1.02c Simultaneous equations: two variables by elimination and substitution

Rearrange the formula \(c = \sqrt{\frac{a + b}{2}}\) to make \(a\) the subject. [3]

Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]

Solve the equation \(\frac{3x + 1}{2x} = 4\). [3]

Rearrange \(y + 5 = x(y + 2)\) to make \(y\) the subject of the formula. [4]

Make \(x\) the subject of the equation \(y = \frac{x + 3}{x - 2}\). [4]

Solve the equation \(\frac{4x + 5}{2x} = -3\). [3]

Make \(a\) the subject of the equation $$2a + 5c = af + 7c.$$ [3]

Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]

Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]

A circle has equation \(x^2 + y^2 = 45\).

1. State the centre and radius of this circle. [2]
2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]

Find the coordinates of the point of intersection of the lines \(y = 5x - 2\) and \(x + 3y = 8\). [4]

Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2x - 5\) and \(6x + 2y = 7\). [4]

\includegraphics{figure_6} Fig. 11 shows a sketch of the curve with equation \(y = (x - 4)^2 - 3\).

1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point. [2]
2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary. [4]
3. The curve is translated by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\). Show that the equation of the translated curve may be written as \(y = x^2 - 12x + 33\). [2]
4. Show that the line \(y = 8 - 2x\) meets the curve \(y = x^2 - 12x + 33\) at just one point, and find the coordinates of this point. [5]

\includegraphics{figure_2} Figure 2 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x^2 - 2x + 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]

A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).

1. Write down an equation of the circle \(C\). [2]
2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
3. Find the distance between these two points. [2]

\includegraphics{figure_9} The diagram shows the curve \(y = 2x^2 + 6x + 7\) and the straight line \(y = 2x + 13\).

1. Find the coordinates of the points where the curve and line intersect. [4]
2. Show that the area of the shaded region bounded by the curve and line is given by $$\int_{-3}^{1} (6 - 4x - 2x^2) dx.$$ [2]
3. Hence find the area of the shaded region. [5]

\includegraphics{figure_3} A is the point with coordinates (1, 4) on the curve \(y = 4x^2\). B is the point with coordinates (0, 1), as shown in Fig. 10.

1. The line through A and B intersects the curve again at the point C. Show that the coordinates of C are \(\left(-\frac{1}{4}, \frac{1}{4}\right)\). [4]
2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = -2x - \frac{1}{4}\). [6]
3. The two tangents intersect at the point D. Find the \(y\)-coordinate of D. [2]

Fig. 7 shows the curve defined implicitly by the equation $$y^2 + y = x^3 + 2x,$$ together with the line \(x = 2\). \includegraphics{figure_7} Find the coordinates of the points of intersection of the line and the curve. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.

It is given that $$\ln x - \ln y = 3$$

1. Express \(x\) in terms of \(y\) in a form not involving logarithms. [3 marks]
2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\) [3 marks]

The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}

1. State the value of \(x\) which is not in the domain of f. [1 mark]
2. Explain how you know that the function f is many-to-one. [2 marks]
3. 1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
   2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
5. Using set notation, state the range of f. [2 marks]