A BIT OF AS MATHS EACH DAY (C1, C2 AND A LITTLE BIT OF S1)
MONDAY / TUESDAY / WEDNESDAY / THURSDAY / FRIDAY / SATURDAY / SUNDAY11th April / 24th March / 25th March / 26th March / 27th March
BACK TO SCHOOL!!
The three events, A, B and C are defined in the same sample space.
The events A and C are mutually exclusive.
The events A and B are independent.
Given that P(A) = 2/5, P(C) = 1/3 and
P(A B) = 5/8 find:
(a) P(A C)
(b) P(B) / HAPPY EASTER!
Get revising!
A little bit, often, makes a big difference
/ MAUNDY THURSDAY
Find the set of values for which:
2x + y = 7 and
x2 + 2y = 19 / GOOD FRIDAY
In the binomial expansion of in descending powers of x, find, in their simplest terms…
(a) the first two terms
(b) the term independent of x. / EASTER SUNDAY
The following table summarises the results of a sales managers analysis of the amounts to the nearest £ of a sample of 750 invoices.
Amount (£), x / Number, f
0-9 / 50
10-19 / 204
20-49 / 165
50-99 / 139
100-149 / 75
150-199 / 62
200-499 / 46
500-749 / 9
Let x represent the midpoint of each class.
The data is coded using the formula with ∑fy = 5021 and ∑fy2 = 115773.5
(a) Using these values, or otherwise, find estimates of the mean and standard deviation.
(b) Explain why the mean and standard deviation may not be the best summary statistics to use with these data.
(c) Calculate estimates of alternative summary statistics which could be used by the sales manager.
(d) Use these estimates to justify your explanation to part (b)
28th March / 29th March / 30th March / 31st March / 1st April / 2nd April / 3rd April
Given that f(x) = x2 – 6x + 18, x≥0,
(a) Express f(x) in the form (x-a)2 + b, where a and b are constants.
The curve C with equation y = f(x), meets the y-axis at P and has a minimum point at Q.
(b) Sketch the graph of C, showing the coordinates of P and Q.
The line y = 41 meets C at the point R.
(c) Find the x-coordinate of R giving your answer in the form p + q√2 where p and q are integers. / (a) (i) Find the value of p for which
(ii) Hence solve the equation
(b) Use logarithms to solve the equation
,
giving your value of x to 4 d.p.
(c) It is given that
Express x in terms of a, giving tour answer in a form not involving logarithms. / (a) The roots of the equation
kx2 + 3x + k = 0 are equal.
Find the possible values of k.
(b) Hence find all the possible solutions to the equation in part (a)
(c) Another equation,
mx2 + 3x + n = 0, also has equal roots and the sum of the two constants, m and n, is given by
m + n = 15/4.
Determine the values of m and n. / (a) Factorise completely
f(x) = x4 – 5x2 + 4.
The diagrams shows the graph of y = f(x).
(b) Write down the coordinates of the points where the graph meets the x-axis.
(c) Calculate the total area of the regions R, S and T, the area between the graph and the x-axis. / A curve is given by the equation
y = x2 + 3x – 10.
(a) Show that the point A, (3, 8), lies on the curve.
(b) A tangent is drawn to the curve at point A, (3, 8). Find the gradient of the tangent and the normal to the curve at this point.
(c) Find the equation of the tangent and the normal to the curve at the point A, giving the equations in the form ax + by + c + 0.
(d) The two lines intercept the
y-axis at B and C. Find the area of triangle BAC. / Pieces made by a certain lathe are subject to three kinds of defects, A, B and C.
A sample of 1000 pieces was inspected and yielded the following results:
- 3.1% had a type A defect
- 3.7% had a type B defect
- 4.2% had a type C defect
- 1.1% had both type A and type B defects
- 1.3% had both type B and type C defects
- 1.0% had both type A and type C defects
- 0.6% had all three types of defect.
(b) What percentage had none of these defects?
(c) Calculate how many pieces had at least one defect.
(d) Find the percentage that had not more than one defect.
A piece is selected at random from this sample.
(e) Given that it had only one defect, find the probability that it was a type A defect.
4th April / 5th April / 6th April / 7th April / 8th April / 9th April / 10th April
Find:
/
The diagram shows a block of wood in the shape of a prism with a right angled triangular cross-section.
The total surface area of the five faces is 144cm2.
(a) Show that xy + x2 = 12 and hence show that the volume of the block, V cm3, is given by
V = 72x – 6x3.
(b) Find the maximum value of V, justifying that this is the maximum value. / The diagram shows part of the curve
y=p cos qx – r, where p, q & r are positive integers and x is measured in radians. The three turning points are (0,1), (π/2,-7) and (π,1).
(i) Find the values of p, q and r.
(ii) Find, in radians to 1 d.p., the coordinates of the points where the graph cuts the x-axis in the diagram. / An arithmetic series has 1st term a and common difference d.
(a) Prove that the sum to n terms of the series is ½n[2a + (n-1)d]
Sean repays a loan over a period of n months. His monthly repayments form an arithmetic sequence. He repays £149 in the 1st month, £147 in the 2nd £145 in the 3rd and so on. He makes his final repayment in the nth month, where n>21.
(b) find the amount Sean repays in the 21st month.
Over n months he repays £5000.
(c) Form a quadratic equation in n and show that it can be written as
n2 + 150n + 5000 = 0
(d) Solve the equation in part (c)
(e) State, with a reason, which of the solutions is not a sensible solution to the repayment problem. / A geometric series has a first term a and a common ratio r.
(a) Prove that
(b) Hence prove that explaining the range of values for which this is valid.
The second term of a series is 4 and the sum to infinity of the series is 25.
(c) Show that 25r2 – 25r + 4 = 0.
(d) Find the two possible values of r.
Given that r takes the larger of its two possible values,
(e) Show that the Sn of the first n terms is given by Sn = 25(1 – rn)
(f) Find the smallest value of n for which Sn exceeds 24. / A local authority is investigating the cost of reconditioning its incinerators. Data from 10 randomly chosen incinerators were collected. The variables monitored were the operating time x (in thousands of hours) since last reconditioning and the reconditioning cost y (in £1000). None of the incinerators had been used for more than 3000 hours since last reconditioning.
The data are summarised below,
Σx = 25.0, Σx2 = 65.68, Σy = 50.0, Σy2 = 260.48, Σxy = 130.64.
(a) Find Sxx, Sxy, Syy
(b) Calculate the product moment correlation coefficient between x and y.
(c) Explain why this value might support the fitting of a linear regression model of the form y = a + bx.
(d) Find the values of a and b.
(e) Give an interpretation of a.
(f) Estimate (i) the reconditioning cost for an operating time of 2400 hours,
(ii) the financial effect of an increase of 1500 hours in operating time.
(g) Suggest why the authority might be cautious about making a prediction of the reconditioning cost of an incinerator which had been operating for 4500 hours since its last reconditioning.
SOLUTION TO EACH QUESTION WILL BE POSTED ON MY TWITTER FEED (@mrchadburn) AND SCHOOLS (@ASCHSMaths) THE DAY AFTER IT APPEARS ON THE CALENDAR