Questions — OCR C1 (333 questions)

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OCR C1 Q2
5 marks Standard +0.3
Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [5]
OCR C1 Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
OCR C1 Q4
6 marks Moderate -0.8
The curve \(C\) has the equation \(y = (x - a)^2\) where \(a\) is a constant. Given that $$\frac{dy}{dx} = 2x - 6,$$ \begin{enumerate}[label=(\roman*)] \item find the value of \(a\), [4] \item describe fully a single transformation that would map \(C\) onto the graph of \(y = x^2\). [2]
OCR C1 Q5
7 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\). \begin{enumerate}[label=(\roman*)] \item Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [4] \item Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR C1 Q6
10 marks Moderate -0.8
\begin{enumerate}[label=(\alph*)] \item Given that \(y = 2^x\), find expressions in terms of \(y\) for
  1. \(2^{x+2}\), [2]
  2. \(2^{3-x}\). [2]
\item Show that using the substitution \(y = 2^x\), the equation $$2^{x+2} + 2^{3-x} = 33$$ can be rewritten as $$4y^2 - 33y + 8 = 0.$$ [2] \item Hence solve the equation $$2^{x+2} + 2^{3-x} = 33.$$ [4]
OCR C1 Q7
11 marks Moderate -0.8
The point \(A\) has coordinates \((4, 6)\). Given that \(OA\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). [4]
Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\). \begin{enumerate}[label=(\roman*)] \setcounter{enumi}{1} \item Find the coordinates of \(B\). [2] \item Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 Q8
12 marks Moderate -0.3
  1. Express \(3x^2 - 12x + 11\) in the form \(a(x + b)^2 + c\). [4]
  2. Sketch the curve with equation \(y = 3x^2 - 12x + 11\), showing the coordinates of the minimum point of the curve. [3]
Given that the curve \(y = 3x^2 - 12x + 11\) crosses the \(x\)-axis at the points \(A\) and \(B\), \begin{enumerate}[label=(\roman*)] \setcounter{enumi}{2} \item find the length \(AB\) in the form \(k\sqrt{3}\). [5]
OCR C1 Q9
13 marks Moderate -0.3
A curve has the equation \(y = x^3 - 5x^2 + 7x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. [4]
The point \(P\) on the curve has coordinates \((3, 3)\).
  1. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [6]
The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  1. Show that triangle \(OQR\), where \(O\) is the origin, has area \(28\frac{1}{8}\). [3]