Questions FP1 (1491 questions)

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AQA FP1 2016 June Q2
5 marks Moderate -0.8
A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).
  1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
  2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]
AQA FP1 2016 June Q3
7 marks Moderate -0.3
The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).
  1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
  2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
    1. Find the gradient of the line. [1 mark]
    2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]
AQA FP1 2016 June Q4
7 marks Moderate -0.3
  1. Given that \(\sin \frac{\pi}{3} = \cos \frac{\pi}{k}\), state the value of the integer \(k\). [1 mark]
  2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$$ giving your answer, in its simplest form, in terms of \(\pi\). [4 marks]
  3. Hence, given that \(\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}\), show that there is only one finite value for \(\tan x\) and state its exact value. [2 marks]
AQA FP1 2016 June Q5
9 marks Standard +0.8
  1. Use the formulae for \(\sum_{r=1}^n r^2\) and \(\sum_{r=1}^n r\) to show that \(\sum_{r=1}^n (6r - 3)^2 = 3n(4n^2 - 1)\). [5 marks]
  2. Hence express \(\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (6r - 3)^2\) as a product of four linear factors in terms of \(n\). [4 marks]
AQA FP1 2016 June Q6
9 marks Standard +0.8
A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).
  1. Show that \(a = 2\). [3 marks]
  2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]
AQA FP1 2016 June Q7
10 marks Standard +0.3
  1. Solve the equation \(x^2 + 4x + 20 = 0\), giving your answers in the form \(c + di\), where \(c\) and \(d\) are integers. [3 marks]
  2. The roots of the quadratic equation $$z^2 + (4 + i + qi)z + 20 = 0$$ are \(w\) and \(w^*\).
    1. In the case where \(q\) is real, explain why \(q\) must be \(-1\). [2 marks]
    2. In the case where \(w = p + 2i\), where \(p\) is real, find the possible values of \(q\). [5 marks]
AQA FP1 2016 June Q8
10 marks Standard +0.3
The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).
    1. Find the matrix \(\mathbf{A}^2\). [1 mark]
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
  2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
  3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
OCR FP1 Q1
6 marks Moderate -0.5
Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]
OCR FP1 Q2
6 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
  1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
  2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]
OCR FP1 Q3
7 marks Moderate -0.8
The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 Q4
6 marks Standard +0.3
Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]
OCR FP1 Q5
7 marks Standard +0.3
  1. Show that $$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$ [2]
  2. Hence find an expression, in terms of \(n\), for $$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$ [4]
  3. Hence write down the value of \(\sum_{r=1}^\infty \frac{1}{(r+1)(r+2)}\). [1]
OCR FP1 Q6
7 marks Standard +0.3
The loci \(C_1\) and \(C_2\) are given by $$|z - 2i| = 2 \quad \text{and} \quad |z + 1| = |z + i|$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C_1\) and \(C_2\). [5]
  2. Hence write down the complex numbers represented by the points of intersection of \(C_1\) and \(C_2\). [2]
OCR FP1 Q7
10 marks Standard +0.3
The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} a & 1 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}\).
  1. Given that \(\mathbf{B}\) is singular, show that \(a = -\frac{2}{3}\). [3]
  2. Given instead that \(\mathbf{B}\) is non-singular, find the inverse matrix \(\mathbf{B}^{-1}\). [4]
  3. Hence, or otherwise, solve the equations \begin{align} -x + y + 3z &= 1,
    2x + y - z &= 4,
    y + 2z &= -1. \end{align} [3]
OCR FP1 Q8
11 marks Moderate -0.3
  1. The quadratic equation \(x^2 - 2x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha\beta\). [2]
    2. Show that \(\alpha^2 + \beta^2 = -4\). [2]
    3. Hence find a quadratic equation which has roots \(\alpha^2\) and \(\beta^2\). [3]
  2. The cubic equation \(x^3 - 12x^2 + ax - 48 = 0\) has roots \(p\), \(2p\) and \(3p\).
    1. Find the value of \(p\). [2]
    2. Hence find the value of \(a\). [2]
OCR FP1 Q9
12 marks Standard +0.3
  1. Write down the matrix \(\mathbf{C}\) which represents a stretch, scale factor \(2\), in the \(x\)-direction. [2]
  2. The matrix \(\mathbf{D}\) is given by \(\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\). Describe fully the geometrical transformation represented by \(\mathbf{D}\). [2]
  3. The matrix \(\mathbf{M}\) represents the combined effect of the transformation represented by \(\mathbf{C}\) followed by the transformation represented by \(\mathbf{D}\). Show that $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
  4. Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]
OCR FP1 2013 January Q1
5 marks Easy -1.2
The matrix A is given by \(A = \begin{pmatrix} a & 1 \\ 1 & a \end{pmatrix}\), where \(a \neq \frac{1}{2}\), and I denotes the \(2 \times 2\) identity matrix. Find
  1. \(2A - 3I\), [3]
  2. \(A^{-1}\). [2]
OCR FP1 2013 January Q2
6 marks Moderate -0.3
Find \(\sum_{r=1}^{n} (r-1)(r+1)\), giving your answer in a fully factorised form. [6]
OCR FP1 2013 January Q3
7 marks Moderate -0.3
The complex number \(2 - i\) is denoted by \(z\).
  1. Find \(|z|\) and \(\arg z\). [2]
  2. Given that \(az + bz^* = 4 - 8i\), find the values of the real constants \(a\) and \(b\). [5]
OCR FP1 2013 January Q4
4 marks Moderate -0.3
The quadratic equation \(x^2 + x + k = 0\) has roots \(\alpha\) and \(\beta\).
  1. Use the substitution \(x = 2u + 1\) to obtain a quadratic equation in \(u\). [2]
  2. Hence, or otherwise, find the value of \(\left(\frac{\alpha - 1}{2}\right)\left(\frac{\beta - 1}{2}\right)\) in terms of \(k\). [2]
OCR FP1 2013 January Q5
6 marks Standard +0.8
By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations \begin{align} 3x + 2y + 4z &= 5,
\lambda y + z &= 1,
x + \lambda y + \lambda z &= 4, \end{align} do not have a unique solution for \(x\), \(y\) and \(z\). [6]
OCR FP1 2013 January Q6
10 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).
  1. Write down the matrix, X, for this transformation. [2]
  2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
  3. Find the matrix that represents transformation Q followed by transformation P. [2]
OCR FP1 2013 January Q7
7 marks Moderate -0.3
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z| = 2\), [2]
    2. \(\arg(z - 3 - i) = \pi\). [3]
  2. Indicate, by shading, the region of the Argand diagram for which $$|z| < 2 \text{ and } 0 < \arg(z - 3 - i) < \pi.$$ [2]
OCR FP1 2013 January Q8
9 marks Standard +0.8
  1. Show that \(\frac{1}{r} - \frac{3}{r+1} + \frac{2}{r+2} = \frac{2-r}{r(r+1)(r+2)}\). [2]
  2. Hence show that \(\sum_{r=1}^{n} \frac{2-r}{r(r+1)(r+2)} = -\frac{n}{(n+1)(n+2)}\). [5]
  3. Find the value of \(\sum_{r=3}^{\infty} \frac{2-r}{r(r+1)(r+2)}\). [2]