DP Mathematics SL Questionbank

• Syllabus

3.4

Sub sections and their related questions

The circular functions \(\sin x\) , \(\cos x\) and \(\tan x\) : their domains and ranges; amplitude, their periodic nature; and their graphs.

• 12N.2.sl.TZ0.5a(i) and (ii): Write down the value of (i)     \(a\) ; (ii)    \(c\) .
• 12N.2.sl.TZ0.5b: Find the value of b .
• 12N.2.sl.TZ0.5c: Find the x-coordinate of R.
• 12M.1.sl.TZ2.3a(i) and (ii): (i)     Write down the value of a . (ii)    Find the value of b .
• 12M.1.sl.TZ2.3b: Write down the gradient of the curve at P.
• 12M.1.sl.TZ2.3c: Write down the equation of the normal to the curve at P.
• 08M.1.sl.TZ1.4a: Write down the period of g.
• 08M.1.sl.TZ1.4b: On the diagram below, sketch the curve of g, for \(0 \le x \le 2\pi \) .
• 08M.1.sl.TZ1.4c: Write down the number of solutions to the equation \(g(x) = 2\) , for \(0 \le x \le 2\pi \) .
• 08M.1.sl.TZ2.9a: (i) Write down the range of the function f . (ii) Consider \(f(x) = 1\) , \(0 \le x \le 2\pi \)...
• 08M.2.sl.TZ2.8a(i), (ii) and (iii): Use the graph to write down an estimate of the value of t when (i)     the depth of water is...
• 12M.1.sl.TZ1.5a: Find the value of a .
• 12M.1.sl.TZ1.5b(i) and (ii): (i)     Show that the period of f is \(\pi \) . (ii)    Hence, find the value of b .
• 12M.1.sl.TZ1.5c: Given that \(0 < c < \pi \)  , write down the value of c .
• 09N.2.sl.TZ0.9b: Consider the graph of \(f\) . Write down (i)     the x-intercept that lies between \(x = 0\) and...
• 09M.2.sl.TZ1.3b: Write down the period of \(h\) .
• 09M.2.sl.TZ1.3c: Write down the range of \(h\) .
• 11M.1.sl.TZ2.10a(i) and (ii): Write down the height of P above ground level after (i)     10 minutes; (ii)    15 minutes.  
• 11M.1.sl.TZ2.10b(i) and (ii): (i)     Show that \(h(8) = 90.5\). (ii)    Find \(h(21)\) .
• 11M.1.sl.TZ2.10c: Sketch the graph of h , for \(0 \le t \le 40\) .
• 11M.1.sl.TZ2.10d: Given that h can be expressed in the form \(h(t) = a\cos bt + c\) , find a , b and c .
• 13M.2.sl.TZ1.10a: Find the maximum height above the ground of the seat.
• 14M.2.sl.TZ1.5a: Find the number of deer in the reserve on 1 May 2014.
• 14M.2.sl.TZ1.9c(i): The function \(f\) can also be written in the form...
• 14M.2.sl.TZ1.9c(ii): The function \(f\) can also be written in the form...
• 14M.2.sl.TZ2.6a: Write down the value of \(r\).
• 14M.2.sl.TZ2.6b(i): Find \(p\).
• 14M.2.sl.TZ2.6b(ii): Find \(q\).
• 13N.1.sl.TZ0.5a: Find the value of \(k\).
• 13N.1.sl.TZ0.5b: Find the minimum value of \(f(x)\).
• 13N.1.sl.TZ0.5c: Let \(g(x) = \sin x\). The graph of g is translated to the graph of \(f\) by the...
• 13M.1.sl.TZ2.5a: \(p\)
• 13M.1.sl.TZ2.5b: \(q\)
• 13M.1.sl.TZ2.5c: \(r\) .
• 13M.2.sl.TZ1.10b: (i)     Show that the period of \(h\) is \(25\) minutes. (ii)     Write down the exact value of...
• 13M.2.sl.TZ1.10c: Find the value of \(a\) .
• 13M.2.sl.TZ1.10d: Sketch the graph of \(h\) , for \(0 \le t \le 50\) .
• 14N.2.sl.TZ0.5a: \(p\);
• 14N.2.sl.TZ0.5b: \(r\);
• 14N.2.sl.TZ0.5c: \(q\).
• 15N.1.sl.TZ0.4a: Write down the amplitude of \(f\).
• 15N.1.sl.TZ0.4b: Find the period of \(f\).
• 15N.1.sl.TZ0.4c: On the following grid, sketch the graph of \(y = f(x)\), for \(0 \le x \le 3\).
• 16M.1.sl.TZ1.3a: (i)     Write down the amplitude of \(f\). (ii)     Find the period of \(f\).
• 16M.1.sl.TZ1.3b: On the following grid sketch the graph of \(f\).
• 16M.1.sl.TZ2.6a: Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).
• 16M.1.sl.TZ2.6b: Hence find the range of \(h\).
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 17M.2.sl.TZ1.8a.i: How much time is there between the first low tide and the next high tide?
• 17M.2.sl.TZ1.8a.ii: Find the difference in height between low tide and high tide.
• 17M.2.sl.TZ1.8b.i: Find the value of \(p\);
• 17M.2.sl.TZ1.8b.ii: Find the value of \(q\);
• 17M.2.sl.TZ1.8b.iii: Find the value of \(r\).
• 17M.2.sl.TZ1.8c: There are two high tides on 12 December 2017. At what time does the second high tide occur?
• 17N.2.sl.TZ0.10a: Show that \(f(2\pi ) = 2\pi \).
• 17N.2.sl.TZ0.10b.i: Find the coordinates of \({{\text{P}}_0}\) and of \({{\text{P}}_1}\).
• 17N.2.sl.TZ0.10b.ii: Find the equation of \(L\).
• 17N.2.sl.TZ0.10c: Show that the distance between the \(x\)-coordinates of \({{\text{P}}_k}\) and...
• 17N.2.sl.TZ0.10d: A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.
• 18M.2.sl.TZ1.10a: Find the coordinates of A.
• 18M.2.sl.TZ1.10b.i: For the graph of \(f\), write down the amplitude.
• 18M.2.sl.TZ1.10c: Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
• 18M.2.sl.TZ1.10d: Find the maximum speed of the ball.
• 18M.2.sl.TZ1.10e: Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
• 18M.2.sl.TZ1.10b.ii: For the graph of \(f\), write down the period.
• 18M.2.sl.TZ2.6a: After 8 minutes, the seat is 117 m above the ground. Find k.
• 18M.2.sl.TZ2.6b: Find the value of a.
• 18M.2.sl.TZ2.6c: Find when the seat is 30 m above the ground for the third time.

Composite functions of the form \(f(x) = a\sin (b(x + c)) + d\) .

• 08N.1.sl.TZ0.10a(i), (ii), (iii) and (iv): (i)     Find the value of a. (ii)    Show that \(b = \frac{\pi }{6}\) . (iii)   Find the value...
• 08N.1.sl.TZ0.10b: The transformation P is given by a horizontal stretch of a scale factor of \(\frac{1}{2}\) ,...
• 08N.1.sl.TZ0.10c: The graph of g is the image of the graph of f under P. Find \(g(t)\) in the form...
• 08N.1.sl.TZ0.10d: The graph of g is the image of the graph of f under P. Give a full geometric description of the...
• 08M.2.sl.TZ2.8a(i), (ii) and (iii): Use the graph to write down an estimate of the value of t when (i)     the depth of water is...
• 08M.2.sl.TZ2.8b(i), (ii) and (iii): The depth of water can be modelled by the function \(y = \cos A(B(t - 1)) + C\) . (i)     Show...
• 10M.1.sl.TZ2.10a(i) and (ii): Solve for \(0 \le x < 2\pi \) (i)     \(6 + 6\sin x = 6\) ; (ii)    \(6 + 6\sin x = 0\) .
• 10M.1.sl.TZ2.10b: Write down the exact value of the x-intercept of f , for \(0 \le x < 2\pi \) .
• 10M.1.sl.TZ2.10c: The area of the shaded region is k . Find the value of k , giving your answer in terms of \(\pi \) .
• 10M.1.sl.TZ2.10d: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 10M.1.sl.TZ2.10e: Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . The graph of f is transformed to...
• 09N.2.sl.TZ0.9b: Consider the graph of \(f\) . Write down (i)     the x-intercept that lies between \(x = 0\) and...
• 09M.2.sl.TZ1.3b: Write down the period of \(h\) .
• 09M.2.sl.TZ1.3c: Write down the range of \(h\) .
• 09M.2.sl.TZ2.10c: Hence write \(f(x)\) in the form \(p\sin (qx + r)\) .
• 10N.2.sl.TZ0.10a: Find the height of a seat above the ground after 15 minutes.
• 10N.2.sl.TZ0.10b: After six minutes, the seat is at point Q. Find its height above the ground at Q.
• 10N.2.sl.TZ0.10c: The height of the seat above ground after t minutes can be modelled by the function...
• 10N.2.sl.TZ0.10d: The height of the seat above ground after t minutes can be modelled by the function...
• 10M.2.sl.TZ1.5a(i), (ii) and (iii): Find the value of (i)     p ; (ii)    q ; (iii)   r.
• 10M.2.sl.TZ1.5b: The equation \(y = k\) has exactly two solutions. Write down the value of k.
• SPNone.1.sl.TZ0.10d: The function \(f(x)\) can be written in the form \(r\cos (x - a)\) . Write down the value of r...
• 11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i)     a ; (ii)    c ; (iii)   d .
• 11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
• 11N.1.sl.TZ0.9c: Find \(f'(x)\) .
• 11N.1.sl.TZ0.9d: At a point R, the gradient is \( - 2\pi \) . Find the x-coordinate of R.
• 11M.2.sl.TZ1.8a: Show that \(a = 4\) .
• 11M.2.sl.TZ1.8b: The wheel turns at a rate of one rotation every 30 seconds. Show that \(b = \frac{\pi }{{15}}\) .
• 11M.2.sl.TZ1.8c: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.2.sl.TZ1.8d: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.1.sl.TZ2.10a(i) and (ii): Write down the height of P above ground level after (i)     10 minutes; (ii)    15 minutes.  
• 11M.1.sl.TZ2.10b(i) and (ii): (i)     Show that \(h(8) = 90.5\). (ii)    Find \(h(21)\) .
• 11M.1.sl.TZ2.10c: Sketch the graph of h , for \(0 \le t \le 40\) .
• 11M.1.sl.TZ2.10d: Given that h can be expressed in the form \(h(t) = a\cos bt + c\) , find a , b and c .
• 14M.2.sl.TZ2.6a: Write down the value of \(r\).
• 14M.2.sl.TZ2.6b(i): Find \(p\).
• 14M.2.sl.TZ2.6b(ii): Find \(q\).
• 13N.1.sl.TZ0.5a: Find the value of \(k\).
• 13N.1.sl.TZ0.5b: Find the minimum value of \(f(x)\).
• 13M.1.sl.TZ2.5a: \(p\)
• 13M.1.sl.TZ2.5b: \(q\)
• 13M.1.sl.TZ2.5c: \(r\) .
• 13M.2.sl.TZ1.10b: (i)     Show that the period of \(h\) is \(25\) minutes. (ii)     Write down the exact value of...
• 13M.2.sl.TZ1.10c: Find the value of \(a\) .
• 13M.2.sl.TZ1.10d: Sketch the graph of \(h\) , for \(0 \le t \le 50\) .
• 14N.2.sl.TZ0.5a: \(p\);
• 14N.2.sl.TZ0.5b: \(r\);
• 14N.2.sl.TZ0.5c: \(q\).
• 15N.1.sl.TZ0.4a: Write down the amplitude of \(f\).
• 15N.1.sl.TZ0.4b: Find the period of \(f\).
• 15N.1.sl.TZ0.4c: On the following grid, sketch the graph of \(y = f(x)\), for \(0 \le x \le 3\).
• 16M.1.sl.TZ1.3a: (i)     Write down the amplitude of \(f\). (ii)     Find the period of \(f\).
• 16M.1.sl.TZ1.3b: On the following grid sketch the graph of \(f\).
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 16N.2.sl.TZ0.10c: (i)     Find \(w\). (ii)     Hence or otherwise, find the maximum positive rate of change of \(g\).
• 17M.2.sl.TZ2.4a: Find the value of \(p\).
• 17M.2.sl.TZ2.4b: Find the value of \(q\).
• 17M.2.sl.TZ2.4c: Use the model to find the depth of the water 10 hours after high tide.
• 18M.2.sl.TZ1.10a: Find the coordinates of A.
• 18M.2.sl.TZ1.10b.i: For the graph of \(f\), write down the amplitude.
• 18M.2.sl.TZ1.10c: Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
• 18M.2.sl.TZ1.10d: Find the maximum speed of the ball.
• 18M.2.sl.TZ1.10e: Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
• 18M.2.sl.TZ1.10b.ii: For the graph of \(f\), write down the period.

Transformations.

• 08N.1.sl.TZ0.10b: The transformation P is given by a horizontal stretch of a scale factor of \(\frac{1}{2}\) ,...
• 08N.1.sl.TZ0.10c: The graph of g is the image of the graph of f under P. Find \(g(t)\) in the form...
• 08N.1.sl.TZ0.10d: The graph of g is the image of the graph of f under P. Give a full geometric description of the...
• 12M.1.sl.TZ1.7a: Show that \(f(x)\) can be expressed as \(1 + \sin 2x\) .
• 12M.1.sl.TZ1.7b: The graph of f is shown below for \(0 \le x \le 2\pi \) . Let \(g(x) = 1 + \cos x\) . On the...
• 12M.1.sl.TZ1.7c: The graph of g can be obtained from the graph of f under a horizontal stretch of scale factor p...
• SPNone.1.sl.TZ0.10d: The function \(f(x)\) can be written in the form \(r\cos (x - a)\) . Write down the value of r...
• 14M.2.sl.TZ1.9c(i): The function \(f\) can also be written in the form...
• 14M.2.sl.TZ1.9c(ii): The function \(f\) can also be written in the form...
• 13N.1.sl.TZ0.5c: Let \(g(x) = \sin x\). The graph of g is translated to the graph of \(f\) by the...
• 14N.2.sl.TZ0.5a: \(p\);
• 14N.2.sl.TZ0.5b: \(r\);
• 14N.2.sl.TZ0.5c: \(q\).
• 16M.1.sl.TZ1.3a: (i)     Write down the amplitude of \(f\). (ii)     Find the period of \(f\).
• 16M.1.sl.TZ1.3b: On the following grid sketch the graph of \(f\).
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 16N.2.sl.TZ0.10c: (i)     Find \(w\). (ii)     Hence or otherwise, find the maximum positive rate of change of \(g\).
• 17M.2.sl.TZ2.4a: Find the value of \(p\).
• 17M.2.sl.TZ2.4b: Find the value of \(q\).
• 17M.2.sl.TZ2.4c: Use the model to find the depth of the water 10 hours after high tide.
• 18M.2.sl.TZ1.10a: Find the coordinates of A.
• 18M.2.sl.TZ1.10b.i: For the graph of \(f\), write down the amplitude.
• 18M.2.sl.TZ1.10c: Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
• 18M.2.sl.TZ1.10d: Find the maximum speed of the ball.
• 18M.2.sl.TZ1.10e: Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.
• 18M.2.sl.TZ1.10b.ii: For the graph of \(f\), write down the period.

Applications.

• 10N.2.sl.TZ0.10a: Find the height of a seat above the ground after 15 minutes.
• 10N.2.sl.TZ0.10b: After six minutes, the seat is at point Q. Find its height above the ground at Q.
• 10N.2.sl.TZ0.10c: The height of the seat above ground after t minutes can be modelled by the function...
• 10N.2.sl.TZ0.10d: The height of the seat above ground after t minutes can be modelled by the function...
• 11M.2.sl.TZ1.8a: Show that \(a = 4\) .
• 11M.2.sl.TZ1.8b: The wheel turns at a rate of one rotation every 30 seconds. Show that \(b = \frac{\pi }{{15}}\) .
• 11M.2.sl.TZ1.8c: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.2.sl.TZ1.8d: In the first rotation, there are two values of t when the bucket is descending at a rate of...
• 11M.1.sl.TZ2.10a(i) and (ii): Write down the height of P above ground level after (i)     10 minutes; (ii)    15 minutes.  
• 11M.1.sl.TZ2.10b(i) and (ii): (i)     Show that \(h(8) = 90.5\). (ii)    Find \(h(21)\) .
• 11M.1.sl.TZ2.10c: Sketch the graph of h , for \(0 \le t \le 40\) .
• 11M.1.sl.TZ2.10d: Given that h can be expressed in the form \(h(t) = a\cos bt + c\) , find a , b and c .
• 13M.2.sl.TZ1.10a: Find the maximum height above the ground of the seat.
• 14M.2.sl.TZ1.5a: Find the number of deer in the reserve on 1 May 2014.
• 13M.2.sl.TZ1.10b: (i)     Show that the period of \(h\) is \(25\) minutes. (ii)     Write down the exact value of...
• 13M.2.sl.TZ1.10c: Find the value of \(a\) .
• 13M.2.sl.TZ1.10d: Sketch the graph of \(h\) , for \(0 \le t \le 50\) .
• 16M.1.sl.TZ1.3a: (i)     Write down the amplitude of \(f\). (ii)     Find the period of \(f\).
• 16M.1.sl.TZ1.3b: On the following grid sketch the graph of \(f\).
• 16M.2.sl.TZ2.4a: Find the height of the seat when \(t = 0\).
• 16M.2.sl.TZ2.4b: The seat first reaches a height of 20 m after \(k\) minutes. Find \(k\).
• 16M.2.sl.TZ2.4c: Calculate the time needed for the seat to complete a full rotation, giving your answer correct to...
• 16N.2.sl.TZ0.10a: (i)     Find the value of \(c\). (ii)     Show that \(b = \frac{\pi }{6}\). (iii)     Find the...
• 16N.2.sl.TZ0.10b: (i)     Write down the value of \(k\). (ii)     Find \(g(x)\).
• 16N.2.sl.TZ0.10c: (i)     Find \(w\). (ii)     Hence or otherwise, find the maximum positive rate of change of \(g\).
• 17M.2.sl.TZ1.8a.i: How much time is there between the first low tide and the next high tide?
• 17M.2.sl.TZ1.8a.ii: Find the difference in height between low tide and high tide.
• 17M.2.sl.TZ1.8b.i: Find the value of \(p\);
• 17M.2.sl.TZ1.8b.ii: Find the value of \(q\);
• 17M.2.sl.TZ1.8b.iii: Find the value of \(r\).
• 17M.2.sl.TZ1.8c: There are two high tides on 12 December 2017. At what time does the second high tide occur?
• 17M.2.sl.TZ2.4a: Find the value of \(p\).
• 17M.2.sl.TZ2.4b: Find the value of \(q\).
• 17M.2.sl.TZ2.4c: Use the model to find the depth of the water 10 hours after high tide.
• 18M.2.sl.TZ2.6a: After 8 minutes, the seat is 117 m above the ground. Find k.
• 18M.2.sl.TZ2.6b: Find the value of a.
• 18M.2.sl.TZ2.6c: Find when the seat is 30 m above the ground for the third time.