of adult males to adult females living in a certain town is 17 :18, and 2/9 of these adults, independent of gender, do not have a driving license.a. Show that the probability that a
A fair eight-sided die is numbered 2, 2, 3, 3, 3, 4, 5 and 6. The die is rolled up to and including the roll on which the first 2 is obtained. Let X represent the number of times the die is rolled.a.
A student wishes to approximate the distribution of X ~B(240, p) by a continuous random variable Y that has a normal distribution.a. Find the values of p for which:i. Approximating X by Y can be
At a store, it is known that 1 out of every 9 customers uses a gift voucher in part-payment for purchases. A randomly selected sample of 72 customers is taken. Use a suitable approximation and
The following histogram summarises the total distance covered on each of 123 taxi journeys provided for customers of Jollicabs during the weekend.a. Find the upper boundary of the range of distances
The masses of 444 newborn babies in the USA and 888 newborn babies in the UK both produce normal curves. For the USA babies, μ = 3.4kg and σ = 200 g; for the UK babies, μ = 3.3kg and σ2 = 36100
X is normally distributed with mean 4 and variance 6. Find the probability that X takes a negative value.
The mass of a certain species of fish caught at sea is normally distributed with mean 5.73kg and variance 2.56 kg2. Find the probability that a randomly selected fish caught at sea has a mass that
The discrete random variable Y ~B(50, 0.6). Use a suitable approximation and continuity correction to find P(Y > 26).
A survey shows that 54% of parents believe mathematics to be the most important subject that their children study. Use a suitable approximation to find the probability that at least 30 out of a
In each of a series of independent trials, a success occurs with a constant probability of 0.9.a. The probability that none of the first n trials results in a failure is less than 0.3. Find the least
The values in two datasets, whose probability distributions are both normal curves, are summarised by the following totals:Σx2 = 35 000, Σx = 12 000 and n = 5000.Σy2 = 72000, Σy = 26 000 and n =
Given that X ~ N(µ, 4/9 µ2) where µ > 0, find P(X < 2µ).
The distance that children at a large school can hop in 15 minutes is normally distributed with mean 199m and variance 3700m2.a. Calculate an estimate of b, given that only 25% of the children hopped
A biased coin is tossed 160 times. The number of heads obtained, H, follows a binomial distribution where E(H) = 100. Find:a. The value of p and the variance of Hb. The approximate probability of
The following stem-and-leaf diagram shows the number of shots taken by 10 players to complete a round of golf.a. Given that the median number of shots is 74.5 and that the mean number of shots is
If T ~N(10, σ2) and P(T > 14.7) = 0.04, find the value of σ.
The daily percentage change in the value of a company’s shares is expected to be normally distributed with mean 0 and standard deviation 0.51. On how many of the next 365 working days should the
To conduct an experiment, a student must fit three capacitors into a circuit. He has eight to choose from but, unknown to him, two are damaged. He fits three randomly selected capacitors into the
One card is selected at random from each of 40 packs. Each pack contains 52 cards and includes 13 clubs. Let C be the number of clubs selected from the 40 packs.a. Show that the variance of C is
The random variable X is such that X ~N(82, 126).i. A value of X is chosen at random and rounded to the nearest whole number. Find the probability that this whole number is 84.ii. Five independent
It is given that V ~N(μ,13) and P(V < 15) = 0.75. Find the value of μ.
The masses, w grams, of a large sample of apples are normally distributed with mean 200 and variance 169. Given that the masses of 3413 apples are in the range 187 < w < 213, calculate an
In a large survey, 55% of the people questioned are in full-time employment. In a random sample of 80 of these people, find:a. The expected number in full-time employmentb. The standard deviation of
a. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.i. Find on how many days of the year (365 days) the daily sales can be
The variable W ~N(μ,σ2). Given that μ = 4σ and P(W < 83) = 0.95, find the value of μ and of σ.
The ages of the children in a gymnastics club are normally distributed with mean 15.2 years and standard deviation σ. Find the value of σ given that 30.5% of the children are less than 13.5 years
A company manufactures rubber and plastic washers in the ratio 4:1. The washers are randomly packed into boxes of 25.a. Find the probability that a randomly selected box contains:i. Exactly 21 rubber
V and W are continuous random variables. V ~N(9, 16) and W ~N(6, σ2). Find the value of σ, given that P(W < 8) = 2 × P(V < 8).
The variable Q ~N(μ, σ2). Given that P(Q < 1.288) = 0.281 and P(Q < 6.472) = 0.591, find the value of μ and of σ, and calculate the P(4 ≤ Q < 5).

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