1.02f Solve quadratic equations: including in a function of unknown

10

1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). [4]

3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where $$v = 2 t ^ { 2 } - 14 t + 20 , \quad t \geqslant 0$$ Find

1. the times when \(P\) is instantaneously at rest,
2. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\)
3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

3

1. Express \(2 \ln x + \ln 3\) as a single logarithm.
2. Hence, given that \(x\) satisfies the equation $$2 \ln x + \ln 3 = \ln ( 5 x + 2 )$$ show that \(x\) is a root of the quadratic equation \(3 x ^ { 2 } - 5 x - 2 = 0\).
3. Solve this quadratic equation, explaining why only one root is a valid solution of $$2 \ln x + \ln 3 = \ln ( 5 x + 2 ) .$$

4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

1. Complete the tree diagram below to show this information. \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
2. Find the probability that the student was awarded a certificate.
3. Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be \(p\), and the probability of passing the test at the second attempt to be ( \(p - 0.2\) ). The percentage of students who complete the course and are awarded a certificate is to be \(95 \%\)
4. Show that \(p\) satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
5. Hence find the value of \(p\), giving your answer to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola with equation \(y = \frac { 1 } { 2 } x ( 10 - x ) , 0 \leqslant x \leqslant 10\) This question concerns rectangles that lie under the parabola in the first quadrant.The bottom edge of each rectangle lies along the \(x\)-axis and the top left vertex lies on the parabola.Some examples are shown in Figure 2. Let the \(x\) coordinate of the top left vertex be \(a\) .

1. Explain why the width,\(w\) ,of such a rectangle must satisfy \(w \leqslant 10 - 2 a\)
2. Find the value of \(a\) that gives the maximum area for such a rectangle. Given that the rectangle must be a square,
3. find the value of \(a\) that gives the maximum area for such a square. Given that the area of the rectangles is fixed as 36
4. find the range of possible values for \(a\) . \includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-16_2255_50_311_1980}

11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$

1. find the value of the discriminant of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
3. Calculate the value of \(c\).

3 Solve the equation \(3 x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } - 2 = 0\).

8

1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.

5 Solve the equation \(x - 8 \sqrt { x } + 13 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

4 By using the substitution \(u = ( 3 x - 2 ) ^ { 2 }\), find the roots of the equation $$( 3 x - 2 ) ^ { 4 } - 5 ( 3 x - 2 ) ^ { 2 } + 4 = 0$$

7

1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).

5 Find the real roots of the equation \(\frac { 3 } { y ^ { 4 } } - \frac { 10 } { y ^ { 2 } } - 8 = 0\).

10

1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?

5 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).

6 Solve the equation \(3 x ^ { \frac { 1 } { 2 } } - 8 x ^ { \frac { 1 } { 4 } } + 4 = 0\).

7 Solve the equation \(x - 6 x ^ { \frac { 1 } { 2 } } + 2 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

4 Solve the equation \(x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } - 6 = 0\).

4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).

9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.

13 \begin{figure}[h] \end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).

1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).

9 Fig. 9 shows a trapezium ABCD , with the lengths in centimetres of three of its sides. \begin{figure}[h] \end{figure} This trapezium has area \(140 \mathrm {~cm} ^ { 2 }\).

1. Show that \(x ^ { 2 } + 2 x - 35 = 0\).
2. Hence find the length of side AB of the trapezium.

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).

1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7

1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.

9

1. The polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$ Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\), and hence find the other two roots.
2. Hence solve the equation $$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$ for \(0 \leqslant x \leqslant 2 \pi\). Give each solution for \(x\) in an exact form.