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 2019 November Q7
7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 3 } { 2 x + 1 } \quad \text { for } x > 0
& \mathrm {~g} : x \mapsto \frac { 1 } { x } + 2 \quad \text { for } x > 0 \end{aligned}$$
  1. Find the range of f and the range of g .
  2. Find an expression for \(\mathrm { fg } ( x )\), giving your answer in the form \(\frac { a x } { b x + c }\), where \(a , b\) and \(c\) are integers.
  3. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\), giving your answer in the same form as for part (ii).
CAIE P1 2019 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-12_560_574_258_781} The diagram shows a sector \(O A C\) of a circle with centre \(O\). Tangents \(A B\) and \(C B\) to the circle meet at \(B\). The arc \(A C\) is of length 6 cm and angle \(A O C = \frac { 3 } { 8 } \pi\) radians.
  1. Find the length of \(O A\) correct to 4 significant figures.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2019 November Q9
9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).
  1. Find the equation of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. Find the coordinates of the stationary point on the curve and, showing all necessary working, determine the nature of this stationary point.
CAIE P1 2019 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1
3
- 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
- 3
5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4
- 2
5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2
2
- 1 \end{array} \right) .$$
  1. Show that \(A B\) is perpendicular to \(B C\).
  2. Show that \(A B C D\) is a trapezium.
  3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.
CAIE P1 2019 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762} The diagram shows a shaded region bounded by the \(y\)-axis, the line \(y = - 1\) and the part of the curve \(y = x ^ { 2 } + 4 x + 3\) for which \(x \geqslant - 2\).
  1. Express \(y = x ^ { 2 } + 4 x + 3\) in the form \(y = ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. Hence, for \(x \geqslant - 2\), express \(x\) in terms of \(y\).
  2. Hence, showing all necessary working, find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-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 2019 November Q1
4 marks
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + a x ) \left( 1 + \frac { x } { 2 } \right) ^ { 6 }\) is 3 . Find the value of the constant \(a\). [4]
CAIE P1 2019 November Q2
2 The point \(M\) is the mid-point of the line joining the points \(( 3,7 )\) and \(( - 1,1 )\). Find the equation of the line through \(M\) which is parallel to the line \(\frac { x } { 3 } + \frac { y } { 2 } = 1\).
CAIE P1 2019 November Q3
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { \sqrt { } x }\), where \(k\) is a constant. The points \(P ( 1 , - 1 )\) and \(Q ( 4,4 )\) lie on the curve. Find the equation of the curve.
CAIE P1 2019 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-06_517_768_262_685} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Points \(A\) and \(B\) lie on the circle and angle \(A O B = 2 \theta\) radians. The tangents to the circle at \(A\) and \(B\) meet at \(T\).
  1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 5\) and \(\theta = 1.2\), find the area of the shaded region.
CAIE P1 2019 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_512_460_258_772}
\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_462_85_260_1279} The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of \(h \mathrm {~cm}\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cone is given by \(V = \frac { 1 } { 3 } \pi \left( 225 h - h ^ { 3 } \right)\).
    [0pt] [The volume of a cone of radius \(r\) and vertical height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Given that \(h\) can vary, find the value of \(h\) for which \(V\) has a stationary value. Determine, showing all necessary working, the nature of this stationary value.
CAIE P1 2019 November Q6
6
  1. Given that \(x > 0\), find the two smallest values of \(x\), in radians, for which \(3 \tan ( 2 x + 1 ) = 1\). Show all necessary working.
  2. The function f : \(x \mapsto 3 \cos ^ { 2 } x - 2 \sin ^ { 2 } x\) is defined for \(0 \leqslant x \leqslant \pi\).
    1. Express \(\mathrm { f } ( x )\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
    2. Find the range of \(f\).
CAIE P1 2019 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-12_784_677_260_735} The diagram shows a three-dimensional shape \(O A B C D E F G\). The base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal rectangles. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. Points \(P\) and \(Q\) are the mid-points of \(O D\) and \(G F\) respectively. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A , C\) and \(D\) are given by \(\overrightarrow { O A } = 6 \mathbf { i } , \overrightarrow { O C } = 8 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 10 \mathbf { k }\).
  1. Express each of the vectors \(\overrightarrow { P B }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Determine whether \(P\) is nearer to \(Q\) or to \(B\).
  3. Use a scalar product to find angle \(B P Q\).
CAIE P1 2019 November Q8
8
  1. Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
    1. Find the distance she runs on the last day of the 21-day period.
    2. Find the total distance she runs in the 21-day period.
  2. The first, second and third terms of a geometric progression are \(x , x - 3\) and \(x - 5\) respectively.
    1. Find the value of \(x\).
    2. Find the fourth term of the progression.
    3. Find the sum to infinity of the progression.
CAIE P1 2019 November Q9
9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R }
& \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.
  1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
  2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
  3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
  4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).
CAIE P1 2019 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-18_979_679_262_731} The diagram shows part of the curve \(y = 1 - \frac { 4 } { ( 2 x + 1 ) ^ { 2 } }\). The curve intersects the \(x\)-axis at \(A\). The normal to the curve at \(A\) intersects the \(y\)-axis at \(B\).
  1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
  2. Find the coordinates of \(B\).
  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 2019 November Q1
1
  1. Expand \(( 1 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).
  2. In the expansion of \(\left( 1 + \left( p x - 2 x ^ { 2 } \right) \right) ^ { 6 }\) the coefficient of \(x ^ { 2 }\) is 48 . Find the value of the positive constant \(p\).
CAIE P1 2019 November Q2
2 The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 } - 6 x + 7\) for \(x > 4\). By first completing the square, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).
CAIE P1 2019 November Q3
3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).
CAIE P1 2019 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-05_360_639_255_753} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). Arc \(O C\) is part of a circle with centre \(A\).
  1. Express angle \(C A O\) in radians in terms of \(\pi\).
  2. Find the area of the shaded region in terms of \(r , \pi\) and \(\sqrt { } 3\), simplifying your answer.
CAIE P1 2019 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-06_462_878_258_635} The dimensions of a cuboid are \(x \mathrm {~cm} , 2 x \mathrm {~cm}\) and \(4 x \mathrm {~cm}\), as shown in the diagram.
  1. Show that the surface area \(S \mathrm {~cm} ^ { 2 }\) and the volume \(V \mathrm {~cm} ^ { 3 }\) are connected by the relation $$S = 7 V ^ { \frac { 2 } { 3 } }$$
  2. When the volume of the cuboid is \(1000 \mathrm {~cm} ^ { 3 }\) the surface area is increasing at \(2 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the volume at this instant.
CAIE P1 2019 November Q6
6 A line has equation \(y = 3 k x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
  2. For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the \(x\)-axis.
CAIE P1 2019 November Q7
7
  1. Show that the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) can be expressed as \(3 x ^ { 2 } - 4 x + 1 = 0\), where \(x = \cos ^ { 2 } \theta\).
  2. Hence solve the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2019 November Q8
8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).
  1. Find the set of values of \(x\) for which f is decreasing.
  2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). Find \(\mathrm { f } ( x )\).
CAIE P1 2019 November Q9
9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.
  1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
  2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
  3. One of these ratios gives a progression which is convergent. Find the sum to infinity.
CAIE P1 2019 November Q10
10 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(X\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 8
- 4
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 10
2