Questions P1 (1401 questions)

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CAIE P1 2023 June Q8
12 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) 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 \(y = f^{-1}(x)\). [2]
  2. Find an expression for \(f^{-1}(x)\). [2]
  3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
  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 \(y = f(x)\), making clear the order in which the transformations are applied. [6]
CAIE P1 2023 June Q9
8 marks Standard +0.3
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. [4]
  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. [4]
CAIE P1 2023 June Q10
13 marks Standard +0.3
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. [5]
  2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
  3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]
CAIE P1 2023 June Q11
8 marks Moderate -0.3
The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
  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. [4]
CAIE P1 2024 June Q1
5 marks Moderate -0.8
  1. Express \(3y^2 - 12y - 15\) in the form \(3(y + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. Hence find the exact solutions of the equation \(3x^4 - 12x^2 - 15 = 0\). [3]
CAIE P1 2024 June Q2
6 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = \text{f}(x)\).
  1. In order to transform the curve \(y = \sin x\) to the curve \(y = \text{f}(x)\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required. [4]
  2. Find f\((x)\) in terms of \(\sin x\). [2]
CAIE P1 2024 June Q3
5 marks Moderate -0.8
The coefficient of \(x^3\) in the expansion of \((3 + ax)^6\) is 160.
  1. Find the value of the constant \(a\). [2]
  2. Hence find the coefficient of \(x^5\) in the expansion of \((3 + ax)^6(1 - 2x)\). [3]
CAIE P1 2024 June Q4
3 marks Easy -1.8
The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)
  1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]
The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).
Chord\(AB\)\(AC\)\(AD\)\(AE\)\(AF\)
Gradient of chord6.25016.25116.26087.2288
  1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
  2. Deduce the value of f\('(2)\) using the values in the table. [1]
CAIE P1 2024 June Q5
6 marks Moderate -0.8
  1. Prove the identity \(\frac{\sin^2 x - \cos x - 1}{1 + \cos x} \equiv -\cos x\). [3]
  2. Hence solve the equation \(\frac{\sin^2 x - \cos x - 1}{2 + 2\cos x} = \frac{1}{4}\) for \(0° \leq x \leq 360°\). [3]
CAIE P1 2024 June Q6
7 marks Moderate -0.3
\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).
  1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
  2. Find an expression for f\(^{-1}(x)\). [3]
  3. Solve the equation f\((x) = 4.5\). [1]
  4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]
CAIE P1 2024 June Q7
8 marks Standard +0.3
\includegraphics{figure_7} In the diagram, \(AOD\) and \(BC\) are two parallel straight lines. Arc \(AB\) is part of a circle with centre \(O\) and radius \(15\text{cm}\). Angle \(BOA = \theta\) radians. Arc \(CD\) is part of a circle with centre \(O\) and radius \(10\text{cm}\). Angle \(COD = \frac{1}{3}\pi\) radians.
  1. Show that \(\theta = 0.7297\), correct to 4 decimal places. [1]
  2. Find the perimeter and the area of the shape \(ABCD\). Give your answers correct to 3 significant figures. [7]
CAIE P1 2024 June Q8
8 marks Moderate -0.3
  1. The first three terms of an arithmetic progression are \(25\), \(4p - 1\) and \(13 - p\), where \(p\) is a constant. Find the value of the tenth term of the progression. [4]
  2. The first three terms of a geometric progression are \(25\), \(4q - 1\) and \(13 - q\), where \(q\) is a positive constant. Find the sum to infinity of the progression. [4]
CAIE P1 2024 June Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\). Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]
CAIE P1 2024 June Q10
8 marks Standard +0.8
The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]
CAIE P1 2024 June Q11
13 marks Standard +0.8
\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.
  1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
  2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]
CAIE P1 2024 June Q1
3 marks Moderate -0.5
The coefficient of \(x^2\) in the expansion of \((1-4x)^6\) is 12 times the coefficient of \(x^2\) in the expansion of \((2+ax)^5\). Find the value of the positive constant \(a\). [3]
CAIE P1 2024 June Q2
5 marks Moderate -0.8
The curve \(y = x^2\) is transformed to the curve \(y = 4(x-3)^2 - 8\). Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied. [5]
CAIE P1 2024 June Q3
6 marks Moderate -0.5
  1. Show that the equation \(\frac{7\tan\theta}{\cos\theta} + 12 = 0\) can be expressed as $$12\sin^2\theta - 7\sin\theta - 12 = 0.$$ [3]
  2. Hence solve the equation \(\frac{7\tan\theta}{\cos\theta} + 12 = 0\) for \(0° < \theta \leqslant 360°\). [3]
CAIE P1 2024 June Q4
7 marks Standard +0.3
The function f is defined as follows: $$f(x) = \sqrt{x-1} \text{ for } x > 1.$$ \begin{enumerate}[label=(\alph*)] \item Find an expression for \(f^{-1}(x)\). [1] \end enumerate} \includegraphics{figure_4} The diagram shows the graph of \(y = g(x)\) where \(g(x) = \frac{1}{x^2+2}\) for \(x \in \mathbb{R}\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item State the range of g and explain whether \(g^{-1}\) exists. [2] \end enumerate} The function h is defined by \(h(x) = \frac{1}{x^2+2}\) for \(x \geqslant 0\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{2} \item Solve the equation \(hf(x) = f\left(\frac{25}{16}\right)\). Give your answer in the form \(a + b\sqrt{c}\), where \(a\), \(b\) and \(c\) are integers. [4] \end enumerate}
CAIE P1 2024 June Q5
9 marks Standard +0.3
The first and second terms of an arithmetic progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\). \begin{enumerate}[label=(\alph*)] \item Given that \(\theta = \frac{1}{4}\pi\), find the exact sum of the first 40 terms of the progression. [4] \end enumerate} The first and second terms of a geometric progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\).
    1. Find the sum to infinity of the progression in terms of \(\theta\). [2]
    2. Given that \(\theta = \frac{1}{3}\pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]
CAIE P1 2024 June Q6
9 marks Moderate -0.3
The curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\). \begin{enumerate}[label=(\alph*)] \item Find the coordinates of \(A\) and \(B\). [4] \end enumerate}
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) and the line \(AB\). It is given that the equation of \(AB\) is \(y = \frac{2x-32}{3}\). Find the area of the shaded region between the curve and the line. [5]
CAIE P1 2024 June Q7
8 marks Standard +0.3
The equation of a circle is \((x-6)^2 + (y+a)^2 = 18\). The line with equation \(y = 2a - x\) is a tangent to the circle.
  1. Find the two possible values of the constant \(a\). [5]
  2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent. [3]
CAIE P1 2024 June Q8
10 marks Moderate -0.8
\includegraphics{figure_8} The diagram shows a symmetrical plate \(ABCDEF\). The line \(ABCD\) is straight and the length of \(BC\) is 2cm. Each of the two sectors \(ABF\) and \(DCE\) is of radius \(r\)cm and each of the angles \(ABF\) and \(DCE\) is equal to \(\frac{1}{4}\pi\) radians.
  1. It is given that \(r = 0.4\)cm.
    1. Show that the length \(EF = 2.4\)cm. [2]
    2. Find the area of the plate. Give your answer correct to 3 significant figures. [4]
  2. It is given instead that the perimeter of the plate is 6cm. Find the value of \(r\). Give your answer correct to 3 significant figures. [4]
CAIE P1 2024 June Q9
8 marks Moderate -0.3
A function f is such that \(f'(x) = 6(2x-3)^2 - 6x\) for \(x \in \mathbb{R}\).
  1. Determine the set of values of \(x\) for which f\((x)\) is decreasing. [4]
  2. Given that f\((1) = -1\), find f\((x)\). [4]
CAIE P1 2024 June Q10
10 marks Standard +0.3
The equation of a curve is \(y = (5-2x)^{\frac{1}{2}} + 5\) for \(x < \frac{5}{2}\).
  1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\). [4]
  2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is \(-3\). Find the equation of the perpendicular bisector of \(AB\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]