Questions P1 (1374 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 June Q8
8 marks Standard +0.3
8 Relative to an origin \(O\), the position vectors of three points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = 3 \mathbf { i } + p \mathbf { j } - 2 p \mathbf { k } , \quad \overrightarrow { O B } = 6 \mathbf { i } + ( p + 4 ) \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = ( p - 1 ) \mathbf { i } + 2 \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.
  1. In the case where \(p = 2\), use a scalar product to find angle \(A O B\).
  2. In the case where \(\overrightarrow { A B }\) is parallel to \(\overrightarrow { O C }\), find the values of \(p\) and \(q\).
CAIE P1 2017 June Q9
9 marks Moderate -0.3
9 The equation of a curve is \(y = 8 \sqrt { } x - 2 x\).
  1. Find the coordinates of the stationary point of the curve.
  2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence, or otherwise, determine the nature of the stationary point.
  3. Find the values of \(x\) at which the line \(y = 6\) meets the curve.
  4. State the set of values of \(k\) for which the line \(y = k\) does not meet the curve.
CAIE P1 2017 June Q10
11 marks Standard +0.2
10 The function f is defined by \(\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) + 4 = 0\), giving your answer correct to 1 decimal place.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2017 June Q1
3 marks Moderate -0.8
1 The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 7 }\) are equal. Find the value of the non-zero constant \(a\).
CAIE P1 2017 June Q2
4 marks Standard +0.8
2 The common ratio of a geometric progression is \(r\). The first term of the progression is \(\left( r ^ { 2 } - 3 r + 2 \right)\) and the sum to infinity is \(S\).
  1. Show that \(S = 2 - r\).
  2. Find the set of possible values that \(S\) can take.
CAIE P1 2017 June Q3
4 marks Moderate -0.8
3 Find the coordinates of the points of intersection of the curve \(y = x ^ { \frac { 2 } { 3 } } - 1\) with the curve \(y = x ^ { \frac { 1 } { 3 } } + 1\). [4]
CAIE P1 2017 June Q4
6 marks Moderate -0.5
4 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l }
CAIE P1 2017 June Q5
6 marks Moderate -0.8
5
1
3 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 5
4
- 3 \end{array} \right) .$$ The point \(P\) lies on \(A B\) and is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\).
  1. Find the position vector of \(P\).
  2. Find the distance \(O P\).
  3. Determine whether \(O P\) is perpendicular to \(A B\). Justify your answer.
    5
  4. Show that the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) may be expressed as \(\cos ^ { 2 } \theta = 2 \sin ^ { 2 } \theta\).
  5. Hence solve the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2017 June Q6
6 marks Standard +0.3
6 The line \(3 y + x = 25\) is a normal to the curve \(y = x ^ { 2 } - 5 x + k\). Find the value of the constant \(k\).
CAIE P1 2017 June Q7
7 marks Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{4782d612-0ec1-418e-8ef3-c871dce82b44-10_611_732_255_705} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(C A D\) is a straight line and \(A B\) is perpendicular to \(C D\).
  1. Find angle \(A B C\) in radians.
  2. Find the area of the shaded region.
    \(8 \quad A ( - 1,1 )\) and \(P ( a , b )\) are two points, where \(a\) and \(b\) are constants. The gradient of \(A P\) is 2 .
  3. Find an expression for \(b\) in terms of \(a\).
  4. \(B ( 10 , - 1 )\) is a third point such that \(A P = A B\). Calculate the coordinates of the possible positions of \(P\).
CAIE P1 2017 June Q9
9 marks Moderate -0.3
9
  1. Express \(9 x ^ { 2 } - 6 x + 6\) in the form \(( a x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
    The function f is defined by \(\mathrm { f } ( x ) = 9 x ^ { 2 } - 6 x + 6\) for \(x \geqslant p\), where \(p\) is a constant.
  2. State the smallest value of \(p\) for which f is a one-one function.
  3. For this value of \(p\), obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain of \(\mathrm { f } ^ { - 1 }\).
  4. State the set of values of \(q\) for which the equation \(\mathrm { f } ( x ) = q\) has no solution.
CAIE P1 2017 June Q10
11 marks Standard +0.3
10
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-16_451_442_269_888} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows part of the curve \(y = x ^ { 2 } - 1\) and the line \(y = h\), where \(h\) is a constant.
    1. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Show that the volume of revolution, \(V\), is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\).
    2. Find, showing all necessary working, the area of the shaded region when \(h = 3\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4782d612-0ec1-418e-8ef3-c871dce82b44-17_257_408_1126_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water is given by \(V = \pi \left( \frac { 1 } { 2 } h ^ { 2 } + h \right)\). Water is poured into the bowl at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at which the height of the water level is increasing when the height of the water level is 3 cm .
CAIE P1 2017 June Q11
11 marks Standard +0.3
11 The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(f ^ { \prime \prime } ( x ) = ( 4 x + 1 ) ^ { - \frac { 1 } { 2 } }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    It is now given that \(f ^ { \prime \prime } ( 0 ) , f ^ { \prime } ( 0 )\) and \(f ( 0 )\) are the first three terms respectively of an arithmetic progression.
  2. Find the value of \(\mathrm { f } ( 0 )\).
  3. Find \(\mathrm { f } ( x )\), and hence find the minimum value of f .
CAIE P1 2018 June Q1
5 marks Moderate -0.8
1
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + a x + 2 x ^ { 2 } \right) ( 1 - 2 x ) ^ { 5 }\) is 12 , find the value of the constant \(a\).
CAIE P1 2018 June Q2
4 marks Moderate -0.5
2 A point is moving along the curve \(y = 2 x + \frac { 5 } { x }\) in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 1\).
CAIE P1 2018 June Q3
6 marks Moderate -0.3
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\). The point \(( 1,1 )\) lies on the curve. Find the coordinates of the point at which the curve intersects the \(x\)-axis.
CAIE P1 2018 June Q4
6 marks Standard +0.3
4
  1. Prove the identity \(( \sin \theta + \cos \theta ) ( 1 - \sin \theta \cos \theta ) \equiv \sin ^ { 3 } \theta + \cos ^ { 3 } \theta\).
  2. Hence solve the equation \(( \sin \theta + \cos \theta ) ( 1 - \sin \theta \cos \theta ) = 3 \cos ^ { 3 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 June Q5
7 marks Moderate -0.3
5
\includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.
  1. Find the equations of \(A C\) and \(O B\).
  2. Find, by calculation, the coordinates of \(B\).
CAIE P1 2018 June Q6
7 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.
  1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.
CAIE P1 2018 June Q7
8 marks Moderate -0.8
7 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 1 \\ 3 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right)$$
  1. Find \(\overrightarrow { A C }\).
  2. The point \(M\) is the mid-point of \(A C\). Find the unit vector in the direction of \(\overrightarrow { O M }\).
  3. Evaluate \(\overrightarrow { A B } \cdot \overrightarrow { A C }\) and hence find angle \(B A C\).
CAIE P1 2018 June Q8
9 marks Moderate -0.3
8
  1. A geometric progression has a second term of 12 and a sum to infinity of 54 . Find the possible values of the first term of the progression.
  2. The \(n\)th term of a progression is \(p + q n\), where \(p\) and \(q\) are constants, and \(S _ { n }\) is the sum of the first \(n\) terms.
    1. Find an expression, in terms of \(p , q\) and \(n\), for \(S _ { n }\).
    2. Given that \(S _ { 4 } = 40\) and \(S _ { 6 } = 72\), find the values of \(p\) and \(q\).
CAIE P1 2018 June Q9
11 marks Moderate -0.3
9 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { 2 } x - 2 \\ & \mathrm {~g} : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 } \end{aligned}$$
  1. Find the points of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\).
  3. Find an expression for \(\mathrm { fg } ( x )\) and deduce the range of fg .
    The function h is defined by \(\mathrm { h } : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 }\) for \(x \geqslant k\).
  4. Find the smallest value of \(k\) for which h has an inverse.
CAIE P1 2018 June Q10
12 marks Standard +0.3
10 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 5 x\) passes through the origin.
  1. Show that the curve has no stationary points.
  2. Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
  3. Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).
    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 June Q1
5 marks Moderate -0.3
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + \frac { x } { 2 } \right) ^ { 6 } + ( a + x ) ^ { 5 }\) is 330 . Find the value of the constant \(a\).
CAIE P1 2018 June Q2
5 marks Moderate -0.5
2 The equation of a curve is \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis.
  2. Find the value of \(k\) for which the line \(y + 2 x = 7\) is a tangent to the curve.