DP Mathematics SL Questionbank

• Syllabus

3.5

Sub sections and their related questions

Solving trigonometric equations in a finite interval, both graphically and analytically.

• 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 ....
• 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.

Equations leading to quadratic equations in \(\sin x\) , \(\cos x\) or \(\tan x\) .

• 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 \(0 \le \theta \le 2\pi \) .
• 09N.1.sl.TZ0.6: Solve \(\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x\) , for \(0 \le x \le 2\pi \) .
• 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.