OCR C1 (Core Mathematics 1)

1. Calculate the discriminant of \(2x^2 + 8x + 8\). [2]
2. State the number of real roots of the equation \(2x^2 + 8x + 8 = 0\). [1]

Find the set of values of \(x\) for which $$(x - 1)(x - 2) < 20.$$ [4]

1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]

Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]

\includegraphics{figure_5} The diagram shows a sketch of the curve with equation \(y = f(x)\). The curve has a maximum at \((-3, 4)\) and a minimum at \((1, -2)\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

1. \(y = 2f(x)\), [3]
2. \(y = -f(x)\). [3]

\(f(x) = 2x^2 - 4x + 1\).

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$f(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = f(x)\). [1]
3. Solve the equation \(f(x) = 3\), giving your answers in exact form. [3]

A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),

1. find the values of \(a\) and \(b\), [6]
2. find the coordinates of the other stationary point of the curve. [3]

The circle \(C\) has the equation $$x^2 + y^2 + 10x - 8y + k = 0,$$ where \(k\) is a constant. Given that the point with coordinates \((-6, 5)\) lies on \(C\),

1. find the value of \(k\), [2]
2. find the coordinates of the centre and the radius of \(C\). [3]

A straight line which passes through the point \(A(2, 3)\) is a tangent to \(C\) at the point \(B\).

1. Find the length \(AB\) in the form \(k\sqrt{5}\). [5]

A curve has the equation \(y = x + \frac{3}{x}\), \(x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate \(1\).

1. Show that the gradient of the curve at \(P\) is \(-2\). [3]
2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = mx + c\). [3]
3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again. [4]

The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\). [2]

The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).

1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]

The line \(l_2\) crosses the \(x\)-axis at the point \(C\).

1. Find the coordinates of \(C\). [1]

The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).

1. Show that \(D\) has coordinates \((5, 4)\). [5]
2. Find the area of triangle \(ACD\). [2]