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 2018 November Q2
2 Showing all necessary working, find \(\int _ { 1 } ^ { 4 } \left( \sqrt { } x + \frac { 2 } { \sqrt { } x } \right) \mathrm { d } x\).
CAIE P1 2018 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-04_540_554_260_792} The diagram shows part of the curve \(y = x \left( 9 - x ^ { 2 } \right)\) and the line \(y = 5 x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5 x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.
  1. Express the length of \(P Q\) in terms of \(t\), simplifying your answer.
  2. Given that \(t\) can vary, find the maximum value of the length of \(P Q\).
CAIE P1 2018 November Q4
4 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 - 3 \cos x \quad \text { for } 0 \leqslant x \leqslant 2 \pi
& \mathrm {~g} : x \mapsto \frac { 1 } { 2 } x \quad \text { for } 0 \leqslant x \leqslant 2 \pi \end{aligned}$$
  1. Solve the equation \(\operatorname { fg } ( x ) = 1\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
CAIE P1 2018 November Q5
5 The first three terms of an arithmetic progression are \(4 , x\) and \(y\) respectively. The first three terms of a geometric progression are \(x , y\) and 18 respectively. It is given that both \(x\) and \(y\) are positive.
  1. Find the value of \(x\) and the value of \(y\).
  2. Find the fourth term of each progression.
    \includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-08_389_716_260_712} The diagram shows a triangle \(A B C\) in which \(B C = 20 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The perpendicular from \(B\) to \(A C\) meets \(A C\) at \(D\) and \(A D = 9 \mathrm {~cm}\). Angle \(B C A = \theta ^ { \circ }\).
  3. By expressing the length of \(B D\) in terms of \(\theta\) in each of the triangles \(A B D\) and \(D B C\), show that \(20 \sin ^ { 2 } \theta = 9 \cos \theta\).
  4. Hence, showing all necessary working, calculate \(\theta\).
CAIE P1 2018 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-10_819_497_262_826} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius 4 units. Points \(A , B\) and \(C\) lie on the circumference of the base such that \(A B\) is a diameter and angle \(B O C = 90 ^ { \circ }\). Points \(P , Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A , B\) and \(C\) respectively. The height of the cylinder is 12 units. The mid-point of \(C R\) is \(M\) and \(N\) lies on \(B Q\) with \(B N = 4\) units. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O B\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
Evaluate \(\overrightarrow { P N } \cdot \overrightarrow { P M }\) and hence find angle \(M P N\).
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-12_483_574_262_788} The diagram shows an isosceles triangle \(A C B\) in which \(A B = B C = 8 \mathrm {~cm}\) and \(A C = 12 \mathrm {~cm}\). The arc \(X C\) is part of a circle with centre \(A\) and radius 12 cm , and the arc \(Y C\) is part of circle with centre \(B\) and radius 8 cm . The points \(A , B , X\) and \(Y\) lie on a straight line.
  1. Show that angle \(C B Y = 1.445\) radians, correct to 4 significant figures.
  2. Find the perimeter of the shaded region.
CAIE P1 2018 November Q9
9 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \in \mathbb { R }\).
  1. Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the range of f .
    The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \leqslant k\).
  3. State the largest value of \(k\) for which g has an inverse.
  4. Given that g has an inverse, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2018 November Q10
10 The equation of a curve is \(y = 2 x + \frac { 12 } { x }\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the line does not meet the curve.
    In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).
  2. Find the coordinates of \(A\) and \(B\).
  3. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\).
CAIE P1 2018 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-18_661_698_260_717} The diagram shows part of the curve \(y = 3 \sqrt { } ( 4 x + 1 ) - 2 x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).
  1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
  2. Find the coordinates of \(M\).
  3. Find, showing all necessary working, the area of the shaded region.
    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 2018 November Q1
1 Find the coefficient of \(\frac { 1 } { x ^ { 3 } }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 7 }\).
CAIE P1 2018 November Q2
2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.
CAIE P1 2018 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-04_467_401_260_872} The diagram shows an arc \(B C\) of a circle with centre \(A\) and radius 5 cm . The length of the arc \(B C\) is 4 cm . The point \(D\) is such that the line \(B D\) is perpendicular to \(B A\) and \(D C\) is parallel to \(B A\).
  1. Find angle \(B A C\) in radians.
  2. Find the area of the shaded region \(B D C\).
CAIE P1 2018 November Q4
4 Two points \(A\) and \(B\) have coordinates \(( - 1,1 )\) and \(( 3,4 )\) respectively. The line \(B C\) is perpendicular to \(A B\) and intersects the \(x\)-axis at \(C\).
  1. Find the equation of \(B C\) and the \(x\)-coordinate of \(C\).
  2. Find the distance \(A C\), giving your answer correct to 3 decimal places.
CAIE P1 2018 November Q5
5 In an arithmetic progression the first term is \(a\) and the common difference is 3 . The \(n\)th term is 94 and the sum of the first \(n\) terms is 1420 . Find \(n\) and \(a\).
CAIE P1 2018 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-08_743_897_260_623} The diagram shows a solid figure \(O A B C D E F G\) with a horizontal rectangular base \(O A B C\) in which \(O A = 8\) units and \(A B = 6\) units. The rectangle \(D E F G\) lies in a horizontal plane and is such that \(D\) is 7 units vertically above \(O\) and \(D E\) is parallel to \(O A\). The sides \(D E\) and \(D G\) have lengths 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. Use a scalar product to find angle \(O B F\), giving your answer in the form \(\cos ^ { - 1 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
CAIE P1 2018 November Q7
7
  1. Show that \(\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } \equiv \frac { 2 ( \tan \theta - \cos \theta ) } { \sin ^ { 2 } \theta }\).
  2. Hence, showing all necessary working, solve the equation $$\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } = 0$$ for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2018 November Q8
8 A curve passes through \(( 0,11 )\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants.
  1. Find the equation of the curve in terms of \(a\) and \(b\).
  2. It is now given that the curve has a stationary point at \(( 2,3 )\). Find the values of \(a\) and \(b\).
CAIE P1 2018 November Q9
9 A curve has equation \(y = 2 x ^ { 2 } - 3 x + 1\) and a line has equation \(y = k x + k ^ { 2 }\), where \(k\) is a constant.
  1. Show that, for all values of \(k\), the curve and the line meet.
  2. State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve.
CAIE P1 2018 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-16_648_823_262_660} The diagram shows part of the curve \(y = 2 ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\) and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\). The curve and the line \(x = \frac { 2 } { 3 }\) intersect at the point \(A\).
  1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Find the equation of the normal to the curve at \(A\), giving your answer in the form \(y = m x + c\).
CAIE P1 2018 November Q11
11
  1. Express \(2 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
    The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 11\) for \(x \leqslant k\).
  2. State the largest value of the constant \(k\) for which f is a one-one function.
  3. For this value of \(k\) find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x \leqslant p\).
  4. With \(k\) now taking the value 1 , find the largest value of the constant \(p\) which allows the composite function fg to be formed, and find an expression for \(\mathrm { fg } ( x )\) whenever this composite function exists.
    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 2019 November Q1
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 6 }\).
CAIE P1 2019 November Q2
2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).
CAIE P1 2019 November Q3
3 The line \(y = a x + b\) is a tangent to the curve \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 3 x + c\) at the point \(( 2,6 )\). Find the values of the constants \(a , b\) and \(c\).
CAIE P1 2019 November Q4
4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run \(x \mathrm {~km}\) on day 1 , and on each subsequent day she will increase the distance by \(10 \%\) of the previous day's distance. On day 21 she will run 20 km .
  1. Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct to 1 decimal place.
  2. Find the total distance she runs over the 21 days.
CAIE P1 2019 November Q5
5
  1. Given that \(4 \tan x + 3 \cos x + \frac { 1 } { \cos x } = 0\), show, without using a calculator, that \(\sin x = - \frac { 2 } { 3 }\).
  2. Hence, showing all necessary working, solve the equation $$4 \tan \left( 2 x - 20 ^ { \circ } \right) + 3 \cos \left( 2 x - 20 ^ { \circ } \right) + \frac { 1 } { \cos \left( 2 x - 20 ^ { \circ } \right) } = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2019 November Q6
6 A straight line has gradient \(m\) and passes through the point ( \(0 , - 2\) ). Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 2 x + 7\) and, for each value of \(m\), find the coordinates of the point where the line touches the curve.