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 2022 June Q10
10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).
  1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
    A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
  2. Find the coordinates of \(A\).
CAIE P1 2022 June Q11
11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).
  1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
    The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
  2. Find the coordinates of \(Q\) in terms of \(m\).
    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 2023 June Q1
1 Solve the equation \(4 \sin \theta + \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2023 June Q2
2
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 2 + 3 x ) ^ { 4 }\).
  2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
  3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + 3 x ) ^ { 4 } ( 1 - 2 x ) ^ { 5 }\).
CAIE P1 2023 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-04_1150_1164_269_484} The diagram shows graphs with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
Describe fully a sequence of two transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to \(y = \mathrm { g } ( x )\).
CAIE P1 2023 June Q4
4 The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius 8 cm . The area of the sector is \(\frac { 16 } { 3 } \pi \mathrm {~cm} ^ { 2 }\). The point \(D\) lies on the \(\operatorname { arc } B C\). Find the perimeter of the segment \(B C D\).
CAIE P1 2023 June Q5
5 marks
5 The line with equation \(y = k x - k\), where \(k\) is a positive constant, is a tangent to the curve with equation \(y = - \frac { 1 } { 2 x }\). Find, in either order, the value of \(k\) and the coordinates of the point where the tangent meets the curve. [5]
CAIE P1 2023 June Q6
6 The first three terms of an arithmetic progression are \(\frac { p ^ { 2 } } { 6 } , 2 p - 6\) and \(p\).
  1. Given that the common difference of the progression is not zero, find the value of \(p\).
  2. Using this value, find the sum to infinity of the geometric progression with first two terms \(\frac { p ^ { 2 } } { 6 }\) and \(2 p - 6\).
CAIE P1 2023 June Q7
7 A curve has equation \(y = 2 + 3 \sin \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant 4 \pi\).
  1. State greatest and least values of \(y\).
  2. Sketch the curve.
    \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
  3. State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for \(0 \leqslant x \leqslant 4 \pi\).
CAIE P1 2023 June Q8
8 The functions f and g are defined as follows, where \(a\) and \(b\) are constants. $$\begin{aligned} & \mathrm { f } ( x ) = 1 + \frac { 2 a } { x - a } \text { for } x > a
& \mathrm {~g} ( x ) = b x - 2 \text { for } x \in \mathbb { R } \end{aligned}$$
  1. Given that \(\mathrm { f } ( 7 ) = \frac { 5 } { 2 }\) and \(\mathrm { gf } ( 5 ) = 4\), find the values of \(a\) and \(b\).
    For the rest of this question, you should use the value of \(a\) which you found in (a).
  2. Find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2023 June Q9
9 Water is poured into a tank at a constant rate of \(500 \mathrm {~cm} ^ { 3 }\) per second. The depth of water in the tank, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\). When the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by the formula \(V = \frac { 4 } { 3 } ( 25 + h ) ^ { 3 } - \frac { 62500 } { 3 }\).
  1. Find the rate at which \(h\) is increasing at the instant when \(h = 10 \mathrm {~cm}\).
  2. At another instant, the rate at which \(h\) is increasing is 0.075 cm per second. Find the value of \(V\) at this instant.
CAIE P1 2023 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).
  1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.
CAIE P1 2023 June Q11
11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).
  1. Find the value of \(a\).
  2. Determine the nature of this stationary point.
  3. Find the equation of the curve.
  4. Find the coordinates of any other stationary points on the curve.
CAIE P1 2023 June Q12
12
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).
  1. Write down the equation of circle \(P\).
  2. Find the equation of the tangent to the circle \(P\) at \(A\).
  3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
  4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
    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 2023 June Q1
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { ( x - 3 ) ^ { 3 } }\) for \(x > 3\). The curve passes through the point \(( 4,5 )\). Find the equation of the curve.
CAIE P1 2023 June Q2
2 The coefficient of \(x ^ { 4 }\) in the expansion of \(( x + a ) ^ { 6 }\) is \(p\) and the coefficient of \(x ^ { 2 }\) in the expansion of \(( a x + 3 ) ^ { 4 }\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\).
CAIE P1 2023 June Q3
3
  1. Express \(4 x ^ { 2 } - 24 x + p\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).
  2. Hence or otherwise find the set of values of \(p\) for which the equation \(4 x ^ { 2 } - 24 x + p = 0\) has no real roots.
CAIE P1 2023 June Q4
4 Solve the equation \(8 x ^ { 6 } + 215 x ^ { 3 } - 27 = 0\).
CAIE P1 2023 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-06_599_1086_274_518} The diagram shows the curve with equation \(y = 10 x ^ { \frac { 1 } { 2 } } - \frac { 5 } { 2 } x ^ { \frac { 3 } { 2 } }\) for \(x > 0\). The curve meets the \(x\)-axis at the points \(( 0,0 )\) and \(( 4,0 )\). Find the area of the shaded region.
The diagram shows a sector \(O A B\) of a circle with centre \(O\). Angle \(A O B = \theta\) radians and \(O P = A P = x\).
  1. Show that the arc length \(A B\) is \(2 x \theta \cos \theta\).
  2. Find the area of the shaded region \(A P B\) in terms of \(x\) and \(\theta\).
CAIE P1 2023 June Q7
7
    1. By first expanding \(( \cos \theta + \sin \theta ) ^ { 2 }\), find the three solutions of the equation $$( \cos \theta + \sin \theta ) ^ { 2 } = 1$$ for \(0 \leqslant \theta \leqslant \pi\).
    2. Hence verify that the only solutions of the equation \(\cos \theta + \sin \theta = 1\) for \(0 \leqslant \theta \leqslant \pi\) are 0 and \(\frac { 1 } { 2 } \pi\).
  2. Prove the identity \(\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } \equiv \frac { \cos \theta + \sin \theta - 1 } { 1 - 2 \sin ^ { 2 } \theta }\).
  3. Using the results of (a) (ii) and (b), solve the equation $$\frac { \sin \theta } { \cos \theta + \sin \theta } + \frac { 1 - \cos \theta } { \cos \theta - \sin \theta } = 2 ( \cos \theta + \sin \theta - 1 )$$ for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2023 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-10_784_913_274_607} The diagram shows the graph of \(y = \mathrm { f } ( x )\) where the function f is defined by $$f ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x \text { for } 0 \leqslant x \leqslant 2 \pi$$
  1. On the diagram above, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

  3. \includegraphics[max width=\textwidth, alt={}, center]{1662cb34-273c-461d-908c-9fe2ffe889b4-11_759_1545_276_331} The diagram above shows part of the graph of the function \(\mathrm { g } ( x ) = 3 + 2 \sin \frac { 1 } { 4 } x\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
    Complete the sketch of the graph of \(\mathrm { g } ( x )\) on the diagram above and hence explain whether the function \(g\) has an inverse.
  4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) to the graph of \(y = \mathrm { f } ( x )\), making clear the order in which the transformations are applied.
CAIE P1 2023 June Q9
9 The second term of a geometric progression is 16 and the sum to infinity is 100 .
  1. Find the two possible values of the first term.
  2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4 ^ { n - 2 }\) multiplied by the \(n\)th term of the other geometric progression.
CAIE P1 2023 June Q10
10 The equation of a circle is \(( x - a ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). The line \(y = \frac { 1 } { 2 } x + 6\) is a tangent to the circle at the point \(P\).
  1. Show that one possible value of \(a\) is 4 and find the other possible value.
  2. For \(a = 4\), find the equation of the normal to the circle at \(P\).
  3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b).
CAIE P1 2023 June Q11
11 The equation of a curve is $$y = k \sqrt { 4 x + 1 } - x + 5$$ where \(k\) is a positive constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the \(x\)-coordinate of the stationary point in terms of \(k\).
  3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan ^ { - 1 } ( 2 )\) with the positive \(x\)-axis.
    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 2023 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-02_778_1061_269_532} The diagram shows the graph of \(y = \mathrm { f } ( x )\), which consists of the two straight lines \(A B\) and \(B C\). The lines \(A ^ { \prime } B ^ { \prime }\) and \(B ^ { \prime } C ^ { \prime }\) form the graph of \(y = \mathrm { g } ( x )\), which is the result of applying a sequence of two transformations, in either order, to \(y = \mathrm { f } ( x )\). State fully the two transformations.