Questions P1 (1401 questions)

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CAIE P1 2007 June Q9
8 marks Moderate -0.8
9 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 4 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ 2 \\ - 4 \end{array} \right) .$$
  1. Given that \(C\) is the point such that \(\overrightarrow { A C } = 2 \overrightarrow { A B }\), find the unit vector in the direction of \(\overrightarrow { O C }\). The position vector of the point \(D\) is given by \(\overrightarrow { O D } = \left( \begin{array} { l } 1 \\ 4 \\ k \end{array} \right)\), where \(k\) is a constant, and it is given that \(\overrightarrow { O D } = m \overrightarrow { O A } + n \overrightarrow { O B }\), where \(m\) and \(n\) are constants.
  2. Find the values of \(m , n\) and \(k\).
CAIE P1 2007 June Q10
12 marks Moderate -0.8
10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).
  1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
  3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
  4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
CAIE P1 2007 June Q11
12 marks Moderate -0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-4_862_892_932_628} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } : x \mapsto \frac { 6 } { 2 x + 3 }\) for \(x \geqslant 0\).
  1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and explain how your answer shows that f is a decreasing function.
  2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Copy the diagram and, on your copy, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs. The function g is defined by \(\mathrm { g } : x \mapsto \frac { 1 } { 2 } x\) for \(x \geqslant 0\).
  4. Solve the equation \(\operatorname { fg } ( x ) = \frac { 3 } { 2 }\).
CAIE P1 2008 June Q1
3 marks Moderate -0.5
1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).
CAIE P1 2008 June Q2
5 marks Moderate -0.3
2
  1. Show that the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\) can be written in the form \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\).
  2. Hence solve the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2008 June Q3
6 marks Moderate -0.8
3
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + x ^ { 2 } \right) ^ { 5 }\).
  2. Hence find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2008 June Q4
7 marks Moderate -0.8
4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).
  1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
  2. Show that one of these points is also the stationary point of \(C\).
CAIE P1 2008 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
CAIE P1 2008 June Q6
7 marks Moderate -0.3
6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).
  1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2008 June Q7
7 marks Standard +0.3
7 The first term of a geometric progression is 81 and the fourth term is 24 . Find
  1. the common ratio of the progression,
  2. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.
  3. Find the sum of the first ten terms of the arithmetic progression.
CAIE P1 2008 June Q8
7 marks Standard +0.8
8 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 4 x - 2 k & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, } \\ \mathrm { g } : x \mapsto \frac { 9 } { 2 - x } & \text { for } x \in \mathbb { R } , x \neq 2 . \end{array}$$
  1. Find the values of \(k\) for which the equation \(\mathrm { fg } ( x ) = x\) has two equal roots.
  2. Determine the roots of the equation \(\operatorname { fg } ( x ) = x\) for the values of \(k\) found in part (i).
CAIE P1 2008 June Q9
8 marks Moderate -0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-3_791_885_1281_630} The diagram shows a curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { k } { x ^ { 3 } }\), where \(k\) is a constant. The curve passes through the points \(( 1,18 )\) and \(( 4,3 )\).
  1. Show, by integration, that the equation of the curve is \(y = \frac { 16 } { x ^ { 2 } } + 2\). The point \(P\) lies on the curve and has \(x\)-coordinate 1.6.
  2. Find the area of the shaded region.
CAIE P1 2008 June Q10
9 marks Moderate -0.3
10 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } + p \mathbf { k }\) respectively.
  1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
  2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
  3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.
CAIE P1 2008 June Q11
9 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).
  1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
  2. Find the coordinates of \(D\).
  3. Find, correct to 1 decimal place, the perimeter of \(A B C D\).
CAIE P1 2009 June Q1
3 marks Standard +0.3
1 Prove the identity \(\frac { \sin x } { 1 - \sin x } - \frac { \sin x } { 1 + \sin x } \equiv 2 \tan ^ { 2 } x\).
CAIE P1 2009 June Q2
4 marks Standard +0.3
2 Find the set of values of \(k\) for which the line \(y = k x - 4\) intersects the curve \(y = x ^ { 2 } - 2 x\) at two distinct points.
CAIE P1 2009 June Q3
5 marks Moderate -0.3
3
  1. Find the first 3 terms in the expansion of \(( 2 + 3 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Hence find the value of the constant \(a\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 5 }\).
CAIE P1 2009 June Q4
6 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_561_1210_895_465} The diagram shows the graph of \(y = a \sin ( b x ) + c\) for \(0 \leqslant x \leqslant 2 \pi\).
  1. Find the values of \(a , b\) and \(c\).
  2. Find the smallest value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) for which \(y = 0\).
CAIE P1 2009 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_385_403_1866_872} The diagram shows a circle with centre \(O\). The circle is divided into two regions, \(R _ { 1 }\) and \(R _ { 2 }\), by the radii \(O A\) and \(O B\), where angle \(A O B = \theta\) radians. The perimeter of the region \(R _ { 1 }\) is equal to the length of the major \(\operatorname { arc } A B\).
  1. Show that \(\theta = \pi - 1\).
  2. Given that the area of region \(R _ { 1 }\) is \(30 \mathrm {~cm} ^ { 2 }\), find the area of region \(R _ { 2 }\), correct to 3 significant figures.
CAIE P1 2009 June Q6
7 marks Moderate -0.3
6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 8 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$$
  1. Find the value of \(\overrightarrow { O A } \cdot \overrightarrow { O B }\) and hence state whether angle \(A O B\) is acute, obtuse or a right angle.
  2. The point \(X\) is such that \(\overrightarrow { A X } = \frac { 2 } { 5 } \overrightarrow { A B }\). Find the unit vector in the direction of \(O X\).
CAIE P1 2009 June Q7
7 marks Moderate -0.8
7
  1. Find the sum to infinity of the geometric progression with first three terms \(0.5,0.5 ^ { 3 }\) and \(0.5 ^ { 5 }\).
  2. The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.
CAIE P1 2009 June Q8
7 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_599_716_1071_717} The diagram shows points \(A , B\) and \(C\) lying on the line \(2 y = x + 4\). The point \(A\) lies on the \(y\)-axis and \(A B = B C\). The line from \(D ( 10 , - 3 )\) to \(B\) is perpendicular to \(A C\). Calculate the coordinates of \(B\) and \(C\).
CAIE P1 2009 June Q9
8 marks Moderate -0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_391_595_1978_774} The diagram shows part of the curve \(y = \frac { 6 } { 3 x - 2 }\).
  1. Find the gradient of the curve at the point where \(x = 2\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).
CAIE P1 2009 June Q10
10 marks Moderate -0.3
10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(0 \leqslant x \leqslant A\), where \(A\) is a constant.
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  2. State the value of \(A\) for which the graph of \(y = \mathrm { f } ( x )\) has a line of symmetry.
  3. When \(A\) has this value, find the range of f . The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant 4\).
  4. Explain why \(g\) has an inverse.
  5. Obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2009 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-4_686_805_950_669} The diagram shows the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\) for \(x \geqslant 0\). The curve has a maximum point at \(A\) and a minimum point on the \(x\)-axis at \(B\). The normal to the curve at \(C ( 2,2 )\) meets the normal to the curve at \(B\) at the point \(D\).
  1. Find the coordinates of \(A\) and \(B\).
  2. Find the equation of the normal to the curve at \(C\).
  3. Find the area of the shaded region.