11 questions · 100 marks · ≈ 2 hours
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By drawing suitable straight lines on the grid below, show, by shading, the region R defined by all of the inequalities y ≤ x x + y ≤ 8 y ≥ 2 Label the region R. y −2 −1 O 1 2 3 4 5 6 7 8 9 x −1 −2 The point P with coordinates (x, y) is a point of the region R. Given that x and y are integers and that P lies on the straight line with equation x − 2y + 2 = 0
write down the coordinates of all possible points P.
E Diagram NOT accurately drawn 12 cm D C M 10 cm A 8 cm B Figure 1 Figure 1 shows a right pyramid ABCDE with horizontal rectangular base ABCD and vertex E. AB =8 cm BC =10 cm EA = EB = EC = ED = 12 cm M is the midpoint of the base.
Calculate the total surface area, in cm² to 3 significant figures, of the pyramid. The point P is the midpoint of AB and the point Q is the midpoint of BC.
Calculate the size, in degrees to one decimal place, of ∠PEQ. Cosine rule: a 2 b 2 c 2 2bc cos A
The table below gives information about the lengths of time, in minutes, that 200 people waited for a train. Time t minutes Frequency 0<t≤5 28 5 < t ≤ 15 74 15 < t ≤ 35 42 35 < t ≤ 50 36 50 < t ≤ 75 20
Find the class interval that contains the median time.
Calculate an estimate for the mean time, in minutes, that these 200 people waited for the train.
On the grid opposite, draw a histogram for the information in the table. Frequency density 0 10 20 30 40 50 60 70 80 Time in minutes
| Time ( minutes) | Frequency |
|---|---|
| 28 | |
| 74 | |
| 42 | |
| 36 | |
| 20 |
Ella sells jars of jam, jars of honey and jars of chutney from her market stall. On Thursday, the total number of jars she sold was 320 55% of the jars were jars of jam. The ratio of the number of jars of honey to the number of jars of chutney that Ella sold on Thursday was 5 : 3
Calculate the number of jars of chutney that Ella sold on Thursday. On Friday, Ella sold 99 jars of strawberry jam. This was of the total number of jars of jam that she sold on Friday. The number of jars of jam that Ella sold increased from Thursday to Friday. Given that this increase = × the number of jars of jam that Ella sold on Thursday, n
find the value of n. Ella increases the price of a jar of jam from 3.20 euros to 3.50 euros.
Calculate the percentage increase in the price of a jar of jam. Ella increases the price of a jar of honey by 6.25% The price of a jar of honey after the price increase is 5.10 euros.
Calculate the price of a jar of honey before the price increase.
On the grid opposite, trapezium B is the image of trapezium A under a single transformation.
Describe fully the single transformation. Trapezium C is the image of trapezium A under a reflection in the line with equation x = −1
On the grid opposite, draw and label trapezium C. Trapezium A is transformed to trapezium D under the transformation with matrix M where 2 0 M
On the grid opposite, draw and label trapezium D. Trapezium D is transformed to trapezium B under the transformation with matrix N.
Find matrix N.
[The inverse of matrix is ]
The point A has coordinates (5, 4) and the point B has coordinates (−7, −1) → 3 The point C is such that BC =
Find the coordinates of the point C. The point D is such that ABCD is a parallelogram with diagonals AC and BD. The length of BC is 5 cm.
Find the area, in cm², of the parallelogram ABCD. Cosine rule : a 2 b 2 c 2 2bc cos A Area of triangle 1 ab sin C
Here is the equation of a curve
Complete the table of values for , giving your values of to 2 decimal places where necessary.
On the grid opposite, plot the points from your completed table and join them to form a smooth curve.
Use your graph to estimate, to one decimal place, the solutions in the interval −2 ≤ x ≤ 2 of the equation x³ − 3x − 1 = 0.5
By drawing a suitable straight line on the grid, find an estimate, to one significant figure, of the gradient of the curve at the point where x = 0.5
Some college students were each asked which of the subjects Mathematics (M), Physics (P) and Chemistry (C) they were studying. Here is some information about their answers. n(M ∩ P) = 21 n(M ∩ C ) = 24 n(P ∩ C ) = 25 n(M ) = 43 n(P) = 50 n([M ∪ P ∪ C ]ʹ) = 25 n([M ∪ P]ʹ ∩ C ) = 8 n(M ∩ P ∩ C ) = x , where x is a positive integer.
Use all the given information to complete the Venn diagram on the opposite page, giving the number of elements in each appropriate subset, in terms of x where necessary. Given that n(C ) = 40
find the total number of college students that were asked. One of these college students is to be chosen at random. Given that the college student studies Chemistry …………… , ……………
find the probability that this student also studies Physics. E M P x C
Figure 2 Figure 2 shows a circle ABCDE with centre O. ∠BAO = 18° ∠AOD = 120° The area of segment BCD, shown shaded in Figure 2, is T cm² Given that the perimeter of the sector AODE is 5(3 + π) cm, calculate the value, to one decimal place, of T. Area of triangle 2 ab sin C
Figure not shown in this reconstruction — the printed paper includes a diagram (not drawn accurately). Labels in the original figure: B, C, O, A, D, E · Figure values: 18°, 120°.
P A O Diagram NOT accurately drawn C Q B T Figure 3 → → Figure 3 shows the triangle OAB in which OA = 4a and OB = 6b C is the point on AB such that AC : CB = 3 : 1 →
Find and simplify an expression for OC in terms of a and b. The point T is such that ABT is a straight line and AC = CT. → 3→ The point P is such that OP = OA Q is the point on OB such that the points P, Q and T are collinear. Given that OQ : QB = n : 1
find the value of n.
The three functions, f, g and h, are defined as f : x ↦ 4x3 + 4x2 − 5x − 3 x−7 g:xα 3 − 2x h : x ↦ 2x + 1
Write down the value of x that must be excluded from any domain of g
Find g(2)
Express the inverse function g−1 in the form g−1(x) = …
Solve the equation g(x) = h(x)
Use the factor theorem to show that (2x + 3) is a factor of f (x)
Hence solve the equation f (x) = 0 Show clear algebraic working.