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 2020 November Q2
2 The first, second and third terms of a geometric progression are \(2 p + 6 , - 2 p\) and \(p + 2\) respectively, where \(p\) is positive. Find the sum to infinity of the progression.
CAIE P1 2020 November Q3
3 The equation of a curve is \(y = 2 x ^ { 2 } + m ( 2 x + 1 )\), where \(m\) is a constant, and the equation of a line is \(y = 6 x + 4\). Show that, for all values of \(m\), the line intersects the curve at two distinct points.
CAIE P1 2020 November Q4
4 The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by $$S _ { n } = n ^ { 2 } + 4 n$$ The \(k\) th term in the progression is greater than 200.
Find the smallest possible value of \(k\).
CAIE P1 2020 November Q5
5 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 4 x - 2 , \quad \text { for } x \in \mathbb { R } ,
& \mathrm {~g} ( x ) = \frac { 4 } { x + 1 } , \quad \text { for } x \in \mathbb { R } , x \neq - 1 \end{aligned}$$
  1. Find the value of fg (7).
  2. Find the values of \(x\) for which \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2020 November Q6
6
  1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) \equiv \frac { 1 } { \tan x }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \cos x } - \tan x \right) \left( \frac { 1 } { \sin x } + 1 \right) = 2 \tan ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2020 November Q7
7 The point \(( 4,7 )\) lies on the curve \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } }\).
  1. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 4\).
  2. Find the equation of the curve.
CAIE P1 2020 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-10_348_700_262_721} In the diagram, \(A B C\) is an isosceles triangle with \(A B = B C = r \mathrm {~cm}\) and angle \(B A C = \theta\) radians. The point \(D\) lies on \(A C\) and \(A B D\) is a sector of a circle with centre \(A\).
  1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 10\) and \(\theta = 0.6\), find the perimeter of the shaded region.
CAIE P1 2020 November Q9
9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.
  1. Find the equation of the circle.
    Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
  2. Show that \(D C\) is a tangent to the circle.
    The other tangent from \(D\) to the circle touches the circle at \(E\).
  3. Find the coordinates of \(E\).
CAIE P1 2020 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-14_378_666_264_737} The diagram shows part of the curve \(y = \frac { 2 } { ( 3 - 2 x ) ^ { 2 } } - x\) and its minimum point \(M\), which lies on the \(x\)-axis.
  1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y \mathrm {~d} x\).
  2. Find, by calculation, the \(x\)-coordinate of \(M\).
  3. Find the area of the shaded region bounded by the curve and the coordinate axes.
CAIE P1 2020 November Q11
2 marks
11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
  1. State the greatest and least values of \(y\).
  2. Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
  3. By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
    1. \(k = - 3\)
    2. \(k = 1\)
    3. \(k = 3\)
      Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2
      & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4
      & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
  4. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
  5. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
    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 2020 November Q1
1
  1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. The curve with equation \(y = x ^ { 2 }\) is transformed to the curve with equation \(y = x ^ { 2 } + 6 x + 5\). Describe fully the transformation(s) involved.
CAIE P1 2020 November Q2
2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).
  1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
  2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.
CAIE P1 2020 November Q3
3 Solve the equation \(3 \tan ^ { 2 } \theta + 1 = \frac { 2 } { \tan ^ { 2 } \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2020 November Q4
4 A curve has equation \(y = 3 x ^ { 2 } - 4 x + 4\) and a straight line has equation \(y = m x + m - 1\), where \(m\) is a constant. Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.
CAIE P1 2020 November Q5
5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).
CAIE P1 2020 November Q6
6 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 1 }\) for \(x > \frac { 1 } { 3 }\).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Show that \(\frac { 2 } { 3 } + \frac { 2 } { 3 ( 3 x - 1 ) }\) can be expressed as \(\frac { 2 x } { 3 x - 1 }\).
  3. State the range of f.
CAIE P1 2020 November Q7
7 The first and second terms of an arithmetic progression are \(\frac { 1 } { \cos ^ { 2 } \theta }\) and \(- \frac { \tan ^ { 2 } \theta } { \cos ^ { 2 } \theta }\), respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that the common difference is \(- \frac { 1 } { \cos ^ { 4 } \theta }\).
  2. Find the exact value of the 13th term when \(\theta = \frac { 1 } { 6 } \pi\).
CAIE P1 2020 November Q8
8 The equation of a curve is \(y = 2 x + 1 + \frac { 1 } { 2 x + 1 }\) for \(x > - \frac { 1 } { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the stationary point and determine the nature of the stationary point.
CAIE P1 2020 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744} In the diagram, arc \(A B\) is part of a circle with centre \(O\) and radius 8 cm . Arc \(B C\) is part of a circle with centre \(A\) and radius 12 cm , where \(A O C\) is a straight line.
  1. Find angle \(B A O\) in radians.
  2. Find the area of the shaded region.
  3. Find the perimeter of the shaded region.
CAIE P1 2020 November Q10
10 A curve has equation \(y = \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } }\) where \(x > 0\) and \(k\) is a positive constant.
  1. It is given that when \(x = \frac { 1 } { 4 }\), the gradient of the curve is 3 . Find the value of \(k\).
  2. It is given instead that \(\int _ { \frac { 1 } { 4 } k ^ { 2 } } ^ { k ^ { 2 } } \left( \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } } \right) \mathrm { d } x = \frac { 13 } { 12 }\). Find the value of \(k\).
CAIE P1 2020 November Q11
11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).
  1. Show that the point \(T ( - 6,6 )\) is outside the circle.
    Two tangents from \(T\) to the circle are drawn.
  2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
    The two tangents touch the circle at \(A\) and \(B\).
  3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
  4. Find the \(x\)-coordinates of \(A\) and \(B\).
    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 2021 November Q1
1
  1. Expand \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }\).
  2. Find the first four terms in the expansion, in ascending powers of \(x\), of \(( 1 + 2 x ) ^ { 6 }\).
  3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }\).
CAIE P1 2021 November Q2
2 A curve has equation \(y = k x ^ { 2 } + 2 x - k\) and a line has equation \(y = k x - 2\), where \(k\) is a constant. Find the set of values of \(k\) for which the curve and line do not intersect.
CAIE P1 2021 November Q3
3 Solve, by factorising, the equation $$6 \cos \theta \tan \theta - 3 \cos \theta + 4 \tan \theta - 2 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2021 November Q4
4 The first term of an arithmetic progression is \(a\) and the common difference is - 4 . The first term of a geometric progression is \(5 a\) and the common ratio is \(- \frac { 1 } { 4 }\). The sum to infinity of the geometric progression is equal to the sum of the first eight terms of the arithmetic progression.
  1. Find the value of \(a\).
    The \(k\) th term of the arithmetic progression is zero.
  2. Find the value of \(k\).