DP Mathematical Studies Questionbank

• Syllabus

6.4

Sub sections and their related questions

Exponential models.

• 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...
• 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 .
• 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.
• 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.
• 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...
• 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...
• 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 \(L\).
• 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.TZ0.13a: 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.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...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 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.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.

Exponential functions and their graphs: \(f\left( x \right) = k{a^x} + c\); \(a \in {\mathbb{Q}^ + }\), \(a \ne 1\), \(k \ne 0\) .

• 10N.1.sl.TZ0.13a: Write down two equations relating p and q.
• 09N.2.sl.TZ0.5A, a: Sketch the graph of y = 2x for \( - 2 \leqslant x \leqslant 3\). Indicate clearly where the curve...
• 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...
• 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...
• 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.
• 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...
• 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...
• 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.TZ1.11c: 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.TZ0.11c: 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...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 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.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.

Exponential functions and their graphs: \(f\left( x \right) = k{a^{ - x}} + c\); \(a \in {\mathbb{Q}^ + }\), \(a \ne 1\), \(k \ne 0\) .

• 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.  
• 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 \(c\) .
• 12M.1.sl.TZ2.12a: Find the value of (i) a ; (ii) b .
• 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 .
• 07N.1.sl.TZ0.5: The following curves are sketches of the graphs of the functions given below, but in a different...
• 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.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...
• 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...
• 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.TZ1.11c: To download a game to the mobile phone, an electrical charge of 2.4 units is needed. Find the...
• 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...
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 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.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.

Concept and equation of a horizontal asymptote.

• 10N.1.sl.TZ0.13c: Write down the equation of the horizontal asymptote to the graph of f (x).
• 11N.1.sl.TZ0.15c: The graph passes through the point (2, 48) . Write down the equation of the horizontal...
• 12M.1.sl.TZ2.12b: Write down the equation of the horizontal asymptote to this graph.
• 09N.2.sl.TZ0.5A, b: Write down the equation of the asymptote of the graph of y = 2x.
• 11M.1.sl.TZ1.15c: The point \((2{\text{, }}1)\) lies on the graph of \(y = f(x)\) . Write down the equation of \(L\) .
• 13M.2.sl.TZ1.3f: The graph of this function has a horizontal asymptote. Write down the equation of this asymptote.
• 07N.2.sl.TZ0.3i.b: Write down the equation of the horizontal asymptote.
• 07N.2.sl.TZ0.3i.c: Find the room temperature.
• 08N.2.sl.TZ0.4b: The following table shows values for \(m\) and \(T(m)\). (i)     Write down the value of...
• 08M.2.sl.TZ2.4i.e: Write down the equation of the horizontal asymptote to the graph of \(y = f(x)\).
• 14M.1.sl.TZ1.11b: The line \(L\) is the horizontal asymptote to the graph. Write down the equation of \(L\).
• 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 \(a = 20\).
• 16N.2.sl.TZ0.3f: Write down the length of MD correct to five significant figures.
• 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.