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 2011 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-2_520_839_1795_653} In the diagram, \(A B C D\) is a parallelogram with \(A B = B D = D C = 10 \mathrm {~cm}\) and angle \(A B D = 0.8\) radians. \(A P D\) and \(B Q C\) are arcs of circles with centres \(B\) and \(D\) respectively.
  1. Find the area of the parallelogram \(A B C D\).
  2. Find the area of the complete figure \(A B Q C D P\).
  3. Find the perimeter of the complete figure \(A B Q C D P\).
  4. Given that $$3 \sin ^ { 2 } x - 8 \cos x - 7 = 0$$ show that, for real values of \(x\), $$\cos x = - \frac { 2 } { 3 }$$
  5. Hence solve the equation $$3 \sin ^ { 2 } \left( \theta + 70 ^ { \circ } \right) - 8 \cos \left( \theta + 70 ^ { \circ } \right) - 7 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2011 November Q6
6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) respectively.
  1. Use a scalar product to find angle \(B O A\). The point \(C\) is the mid-point of \(A B\). The point \(D\) is such that \(\overrightarrow { O D } = 2 \overrightarrow { O B }\).
  2. Find \(\overrightarrow { D C }\).
CAIE P1 2011 November Q7
7
  1. A straight line passes through the point \(( 2,0 )\) and has gradient \(m\). Write down the equation of the line.
  2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x ^ { 2 } - 4 x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve.
  3. Express \(x ^ { 2 } - 4 x + 5\) in the form \(( x + a ) ^ { 2 } + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve.
CAIE P1 2011 November Q8
8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P ( 3 , - 10 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } + k x - 12\), where \(k\) is a constant.
  1. Show that \(k = - 2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and determine the nature of each of the stationary points \(P\) and \(Q\).
  3. Find \(\mathrm { f } ( x )\).
CAIE P1 2011 November Q9
9 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 3 & \text { for } x \leqslant 0
\mathrm {~g} : x \mapsto x ^ { 2 } - 6 x & \text { for } x \leqslant 3 \end{array}$$
  1. Express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) and solve the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).
  2. On the same diagram sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), showing the coordinates of their point of intersection and the relationship between the graphs.
  3. Find the set of values of \(x\) which satisfy \(\operatorname { gf } ( x ) \leqslant 16\).
CAIE P1 2011 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-4_764_929_255_609} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt { } ( x + 1 )\), meeting at \(( - 1,0 )\) and \(( 0,1 )\).
  1. Find the area of the shaded region.
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
CAIE P1 2012 November Q1
1 The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first \(n\) terms in \(n\). Find the value of the positive integer \(n\).
CAIE P1 2012 November Q2
2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { x ^ { 3 } } - 1\) and the point \(( 2,4 )\) lies on the curve. Find the equation of the curve.
CAIE P1 2012 November Q3
3 An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday.
CAIE P1 2012 November Q4
4
  1. Find the first 3 terms in the expansion of \(\left( 2 x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }\).
CAIE P1 2012 November Q5
5 A curve has equation \(y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.
CAIE P1 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-2_526_659_1336_742} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) on \(O A\) is such that \(B C\) is perpendicular to \(O A\). The point \(D\) is on \(B C\) and the circular arc \(A D\) has centre \(C\).
  1. Find \(A C\) in terms of \(r\) and \(\theta\).
  2. Find the perimeter of the shaded region \(A B D\) when \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 4\), giving your answer as an exact value.
CAIE P1 2012 November Q7
7
  1. Solve the equation \(2 \cos ^ { 2 } \theta = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  2. The smallest positive solution of the equation \(2 \cos ^ { 2 } ( n \theta ) = 3 \sin ( n \theta )\), where \(n\) is a positive integer, is \(10 ^ { \circ }\). State the value of \(n\) and hence find the largest solution of this equation in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2012 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-3_629_853_251_644} The diagram shows the curve \(y ^ { 2 } = 2 x - 1\) and the straight line \(3 y = 2 x - 1\). The curve and straight line intersect at \(x = \frac { 1 } { 2 }\) and \(x = a\), where \(a\) is a constant.
  1. Show that \(a = 5\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2012 November Q9
9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. The position vectors of points \(C\) and \(D\) relative to \(O\) are \(3 \mathbf { a }\) and \(2 \mathbf { b }\) respectively. It is given that $$\mathbf { a } = \left( \begin{array} { l } 2
1
2 \end{array} \right) \quad \text { and } \quad \mathbf { b } = \left( \begin{array} { l } 4
0
6 \end{array} \right) .$$
  1. Find the unit vector in the direction of \(\overrightarrow { C D }\).
  2. The point \(E\) is the mid-point of \(C D\). Find angle \(E O D\).
CAIE P1 2012 November Q10
10 The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } + c\) and hence state the coordinates of the vertex of the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \leqslant 1\).
  2. State the range of g .
  3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).
CAIE P1 2012 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-4_885_967_255_589} The diagram shows the curve \(y = ( 6 x + 2 ) ^ { \frac { 1 } { 3 } }\) and the point \(A ( 1,2 )\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).
  1. Find the equation of the tangent \(A B\) and the equation of the normal \(A C\).
  2. Find the distance \(B C\).
  3. Find the coordinates of the point of intersection, \(E\), of \(O A\) and \(B C\), and determine whether \(E\) is the mid-point of \(O A\).
CAIE P1 2012 November Q1
1 In the expansion of \(\left( x ^ { 2 } - \frac { a } { x } \right) ^ { 7 }\), the coefficient of \(x ^ { 5 }\) is - 280 . Find the value of the constant \(a\).
CAIE P1 2012 November Q2
2 A function f is such that \(\mathrm { f } ( x ) = \sqrt { } \left( \frac { x + 3 } { 2 } \right) + 1\), for \(x \geqslant - 3\). Find
  1. \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants,
  2. the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2012 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-2_485_755_751_696} The diagram shows a plan for a rectangular park \(A B C D\), in which \(A B = 40 \mathrm {~m}\) and \(A D = 60 \mathrm {~m}\). Points \(X\) and \(Y\) lie on \(B C\) and \(C D\) respectively and \(A X , X Y\) and \(Y A\) are paths that surround a triangular playground. The length of \(D Y\) is \(x \mathrm {~m}\) and the length of \(X C\) is \(2 x \mathrm {~m}\).
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the playground is given by $$A = x ^ { 2 } - 30 x + 1200$$
  2. Given that \(x\) can vary, find the minimum area of the playground.
CAIE P1 2012 November Q4
4 The line \(y = \frac { x } { k } + k\), where \(k\) is a constant, is a tangent to the curve \(4 y = x ^ { 2 }\) at the point \(P\). Find
  1. the value of \(k\),
  2. the coordinates of \(P\).
CAIE P1 2012 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-3_602_751_255_696} The diagram shows a triangle \(A B C\) in which \(A\) has coordinates ( 1,3 ), \(B\) has coordinates ( 5,11 ) and angle \(A B C\) is \(90 ^ { \circ }\). The point \(X ( 4,4 )\) lies on \(A C\). Find
  1. the equation of \(B C\),
  2. the coordinates of \(C\).
CAIE P1 2012 November Q6
6
  1. Show that the equation \(2 \cos x = 3 \tan x\) can be written as a quadratic equation in \(\sin x\).
  2. Solve the equation \(2 \cos 2 y = 3 \tan 2 y\), for \(0 ^ { \circ } \leqslant y \leqslant 180 ^ { \circ }\).
CAIE P1 2012 November Q7
7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1
0
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k
- k
2 k \end{array} \right)$$ where \(k\) is a constant.
  1. In the case where \(k = 2\), calculate angle \(A O B\).
  2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.
CAIE P1 2012 November Q8
8
  1. In a geometric progression, all the terms are positive, the second term is 24 and the fourth term is \(13 \frac { 1 } { 2 }\). Find
    1. the first term,
    2. the sum to infinity of the progression.
  2. A circle is divided into \(n\) sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are \(3 ^ { \circ }\) and \(5 ^ { \circ }\). Find the value of \(n\).