Questions — CAIE P1 (1202 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 2019 March Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-16_611_529_260_808} The diagram shows part of the curve with equation \(y = \sqrt { } \left( x ^ { 3 } + x ^ { 2 } \right)\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
  1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. \(P\) is the point on the curve with \(x\)-coordinate 3 . Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis.
CAIE P1 2019 March Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-18_645_490_262_824} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } }\).
  1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(A B\).
  2. The tangent to the curve at a point \(T\) is parallel to \(A B\). Find the coordinates of \(T\).
  3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(A B\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2002 November Q1
1 Find the value of the term which is independent of \(x\) in the expansion of \(\left( x + \frac { 3 } { x } \right) ^ { 4 }\).
CAIE P1 2002 November Q2
2 A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8 . Find
  1. the first term and the common ratio of the progression,
  2. the sum to infinity of the progression.
CAIE P1 2002 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-2_330_634_753_758} In the diagram, \(O P Q\) is a sector of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(Q O P = \theta\) radians. The tangent to the circle at \(Q\) meets \(O P\) extended at \(R\).
  1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region is given by \(A = \frac { 1 } { 2 } r ^ { 2 } ( \tan \theta - \theta )\).
  2. In the case where \(\theta = 0.8\) and \(r = 15\), evaluate the length of the perimeter of the shaded region.
CAIE P1 2002 November Q4
4 The gradient at any point \(( x , y )\) on a curve is \(\sqrt { } ( 1 + 2 x )\). The curve passes through the point \(( 4,11 )\). Find
  1. the equation of the curve,
  2. the point at which the curve intersects the \(y\)-axis.
CAIE P1 2002 November Q5
5
  1. Show that the equation \(3 \tan \theta = 2 \cos \theta\) can be expressed as $$2 \sin ^ { 2 } \theta + 3 \sin \theta - 2 = 0$$
  2. Hence solve the equation \(3 \tan \theta = 2 \cos \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2002 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-3_602_570_260_790} In the diagram, triangle \(A B C\) is right-angled and \(D\) is the mid-point of \(B C\). Angle \(D A C = 30 ^ { \circ }\) and angle \(B A D = x ^ { \circ }\). Denoting the length of \(A D\) by \(l\),
  1. express each of \(A C\) and \(B C\) exactly in terms of \(l\), and show that \(A B = \frac { 1 } { 2 } l \sqrt { } 7\),
  2. show that \(x = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { } 3 } \right) - 30\).
CAIE P1 2002 November Q7
7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2
- 2
1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2
6
3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p
p
p + 1 \end{array} \right)\), find
  1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
  2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.
CAIE P1 2002 November Q8
8 A curve has equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant.
  1. Write down an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the \(x\)-coordinates of the two stationary points on the curve.
  3. Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.
CAIE P1 2002 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_719_958_264_589} The diagram shows a rectangle \(A B C D\), where \(A\) is \(( 3,2 )\) and \(B\) is \(( 1,6 )\).
  1. Find the equation of \(B C\). Given that the equation of \(A C\) is \(y = x - 1\), find
  2. the coordinates of \(C\),
  3. the perimeter of the rectangle \(A B C D\).
CAIE P1 2002 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_595_800_1548_669} The diagram shows the points \(A ( 1,2 )\) and \(B ( 4,4 )\) on the curve \(y = 2 \sqrt { } x\). The line \(B C\) is the normal to the curve at \(B\), and \(C\) lies on the \(x\)-axis. Lines \(A D\) and \(B E\) are perpendicular to the \(x\)-axis.
  1. Find the equation of the normal \(B C\).
  2. Find the area of the shaded region.
CAIE P1 2002 November Q11
11
  1. Express \(2 x ^ { 2 } + 8 x - 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. For the curve \(y = 2 x ^ { 2 } + 8 x - 10\), state the least value of \(y\) and the corresponding value of \(x\).
  3. Find the set of values of \(x\) for which \(y \geqslant 14\). Given that \(\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10\) for the domain \(x \geqslant k\),
  4. find the least value of \(k\) for which f is one-one,
  5. express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) in this case.
CAIE P1 2003 November Q1
1 Find the coordinates of the points of intersection of the line \(y + 2 x = 11\) and the curve \(x y = 12\).
CAIE P1 2003 November Q2
2
  1. Show that the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\) may be written in the form \(4 x ^ { 2 } + 7 x - 2 = 0\), where \(x = \sin ^ { 2 } \theta\).
  2. Hence solve the equation \(4 \sin ^ { 4 } \theta + 5 = 7 \cos ^ { 2 } \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2003 November Q3
3
  1. A debt of \(
    ) 3726\( is repaid by weekly payments which are in arithmetic progression. The first payment is \)\\( 60\) and the debt is fully repaid after 48 weeks. Find the third payment.
  2. Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4 .
CAIE P1 2003 November Q4
4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 4 x + 1\). The curve passes through the point \(( 1,5 )\).
  1. Find the equation of the curve.
  2. Find the set of values of \(x\) for which the gradient of the curve is positive.
CAIE P1 2003 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-2_594_778_1360_682} The diagram shows a trapezium \(A B C D\) in which \(B C\) is parallel to \(A D\) and angle \(B C D = 90 ^ { \circ }\). The coordinates of \(A , B\) and \(D\) are \(( 2,0 ) , ( 4,6 )\) and \(( 12,5 )\) respectively.
  1. Find the equations of \(B C\) and \(C D\).
  2. Calculate the coordinates of \(C\).
CAIE P1 2003 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_293_502_269_826} The diagram shows the sector \(O P Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(P O Q\) is \(\theta\) radians and the perimeter of the sector is 20 cm .
  1. Show that \(\theta = \frac { 20 } { r } - 2\).
  2. Hence express the area of the sector in terms of \(r\).
  3. In the case where \(r = 8\), find the length of the chord \(P Q\).
CAIE P1 2003 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_636_1047_1153_550} The diagram shows a triangular prism with a horizontal rectangular base \(A D F C\), where \(C F = 12\) units and \(D F = 6\) units. The vertical ends \(A B C\) and \(D E F\) are isosceles triangles with \(A B = B C = 5\) units. The mid-points of \(B E\) and \(D F\) are \(M\) and \(N\) respectively. The origin \(O\) is at the mid-point of \(A C\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O C , O N\) and \(O B\) respectively.
  1. Find the length of \(O B\).
  2. Express each of the vectors \(\overrightarrow { M C }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Evaluate \(\overrightarrow { M C } \cdot \overrightarrow { M N }\) and hence find angle \(C M N\), giving your answer correct to the nearest degree.
CAIE P1 2003 November Q8
8 A solid rectangular block has a base which measures \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\). The height of the block is \(y \mathrm {~cm}\) and the volume of the block is \(72 \mathrm {~cm} ^ { 3 }\).
  1. Express \(y\) in terms of \(x\) and show that the total surface area, \(A \mathrm {~cm} ^ { 2 }\), of the block is given by $$A = 4 x ^ { 2 } + \frac { 216 } { x }$$ Given that \(x\) can vary,
  2. find the value of \(x\) for which \(A\) has a stationary value,
  3. find this stationary value and determine whether it is a maximum or a minimum.
CAIE P1 2003 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733} The diagram shows points \(A ( 0,4 )\) and \(B ( 2,1 )\) on the curve \(y = \frac { 8 } { 3 x + 2 }\). The tangent to the curve at \(B\) crosses the \(x\)-axis at \(C\). The point \(D\) has coordinates \(( 2,0 )\).
  1. Find the equation of the tangent to the curve at \(B\) and hence show that the area of triangle \(B D C\) is \(\frac { 4 } { 3 }\).
  2. Show that the volume of the solid formed when the shaded region \(O D B A\) is rotated completely about the \(x\)-axis is \(8 \pi\).
CAIE P1 2003 November Q10
10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 5 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto \frac { 4 } { 2 - x } , \quad x \in \mathbb { R } , \quad x \neq 2 . \end{aligned}$$
  1. Find the value of \(x\) for which \(\mathrm { fg } ( x ) = 7\).
  2. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
  3. Show that the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no real roots.
  4. Sketch, on a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between these two graphs.
CAIE P1 2004 November Q1
1 Find the coefficient of \(x\) in the expansion of \(\left( 3 x - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2004 November Q2
2 Find
  1. the sum of the first ten terms of the geometric progression \(81,54,36 , \ldots\),
  2. the sum of all the terms in the arithmetic progression \(180,175,170 , \ldots , 25\).