No questions found
(a) Translate triangle $T$ by the vector $\begin{pmatrix} -2 \\ 2 \end{pmatrix}$. [2]
(b) Reflect triangle $T$ in the line $y = 0.5$. [2]
(c) Describe fully the single transformation that maps triangle $P$ onto triangle $T$.
............................................................................................................................
............................................................................................................................ [3]
(d) Enlarge triangle $P$ with scale factor $-2$, centre $(3, -1)$. [2]
547
553
534
(a) The cumulative frequency curve shows the marks for 300 students in a history test.
(i) Find an estimate for the median. ....................................................... [1]
(ii) Estimate the number of students with a mark of more than 20. ....................................................... [2]
(iii) 70% of the students pass the test. Find the pass mark. ....................................................... [2]
(b) The table shows the marks for 100 students in a geography test.
[Table_1]
Calculate an estimate of the mean.
....................................................... [2]
(c) The table shows the marks for 9 students in chemistry and in physics.
[Table_2]
(i) Find the equation of the regression line for $y$ in terms of $x$.
$$y = \text{.......................................................}$$ [2]
(ii) What type of correlation is seen in this data?
....................................................... [1]
(iii) Use your answer to part (c)(i) to estimate the physics mark for a student with a mark of 30 in chemistry.
....................................................... [1]
521
505
504
Given \( f(x) = 2x + 4 - \frac{1}{x^2} \)
(a) On the diagram, sketch the graph of \( y = f(x) \) for values of \( x \) between \(-3\) and \(1\). [3]
(b) Write down the equation of the asymptote of the graph.
................................................ [1]
(c) Find the coordinates of the local maximum.
(.......................... , ..........................) [1]
(d) \( g(x) = x^3 - 5x \) for \(-3 \leq x \leq 1\).
Solve \( f(x) \leq g(x) \).
................................................ [4]
554
582
579
(a) $216 is shared in the ratio \( 5 : 1 \).
Work out the larger share.
$ ................................................
(b) Luis shares some money between Ali, Betty and Clare in the ratio \( 3 : 4 : 6 \). Ali receives $171.
Find the total amount of money Luis shared.
$ ................................................
(c) Farima invests $1400 in a savings account paying simple interest at a rate of 2.5\% per year.
Calculate the total amount in the account at the end of 3 years.
$ ................................................
(d) Emir invests $3000 at a rate of 2\% per year compound interest.
(i) Calculate the value of Emir’s investment at the end of 4 years.
$ ................................................
(ii) Find the number of complete years until Emir’s investment is first worth more than $4000.
................................................
543
593
532
A sequence of patterns is made using grey tiles and white tiles.
(a) Complete the table.
[Table_1]
Pattern number | 1 | 2 | 3 | 4 | n
Number of grey tiles | 6 | 10 | | |
Number of white tiles | 0 | 2 | | |
[6]
(b) Find and simplify an expression for the total number of tiles in Pattern $n$.
.................................................. [1]
(c) Pattern $k$ has a total of 600 tiles.
Find the number of grey tiles in Pattern $k$.
.................................................. [4]
(d) The tiles in a pattern are put in a bag.
The probability of taking a grey tile from the bag at random is $\frac{5}{12}$.
A tile is taken from the bag at random and replaced. This is repeated 3 times.
Find the probability that all 3 tiles are white.
.................................................. [2]
(e) All the grey tiles from Pattern 4 are put in a bag.
Two tiles are taken from the bag at random without replacement.
Find the probability that one tile came from a corner of the pattern and the other did not.
.................................................. [3]
599
598
545
(a) The diagram shows a circle, centre $O$, with radius 5 cm.
$BA$ and $BC$ are tangents to the circle at $A$ and $C$.
Angle $ABC = 30^{\circ}$.
Calculate the area of the shaded minor segment.
........................................ $\text{cm}^2$ [4]
(b) The circle, centre $O$, has radius 12 cm.
Angle $DOE = 40^{\circ}$.
The minor sector $DOE$ is removed.
The major sector is formed into a cone by joining $OD$ to $OE$.
Calculate the height of the cone.
........................................ cm [5]
583
514
567
Abbi makes wooden boards in three sizes, small, medium and large.
They are all cuboids.
The medium board has height 2 cm, width 23 cm and length 50 cm.
(a) Calculate the volume of the medium board.
........................................... $\text{cm}^3$ [2]
(b) The small board is mathematically similar to the large board.
The small board has a volume of $287.5 \text{ cm}^3$ and a height of $1.15 \text{ cm}$.
The large board has a volume of $18400 \text{ cm}^3$.
(i) Find the height of the large board.
............................................ cm [3]
(ii) Is the medium board mathematically similar to the large board?
Explain how you decide.
......................... because ...........................................................................................................
............................................................................................................................... [3]
500
591
583
(a) \( A \) is the point \((-11, 7)\) and \( B \) is the point \((8, -13)\).
Find the length of \( AB \).
................................................... \([3]\)
(b) \( P \) is the point \((2, -5)\) and \( Q \) is the point \((6, 11)\).
Line \( L \) is perpendicular to \( PQ \) and crosses \( PQ \) at point \( R \).
The ratio \( PR : RQ = 3 : 1 \).
Find the equation of line \( L \).
................................................... \([6]\)
529
585
573
(a) \; f(x) = 2x + 3 \quad g(x) = 2 - 4x \quad h(x) = 3^x
(i) \; \text{Find} \; f(5). \; \text{................................................ [1]}
(ii) \; \text{Find and simplify} \; g(f(x)). \; \text{................................................ [2]}
(iii) \; \text{Find} \; g^{-1}(x). \; g^{-1}(x) = \; \text{................................................ [2]}
(iv) \; \text{Solve} \; h(x) = 48. \; \text{................................................ [2]}
(b)
(i) \; \text{The diagram shows a sketch of the graph of} \; y = j(x).
\text{On the same diagram, sketch the graph of} \; y = j(x + 2). \; \text{[1]}
(ii) \; \text{The diagram shows the graphs of} \; y = k(x) \; \text{and} \; y = m(x).
\text{Write} \; k(x) \; \text{in terms of} \; m(x). \; k(x) = \; \text{................................................ [1]}
560
579
510
(a) Simplify fully. $$\frac{4x^2y}{3} \div \frac{x}{12y}$$
(b) Write as a single fraction in its simplest form. $$\frac{1}{x-3} - \frac{x-3}{2}$$
(c) The $n$th term of a sequence is $an^2 + bn - 5$.
The second term of this sequence is $-3$ and the third term is $4$.
Find the value of $a$ and the value of $b$.
You must show all your working.
545
566
510
The diagram shows the symmetrical cross-section of a ditch containing water. The angle between the base and each side of the ditch is 110\degree. The width of the base is 0.9 m and the depth of the water is 2.1 m. The ditch is 100 m long.
(a) Calculate the volume of water in the ditch. ............................................ m^3 [4]
(b) On a different day, the ditch contains 300 m^3 of water. Water is pumped out of the ditch at a rate of 4.2 litres per second. Calculate the time taken to empty the ditch completely. Give your answer in hours and minutes, correct to the nearest minute. .................. h .................. min [4]
584
581
574
(a) Calculate the area of triangle $BCD$.
................................................... $m^2$ [2]
(b) Calculate angle $ADB$.
Angle $ADB = ...................................................$ [6]
527
538
518
