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25 students each record the number of logic problems they solve in one hour. The table shows the results.
[Table_1]
(a) Find
(i) the range
..................................................... [1]
(ii) the mode
..................................................... [1]
(iii) the median
..................................................... [1]
(iv) the interquartile range
..................................................... [2]
(v) the mean.
..................................................... [2]
(b) Nabile draws a pie chart. Calculate the angle that represents 7 logic problems solved.
..................................................... [2]
(c) Shabana draws a bar chart using these results. The bar that represents 4 logic problems solved has a height of 4.5 cm. Calculate the height of the bar that represents 5 logic problems solved.
..................................................... cm [2]
558
568
591
(a) Calculate the volume of each shape.
(i) A cuboid with a square base of side 5 cm and height 3 cm.
.......................................... cm^3 [2]
(ii) A sphere with radius 4 cm.
.......................................... cm^3 [2]
(b) A cylinder has volume 120 cm^3 and height 6 cm.
Calculate its radius.
.......................................... cm [2]
(c) A cone has volume 120 cm^3 and height 6 cm.
Calculate the length of its sloping edge.
.......................................... cm [3]
543
564
527
Given $f(x) = |\cos x^\circ|$ for $0 \leq x \leq 360$.
(a) On the diagram, sketch the graph of $y = f(x)$. [2]
(b) Find the zeros of $f(x)$. [2]
...................................................
(c)
(i) Solve the equation $f(x) = 0.5$. [2]
.......................................................................
(ii) Solve the inequality $f(x) < 0.5$. [2]
..............................................................................
(iii) On the diagram, shade the regions that satisfy the inequalities $y < 0.5$ and $y > f(x)$. [1]
(d) The equation $f(x) = k$ has four solutions.
Complete the statement to show the range of possible values of $k$.
................. $<$ $k$ $<$ ................. [1]
554
512
551
(a) Alex invests $650 at a rate of 2\% per year compound interest.
(i) Calculate the value of this investment at the end of 10 years.
$ \text{\$ ................................................} $ [2]
(ii) Calculate the number of complete years it takes for the value of this investment of $650 to be first greater than $1000.
................................................ [4]
(b) 2 years ago Chris invested $x at a rate of 3\% per year compound interest.
The value of this investment is now $607.90 correct to the nearest cent.
Calculate the value of x.
x = ................................................ [2]
(c) Sam invested $200 at a rate of $r\%$ per year compound interest.
At the end of 18 years, the value of this investment is $247.90 correct to the nearest cent.
Find the value of r.
$r = \text{................................................} $ [3]
565
572
515
(a) The equation of line $L$ is $y = 4x + 7$.
(i) Write down the gradient of line $L$.
.............................................. [1]
(ii) Write down the coordinates of the point where line $L$ cuts the $y$-axis.
(..................... , .....................) [1]
(b) $A$ is the point $(3, 1)$ and $B$ is the point $(11, 5)$.
(i) Calculate the length of $AB$.
.............................................. [3]
(ii) Find the equation of the perpendicular bisector of the line $AB$.
Give your answer in the form $y = mx + c$.
$y = ..............................................$ [5]
587
556
511
Given functions:
\( f(x) = 3 - 2x \), \( g(x) = x + 1 \), \( h(x) = (x + 1)^2 \), \( j(x) = \tan x^\circ \) for \( 0 < x < 180 \).
(a) Find \( f(-1.5) \). [1]
(b) Find \( h(h(2)) \). [2]
(c) Find \( g(f(x)) \), giving your answer in its simplest form. [2]
(d) Find \( f^{-1}(x) \).
\( f^{-1}(x) = \text{..................................................} \) [2]
(e) Find \( x \) when \( j^{-1}(x) = 75 \). [2]
547
559
500
(a)
The diagram shows a shape $AVBCD$.
$ABCD$ is a square of side $12 ext{ cm}$.
$M$ is the mid-point of $AB$ and $N$ is the mid-point of $DC$.
Triangle $AVB$ is isosceles with $AV = VB = 10 ext{ cm}$.
The arc $CD$ is part of a circle with centre $M$.
(i) Calculate angle $CMN$.
Angle $CMN = .........................................................$ [2]
(ii) Calculate the length of $CM$.
$CM = ......................................................... ext{ cm}$ [2]
(iii) Calculate the perimeter of the shape $AVBCD$.
......................................................... cm [3]
(iv) Calculate the area of the shape $AVBCD$.
......................................................... ext{ cm}^2 [5]
(b) Two solids are mathematically similar with volumes $240 ext{ cm}^3$ and $810 ext{ cm}^3$.
The surface area of the larger solid is $558 ext{ cm}^2$.
Calculate the surface area of the smaller solid.
......................................................... ext{ cm}^2 [3]
567
517
511
(a) The cost of a television is $t and the cost of a computer is $c. The total cost of 2 televisions and 1 computer is $1470. The total cost of 3 televisions and 2 computers is $2480.
Use simultaneous equations to find the cost of a television. You must show all your working.
$ .................................................. [4]
(b) Jono spends $9.69 on bags of potatoes. When the cost of a bag is $x$ cents he can buy 2 more bags than when the cost of a bag is $(x + 6)$ cents.
(i) Show that $x^2 + 6x - 2907 = 0$. [3]
(ii) Solve the equation $x^2 + 6x - 2907 = 0$.
$x = ..............$ or $x = ..............$ [2]
(iii) Find the number of bags Jono can buy for $9.69 when the cost of one bag is $x$ cents.
.................................................. [1]
547
573
525
(a) $B$ is due east of $A$.
Find the bearing of $A$ from $C$. ..................................................... [2]
(b) Calculate the area of triangle $ABC$.
...................................... $m^2$ [2]
(c) Calculate angle $CAD$.
Angle $CAD = ....................................................$ [4]
(d) Calculate the length of the straight line $BD$
.................................................... $m$ [3]
561
530
563
(a) (i) In the Venn diagram, shade the region $P \cup Q'$.
[1]
(ii) Use set notation to describe the shaded region in the Venn diagram.
..................................................... [1]
(b) 20 students are asked if they like swimming $(S)$ and if they like tennis $(T)$.
The Venn diagram shows the results.
(i) How many students like swimming or tennis but not both?
..................................................... [1]
(ii) Find $n(S \cup T)$.
..................................................... [1]
(iii) One of the 20 students is chosen at random.
Find the probability that this student likes swimming and tennis.
..................................................... [1]
(iv) Two of the 20 students are chosen at random.
Find the probability that they both like tennis.
..................................................... [2]
(v) Two of the students who like swimming are chosen at random.
Find the probability that
(a) they both like tennis
..................................................... [2]
(b) one likes swimming only and one likes swimming and tennis.
..................................................... [3]
581
530
567
(a) (i) Write 0.000 021 in standard form.
.............................................. [1]
(ii) Calculate $\left(7.3 \times 10^{-11}\right) \times \left(4.7 \times 10^{-7}\right)$, giving your answer in standard form.
.............................................. [1]
(iii) Calculate $\left(3.2 \times 10^{-200}\right) \div \left(4 \times 10^{-100}\right)$, giving your answer in standard form.
.............................................. [2]
(iv) Simplify $\left(5 \times 10^9\right)^2$, giving your answer in standard form.
.............................................. [2]
(b) $y = 10^x$
Write $x$ in terms of $y$.
$x = ..............................................$ [1]
(c) Solve $7^x = 14$.
$x = ..............................................$ [1]
(d) $\log y = 1 + 3 \log x - \frac{1}{2} \log w$
Find $y$ in terms of $x$ and $w$.
$y = ..............................................$ [4]
541
574
547
