No questions found
Sancha flew from Santiago to Paris, a distance of 11 585 km. The average speed of the flight was 852.9 km/h.
(a) Find the length of time for the flight. Give your answer in hours and minutes.
Answer(a) .............................. h .................. min [3]
(b) The journey back from Paris to Santiago took 14 hours 30 minutes. The plane left Paris at 23 20. The local time in Santiago is 6 hours behind the local time in Paris.
Find the local time this plane arrived in Santiago.
Answer(b) ......................................................... [2]
(c) Find the overall average speed for the total journey from Santiago to Paris and back to Santiago.
Answer(c) ......................................................... km/h [3]
503
515
520
(a) (i) Rotate triangle $A$ through $90^\circ$ anticlockwise about the origin.
Label the image $C$.
[2]
(ii) Reflect triangle $C$ in the $x$-axis.
Label the image $D$.
[2]
(iii) Describe fully the \textit{single} transformation that is equivalent to a rotation through $90^\circ$ anticlockwise about the origin followed by a reflection in the $x$-axis.
\textit{Answer(a)(iii)} ..............................................................
.............................................................................................................
.............................................................................................................
[2]
(b) Describe fully the \textit{single} transformation that maps triangle $A$ onto triangle $B$.
\textit{Answer(b)} .........................................................................
.............................................................................................................
.............................................................................................................
[3]
574
593
511
Sinitta makes necklaces.
Each necklace costs Sinitta $56 to make.
They are sold through an internet shop at a selling price of $80.
(a) (i) The internet shop charges her 7% of the selling price.
Find the amount that Sinitta receives from the shop for a necklace.
Answer (a)(i) $ ............................................................... [2]
(ii) The shop increases the charge to 12% of the selling price of $80.
Calculate the percentage reduction in Sinitta’s \textit{profit}.
Answer (a)(ii) .......................................... \% [4]
(b) Sinitta also makes silver rings.
Each ring contains 22 g of silver.
In the last year the cost of silver has increased by 8% to $143.10 per 100 grams.
(i) Find the cost of each 100 g of silver before the increase.
Answer (b)(i) $ ............................................................... [2]
(ii) Find the increase in the cost of the silver in a ring.
Answer (b)(ii) $ ............................................................... [2]
529
578
523
P is the point (0, 4), Q is the point (6, 0) and R is the point (2, 7).
(a) S is the point such that $\overrightarrow{RS} = \overrightarrow{QP}$.
Find the co-ordinates of S.
Answer(a) (\text{....................., .....................}) [2]
(b) Calculate $|\overrightarrow{QP}|$.
Answer(b) .................................................... [2]
(c) Find the equation of the line PQ.
Answer(c) .................................................... [2]
(d) Write down the co-ordinates of N, the midpoint of PQ.
Answer(d) (\text{....................., .....................}) [1]
(e) Find the equation of the perpendicular bisector of PQ.
Answer(e) .................................................... [3]
(f) A and B are points on the perpendicular bisector of PQ such that $AN \neq BN$.
What is the mathematical name given to the quadrilateral PAQB?
Answer(f) .................................................... [1]
561
597
567
The diagram shows a rectangle, with sides 40 cm and 30 cm, made from a metal sheet.
A square of side $x$ cm is cut from each of the four corners of the rectangle.
The remaining shape is folded up to make a rectangular open box with $ABCD$ as the base.
The height of the box is $x$ cm.
(a) Show that the volume of the box is $1200x - 140x^2 + 4x^3$. [3]
(b) On the diagram, sketch the graph of $y = 1200x - 140x^2 + 4x^3$ for $0 \leq x \leq 25$. [2]
(c) Solve the equation $1200x - 140x^2 + 4x^3 = 2000$.
\text{Answer(c) } x = \text{............................... or } x = \text{............................... or } x = \text{..............................} [3]
(d) Which solution to part (c) is not a possible value of $x$ when the volume of the box is $2000 \text{ cm}^3$? Give a reason for your answer.
\text{Answer(d) ...........................................................................................................................} [1]
(e) What is the maximum volume of the box?
For this volume what is the length of the box?
\text{Answer(e) Maximum volume = .................................................. cm}^3 \text{ }
\text{Length = ............................................... cm} [2]
507
543
548
(a) (i) Find an expression for the $n$th term of this sequence.
2, 6, 10, 14, ...
Answer(a)(i) .............................................................. [2]
(ii) Use your answer to part (a)(i) to find an expression for $u$, the $n$th term of this sequence.
$2 \times 10^{2}$, $6 \times 10^{3}$, $10 \times 10^{4}$, $14 \times 10^{5}$, ...
Answer(a)(ii) $u = $ .............................................................. [1]
(b) The $n$th term, $t$, of another sequence, is given by $t = 2 \times 10^{(3 - 2n)}$.
(i) Write down the first 4 terms in this sequence, giving your answers in standard form.
Answer(b)(i) $............................... , ............................... , ............................... , ...............................$ [2]
(ii) Using your answer to part (a)(ii), find and simplify an expression for $\frac{u}{t}$.
Answer(b)(ii) .............................................................. [3]
525
505
511
The diagram shows a field $ABCD$ with a path from $A$ to $C$.
$AC = 150\text{m}$, $AD = 120\text{m}$ and $CD = 235\text{m}$.
Angle $ABC = 90^{\circ}$, angle $BAC = 55^{\circ}$ and the bearing of $B$ from $A$ is $070^{\circ}$.
(a) Calculate the length of $AB$.
Answer(a) .......................................................... $\text{m}$ [2]
(b) Calculate the bearing of $D$ from $A$.
Answer(b) .......................................................... [4]
(c) Calculate the area of the field $ABCD$.
Answer(c) .......................................................... $\text{m}^2$ [3]
546
529
514
100 light bulbs were tested. The length of life, $t$, in thousands of hours was recorded. The results are shown in this table.
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Length of life (}t\text{) in thousands of hours} & 4 < t \leq 5 & 5 < t \leq 6 & 6 < t \leq 7 & 7 < t \leq 8 & 8 < t \leq 9 & 9 < t \leq 10 & 10 < t \leq 12 \\ \hline \text{Frequency} & 8 & 21 & 31 & 23 & 10 & 5 & 2 \\ \hline \end{array}\]
(a) Calculate an estimate of the mean value of $t$.
Answer(a) ........................................................... [2]
(b) Draw a cumulative frequency curve for the length of life of the light bulbs.
[5]
(c) Use your graph to estimate
(i) the number of light bulbs that lasted longer than 8500 hours,
Answer(c)(i) ........................................................... [2]
(ii) the interquartile range.
Answer(c)(ii) ................................................ hours [2]
578
597
515
(a)
The diagram shows two similar triangles $EAB$ and $ECD$.
$AB = 20 \text{ cm}$, $CD = 15 \text{ cm}$, $AC = 40 \text{ cm}$ and angle $CAB = 90^\circ$.
(i) Show that $EC = 120 \text{ cm}$. [2]
(ii) Find $ED$. [2]
Answer (a)(ii) .................................................. cm [2]
(iii) Find $DB$. [2]
Answer (a)(iii) .................................................. cm [2]
(b)
The diagram shows an open waste paper bin made from metal.
The radius of the circular top is $20 \text{ cm}$.
The radius of the circular base is $15 \text{ cm}$.
The perpendicular height of the bin is $40 \text{ cm}$.
Using answers from part (a), calculate
(i) the volume of the waste paper bin. [3]
Answer (b)(i) .................................................. \text{cm}^3 [3]
(ii) the area of metal needed to make the bin. [4]
Answer (b)(ii) .................................................. \text{cm}^2 [4]
544
538
511
Tricia has 2 bags.
In the first bag there are 6 white balls and 4 red balls.
In the second bag there are 4 blue balls, 3 white balls and 2 red balls.
She takes a ball at random out of the first bag.
She then takes a ball at random out of the second bag.
(a) Complete the tree diagram to show the probability of all the possible outcomes for the two balls. [2]
(b) Calculate the probability that Tricia’s two balls are
(i) both white, Answer(b)(i) ............................................... [2]
(ii) one white and one red, Answer(b)(ii) ............................................... [3]
(iii) of different colours. Answer(b)(iii) ............................................... [3]
549
581
573
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $x = -6$ and $x = 4$. [3]
(b) Write down the equations of the asymptotes of the graph of $y = f(x)$.
Answer(b) ..............................................................
.............................................................. [2]
(c) Find the range of values for $y$ when $x \geq 0$.
Answer(c) ..................................................................... [2]
(d) On this diagram, sketch the graph of $y = \left| \frac{(1 - 2x)}{(x + 3)} \right|$. [2]
(e) Solve $\left| \frac{(1 - 2x)}{(x + 3)} \right| = 6$.
Answer(e) $x = .........................$ or $x = ......................$ [2]
514
581
588
Given $f(x) = 3x - 1$ and $g(x) = 4 - 2x$:
(a) Find
(i) $g(3)$,
Answer(a)(i) $\text{.........................}$ [1]
(ii) $f(g(3))$.
Answer(a)(ii) $\text{.........................}$ [1]
(b) Find and simplify expressions for
(i) $g(f(x))$,
Answer(b)(i) $\text{.........................}$ [2]
(ii) $g^{-1}(x)$,
Answer(b)(ii) $\text{.........................}$ [2]
(iii) $\frac{2}{f(x)} - \frac{3}{g(x)}$.
Answer(b)(iii) $\text{.........................}$ [3]
547
600
543
