Questions P1 (1401 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
CAIE P1 2011 June Q6
7 marks Standard +0.3
6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$
  1. Show that \(z = 3 x + \frac { 1200 } { x }\).
  2. Find the stationary value of \(z\) and determine its nature.
CAIE P1 2011 June Q7
7 marks Moderate -0.8
7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { ( 1 + 2 x ) ^ { 2 } }\) and the point \(\left( 1 , \frac { 1 } { 2 } \right)\) lies on the curve.
  1. Find the equation of the curve.
  2. Find the set of values of \(x\) for which the gradient of the curve is less than \(\frac { 1 } { 3 }\).
CAIE P1 2011 June Q8
7 marks Standard +0.8
8 A television quiz show takes place every day. On day 1 the prize money is \(\\) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(\\) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity
  1. if Model 1 is used,
  2. if Model 2 is used.
CAIE P1 2011 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).
  1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
  2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).
CAIE P1 2011 June Q10
10 marks Moderate -0.8
10
  1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
  2. Find the coordinates of \(Q\).
  3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).
CAIE P1 2011 June Q11
11 marks Moderate -0.8
11 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 \\ & \mathrm {~g} : x \mapsto x ^ { 2 } - 2 \end{aligned}$$
  1. Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
  2. Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
  3. Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
  4. Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\). The function h is defined by $$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
  5. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2011 June Q1
5 marks Moderate -0.5
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 1 - 2 x ) ^ { 6 }\), where \(a\) is positive, is 90 . Find the value of \(a\).
CAIE P1 2011 June Q2
5 marks Standard +0.3
2 Find the set of values of \(m\) for which the line \(y = m x + 4\) intersects the curve \(y = 3 x ^ { 2 } - 4 x + 7\) at two distinct points.
CAIE P1 2011 June Q3
5 marks Standard +0.3
3 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Given that \(P Q = \sqrt { } ( 45 )\) and that the gradient of the line \(P Q\) is \(- \frac { 1 } { 2 }\), find the values of \(a\) and \(b\).
CAIE P1 2011 June Q4
7 marks Easy -1.2
4
  1. Differentiate \(\frac { 2 x ^ { 3 } + 5 } { x }\) with respect to \(x\).
  2. Find \(\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\) and hence find the value of \(\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\).
CAIE P1 2011 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-2_748_1155_1146_495} In the diagram, \(O A B C D E F G\) is a rectangular block in which \(O A = O D = 6 \mathrm {~cm}\) and \(A B = 12 \mathrm {~cm}\). The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is the mid-point of \(D G , Q\) is the centre of the square face \(C B F G\) and \(R\) lies on \(A B\) such that \(A R = 4 \mathrm {~cm}\).
  1. Express each of the vectors \(\overrightarrow { P Q }\) and \(\overrightarrow { R Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(R Q P\).
CAIE P1 2011 June Q6
8 marks Standard +0.3
6
  1. A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term.
  2. An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms.
CAIE P1 2011 June Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-3_462_956_258_593} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius 6 cm , and angle \(A O B = \frac { 1 } { 3 } \pi\) radians. The line \(A X\) is a tangent to the circle at \(A\), and \(O B X\) is a straight line.
  1. Show that the exact length of \(A X\) is \(6 \sqrt { } 3 \mathrm {~cm}\). Find, in terms of \(\pi\) and \(\sqrt { } 3\),
  2. the area of the shaded region,
  3. the perimeter of the shaded region.
CAIE P1 2011 June Q8
7 marks Standard +0.3
8
  1. Prove the identity \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } \equiv \frac { 1 - \cos \theta } { 1 + \cos \theta }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } = \frac { 2 } { 5 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2011 June Q9
11 marks Moderate -0.3
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { } x } - 1\) and \(P ( 9,5 )\) is a point on the curve.
  1. Find the equation of the curve.
  2. Find the coordinates of the stationary point on the curve.
  3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and determine the nature of the stationary point.
  4. The normal to the curve at \(P\) makes an angle of \(\tan ^ { - 1 } k\) with the positive \(x\)-axis. Find the value of \(k\).
CAIE P1 2011 June Q10
12 marks Moderate -0.8
10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto 2 ( x - 1 ) ^ { 3 } + 8 , \quad x > 1 . \end{aligned}$$
  1. Evaluate fg(2).
  2. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
  3. Obtain an expression for \(\mathrm { g } ^ { \prime } ( x )\) and use your answer to explain why g has an inverse.
  4. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
CAIE P1 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\sin 2 x = 2 \cos 2 x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2012 June Q2
4 marks Standard +0.3
2 Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 2 x ^ { 3 } - \frac { 1 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2012 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_584_659_575_742} In the diagram, \(A B C\) is an equilateral triangle of side 2 cm . The mid-point of \(B C\) is \(Q\). An arc of a circle with centre \(A\) touches \(B C\) at \(Q\), and meets \(A B\) at \(P\) and \(A C\) at \(R\). Find the total area of the shaded regions, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2012 June Q4
5 marks Moderate -0.3
4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, \(M \mathrm {~kg}\), and radius, \(r \mathrm {~cm}\), are related by the formula \(M = k r ^ { 3 }\), where \(k\) is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of \(k\) and the rate at which the mass is increasing on this day.
CAIE P1 2012 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_636_947_1738_598} The diagram shows the curve \(y = 7 \sqrt { } x\) and the line \(y = 6 x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).
  1. For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(k\) for which \(y = 6 x + k\) is a tangent to the curve \(y = 7 \sqrt { } x\).
CAIE P1 2012 June Q6
7 marks Moderate -0.8
6 Two vectors \(\mathbf { u }\) and \(\mathbf { v }\) are such that \(\mathbf { u } = \left( \begin{array} { c } p ^ { 2 } \\ - 2 \\ 6 \end{array} \right)\) and \(\mathbf { v } = \left( \begin{array} { c } 2 \\ p - 1 \\ 2 p + 1 \end{array} \right)\), where \(p\) is a constant.
  1. Find the values of \(p\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
  2. For the case where \(p = 1\), find the angle between the directions of \(\mathbf { u }\) and \(\mathbf { v }\).
CAIE P1 2012 June Q7
7 marks Standard +0.3
7
  1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
  2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
    1. Find the set of values of \(\theta\) for which the progression is convergent.
    2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).
CAIE P1 2012 June Q8
8 marks Moderate -0.3
8 The function \(\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k\) is defined for the domain \(x \geqslant p\), where \(k\) and \(p\) are constants.
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b + k\), where \(a\) and \(b\) are constants.
  2. State the range of f in terms of \(k\).
  3. State the smallest value of \(p\) for which f is one-one.
  4. For the value of \(p\) found in part (iii), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain \(\mathrm { f } ^ { - 1 }\), giving your answers in terms of \(k\).
CAIE P1 2012 June Q9
9 marks Standard +0.3
9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).
  1. Find the equation of \(M B\) and the coordinates of \(B\).
  2. Show that \(A B\) is perpendicular to \(B C\).
  3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).