Questions — CAIE (7279 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P1 2020 Specimen Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3986478a-062a-4bc6-bce2-85408b51a0b2-14_716_912_258_571} Th id ag am sh s a circle with cen re \(A\) ad rad s r. Diameters \(C A D\) ad \(B A E\) are \(\mathbf { p }\) re \(\dot { \text { d } }\) ch ar to each b r. A larg r circle \(\mathbf { h }\) s cen re \(B\) a¢ sses th \(\mathbf { g } \quad C\) ad \(D\).
  1. Sth that th rad s 6 th larg r circle is \(r \sqrt { 2 }\).
  2. Fid b area \(\mathbf { 6 }\) th sh d d eg i r it erms \(\mathbf { 6 } r\).
CAIE P1 2020 Specimen Q10
10 Th circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - \quad\) Ch s cen re \(C\) a¢ sses the id s \(A\) ad \(B\).
  1. State th co ida tesg \(C\). It is \(\dot { \mathbf { g } } \dot { \mathrm { n } }\) the t th mid oin, \(D , 6 \quad A B \mathbf { h }\) s co \(\dot { \mathrm { d } } \mathbf { a }\) tes \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
  2. Fid eq tin \(A B , \dot { \mathrm {~g} } \dot { \mathrm { v } }\) as wer in th fo \(\mathrm { m } y = m x + c\).
  3. Fidy calch atitil b \(x\)-co dia tes \(6 A\) ad \(B\).
CAIE P1 2020 Specimen Q11
11 Th fo tin f is d fin f \(\mathbf { o } x \in \mathbb { R } , \quad x \mapsto x ^ { 2 } + a x + b , \mathrm { w } \mathbf { b }\) re \(a\) ad \(b\) are \(\mathrm { c } \mathbf { B }\) tan s .
  1. It is g it h t \(a = 6\) d \(b = 8\) Fid b rag 6 f .
  2. It is g ven in tead th \(\mathrm { t } a = 5\) ad th t th ro s \(\mathbf { 6 }\) the eq tin \(\mathrm { f } ( x ) = 0\) are \(k\) ad \(z k\), wh re \(k\) is a CB tan. Fid b le s \(6 b\) ad \(k\).
  3. Sha th tif the equ tif \(( x + a ) = a \mathbf { h }\) so eal ro sth \(\mathrm { n } a ^ { 2 } < ( 4 b - a )\).
CAIE P1 2020 Specimen Q12
12
\includegraphics[max width=\textwidth, alt={}, center]{3986478a-062a-4bc6-bce2-85408b51a0b2-20_542_1003_260_539} Th d ag am sw s th cn \& with equ tin \(y = x \left( x - \mathcal { P } ^ { 2 } \right.\). Th min mm \(\dot { \mathbf { p } } n\) n th cn \(\mathbf { t }\) s co dia tes \(( a , \emptyset )\) ad l \(x\)-co id a te of th max \(\mathrm { mm } \dot { \mathrm { p } } \quad \mathrm { n }\) is \(b , \mathrm { w } \mathbf { b }\) re \(a\) ad \(b\) are \(\mathrm { c } \mathbf { n }\) tan s .
  1. State the le \(6 a\).
  2. Calch ate th \& le 6 b.
  3. Fid b area 6 th sh d d eg n
  4. Th g ad en, \(\frac { \mathrm { dy } } { \mathrm { dx } } , 6\) th cn a sa min mm \& le \(m\). Calch ate th le \(6 m\). If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n
CAIE P1 2002 June Q1
1 The line \(x + 2 y = 9\) intersects the curve \(x y + 18 = 0\) at the points \(A\) and \(B\). Find the coordinates of \(A\) and \(B\).
CAIE P1 2002 June Q2
2
  1. Show that \(\sin x \tan x\) may be written as \(\frac { 1 - \cos ^ { 2 } x } { \cos x }\).
  2. Hence solve the equation \(2 \sin x \tan x = 3\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2002 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-2_629_659_715_740} The diagram shows the curve \(y = 3 \sqrt { } x\) and the line \(y = x\) intersecting at \(O\) and \(P\). Find
  1. the coordinates of \(P\),
  2. the area of the shaded region.
CAIE P1 2002 June Q4
4 A progression has a first term of 12 and a fifth term of 18.
  1. Find the sum of the first 25 terms if the progression is arithmetic.
  2. Find the 13th term if the progression is geometric.
CAIE P1 2002 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.
  1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Hence find the angle \(O M C ^ { \prime }\).
CAIE P1 2002 June Q6
6 The function f , where \(\mathrm { f } ( x ) = a \sin x + b\), is defined for the domain \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2\) and that \(\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8\),
  1. find the values of \(a\) and \(b\),
  2. find the values of \(x\) for which \(\mathrm { f } ( x ) = 0\), giving your answers in radians correct to 2 decimal places,
  3. sketch the graph of \(y = \mathrm { f } ( x )\).
CAIE P1 2002 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).
  1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
  2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
  3. Find the area of the new cross-section (shown shaded in the diagram).
CAIE P1 2002 June Q8
8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).
  1. Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$ Given that \(r\) can vary,
  2. find the value of \(r\) for which \(V\) has a stationary value,
  3. find this stationary value and determine whether it is a maximum or a minimum.
CAIE P1 2002 June Q9
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\) and \(P ( 1,5 )\) is a point on the curve.
  1. The normal to the curve at \(P\) crosses the \(x\)-axis at \(Q\). Find the coordinates of \(Q\).
  2. Find the equation of the curve.
  3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 1\).
CAIE P1 2002 June Q10
10 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 3 x + 2 , & x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto \frac { 6 } { 2 x + 3 } , & x \in \mathbb { R } , x \neq - 1.5 . \end{array}$$
  1. Find the value of \(x\) for which \(\operatorname { fg } ( x ) = 3\).
  2. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs.
  3. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\), and solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2003 June Q1
1 Find the value of the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
CAIE P1 2003 June Q2
2 Find all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) which satisfy the equation \(\sin 3 x + 2 \cos 3 x = 0\).
CAIE P1 2003 June Q3
3
  1. Differentiate \(4 x + \frac { 6 } { x ^ { 2 } }\) with respect to \(x\).
  2. Find \(\int \left( 4 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x\).
CAIE P1 2003 June Q4
4 In an arithmetic progression, the 1 st term is - 10 , the 15th term is 11 and the last term is 41 . Find the sum of all the terms in the progression.
CAIE P1 2003 June Q5
5 The function f is defined by \(\mathrm { f } : x \mapsto a x + b\), for \(x \in \mathbb { R }\), where \(a\) and \(b\) are constants. It is given that \(f ( 2 ) = 1\) and \(f ( 5 ) = 7\).
  1. Find the values of \(a\) and \(b\).
  2. Solve the equation \(\operatorname { ff } ( x ) = 0\).
CAIE P1 2003 June Q6
6
  1. Sketch the graph of the curve \(y = 3 \sin x\), for \(- \pi \leqslant x \leqslant \pi\). The straight line \(y = k x\), where \(k\) is a constant, passes through the maximum point of this curve for \(- \pi \leqslant x \leqslant \pi\).
  2. Find the value of \(k\) in terms of \(\pi\).
  3. State the coordinates of the other point, apart from the origin, where the line and the curve intersect.
CAIE P1 2003 June Q7
7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).
  1. Find the equation of \(L _ { 2 }\).
  2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).
CAIE P1 2003 June Q8
8 The points \(A , B , C\) and \(D\) have position vectors \(3 \mathbf { i } + 2 \mathbf { k } , 2 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , 2 \mathbf { j } + 7 \mathbf { k }\) and \(- 2 \mathbf { i } + 10 \mathbf { j } + 7 \mathbf { k }\) respectively.
  1. Use a scalar product to show that \(B A\) and \(B C\) are perpendicular.
  2. Show that \(B C\) and \(A D\) are parallel and find the ratio of the length of \(B C\) to the length of \(A D\).
CAIE P1 2003 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{8214ccb9-0894-4c3c-a8d9-d8f8749fdbe1-3_321_636_267_758} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 8 cm . Angle \(A O B = \theta\) radians.
  1. In the case where \(\theta = 1\), calculate the area of the sector BOC.
  2. Find the value of \(\theta\) for which the perimeter of sector \(A O B\) is one half of the perimeter of sector BOC.
  3. In the case where \(\theta = \frac { 1 } { 3 } \pi\), show that the exact length of the perimeter of triangle \(A B C\) is \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).
CAIE P1 2003 June Q10
10 The equation of a curve is \(y = \sqrt { } ( 5 x + 4 )\).
  1. Calculate the gradient of the curve at the point where \(x = 1\).
  2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(x\) has the constant value 0.03 units per second. Find the rate of increase of \(y\) at the instant when \(x = 1\).
  3. Find the area enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).
CAIE P1 2003 June Q11
11 The equation of a curve is \(y = 8 x - x ^ { 2 }\).
  1. Express \(8 x - x ^ { 2 }\) in the form \(a - ( x + b ) ^ { 2 }\), stating the numerical values of \(a\) and \(b\).
  2. Hence, or otherwise, find the coordinates of the stationary point of the curve.
  3. Find the set of values of \(x\) for which \(y \geqslant - 20\). The function g is defined by \(\mathrm { g } : x \mapsto 8 x - x ^ { 2 }\), for \(x \geqslant 4\).
  4. State the domain and range of \(\mathrm { g } ^ { - 1 }\).
  5. Find an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).