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.