DP Mathematics SL Questionbank

• Syllabus

Topic 3 - Circular functions and trigonometry

Sub sections and their related questions

3.1

• 12N.2.sl.TZ0.8a: Find the length of the chord [AB].
• 12N.2.sl.TZ0.8b: Find the area of triangle AOB.
• 12N.2.sl.TZ0.8c: Angle BOC is 2.4 radians. Find the length of arc ADC.
• 12N.2.sl.TZ0.8d: Angle BOC is 2.4 radians. Find the area of the shaded region.
• 12N.2.sl.TZ0.8e: Angle BOC is 2.4 radians. The shaded region is to be painted red. Red paint is sold in cans...
• 12M.2.sl.TZ2.10a: Use the cosine rule to show that ${\rm{PQ}} = 2r\sin \theta$ .
• 12M.2.sl.TZ2.10b: Let l be the length of the arc PRQ . Given that $1.3{\rm{PQ}} - l = 0$ , find the value of...
• 12M.2.sl.TZ2.10c(i) and (ii): Consider the function $f(\theta ) = 2.6\sin \theta  - 2\theta$ , for...
• 12M.2.sl.TZ2.10d: Use the graph of f to find the values of $\theta$ for which $l < 1.3{\rm{PQ}}$ .
• 08M.2.sl.TZ1.3a: Find the value of r.
• 08M.2.sl.TZ1.3b: Find the perimeter of sector OABC.
• 08M.2.sl.TZ1.3c: Find the area of sector OABC.
• 12M.2.sl.TZ1.9a(i) and (ii): (i)     Show that ${p^2}(41 - 40\cos 0.7) = 36$ . (ii)    Find p .
• 12M.2.sl.TZ1.9b: Write down the length of BD.
• 12M.2.sl.TZ1.9c: Find ${\rm{A}}\widehat {\rm{D}}{\rm{B}}$ .
• 12M.2.sl.TZ1.9d(i) and (ii): (i)     Show that ${\rm{C}}\widehat {\rm{B}}{\rm{D}} = 1.29$ radians, correct to 2 decimal...
• 10N.1.sl.TZ0.3a: Find the length of the arc ACB .
• 10N.1.sl.TZ0.3b: Find the area of the shaded region.
• 09N.2.sl.TZ0.8c: Find the area of sector OABC.
• 09M.2.sl.TZ1.2b: Find the area of the shaded region.
• 10M.2.sl.TZ2.8a: Find the size of angle AOC .
• 10M.2.sl.TZ2.8b: Hence find the area of the shaded region.
• 10M.2.sl.TZ2.8c: The area of sector OCDE is $45{\text{ c}}{{\text{m}}^2}$. Find the size of angle COE .
• 10M.2.sl.TZ2.8d: Find EF .
• SPNone.2.sl.TZ0.7a: Find an expression for the area of the shaded region.
• 11N.2.sl.TZ0.3a: Find the value of $\theta$ .
• 11N.2.sl.TZ0.3b: Find the area of the shaded region.
• 13M.2.sl.TZ2.7b: The area of the shaded region is $25$ cm2 . Find the value of $r$ .
• 14M.2.sl.TZ2.1a(i): Find the length of the arc ${\text{ABC}}$.
• 14M.2.sl.TZ2.1a(ii): Find the perimeter of the shaded sector.
• 14M.2.sl.TZ2.1b: Find the area of the shaded sector.
• 13N.2.sl.TZ0.8c: Hence or otherwise, find the length of arc ${\text{ABC}}$.
• 14N.2.sl.TZ0.3a: Find the length of arc $ACB$.
• 15M.1.sl.TZ1.2a: Find the length of ${\text{arc ACB}}$.
• 15M.1.sl.TZ1.2b: Find the perimeter of the shaded region.
• 15M.2.sl.TZ2.10b: When $\theta  = \alpha$, the area of the square $ABCD$ is equal to the area of the sector...
• 15N.2.sl.TZ0.1a: Find the length of arc $ABC$.
• 15N.2.sl.TZ0.1b: Find the area of the shaded region.
• 16M.1.sl.TZ2.5a: Find ${\rm{A\hat BC}}$.
• 16M.1.sl.TZ2.5b: Find the exact area of the sector BDC.
• 16N.2.sl.TZ0.3a: Write down the exact value of $\theta$ in radians.
• 16N.2.sl.TZ0.3b: Find the value of $r$.
• 17M.2.sl.TZ1.5: The following diagram shows the chord [AB] in a circle of radius 8 cm, where...
• 17M.2.sl.TZ2.1a: Find the length of arc ABC.
• 17M.2.sl.TZ2.1b: Find the perimeter of sector OABC.
• 17M.2.sl.TZ2.1c: Find the area of sector OABC.
• 17N.1.sl.TZ0.4a: Show that ${\text{AC}} = 7{\text{ cm}}$.
• 17N.1.sl.TZ0.4b: The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the...
• 18M.2.sl.TZ1.3a: Find the area of the shaded region, in terms of θ.
• 18M.2.sl.TZ1.3b: The area of the shaded region is 12 cm2. Find the value of θ.
• 18M.1.sl.TZ2.4: The following diagram shows a circle with centre O and radius r cm. The points A and B lie on...

3.2

• 08M.1.sl.TZ1.2a(i) and (ii): Write down an expression for (i)     $\sin 140^\circ$ ; (ii)    $\cos 70^\circ$ .
• 08M.1.sl.TZ1.2c: Find an expression for $\tan 140^\circ$ .
• 10M.1.sl.TZ1.9a: Use the quotient rule to show that $f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}$ .
• 10M.1.sl.TZ1.9b: Find $f''(x)$ .
• 10M.1.sl.TZ1.9c: Find the value of p and of q.
• 10M.1.sl.TZ1.9d: Use information from the table to explain why there is a point of inflexion on the graph of f...
• 10M.1.sl.TZ2.4a: Find $f\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4b: Find $(g \circ f)\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4c: Given that $(g \circ f)(x)$ can be written as $\cos (kx)$ , find the value of k,...
• 09M.1.sl.TZ1.8a: Write down an expression in terms of $\theta$ for (i)     $x$ ; (ii)    $y$ .
• 09M.1.sl.TZ2.7: Let $f(x) = \sqrt 3 {{\rm{e}}^{2x}}\sin x + {{\rm{e}}^{2x}}\cos x$ , for $0 \le x \le \pi$ ....
• 09M.1.sl.TZ2.8b: Let $h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)$ . Find the exact value...
• SPNone.1.sl.TZ0.6b: Hence or otherwise, solve the equation $6\sin x\cos x = \frac{3}{2}$ , for...
• SPNone.1.sl.TZ0.10c: Find the maximum value of $f(x)$ .
• 11M.1.sl.TZ2.4: Let $h(x) = \frac{{6x}}{{\cos x}}$ . Find $h'(0)$ .
• 14M.1.sl.TZ1.6: Let $\int_\pi ^a {\cos 2x{\text{d}}x}  = \frac{1}{2}{\text{, where }}\pi  < a < 2\pi$....
• 16M.1.sl.TZ2.5a: Find ${\rm{A\hat BC}}$.
• 16M.1.sl.TZ2.5b: Find the exact area of the sector BDC.
• 17M.1.sl.TZ1.3: The following diagram shows triangle PQR. Find PR.
• 17M.1.sl.TZ1.10b: Given that $\tan \theta  > 0$, find $\tan \theta$.
• 17M.1.sl.TZ2.7: Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) =  - 1$, for...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
• 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
• 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
• 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.

3.3

• 12N.1.sl.TZ0.5a: Let $\sin {100^ \circ } = m$. Find an expression for $\cos {100^ \circ }$ in terms of m.
• 12N.1.sl.TZ0.5b: Let $\sin {100^ \circ } = m$ . Find an expression for $\tan {100^ \circ }$ in terms of m.
• 12N.1.sl.TZ0.5c: Let $\sin {100^ \circ } = m$. Find an expression for $\sin {200^ \circ }$ in terms of m.
• 08N.1.sl.TZ0.7a: Show that $f(x) = \sin x$ .
• 08N.1.sl.TZ0.7b: Let $\sin x = \frac{2}{3}$ . Show that $f(2x) = - \frac{{4\sqrt 5 }}{9}$ .
• 08M.1.sl.TZ1.2b: Find an expression for $\cos 140^\circ$ .
• 08M.1.sl.TZ2.4a: Given that $\cos A = \frac{1}{3}$ and $0 \le A \le \frac{\pi }{2}$ , find $\cos 2A$ .
• 08M.1.sl.TZ2.4b: Given that $\sin B = \frac{2}{3}$  and $\frac{\pi }{2} \le B \le \pi$ , find $\cos B$ .
• 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...
• 10N.1.sl.TZ0.5a: Show that $4 - \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin \theta + 3$ .
• 10N.1.sl.TZ0.5b: Hence, solve the equation $4 - \cos 2\theta + 5\sin \theta = 0$ for $0 \le \theta \le 2\pi$ .
• 10M.1.sl.TZ1.4a: Write down the value of $\tan \theta$ .
• 10M.1.sl.TZ1.4b(i) and (ii): Find the value of (i)     $\sin 2\theta$ ; (ii)    $\cos 2\theta$ .
• 10M.1.sl.TZ2.4a: Find $f\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4b: Find $(g \circ f)\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4c: Given that $(g \circ f)(x)$ can be written as $\cos (kx)$ , find the value of k,...
• 09N.1.sl.TZ0.6: Solve $\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x$ , for $0 \le x \le 2\pi$ .
• 09M.1.sl.TZ1.8b: Let the area of the rectangle be A. Show that $A = 18\sin 2\theta$ .
• 09M.1.sl.TZ1.9c: (i)     Find ${\rm{sinR}}\widehat {\rm{P}}{\rm{Q}}$ . (ii)    Hence, find the area of triangle...
• SPNone.1.sl.TZ0.6a: Find the value of a and of b .
• 11N.1.sl.TZ0.6a: Find $\cos \theta$ .
• 11N.1.sl.TZ0.6b: Find $\tan 2\theta$ .
• 11M.2.sl.TZ2.10a: Show that the area of the window is given by $y = 4\sin \theta + 2\sin 2\theta$ .
• 11M.2.sl.TZ2.10b: Zoe wants a window to have an area of $5{\text{ }}{{\text{m}}^2}$. Find the two possible values...
• 11M.2.sl.TZ2.10c: John wants two windows which have the same area A but different values of $\theta$ . Find all...
• 14M.1.sl.TZ2.1a: Show that $\cos A = \frac{{12}}{{13}}$.
• 14M.1.sl.TZ2.1b: Find $\cos 2A$.
• 15M.1.sl.TZ1.5a: find the value of $\cos x;$
• 15M.1.sl.TZ1.5b: find the value of $\cos 2x.$
• 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.1.sl.TZ0.2a: Find $\cos \theta$.
• 16N.1.sl.TZ0.2b: Find $\cos 2\theta$.
• 17M.1.sl.TZ1.10a: Show that $\cos \theta  = \frac{3}{4}$.
• 17M.1.sl.TZ1.10b: Given that $\tan \theta  > 0$, find $\tan \theta$.
• 17M.1.sl.TZ1.10c: Let $y = \frac{1}{{\cos x}}$, for $0 < x < \frac{\pi }{2}$. The graph of $y$between...
• 17M.1.sl.TZ2.7: Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) =  - 1$, for...
• 17N.1.sl.TZ0.6: Let $f(x) = 15 - {x^2}$, for $x \in \mathbb{R}$. The following diagram shows part of the...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
• 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
• 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
• 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.

3.4

• 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.
• 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.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...
• 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...
• 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 .
• 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...
• 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 .
• 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 $h$.
• 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.
• 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.

3.5

• 12M.2.sl.TZ2.10a: Use the cosine rule to show that ${\rm{PQ}} = 2r\sin \theta$ .
• 12M.2.sl.TZ2.10b: Let l be the length of the arc PRQ . Given that $1.3{\rm{PQ}} - l = 0$ , find the value of...
• 12M.2.sl.TZ2.10c(i) and (ii): Consider the function $f(\theta ) = 2.6\sin \theta  - 2\theta$ , for...
• 12M.2.sl.TZ2.10d: Use the graph of f to find the values of $\theta$ for which $l < 1.3{\rm{PQ}}$ .
• 08M.2.sl.TZ2.8c: A sailor knows that he cannot sail past P when the depth of the water is less than 12 m ....
• 10N.1.sl.TZ0.5a: Show that $4 - \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin \theta + 3$ .
• 10N.1.sl.TZ0.5b: Hence, solve the equation $4 - \cos 2\theta + 5\sin \theta = 0$ for $0 \le \theta \le 2\pi$ .
• 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.1.sl.TZ0.6: Solve $\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x$ , for $0 \le x \le 2\pi$ .
• 09M.1.sl.TZ2.7: Let $f(x) = \sqrt 3 {{\rm{e}}^{2x}}\sin x + {{\rm{e}}^{2x}}\cos x$ , for $0 \le x \le \pi$ ....
• 09M.2.sl.TZ2.10e: Write down the two values of k for which the equation $f(x) = k$ has exactly two solutions.
• 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.TZ0.6b: Hence or otherwise, solve the equation $6\sin x\cos x = \frac{3}{2}$ , for...
• SPNone.2.sl.TZ0.7b: The chord [AB] divides the area of the circle in the ratio 1:7. Find the value of $\theta$ .
• 11M.1.sl.TZ1.6: Solve the equation $2\cos x = \sin 2x$ , for $0 \le x \le 3\pi$ .
• 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.2.sl.TZ2.10a: Show that the area of the window is given by $y = 4\sin \theta + 2\sin 2\theta$ .
• 11M.2.sl.TZ2.10b: Zoe wants a window to have an area of $5{\text{ }}{{\text{m}}^2}$. Find the two possible values...
• 11M.2.sl.TZ2.10c: John wants two windows which have the same area A but different values of $\theta$ . Find all...
• 13M.2.sl.TZ1.10e: In one rotation of the wheel, find the probability that a randomly selected seat is at least...
• 14M.2.sl.TZ2.6c: Solve $f(x) = 7$.
• 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...
• 17M.1.sl.TZ2.7: Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) =  - 1$, for...
• 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.1.sl.TZ0.6: Let $f(x) = 15 - {x^2}$, for $x \in \mathbb{R}$. The following diagram shows part of the...
• 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
• 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
• 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
• 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.
• 18M.2.sl.TZ1.3a: Find the area of the shaded region, in terms of θ.
• 18M.2.sl.TZ1.3b: The area of the shaded region is 12 cm2. Find the value of θ.
• 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.

3.6

• 12N.2.sl.TZ0.8a: Find the length of the chord [AB].
• 12N.2.sl.TZ0.8b: Find the area of triangle AOB.
• 12N.2.sl.TZ0.8c: Angle BOC is 2.4 radians. Find the length of arc ADC.
• 12N.2.sl.TZ0.8d: Angle BOC is 2.4 radians. Find the area of the shaded region.
• 12N.2.sl.TZ0.8e: Angle BOC is 2.4 radians. The shaded region is to be painted red. Red paint is sold in cans...
• 12M.2.sl.TZ2.1a: Find ${\rm{R\hat PQ}}$ .
• 12M.2.sl.TZ2.1b: Find PR .
• 12M.2.sl.TZ2.1c: Find the area of $\Delta {\rm{PQR}}$ .
• 12M.2.sl.TZ2.10a: Use the cosine rule to show that ${\rm{PQ}} = 2r\sin \theta$ .
• 12M.2.sl.TZ2.10b: Let l be the length of the arc PRQ . Given that $1.3{\rm{PQ}} - l = 0$ , find the value of...
• 12M.2.sl.TZ2.10c(i) and (ii): Consider the function $f(\theta ) = 2.6\sin \theta  - 2\theta$ , for...
• 12M.2.sl.TZ2.10d: Use the graph of f to find the values of $\theta$ for which $l < 1.3{\rm{PQ}}$ .
• 08N.2.sl.TZ0.6a: Find the distance the second ship will travel.
• 08N.2.sl.TZ0.6b: Find the bearing of the course taken by the second ship.
• 08N.2.sl.TZ0.8d: Hence, or otherwise, find the area of the parallelogram.
• 08M.2.sl.TZ1.2a: Find ${\rm{P}}\widehat {\rm{R}}{\rm{Q}}$ .
• 08M.2.sl.TZ1.2b: Find the area of triangle PQR.
• 08M.1.sl.TZ2.10a: Find the area of the triangle OPB, in terms of $\theta$ .
• 08M.1.sl.TZ2.10b: Explain why the area of triangle OPA is the same as the area triangle OPB.
• 12M.2.sl.TZ1.9a(i) and (ii): (i)     Show that ${p^2}(41 - 40\cos 0.7) = 36$ . (ii)    Find p .
• 12M.2.sl.TZ1.9b: Write down the length of BD.
• 12M.2.sl.TZ1.9c: Find ${\rm{A}}\widehat {\rm{D}}{\rm{B}}$ .
• 12M.2.sl.TZ1.9d(i) and (ii): (i)     Show that ${\rm{C}}\widehat {\rm{B}}{\rm{D}} = 1.29$ radians, correct to 2 decimal...
• 09N.2.sl.TZ0.8a: Find AD.
• 09N.2.sl.TZ0.8b: Find OD.
• 09N.2.sl.TZ0.8d: Find the area of region ABCD.
• 09M.1.sl.TZ1.9c: (i)     Find ${\rm{sinR}}\widehat {\rm{P}}{\rm{Q}}$ . (ii)    Hence, find the area of triangle...
• 09M.2.sl.TZ1.2a: Find AB.
• 09M.2.sl.TZ2.4a: Find the size of angle ACB.
• 09M.2.sl.TZ2.4b: Find the size of angle CAD.
• 10N.2.sl.TZ0.6a: Find ${\rm{A}}\widehat {\rm{C}}{\rm{B}}$ .
• 10N.2.sl.TZ0.6b: Find AB.
• 10M.2.sl.TZ1.8a: Use the cosine rule to show that ${\rm{AC}} = \sqrt {41 - 40\cos x}$ .
• 10M.2.sl.TZ1.8b: Use the sine rule in triangle ABC to find another expression for AC.
• 10M.2.sl.TZ1.8c: (i)     Hence, find x, giving your answer to two decimal places. (ii)    Find AC .
• 10M.2.sl.TZ1.8d(i) and (ii): (i)     Find y. (ii)    Hence, or otherwise, find the area of triangle ACD.
• 10M.2.sl.TZ2.8a: Find the size of angle AOC .
• 10M.2.sl.TZ2.8b: Hence find the area of the shaded region.
• 10M.2.sl.TZ2.8c: The area of sector OCDE is $45{\text{ c}}{{\text{m}}^2}$. Find the size of angle COE .
• 10M.2.sl.TZ2.8d: Find EF .
• SPNone.2.sl.TZ0.6: Find the height of the building.
• SPNone.2.sl.TZ0.7a: Find an expression for the area of the shaded region.
• 11N.2.sl.TZ0.4a: Find the two possible values of ${\rm{A}}\widehat {\rm{C}}{\rm{B}}$ .
• 11N.2.sl.TZ0.4b: Hence, find ${\rm{A}}\widehat {\rm{B}}{\rm{C}}$ , given that it is acute.
• 11M.2.sl.TZ1.1a: Find AC .
• 11M.2.sl.TZ1.1b: Find ${\rm{B}}\widehat {\rm{A}}{\rm{C}}$ .
• 11M.2.sl.TZ2.5a: Complete the diagram, showing clearly all the information above.
• 11M.2.sl.TZ2.5b: Find x .
• 13M.2.sl.TZ1.8a: Find AC.
• 13M.2.sl.TZ1.8b: (i)     Find $A\hat CD$ . (ii)     Hence, find $A\hat CB$ .
• 13M.2.sl.TZ2.3a: Find $x$ .
• 13M.2.sl.TZ2.3b: Find BC.
• 14M.2.sl.TZ1.1a: Find AC.
• 14M.2.sl.TZ1.1b: Find ${\rm{B\hat CA}}$.
• 14M.2.sl.TZ2.5a: Find the two possible values for $\hat A$.
• 14M.2.sl.TZ2.5b: Given that $\hat A$ is obtuse, find ${\text{BC}}$.
• 13N.2.sl.TZ0.8a: Find ${\rm{B\hat AC}}$.
• 13N.2.sl.TZ0.8b: Find ${\text{AC}}$.
• 13N.2.sl.TZ0.8c: Hence or otherwise, find the length of arc ${\text{ABC}}$.
• 13M.2.sl.TZ1.8c: Find the area of triangle ADC.
• 13M.2.sl.TZ1.8d: Hence or otherwise, find the total area of the shaded regions.
• 14N.1.sl.TZ0.7: The following diagram shows triangle $ABC$. Let...
• 14N.2.sl.TZ0.3b: Find $AB$.
• 15M.2.sl.TZ2.1a: Find $AC$.
• 15M.2.sl.TZ2.1b: Find the area of triangle $ABC$.
• 15M.2.sl.TZ2.10a: Show that the area of the square $ABCD$ is $2{r^2}(1 - \cos \theta )$.
• 15N.2.sl.TZ0.8a: Find $AC$.
• 15N.2.sl.TZ0.8b: Find the area of triangle $ABC$.
• 15N.2.sl.TZ0.8c: The area of triangle $ACD$ is half the area of triangle $ABC$. Find the possible values of...
• 15N.2.sl.TZ0.8d: Given that $\theta$ is obtuse, find $CD$.
• 16M.2.sl.TZ1.3b: Find the distance from Town A to Town C.
• 16M.2.sl.TZ1.3c: Use the sine rule to find ${\rm{A\hat CB}}$.
• 16M.2.sl.TZ1.3a: Find ${\rm{A\hat BC}}$.
• 16M.1.sl.TZ2.5a: Find ${\rm{A\hat BC}}$.
• 16M.1.sl.TZ2.5b: Find the exact area of the sector BDC.
• 16M.2.sl.TZ2.2a: Find BD.
• 16M.2.sl.TZ2.2b: Find ${\rm{D\hat BC}}$.
• 16N.1.sl.TZ0.8c: Write down an expression in terms of $\theta$ for (i)     angle ADB; (ii)     area of...
• 16N.2.sl.TZ0.3c: Find AB.
• 17M.1.sl.TZ1.3: The following diagram shows triangle PQR. Find PR.
• 17M.1.sl.TZ2.9a: Find the coordinates of A.
• 17M.1.sl.TZ2.9b.i: Find $\overrightarrow {{\text{AB}}}$;
• 17M.1.sl.TZ2.9c: Find $\cos {\rm{B\hat AC}}$.
• 17M.1.sl.TZ2.9d: Hence or otherwise, find the distance between ${P_1}$ and ${P_2}$ two seconds after they...
• 17M.1.sl.TZ2.9b.ii: Find $\left| {\overrightarrow {{\text{AB}}} } \right|$.
• 17M.2.sl.TZ1.5: The following diagram shows the chord [AB] in a circle of radius 8 cm, where...
• 17M.2.sl.TZ2.9a: Find the bearing of A from E.
• 17M.2.sl.TZ2.9b: Finds CE.
• 17M.2.sl.TZ2.9c: Find DE.
• 17M.2.sl.TZ2.9d: When the ship reaches D, it changes direction and travels directly to the island at 50 km per...
• 17N.1.sl.TZ0.4a: Show that ${\text{AC}} = 7{\text{ cm}}$.
• 17N.1.sl.TZ0.4b: The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the...
• 17N.2.sl.TZ0.1a: Find BC.
• 17N.2.sl.TZ0.1b: Find the area of triangle ABC.
• 18M.2.sl.TZ1.6: Triangle ABC has a = 8.1 cm, b = 12.3 cm and area 15 cm2. Find the largest possible perimeter of...
• 18M.2.sl.TZ2.2a: Find DB.
• 18M.2.sl.TZ2.2b: Find DC.
• 18M.2.sl.TZ2.8a.i: Find $\mathop {{\text{PQ}}}\limits^ \to$.
• 18M.2.sl.TZ2.8a.ii: Find $\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|$.
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.