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 2017 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2
- 2
- 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6
1
1 \end{array} \right)\).
  1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
    The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
  2. Find the position vector of \(C\).
  3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.
CAIE P1 2017 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-18_551_689_260_726} The diagram shows part of the curve \(y = \sqrt { } ( 5 x - 1 )\) and the normal to the curve at the point \(P ( 2,3 )\). This normal meets the \(x\)-axis at \(Q\).
  1. Find the equation of the normal at \(P\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2017 November Q1
1 An arithmetic progression has first term - 12 and common difference 6 . The sum of the first \(n\) terms exceeds 3000 . Calculate the least possible value of \(n\).
CAIE P1 2017 November Q2
2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.
CAIE P1 2017 November Q3
3
  1. Find the term independent of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }\).
  2. Find the value of \(a\) for which there is no term independent of \(x\) in the expansion of $$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$
CAIE P1 2017 November Q4
4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.
CAIE P1 2017 November Q5
3 marks
5
  1. Show that the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0\).
    [0pt] [3]
  2. Hence solve the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2017 November Q6
6 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } - 1 } \text { for } x < - 1
& \mathrm {~g} ( x ) = x ^ { 2 } + 1 \text { for } x > 0 \end{aligned}$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Solve the equation \(\operatorname { gf } ( x ) = 5\).
CAIE P1 2017 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-10_401_561_260_790} The diagram shows a rectangle \(A B C D\) in which \(A B = 5\) units and \(B C = 3\) units. Point \(P\) lies on \(D C\) and \(A P\) is an arc of a circle with centre \(B\). Point \(Q\) lies on \(D C\) and \(A Q\) is an arc of a circle with centre \(D\).
  1. Show that angle \(A B P = 0.6435\) radians, correct to 4 decimal places.
  2. Calculate the areas of the sectors \(B A P\) and \(D A Q\).
  3. Calculate the area of the shaded region.
CAIE P1 2017 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-12_485_570_262_790} The diagram shows parts of the graphs of \(y = 3 - 2 x\) and \(y = 4 - 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).
  1. Find by calculation the \(x\)-coordinates of \(A\) and \(B\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2017 November Q9
9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8
- 6
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10
3
- 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2
- 3
- 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.
  1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
  2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).
CAIE P1 2017 November Q10
10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x\), where \(a\) and \(b\) are positive constants.
  1. Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.
  2. It is now given that the curve has a stationary point at \(( - 2 , - 3 )\) and that the gradient of the curve at \(x = 1\) is 9 . Find \(\mathrm { f } ( x )\).
CAIE P1 2017 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.
  1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
  2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
  3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.
CAIE P1 2018 November Q1
1 Showing all necessary working, solve the equation \(4 x - 11 x ^ { \frac { 1 } { 2 } } + 6 = 0\).
CAIE P1 2018 November Q2
2 A line has equation \(y = x + 1\) and a curve has equation \(y = x ^ { 2 } + b x + 5\). Find the set of values of the constant \(b\) for which the line meets the curve.
CAIE P1 2018 November Q3
3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.
  1. Find the equation of the line through the origin parallel to \(A B\).
  2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).
CAIE P1 2018 November Q4
4 The first term of a series is 6 and the second term is 2 .
  1. For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
  2. For the case where the series is a geometric progression, find the sum to infinity.
CAIE P1 2018 November Q5
5
  1. Show that the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ may be expressed as \(9 \cos ^ { 2 } \theta - 22 \cos \theta + 4 = 0\).
  2. Hence solve the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 November Q6
6 A curve has a stationary point at \(\left( 3,9 \frac { 1 } { 2 } \right)\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + a ^ { 2 } x\), where \(a\) is a non-zero constant.
  1. Find the value of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-08_67_1569_461_328}
  2. Find the equation of the curve.
  3. Determine, showing all necessary working, the nature of the stationary point.
CAIE P1 2018 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-10_503_853_260_641} The diagram shows part of the curve with equation \(y = k \left( x ^ { 3 } - 7 x ^ { 2 } + 12 x \right)\) for some constant \(k\). The curve intersects the line \(y = x\) at the origin \(O\) and at the point \(A ( 2,2 )\).
  1. Find the value of \(k\).
  2. Verify that the curve meets the line \(y = x\) again when \(x = 5\).
  3. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2018 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-12_595_748_260_699} The diagram shows a solid figure \(O A B C D E F\) having a horizontal rectangular base \(O A B C\) with \(O A = 6\) units and \(A B = 3\) units. The vertical edges \(O F , A D\) and \(B E\) have lengths 6 units, 4 units and 4 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O F\) respectively.
  1. Find \(\overrightarrow { D F }\).
  2. Find the unit vector in the direction of \(\overrightarrow { E F }\).
  3. Use a scalar product to find angle \(E F D\).
CAIE P1 2018 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-14_465_677_260_733} The diagram shows a triangle \(O A B\) in which angle \(A B O\) is a right angle, angle \(A O B = \frac { 1 } { 5 } \pi\) radians and \(A B = 5 \mathrm {~cm}\). The arc \(B C\) is part of a circle with centre \(A\) and meets \(O A\) at \(C\). The arc \(C D\) is part of a circle with centre \(O\) and meets \(O B\) at \(D\). Find the area of the shaded region.
CAIE P1 2018 November Q10
10 A curve has equation \(y = \frac { 1 } { 2 } ( 4 x - 3 ) ^ { - 1 }\). The point \(A\) on the curve has coordinates \(\left( 1 , \frac { 1 } { 2 } \right)\).
  1. (a) Find and simplify the equation of the normal through \(A\).
    (b) Find the \(x\)-coordinate of the point where this normal meets the curve again.
  2. A point is moving along the curve in such a way that as it passes through \(A\) its \(x\)-coordinate is decreasing at the rate of 0.3 units per second. Find the rate of change of its \(y\)-coordinate at \(A\).
CAIE P1 2018 November Q11
11
  1. The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1\) for \(x < a\), where \(a\) is a constant.
    1. State the greatest possible value of \(a\).
    2. It is given that \(a\) takes this greatest possible value. State the range of f and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  2. The function g is defined by \(\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }\) for \(x \geqslant 0\).
    1. Show that \(\operatorname { gg } ( 2 x )\) can be expressed in the form \(( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c\), where \(b\) and \(c\) are constants to be found.
    2. Hence expand \(\operatorname { gg } ( 2 x )\) completely, simplifying your answer.
      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 \(\frac { 1 } { x ^ { 2 } }\) in the expansion of \(\left( 3 x + \frac { 2 } { 3 x ^ { 2 } } \right) ^ { 7 }\).