• Syllabus

Topic 6 - Mathematical models

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Sub sections and their related questions

6.1

• 10M.1.sl.TZ1.14a: Use the mapping diagram to write down two equations in terms of a and b.
• 10M.1.sl.TZ2.13d: The axis of symmetry is x = 2.5.x Write down the range of y.
• 10M.2.sl.TZ1.3a: Write down the values of x where the graph of f (x) intersects the x-axis.
• 10N.1.sl.TZ0.10d: The minimum value for y is – 4. Write down the range of f (x).
• 11N.1.sl.TZ0.13b: Sketch the graph of \(y = f(x)\) for the domain \( - 1 \leqslant x \leqslant 3\).
• 12N.1.sl.TZ0.10a: It is given that f (2) = −5 . Find the value of a .
• 12N.1.sl.TZ0.10c: Write down the range of this quadratic function.
• 12N.2.sl.TZ0.5h: Write down the interval for which g(x) is increasing in the domain 0 < x < 5 .
• 12M.1.sl.TZ1.14a: Find the value of (i) p ; (ii) q .
• 12M.1.sl.TZ1.14b: Write down the value of r .
• 12M.1.sl.TZ1.14c: Find the value of s .
• 12M.2.sl.TZ1.5b: Find the value of f (−1).
• 12M.1.sl.TZ2.11b: The domain of f is  0 ≤ x ≤ 6. Find the range of f .
• 09N.1.sl.TZ0.11a: find the range of \(f\).
• 09M.1.sl.TZ1.7c: Write down the range of \(f (x)\).
• 11M.2.sl.TZ1.3a: Calculate \(f(2)\) .
• 11M.2.sl.TZ1.3f: Find the range of \(f\) .
• 13M.2.sl.TZ1.4a: Calculate \(f (1)\).
• 13M.2.sl.TZ1.4d: Find the range of \(f\).
• 11M.2.sl.TZ2.5g: Using your sketch from (e), determine the range of the function \(f(x)\) for...
• SPM.1.sl.TZ0.7a: On the grid below sketch the graph of the function \(f(x) = 2{(1.6)^x}\) for the domain...
• SPM.1.sl.TZ0.7c: On the grid draw the graph of the function \(g(x) = 5 - 2x\) for the domain...
• SPM.2.sl.TZ0.4e: Some sea creatures have adapted so they can see in low intensity light and cannot tolerate too...
• 08M.1.sl.TZ1.11b: A function \(f\) is given by...
• 08M.1.sl.TZ2.10b: A function is defined as \(f(x) = 1 + {x^3}\) for...
• 13N.1.sl.TZ0.12a: Using the mapping diagram, write down two equations in terms of \(a\) and \(b\).
• 13N.2.sl.TZ0.4a: Find \(f( - 2)\).
• 14N.1.sl.TZ0.11a: Write down (i)     the \(x\)-intercept of the graph of \(y = {\text{ }}f(x)\); (ii)     the...
• 14N.1.sl.TZ0.11b: Solve \(f(x) = g(x)\).
• 14N.1.sl.TZ0.11c: Write down the interval for the values of \(x\) for which \(f(x) > g(x)\).
• 15M.1.sl.TZ1.14c: The vertex of the function is \((3,{\text{ }}27)\). Write down the range of \(f\).
• 15M.1.sl.TZ2.15d: The range of \(A(x)\) is \(m \leqslant A(x) \leqslant n\). Hence write down the value of \(m\)...
• 15M.2.sl.TZ1.5f: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Sketch the graph of...
• 15M.2.sl.TZ2.5a: Find \(f( - 2)\).
• 16M.1.sl.TZ1.14a: A population of \(200\) rabbits was introduced to an island. One week later the number of rabbits...
• 16M.1.sl.TZ1.14b: Calculate the number of rabbits on the island after 10 weeks.
• 16M.1.sl.TZ1.14c: An ecologist estimates that the island has enough food to support a maximum population of 1000...
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16N.1.sl.TZ0.9a: Write down the equation of the axis of symmetry for this graph.
• 16N.1.sl.TZ0.9b: Find the value of \(b\).
• 16N.1.sl.TZ0.9c: Write down the range of \(f(x)\).
• 16M.1.sl.TZ2.8a: Consider the curve \(y = 1 + \frac{1}{{2x}},\,\,x \ne 0.\) For this curve, write down i)  ...
• 16M.1.sl.TZ2.8b: Sketch the curve for \( - 2 \leqslant x \leqslant 4\) on the axes below.  
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17M.1.sl.TZ1.12a: Write down the domain of the function.
• 17M.1.sl.TZ1.12b.i: Draw the line \(y =  - 6\) on the axes.
• 17M.1.sl.TZ1.12b.ii: Write down the number of solutions to \(f(x) =  - 6\).
• 17M.1.sl.TZ1.12c: Find the range of values of \(k\) for which \(f(x) = k\) has no solution.
• 17M.1.sl.TZ2.14b: Use your graphic display calculator to find how long it will take for Jashanti to have saved...
• 259485: This is an example question for the example test. You can delete this question.
• 18M.1.sl.TZ1.15a: Find the range of f.
• 18M.1.sl.TZ1.15b: Write down the x-coordinate of P and the x-coordinate of Q.
• 18M.1.sl.TZ1.15c: Write down the values of x for which \(f\left( x \right) > g\left( x \right)\).
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

6.2

• 10M.1.sl.TZ1.15b: Draw the line y = 5 on your sketch.
• 12N.2.sl.TZ0.5f: (i) Sketch the graph of y = g(x) for −2 ≤ x ≤ 5 and −15 ≤ y ≤ 25, indicating clearly your answer...
• 12M.1.sl.TZ2.6b: Use the model to calculate the boiling point of water at a height of 1.37 km above sea level.
• 12M.1.sl.TZ2.6c: Water boils at the top of Mt. Everest at 70 °C. Use the model to calculate the height above sea...
• 07M.1.sl.TZ0.12c: Draw a sketch, in the space provided, to show how the prices can be found graphically.
• 09M.1.sl.TZ2.2a: Write down the fixed cost.
• 09M.1.sl.TZ2.2b: It takes \(3 \frac{1}{2}\) hours to complete a job for Paula. Find the total cost.
• 09M.1.sl.TZ2.2c: Steve received a bill for 510 AUD. Calculate the time it took the plumber to complete the job.
• SPM.1.sl.TZ0.7c: On the grid draw the graph of the function \(g(x) = 5 - 2x\) for the domain...
• SPM.1.sl.TZ0.13a: Using the model, write down a second equation in \(a\) and \(b\) .
• SPM.1.sl.TZ0.13b: Using your graphic display calculator or otherwise, find the value of \(a\) and of \(b\) .
• SPM.1.sl.TZ0.13c: Use the model to predict the temperature of the liquid \(60{\text{ seconds}}\) after the start of...
• 07N.2.sl.TZ0.1ii.b: On the same diagram, sketch the line \(y = 7 − 2x\) and find the coordinates of the point of...
• 08M.2.sl.TZ1.1c: On the same diagram as part (a) sketch the graph of \(g(x) = x + 0.5\) .
• 08M.2.sl.TZ1.4i.d: (i)     Sketch the graphs of the two equations from parts (a) and (b). (ii)    Write down the...
• 08M.2.sl.TZ2.4i.f: (i) Draw and label the line \( y = 1\) on your graph. (ii) The equation \(f(x) = 1\) has two...
• 14N.1.sl.TZ0.11a: Write down (i)     the \(x\)-intercept of the graph of \(y = {\text{ }}f(x)\); (ii)     the...
• 14N.1.sl.TZ0.11b: Solve \(f(x) = g(x)\).
• 14N.1.sl.TZ0.11c: Write down the interval for the values of \(x\) for which \(f(x) > g(x)\).
• 14N.2.sl.TZ0.1g: A forest ranger uses her own formula for estimating the air temperature. She counts the number of...
• 14N.2.sl.TZ0.1h: A cricket makes 20 chirps in 15 seconds. For this chirp rate (i)     calculate an estimate for...
• 15M.1.sl.TZ1.12a: At 150°C the length of the iron bar is 180 mm. Write down an equation that shows this information.
• 15M.1.sl.TZ1.12b: At 210°C the length of the iron bar is 181.5 mm. Write down an equation that shows this second...
• 15M.2.sl.TZ2.6d: All the bicycles that are produced are sold. The bicycles are sold for 304 USD each. Explain why...
• 16M.1.sl.TZ1.13a: The golden ratio, \(r\) , was considered by the Ancient Greeks to be the perfect ratio...
• 16M.1.sl.TZ1.13b: Phidias is designing rectangular windows with adjacent sides of length \(x\) metres...
• 16M.1.sl.TZ1.13c: Phidias designs the windows so that the ratio between the longer side, \(y\) , and the...
• 16M.1.sl.TZ1.13d: Find the value of \(x\) .
• 16M.2.sl.TZ1.5a: Antonio and Barbara start work at the same company on the same day. They each earn an annual...
• 16M.2.sl.TZ1.5b: Write down an expression for i)     Antonio’s annual salary during his \(n\) th year of...
• 16M.2.sl.TZ1.5c: Determine the number of years for which Antonio’s annual salary is greater than or equal to...
• 16M.2.sl.TZ1.5d: Both Antonio and Barbara plan to work at the company for a total of \(15\) years. i)  ...
• 16N.1.sl.TZ0.8a: Use the above information to write down an equation in \(x\) and \(y\).
• 16N.1.sl.TZ0.8b: Use the information about the cost of tickets to write down a second equation in \(x\) and \(y\).
• 16N.1.sl.TZ0.8c: Find the value of \(x\) and the value of \(y\).
• 16N.2.sl.TZ0.1a: On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours...
• 16N.2.sl.TZ0.1b: (i)     \({\bar x}\), the mean number of hours spent on social media; (ii)     \({\bar y}\), the...
• 16N.2.sl.TZ0.1c: Plot the point \((\bar x,{\text{ }}\bar y)\) on your scatter diagram and label this point M.
• 16N.2.sl.TZ0.1d: Write down the value of \(r\), the Pearson’s product–moment correlation coefficient, for these data.
• 16N.2.sl.TZ0.1h: Write down a reason why this estimate is not reliable.
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17M.1.sl.TZ2.14a: Write down the amount of money Jashanti saves per month.
• 17M.1.sl.TZ2.14c: Calculate how much extra money Jashanti needs.
• 17M.2.sl.TZ1.3a: Calculate \(f(1)\).
• 17M.2.sl.TZ1.3b: Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and...
• 17M.2.sl.TZ1.3c: Write down the equation of the vertical asymptote.
• 17M.2.sl.TZ1.3d: Write down the coordinates of the \(x\)-intercept.
• 17M.2.sl.TZ1.3e: Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).
• 17M.2.sl.TZ1.3f: Find the solution of \(f(x) = g(x)\).
• 17N.1.sl.TZ0.6a: Use this information to write down an equation involving \(x\) and \(y\).
• 17N.1.sl.TZ0.6b: Use this ratio to write down \(y\) in terms of \(x\).
• 17N.1.sl.TZ0.6c: Find the value of \(x\) and of \(y\).
• 18M.1.sl.TZ2.6a: Find the value of a.
• 18M.1.sl.TZ2.6b: Find the coordinates of R.
• 18M.1.sl.TZ2.6c: Line L3 is parallel to line L2 and passes through the point (2, 3). Find the equation of line...

6.3

• 10M.1.sl.TZ1.14c: Calculate the x-coordinate of the vertex of the graph of f (x).
• 10M.1.sl.TZ2.13a: Write down the value of q.
• 10M.1.sl.TZ2.13b: The axis of symmetry is x = 2.5. Find the value of r.
• 10M.1.sl.TZ2.13c: The axis of symmetry is x = 2.5. Write down the minimum value of y.
• 10N.1.sl.TZ0.10a: Write down the solutions to the equation f (x) = 0.
• 10N.1.sl.TZ0.10b: Write down the equation of the axis of symmetry of the graph of f (x).
• 10N.1.sl.TZ0.10c: The equation f (x) = 12 has two solutions. One of these solutions is x = 6. Use the symmetry of...
• 10N.1.sl.TZ0.10d: The minimum value for y is – 4. Write down the range of f (x).
• 11N.1.sl.TZ0.13a: (i) Find the value of \(k\) . (ii) Calculate the \(y\)-coordinate of this minimum point.
• 11N.1.sl.TZ0.13b: Sketch the graph of \(y = f(x)\) for the domain \( - 1 \leqslant x \leqslant 3\).
• 12N.1.sl.TZ0.10a: It is given that f (2) = −5 . Find the value of a .
• 12N.1.sl.TZ0.10b: Find the equation of the axis of symmetry of the graph of y = f (x) .
• 12M.1.sl.TZ1.4a: Write down the equation of the axis of symmetry.
• 12M.1.sl.TZ1.4c: The graph of y = f (x) intersects the x-axis for a second time at point D. Write down the...
• 12M.1.sl.TZ2.11a: Find the value of (i) b ; (ii) c .
• 12M.1.sl.TZ2.11b: The domain of f is  0 ≤ x ≤ 6. Find the range of f .
• 09N.1.sl.TZ0.13a: Calculate the value of p.
• 09N.1.sl.TZ0.13b: Given that the quadratic function has an equation \(y = -x^2 + bx + c\) where...
• 09N.2.sl.TZ0.5A, c: On the same axes sketch the graph of y = 3 + 2x − x2. Indicate clearly where this curve...
• 09N.2.sl.TZ0.5A, e: Write down the maximum value of the function f (x) = 3 + 2x − x2.
• 09N.2.sl.TZ0.5B, a: The curve y = px2 + qx − 4 passes through the point (2, –10). Use the above information to write...
• 11M.1.sl.TZ1.6a: Find the coordinates of B .
• 11M.1.sl.TZ1.6b: Another point, C , which lies on the graph of \(y = f(x)\) has the same \(y\)-coordinate as A...
• 09M.1.sl.TZ1.7a: Write down the equation of the axis of symmetry.
• 09M.1.sl.TZ1.7b: Write down the coordinates of the minimum point.
• 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
• 09M.1.sl.TZ2.14a: Find the value of h.
• 09M.1.sl.TZ2.14b: Find the equation of the curve.
• 13M.1.sl.TZ1.9a: Write down the value of c.
• 13M.1.sl.TZ1.9b: Find the value of b.
• 13M.1.sl.TZ1.9c: Find the x-intercepts of the graph of f .
• 13M.1.sl.TZ1.9d: Write down \(f (x)\) in the form \(f (x) = −(x − p) (x + q)\).
• 13M.2.sl.TZ2.5a: The coordinates of point A are (75, 450). Determine whether point A is on the bicycle track. Give...
• SPM.1.sl.TZ0.15a: Find \(P(x)\) .
• 07M.2.sl.TZ0.3ii.a: Using your graphic display calculator or otherwise, find the value of a.
• 07N.2.sl.TZ0.5f: State the equation of the axis of symmetry of P.
• 07N.2.sl.TZ0.5g: Find the coordinates of the vertex of P and state the gradient of the curve at this point.
• 08N.1.sl.TZ0.13c: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08N.1.sl.TZ0.13d: The diagram below shows the graph of the function \(f(x) = {x^2} - kx\) for a particular value of...
• 08M.2.sl.TZ1.5i.c: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Find the equation of the axis of symmetry on...
• 08M.2.sl.TZ1.5i.d: Consider a function \(f(x) = 3{x^2} + 13x - 10\) . Calculate the minimum value of this function.
• 14M.2.sl.TZ2.6a: Write down the value of \(c\).
• 14M.2.sl.TZ2.6b: Find the value of \(a\).
• 14M.2.sl.TZ2.6c: Hence write down the equation of the quadratic function which models the edge of the water tank.
• 14M.2.sl.TZ2.6d: The water tank is shown below. It is partially filled with water. Calculate the value of y...
• 14M.2.sl.TZ2.6e: The water tank is shown below. It is partially filled with water. State what the value of...
• 14M.2.sl.TZ2.6f: The water tank is shown below. It is partially filled with water. Find the value of \(x\) when...
• 14M.1.sl.TZ1.13a: Write down the value of \(c\).
• 14M.1.sl.TZ1.13b: By using the coordinates of the vertex, B, or otherwise, write down two equations in \(a\) and...
• 14N.1.sl.TZ0.14b: The graph of the quadratic function intersects the \(x\)-axis at the point \({\text{N}}(2, 0)\)....
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 14N.1.sl.TZ0.14a: Draw the axis of symmetry on the following axes. The graph of the quadratic function...
• 15M.1.sl.TZ1.14a: Calculate the value of \(q\).
• 15M.1.sl.TZ1.14b: The vertex of the function is \((3,{\text{ }}27)\). Find the value of \(p\).
• 15M.1.sl.TZ1.14c: The vertex of the function is \((3,{\text{ }}27)\). Write down the range of \(f\).
• 15M.1.sl.TZ2.15b: Site T has the same area as site S, but a different width. Find the width of T.
• 15M.1.sl.TZ2.15c: When the width of the construction site is \(b\) metres, the site has a maximum area. (i)    ...
• 15M.1.sl.TZ2.15d: The range of \(A(x)\) is \(m \leqslant A(x) \leqslant n\). Hence write down the value of \(m\)...
• 16M.1.sl.TZ1.11a: Consider the function \(f(x) = a{x^2} + c\). Find \(f'(x)\)  
• 16M.1.sl.TZ1.11b: Point \({\text{A}}( - 2,\,5)\)  lies on the graph of \(y = f(x)\) . The gradient of the tangent...
• 16M.1.sl.TZ1.11c: Find the value of \(c\) .
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17M.1.sl.TZ1.15a: Write down the value of \(c\).
• 17M.1.sl.TZ1.15b: Find the value of \(a\) and of \(b\).
• 17M.1.sl.TZ1.15c: Write down the second \(x\)-intercept of the function.
• 17M.1.sl.TZ2.15: Consider the following graphs of quadratic functions. The equation of each of the quadratic...
• 17M.2.sl.TZ1.6a: Find \(g'(x)\).
• 17M.2.sl.TZ1.6b.i: Show that \(k = 6\).
• 17M.2.sl.TZ1.6b.ii: Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in...
• 17M.2.sl.TZ1.6c: Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the...
• 17M.2.sl.TZ1.6d.i: Find \(g’( - 1)\).
• 17M.2.sl.TZ1.6d.ii: Hence justify that \(g\) is decreasing at \(x =  - 1\).
• 17M.2.sl.TZ1.6e: Find the \(y\)-coordinate of the local minimum.
• 17M.2.sl.TZ2.5a: Show that the width of the plot, in metres, is given by \(100 - x\).
• 17M.2.sl.TZ2.5b: Write down the area of the plot in terms of \(x\).
• 17M.2.sl.TZ2.5c: Find the value of \(x\) that maximizes the area of the plot.
• 17M.2.sl.TZ2.5d: Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.
• 17M.2.sl.TZ2.5e.i: Find the amount of money that Violeta would have after 6 years. Give your answer correct to two...
• 17M.2.sl.TZ2.5f: Find the lowest possible value for \(r\).
• 17M.2.sl.TZ2.5e.ii: Find how long it would take for the interest earned to be 2000 BGN.
• 17N.1.sl.TZ0.11a: Find the equation of the axis of symmetry of the graph of \(y = f(x)\).
• 17N.1.sl.TZ0.11b: Write down the value of \(c\).
• 17N.1.sl.TZ0.11c: Find the value of \(a\) and of \(b\).
• 17N.1.sl.TZ0.15a: Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of...
• 17N.1.sl.TZ0.15b: Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese...
• 17N.1.sl.TZ0.15c: Write down an expression for \(W\) in terms of \(p\).
• 17N.1.sl.TZ0.15d: Find the price, \(p\), that will give Maria the highest weekly profit.
• 18M.1.sl.TZ1.12a: Using only this information, write down an equation in terms of a and b.
• 18M.1.sl.TZ1.12b: Using this information, write down a second equation in terms of a and b.
• 18M.1.sl.TZ1.12c: Hence find the value of a and of b.
• 18M.1.sl.TZ1.12d: The graph intersects the x-axis at a second point, P. Find the x-coordinate of P.
• 18M.1.sl.TZ2.13a: Find the cost of producing 70 shirts.
• 18M.1.sl.TZ2.13b: Find the value of s.
• 18M.1.sl.TZ2.13c: Find the number of shirts produced when the cost of production is lowest.
• 18M.1.sl.TZ2.14a: Find f'(x)
• 18M.1.sl.TZ2.14b: Find the gradient of the graph of f at \(x =  - \frac{1}{2}\).
• 18M.1.sl.TZ2.14c: Find the x-coordinate of the point at which the normal to the graph of f has...

6.4

• 10M.1.sl.TZ1.15a.i: On the axes below sketch the graph of f (x) and show the behaviour of the curve as x increases.  
• 10M.1.sl.TZ2.12a: Find the number of teenagers who started the rumour.
• 10M.1.sl.TZ2.12b: Write down the number of teenagers who have heard the rumour five hours after it is first started.
• 10M.1.sl.TZ2.12c: Determine the length of time it would take for 150 teenagers to have heard the rumour. Give your...
• 10N.1.sl.TZ0.13a: Write down two equations relating p and q.
• 10N.1.sl.TZ0.13c: Write down the equation of the horizontal asymptote to the graph of f (x).
• 11N.1.sl.TZ0.15a: Find the value of \(k\) .
• 11N.1.sl.TZ0.15b: The graph passes through the point (2, 48) . Find the value of \(c\) .
• 11N.1.sl.TZ0.15c: The graph passes through the point (2, 48) . Write down the equation of the horizontal...
• 12M.1.sl.TZ1.14a: Find the value of (i) p ; (ii) q .
• 12M.1.sl.TZ1.14b: Write down the value of r .
• 12M.1.sl.TZ1.14c: Find the value of s .
• 12M.1.sl.TZ2.12a: Find the value of (i) a ; (ii) b .
• 12M.1.sl.TZ2.12b: Write down the equation of the horizontal asymptote to this graph.
• 09N.1.sl.TZ0.11a: find the range of \(f\).
• 09N.1.sl.TZ0.11b: find the value of \(x\) given that \(f (x) =162\).
• 09N.2.sl.TZ0.5A, a: Sketch the graph of y = 2x for \( - 2 \leqslant x \leqslant 3\). Indicate clearly where the curve...
• 09N.2.sl.TZ0.5A, b: Write down the equation of the asymptote of the graph of y = 2x.
• 09M.1.sl.TZ1.14a: Use the above information to write down two equations in p and q ;
• 09M.1.sl.TZ1.14b: Use the above information to calculate the value of p and of q ;
• 09M.1.sl.TZ1.14c: Use the above information to find the number of cells in the culture at 22:00 on Monday.
• 11M.1.sl.TZ1.15c: The point \((2{\text{, }}1)\) lies on the graph of \(y = f(x)\) . Write down the equation of \(L\) .
• 11M.1.sl.TZ1.15b: The point \((2{\text{, }}1)\) lies on the graph of \(y = f(x)\) . Calculate the value of \(a\) .
• 11M.1.sl.TZ1.15a: Find the \(y\)-coordinate of A .
• 09M.2.sl.TZ2.2ii, b: Using the values in the table above, draw the graph of N for 0 ≤ t ≤ 20. Use 1 cm to represent 2...
• 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
• 09M.2.sl.TZ2.2ii, a, i: Write down the value of p.
• 11M.1.sl.TZ2.14a: Write down the value of the car when Shiyun bought it.
• 11M.1.sl.TZ2.14b: Calculate the value of the car three years after Shiyun bought it. Give your answer correct to...
• 11M.1.sl.TZ2.14c: Calculate the time for the car to depreciate to half of its value since Shiyun bought it.
• 13M.2.sl.TZ1.3d: The function that models the change in temperature of the coffee is y = p (2−t )+ q. (i) Use the...
• 13M.2.sl.TZ1.3f: The graph of this function has a horizontal asymptote. Write down the equation of this asymptote.
• SPM.1.sl.TZ0.7a: On the grid below sketch the graph of the function \(f(x) = 2{(1.6)^x}\) for the domain...
• 13M.1.sl.TZ2.13a: Write down the number of bacteria in the colony at time \(t = 0\).
• 13M.1.sl.TZ2.13b: Calculate the number of bacteria present at 2 hours and 30 minutes. Give your answer correct to...
• 13M.1.sl.TZ2.13c: Calculate the time, in hours, for the number of bacteria to reach 5500.
• SPM.2.sl.TZ0.4a: Calculate the value of \(k\) .
• SPM.2.sl.TZ0.4b: Find the intensity of light at a depth \(25{\text{ m}}\) below the surface.
• SPM.2.sl.TZ0.4c: To be able to see clearly, a diver needs the intensity of light to be at least \(65\% \). Using...
• SPM.2.sl.TZ0.4d: The table below gives the intensity of light (correct to the nearest integer) at different...
• SPM.2.sl.TZ0.4e: Some sea creatures have adapted so they can see in low intensity light and cannot tolerate too...
• 07N.2.sl.TZ0.3i.a: Find the initial temperature of the coffee.
• 07N.2.sl.TZ0.3i.b: Write down the equation of the horizontal asymptote.
• 07N.2.sl.TZ0.3i.c: Find the room temperature.
• 07N.2.sl.TZ0.3i.d: Find the temperature of the coffee after 10 minutes.
• 07N.2.sl.TZ0.3i.e: Find the temperature of Robert’s coffee after being heated in the microwave for 30 seconds after...
• 07N.2.sl.TZ0.3i.f: Calculate the length of time it would take a similar cup of coffee, initially at 20°C, to be...
• 08N.2.sl.TZ0.4a: Show that the temperature of the water when it is removed from the cooker is...
• 08N.2.sl.TZ0.4b: The following table shows values for \(m\) and \(T(m)\). (i)     Write down the value of...
• 08M.1.sl.TZ1.15a: Find the value of \(a\).
• 08M.1.sl.TZ1.15b: Calculate how many bacteria there will be after one day.
• 08M.1.sl.TZ1.15c: Find how long it will take for there to be two million bacteria.
• 08M.2.sl.TZ2.4i.a: Given that \(f(1) = 2\), show that \(k = 4\).
• 08M.2.sl.TZ2.4i.b: Write down the values of \(q\) and \(r\) for the following table.
• 08M.2.sl.TZ2.4i.d: Using your graphic display calculator, find the coordinates of \({\text{M}}\), the maximum point...
• 08M.2.sl.TZ2.4i.e: Write down the equation of the horizontal asymptote to the graph of \(y = f(x)\).
• 09M.2.sl.TZ2.2ii, a, ii: Write down the value of q.
• 14M.1.sl.TZ2.11a: Write down the amount of the drug in the bloodstream at \(t = 0\).
• 14M.1.sl.TZ2.11b: Calculate the amount of the drug in the bloodstream after 3 hours.
• 14M.1.sl.TZ2.11c: Use your graphic display calculator to determine the time it takes for the amount of the drug in...
• 13N.1.sl.TZ0.15a: Calculate the number of computers infected after \(6\) hours.
• 13N.1.sl.TZ0.15b: Calculate the time for the number of infected computers to be greater than...
• 14M.1.sl.TZ1.11a: Write down the amount of electrical charge in the battery at \(t = 0\).
• 14M.1.sl.TZ1.11b: The line \(L\) is the horizontal asymptote to the graph. Write down the equation of \(L\).
• 14M.1.sl.TZ1.11c: To download a game to the mobile phone, an electrical charge of 2.4 units is needed. Find the...
• 14N.1.sl.TZ0.11a: Write down (i)     the \(x\)-intercept of the graph of \(y = {\text{ }}f(x)\); (ii)     the...
• 14N.1.sl.TZ0.11b: Solve \(f(x) = g(x)\).
• 14N.1.sl.TZ0.11c: Write down the interval for the values of \(x\) for which \(f(x) > g(x)\).
• 14N.1.sl.TZ0.13a: Write down the temperature of the potato at the moment it is placed in the oven.
• 14N.1.sl.TZ0.13b: Find the temperature of the potato half an hour after it has been placed in the oven.
• 14N.1.sl.TZ0.13c: After the potato has been in the oven for \(k\) minutes, its temperature is 40°C. Find the value...
• 15M.1.sl.TZ2.14a: Find the value of \(a\).
• 15M.1.sl.TZ2.14b: After \(4\) months \(90\) fish were counted. Find the value of \(b\).
• 15M.1.sl.TZ2.14c: The number of fish in the pond will not decrease below \(p\). Write down the value of \(p\).
• 15M.2.sl.TZ1.6a: Explain why \(a = 20\).
• 15M.2.sl.TZ1.6b: Initially, at \(m = 0\), the temperature of the water is 100°C. Find the value of \(b\).
• 15M.2.sl.TZ1.6c: After being placed in the room for one minute, the temperature of the water is 84°C. Show that...
• 15M.2.sl.TZ1.6d: After being placed in the room for one minute, the temperature of the water is 84°C. Find the...
• 15M.2.sl.TZ1.6e: After being placed in the room for one minute, the temperature of the water is 84°C. Find the...
• 16M.1.sl.TZ1.14a: A population of \(200\) rabbits was introduced to an island. One week later the number of rabbits...
• 16M.1.sl.TZ1.14b: Calculate the number of rabbits on the island after 10 weeks.
• 16M.1.sl.TZ1.14c: An ecologist estimates that the island has enough food to support a maximum population of 1000...
• 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)  
• 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
• 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
• 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i)    ...
• 16N.1.sl.TZ0.15a: Write down, and simplify, an expression for the car’s value when Gabriella purchased it.
• 16N.1.sl.TZ0.15b: Find the value of \(k\).
• 16N.1.sl.TZ0.15c: Find the value of \(n\).
• 16M.1.sl.TZ2.13a: A population of mosquitoes decreases exponentially. The size of the population, \(P\) , after...
• 16M.1.sl.TZ2.13b: Find the size of the population after \(4\) days.
• 16M.1.sl.TZ2.13d: The population will stabilize when it reaches a size of \(k\) . Write down the value of \(k\) .
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 17M.1.sl.TZ2.14a: Write down the amount of money Jashanti saves per month.
• 17M.1.sl.TZ2.14b: Use your graphic display calculator to find how long it will take for Jashanti to have saved...
• 17M.1.sl.TZ2.14c: Calculate how much extra money Jashanti needs.
• 17M.2.sl.TZ1.3a: Calculate \(f(1)\).
• 17M.2.sl.TZ1.3b: Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and...
• 17M.2.sl.TZ1.3c: Write down the equation of the vertical asymptote.
• 17M.2.sl.TZ1.3d: Write down the coordinates of the \(x\)-intercept.
• 17M.2.sl.TZ1.3e: Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).
• 17M.2.sl.TZ1.3f: Find the solution of \(f(x) = g(x)\).
• 17M.2.sl.TZ1.4a: Calculate the volume of this pan.
• 17M.2.sl.TZ1.4b: Find the radius of the sphere in cm, correct to one decimal place.
• 17M.2.sl.TZ1.4c: Find the value of \(a\).
• 17M.2.sl.TZ1.4d: Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.
• 17M.2.sl.TZ1.4e: Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can...
• 17M.2.sl.TZ1.4f: In the context of this model, state what the value of 19 represents.
• 17N.1.sl.TZ0.12a: Write down what the value of 150 represents in the context of the question.
• 17N.1.sl.TZ0.12b: Find the value of \(a\).
• 17N.1.sl.TZ0.12c: Find the total time that the baking tin is in the oven.
• 18M.1.sl.TZ1.9a.i: Find the number of fruit flies which were placed in the container.
• 18M.1.sl.TZ1.9a.ii: Find the number of fruit flies that are in the container after 6 days.
• 18M.1.sl.TZ1.9b: The maximum capacity of the container is 8000 fruit flies. Find the number of days until the...
• 18M.1.sl.TZ2.10a: Find the value of A.
• 18M.1.sl.TZ2.10b: Interpret what A represents in this context.
• 18M.1.sl.TZ2.10c: Find the time since the experiment began for the bacteria population to be equal to 40A.

6.5

• 10M.2.sl.TZ1.3a: Write down the values of x where the graph of f (x) intersects the x-axis.
• 10N.2.sl.TZ0.5a: Write down f (0).
• 12N.2.sl.TZ0.5a: Write down the equation of the vertical asymptote of the graph of y = g(x) .
• 12M.2.sl.TZ1.5a: Sketch the graph of y = f (x) for −3 ≤ x ≤ 6 and −10 ≤ y ≤ 10 showing clearly the axes intercepts...
• 11N.2.sl.TZ0.4a: Write down (i)     the equation of the vertical asymptote to the graph of \(y = f (x)\)...
• 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
• 07N.2.sl.TZ0.1ii.a: Sketch the curve of the function \(f (x) = x^3 − 2x^2 + x − 3\) for values of \(x\) from −2 to 4,...
• 11M.2.sl.TZ1.3b: Sketch the graph of the function \(y = f(x)\) for \( - 5 \leqslant x \leqslant 5\) and...
• 09M.2.sl.TZ2.5b: Consider the graph of f. The graph of f passes through the point P(1, 4). Find the value of c.
• 09M.2.sl.TZ2.5c, i: There is a local minimum at the point Q. Find the coordinates of Q.
• 13M.2.sl.TZ1.4a: Calculate \(f (1)\).
• 11M.2.sl.TZ2.5a: Write down the equation of the vertical asymptote.
• 07M.2.sl.TZ0.3i.a: Write down the equation of the vertical asymptote.
• 08N.2.sl.TZ0.5a: (i)     Write down the value of \(y\) when \(x\) is \(2\). (ii)    Write down the coordinates of...
• 08N.2.sl.TZ0.5b: Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\)....
• 08M.2.sl.TZ1.1b: Write down the equation of the vertical asymptote.
• 09M.1.sl.TZ1.15c, ii: Write down the value of \(f (x)\) at this point.
• 14M.2.sl.TZ2.5d: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 13N.2.sl.TZ0.4a: Find \(f( - 2)\).
• 15M.2.sl.TZ2.5a: Find \(f( - 2)\).
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)  
• 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
• 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
• 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i)    ...
• 16M.1.sl.TZ2.8a: Consider the curve \(y = 1 + \frac{1}{{2x}},\,\,x \ne 0.\) For this curve, write down i)  ...
• 16M.1.sl.TZ2.8b: Sketch the curve for \( - 2 \leqslant x \leqslant 4\) on the axes below.  
• 16N.2.sl.TZ0.3a: Show that A lies on \({L_1}\).
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 16M.2.sl.TZ2.5a: Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\)....
• 16M.2.sl.TZ2.5b: (i)     State whether \(x\) can have a value of \(5\). Give a reason for your answer. (ii)  ...
• 16M.2.sl.TZ2.5c: Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given...
• 16M.2.sl.TZ2.5d: Find \(\frac{{dV}}{{dx}}.\)
• 16M.2.sl.TZ2.5e: Using your answer from part (d), find the value of \(x\) that maximizes the volume of the tray.
• 16M.2.sl.TZ2.5f: Calculate the maximum volume of the tray.
• 16M.2.sl.TZ2.5g: Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found...
• 17M.2.sl.TZ1.3a: Calculate \(f(1)\).
• 17M.2.sl.TZ1.3b: Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and...
• 17M.2.sl.TZ1.3c: Write down the equation of the vertical asymptote.
• 17M.2.sl.TZ1.3d: Write down the coordinates of the \(x\)-intercept.
• 17M.2.sl.TZ1.3e: Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).
• 17M.2.sl.TZ1.3f: Find the solution of \(f(x) = g(x)\).
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 17N.1.sl.TZ0.14a: Write down the derivative of \(f\).
• 17N.1.sl.TZ0.14b: Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.
• 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
• 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
• 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
• 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
• 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
• 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
• 18M.1.sl.TZ1.15a: Find the range of f.
• 18M.1.sl.TZ1.15b: Write down the x-coordinate of P and the x-coordinate of Q.
• 18M.1.sl.TZ1.15c: Write down the values of x for which \(f\left( x \right) > g\left( x \right)\).
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 18M.1.sl.TZ2.11a: Write down the equation of the vertical asymptote.
• 18M.1.sl.TZ2.11b: Write down the equation of the horizontal asymptote.
• 18M.1.sl.TZ2.11c: Calculate the value of x for which f(x) = 0 .
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).

6.6

• 10M.1.sl.TZ1.15a.i: On the axes below sketch the graph of f (x) and show the behaviour of the curve as x increases.  
• 10M.2.sl.TZ1.3d: Let P be the point where the graph of f (x) intersects the y axis. Write down the coordinates of P.
• 11N.1.sl.TZ0.13b: Sketch the graph of \(y = f(x)\) for the domain \( - 1 \leqslant x \leqslant 3\).
• 10N.2.sl.TZ0.5e: Sketch the graph of f (x) for \( - 5 \leqslant x \leqslant 7\) and...
• 12N.2.sl.TZ0.5f: (i) Sketch the graph of y = g(x) for −2 ≤ x ≤ 5 and −15 ≤ y ≤ 25, indicating clearly your answer...
• 12M.1.sl.TZ1.4b: Sketch the graph of y = f (x) on the axes below for 0 ≤ x ≤ 4 . Mark clearly on the sketch the...
• 12M.2.sl.TZ1.5a: Sketch the graph of y = f (x) for −3 ≤ x ≤ 6 and −10 ≤ y ≤ 10 showing clearly the axes intercepts...
• 12M.2.sl.TZ1.5h: Sketch the tangent to the graph of f (x) at x = −1 on your diagram for (a).
• 12M.2.sl.TZ1.5i: P and Q are points on the curve such that the tangents to the curve at these points are...
• 11N.2.sl.TZ0.5c: The sketch below shows the graph of the function, \(h(t)\) , for the height above ground of C,...
• 09M.2.sl.TZ1.5f: Sketch the graph of the function f, for −3 ≤ x ≤ 6 and −7 ≤ y ≤ 15. Indicate clearly the point P...
• 09M.2.sl.TZ2.2ii, b: Using the values in the table above, draw the graph of N for 0 ≤ t ≤ 20. Use 1 cm to represent 2...
• 13M.2.sl.TZ1.3c: Use the seven results in the table to draw a graph that shows how the temperature of the coffee...
• 11M.2.sl.TZ1.3b: Sketch the graph of the function \(y = f(x)\) for \( - 5 \leqslant x \leqslant 5\) and...
• 09M.2.sl.TZ2.2ii, c: Use your graph to estimate the number of days for the population of fleas to reach 55.
• 11M.2.sl.TZ2.5e: Sketch the graph of \(f(x)\) for \( - 10 \leqslant x \leqslant 10\) and...
• SPM.1.sl.TZ0.7a: On the grid below sketch the graph of the function \(f(x) = 2{(1.6)^x}\) for the domain...
• SPM.1.sl.TZ0.5b: where the function attains its least value;
• SPM.1.sl.TZ0.5c: where the function attains its greatest value;
• SPM.2.sl.TZ0.4d: The table below gives the intensity of light (correct to the nearest integer) at different...
• 07M.2.sl.TZ0.3ii.d: Draw a graph showing the path of the football from the point where it is kicked to the point...
• 07M.2.sl.TZ0.3ii.e: The goal posts are 35 m from the point where the ball is kicked. At what height does the ball...
• 08N.2.sl.TZ0.4b: The following table shows values for \(m\) and \(T(m)\). (i)     Write down the value of...
• 08N.2.sl.TZ0.4c: Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) . The function...
• 08N.2.sl.TZ0.4d: Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) . The function...
• 08N.2.sl.TZ0.4e: Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) . The function...
• 08M.2.sl.TZ1.1a: Sketch the graph of the function \(f(x)\), for \( - 10 \leqslant x \leqslant 10\) . Indicating...
• 08M.2.sl.TZ2.4i.c: As \(x\) increases from \( - 1\), the graph of \(y = f(x)\) reaches a maximum value and then...
• 10M.1.sl.TZ1.15a.ii: Write down the coordinates of any intercepts with the axes.
• 14M.2.sl.TZ2.5d: The parcel is tied up using a length of string that fits exactly around the parcel, as shown in...
• 13N.1.sl.TZ0.10a: Sketch the graphs of \(y = f(x)\) and \(y = g(x)\) on the axes below. Indicate clearly the points...
• 13N.2.sl.TZ0.4d: The graph of the function \(f(x)\) has a local minimum at the point where \(x =  - 2\). Sketch...
• 14N.1.sl.TZ0.11c: Write down the interval for the values of \(x\) for which \(f(x) > g(x)\).
• 14N.1.sl.TZ0.14c: (i)     Find the value of \(b\) and the value of \(c\). (ii)     Draw the graph of the function...
• 15M.2.sl.TZ1.5f: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Sketch the graph of...
• 15M.2.sl.TZ1.5g: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Write down the coordinates of...
• 15M.2.sl.TZ2.5e: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Sketch the graph of \(f\) for...
• 15M.2.sl.TZ2.5f: Let \(T\) be the tangent to the graph of \(f\) at \(x =  - 2\). Draw \(T\) on your sketch.
• 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)  
• 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
• 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
• 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i)    ...
• 16M.1.sl.TZ2.8a: Consider the curve \(y = 1 + \frac{1}{{2x}},\,\,x \ne 0.\) For this curve, write down i)  ...
• 16M.1.sl.TZ2.8b: Sketch the curve for \( - 2 \leqslant x \leqslant 4\) on the axes below.  
• 17M.1.sl.TZ1.12a: Write down the domain of the function.
• 17M.1.sl.TZ1.12b.i: Draw the line \(y =  - 6\) on the axes.
• 17M.1.sl.TZ1.12b.ii: Write down the number of solutions to \(f(x) =  - 6\).
• 17M.1.sl.TZ1.12c: Find the range of values of \(k\) for which \(f(x) = k\) has no solution.
• 17M.1.sl.TZ2.13a: Write down the value of \(f(1)\).
• 17M.1.sl.TZ2.13b: Find the equation of \(N\). Give your answer in the form \(ax + by + d = 0\) where \(a\), \(b\),...
• 17M.1.sl.TZ2.13c: Draw the line \(N\) on the diagram above.
• 17N.2.sl.TZ0.5a: Find the exact value of each of the zeros of \(f\).
• 17N.2.sl.TZ0.5b.i: Expand the expression for \(f(x)\).
• 17N.2.sl.TZ0.5b.ii: Find \(f’(x)\).
• 17N.2.sl.TZ0.5c: Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.
• 17N.2.sl.TZ0.5d: Draw the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and...
• 17N.2.sl.TZ0.5e: Write down the coordinates of the point of intersection.
• 18M.2.sl.TZ1.4a: Find the value of k.
• 18M.2.sl.TZ1.4b: Using your value of k , find f ′(x).
• 18M.2.sl.TZ1.4c: Use your answer to part (b) to show that the minimum value of f(x) is −22 .
• 18M.2.sl.TZ1.4d: Write down the two values of x which satisfy f (x) = 0.
• 18M.2.sl.TZ1.4e: Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.Clearly indicate the minimum...
• 18M.2.sl.TZ2.6a: Sketch the curve for −1 < x < 3 and −2 < y < 12.
• 18M.2.sl.TZ2.6b: A teacher asks her students to make some observations about the curve. Three students...
• 18M.2.sl.TZ2.6c: Find the value of y when x = 1 .
• 18M.2.sl.TZ2.6d: Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).
• 18M.2.sl.TZ2.6e: Show that the stationary points of the curve are at x = 1 and x = 2.
• 18M.2.sl.TZ2.6f: Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

6.7

• 10M.1.sl.TZ1.15c: Write down the number of solutions to the equation f (x) = 5 .
• 10M.2.sl.TZ1.5B.f: Tn is the sum of the first n terms of the geometric sequence. Use your graphic display...
• 12N.2.sl.TZ0.5e: Using your graphic display calculator find the coordinates of the point where the graph of y =...
• 10M.2.sl.TZ2.3d: Find the number of complete years it will take for Ying’s investment to exceed Ruby’s investment.
• 12M.2.sl.TZ1.5c: Write down the coordinates of the y-intercept of the graph of f (x).
• 12M.2.sl.TZ1.5i: P and Q are points on the curve such that the tangents to the curve at these points are...
• 09N.1.sl.TZ0.11b: find the value of \(x\) given that \(f (x) =162\).
• 11N.2.sl.TZ0.4a: Write down (i)     the equation of the vertical asymptote to the graph of \(y = f (x)\)...
• 11N.2.sl.TZ0.4g: The line, \(L\) , passes through the point A and is perpendicular to the tangent at A. \(L\)...
• 09N.2.sl.TZ0.5A, d: Using your graphic display calculator, solve the equation 3 + 2x − x2 = 2x.
• 09M.2.sl.TZ2.5e: T intersects the graph again at R. Use your graphic display calculator to find the coordinates of R.
• 13M.1.sl.TZ2.13c: Calculate the time, in hours, for the number of bacteria to reach 5500.
• 13M.2.sl.TZ2.4f: Each lap Paco runs again takes him 10 seconds longer than his previous lap. After a certain...
• 13M.2.sl.TZ2.5f: Use your graphic display calculator to find the coordinates of the point where Scott first...
• SPM.1.sl.TZ0.7b: Write down the coordinates of the \(y\)-intercept of the graph of \(y = f(x)\) .
• SPM.1.sl.TZ0.7d: Use your graphic display calculator to solve \(f(x) = g(x)\) .
• SPM.1.sl.TZ0.14c: Write down the coordinates of the second point where this tangent intersects the graph of...
• SPM.2.sl.TZ0.4c: To be able to see clearly, a diver needs the intensity of light to be at least \(65\% \). Using...
• 08N.2.sl.TZ0.1e: Each January, one of these two schools, the one that has more students, is given extra money to...
• 08N.2.sl.TZ0.4b: The following table shows values for \(m\) and \(T(m)\). (i)     Write down the value of...
• 08N.2.sl.TZ0.4e: Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) . The function...
• 08M.1.sl.TZ1.15c: Find how long it will take for there to be two million bacteria.
• 08M.2.sl.TZ1.1d: Using your graphical display calculator write down the coordinates of one of the points of...
• 08M.2.sl.TZ1.4i.d: (i)     Sketch the graphs of the two equations from parts (a) and (b). (ii)    Write down the...
• 08M.2.sl.TZ2.4i.f: (i) Draw and label the line \( y = 1\) on your graph. (ii) The equation \(f(x) = 1\) has two...
• 09M.2.sl.TZ1.5g, ii: T intersects the graph of f at a second point. Write down the x-coordinate of this point of...
• 14M.1.sl.TZ2.11c: Use your graphic display calculator to determine the time it takes for the amount of the drug in...
• 13N.1.sl.TZ0.10b: Use your graphic display calculator to solve \(f(x) = g(x)\).
• 14M.1.sl.TZ1.11c: To download a game to the mobile phone, an electrical charge of 2.4 units is needed. Find the...
• 14M.2.sl.TZ1.3e: \(k\) is the smallest value of \(n\) for which \({v_n}\) is greater than \({u_n}\). Calculate...
• 14N.1.sl.TZ0.9c: Use your graphic display calculator to find the value of \(x\) and the value of \(y\).
• 14N.1.sl.TZ0.11b: Solve \(f(x) = g(x)\).
• 15M.2.sl.TZ1.5g: The graph of \(y = f(x)\) has a local minimum point at \(x = 4\). Write down the coordinates of...
• 15M.2.sl.TZ1.6e: After being placed in the room for one minute, the temperature of the water is 84°C. Find the...
• 15M.2.sl.TZ2.5g: The tangent, \(T\), intersects the graph of \(f\) at a second point, P. Use your graphic display...
• 16M.1.sl.TZ1.13a: The golden ratio, \(r\) , was considered by the Ancient Greeks to be the perfect ratio...
• 16M.1.sl.TZ1.13b: Phidias is designing rectangular windows with adjacent sides of length \(x\) metres...
• 16M.1.sl.TZ1.13c: Phidias designs the windows so that the ratio between the longer side, \(y\) , and the...
• 16M.1.sl.TZ1.13d: Find the value of \(x\) .
• 16M.1.sl.TZ1.14a: A population of \(200\) rabbits was introduced to an island. One week later the number of rabbits...
• 16M.1.sl.TZ1.14b: Calculate the number of rabbits on the island after 10 weeks.
• 16M.1.sl.TZ1.14c: An ecologist estimates that the island has enough food to support a maximum population of 1000...
• 16M.1.sl.TZ1.15a: A company sells fruit juices in cylindrical cans, each of which has a volume of...
• 16M.1.sl.TZ1.15b: Calculate the value of \(r\) that minimizes the surface area of a can.
• 16M.2.sl.TZ1.3a: A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the...
• 16M.2.sl.TZ1.3b: Find the height of the flare \(15\) seconds after it was fired.
• 16M.2.sl.TZ1.3c: The flare fell into the sea \(k\) seconds after it was fired. Find the value of \(k\) .
• 16M.2.sl.TZ1.3d: Find \(h'\,(t)\,.\)
• 16M.2.sl.TZ1.3e: i)     Show that the flare reached its maximum height \(40\) seconds after being fired. ii)  ...
• 16M.2.sl.TZ1.3f: The nearest coastguard can see the flare when its height is more than \(40\) metres above sea...
• 16M.2.sl.TZ1.6a: A function, \(f\) , is given by \[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\] Calculate \(f(0)\)  
• 16M.2.sl.TZ1.6b: Use your graphic display calculator to solve \(f(x) = 0.\)
• 16M.2.sl.TZ1.6c: Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and...
• 16M.2.sl.TZ1.6d: The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\) i)    ...
• 16N.1.sl.TZ0.15a: Write down, and simplify, an expression for the car’s value when Gabriella purchased it.
• 16N.1.sl.TZ0.15b: Find the value of \(k\).
• 16N.1.sl.TZ0.15c: Find the value of \(n\).
• 16M.1.sl.TZ2.13a: A population of mosquitoes decreases exponentially. The size of the population, \(P\) , after...
• 16M.1.sl.TZ2.13b: Find the size of the population after \(4\) days.
• 16M.1.sl.TZ2.13d: The population will stabilize when it reaches a size of \(k\) . Write down the value of \(k\) .
• 16M.1.sl.TZ2.15a: Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown...
• 16M.1.sl.TZ2.15b: There are two points at which the gradient of the graph of \(f\) is \(11\). Find...
• 16N.2.sl.TZ0.3a: Show that A lies on \({L_1}\).
• 16N.2.sl.TZ0.3b: Find the coordinates of M.
• 16N.2.sl.TZ0.3c: Find the length of AC.
• 16N.2.sl.TZ0.3d: Show that the equation of \({L_2}\) is \(2y + x - 19 = 0\).
• 16N.2.sl.TZ0.3e: Find the coordinates of D.
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 16N.2.sl.TZ0.3g: Find the area of ABCD.
• 16N.2.sl.TZ0.6a: Write down a formula for \(A\), the surface area to be coated.
• 16N.2.sl.TZ0.6b: Express this volume in \({\text{c}}{{\text{m}}^3}\).
• 16N.2.sl.TZ0.6c: Write down, in terms of \(r\) and \(h\), an equation for the volume of this water container.
• 16N.2.sl.TZ0.6e: Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).
• 16N.2.sl.TZ0.6f: Using your answer to part (e), find the value of \(r\) which minimizes \(A\).
• 16N.2.sl.TZ0.6g: Find the value of this minimum area.
• 16N.2.sl.TZ0.6h: Find the least number of cans of water-resistant material that will coat the area in part (g).
• 17M.1.sl.TZ1.15a: Write down the value of \(c\).
• 17M.1.sl.TZ1.15b: Find the value of \(a\) and of \(b\).
• 17M.1.sl.TZ1.15c: Write down the second \(x\)-intercept of the function.
• 17M.1.sl.TZ2.14a: Write down the amount of money Jashanti saves per month.
• 17M.1.sl.TZ2.14b: Use your graphic display calculator to find how long it will take for Jashanti to have saved...
• 17M.1.sl.TZ2.14c: Calculate how much extra money Jashanti needs.
• 17M.2.sl.TZ2.5a: Show that the width of the plot, in metres, is given by \(100 - x\).
• 17M.2.sl.TZ2.5b: Write down the area of the plot in terms of \(x\).
• 17M.2.sl.TZ2.5c: Find the value of \(x\) that maximizes the area of the plot.
• 17M.2.sl.TZ2.5d: Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.
• 17M.2.sl.TZ2.5e.i: Find the amount of money that Violeta would have after 6 years. Give your answer correct to two...
• 17M.2.sl.TZ2.5f: Find the lowest possible value for \(r\).
• 17M.2.sl.TZ2.6a: Write down the \(y\)-intercept of the graph.
• 17M.2.sl.TZ2.6b: Find \(f'(x)\).
• 17M.2.sl.TZ2.6c.i: Show that \(a = 8\).
• 17M.2.sl.TZ2.6d.i: Write down the \(x\)-coordinates of these two points;
• 17M.2.sl.TZ2.6e: Write down the range of \(f(x)\).
• 17M.2.sl.TZ2.6f: Write down the number of possible solutions to the equation \(f(x) = 5\).
• 17M.2.sl.TZ2.6g: The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible...
• 17M.2.sl.TZ2.5e.ii: Find how long it would take for the interest earned to be 2000 BGN.
• 17M.2.sl.TZ2.6c.ii: Find \(f(2)\).
• 17M.2.sl.TZ2.6d.ii: Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.
• 16N.2.sl.TZ0.6d: Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).