DP Mathematics SL Questionbank

• Syllabus

2.7

Sub sections and their related questions

Solving equations, both graphically and analytically.

• 08M.2.sl.TZ1.4b: Solve the equation \(f(x) = 1\) .
• 08M.2.sl.TZ1.10c: There are two values of x for which the gradient of f is equal to the gradient of g. Find both...
• 08M.2.sl.TZ2.10a(i) and (ii): (i)     Find the number of taxis in the city at the end of 2005. (ii)    Find the year in...
• 08M.2.sl.TZ2.10b(i) and (ii): At the end of 2000 there were \(25600\) people in the city who used taxis. After n years the...
• 08M.2.sl.TZ2.10c(i) and (ii): Let R be the ratio of the number of people using taxis in the city to the number of taxis. The...
• 12M.2.sl.TZ1.2a(i) and (ii): (i)     Write down the coordinates of the vertex. (ii)    Hence or otherwise, express the...
• 12M.2.sl.TZ1.2b: Solve the equation \(f(x) = 0\) .
• 09M.2.sl.TZ1.6b: Find the smallest value of n for which \({S_n} > 40\) .
• 09M.2.sl.TZ2.3: Solve the equation \({{\rm{e}}^x} = 4\sin x\) , for \(0 \le x \le 2\pi \) .
• 09M.2.sl.TZ2.10f: Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of x, between \(0\) and...
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• SPNone.2.sl.TZ0.10b: Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for the data....
• 11M.2.sl.TZ1.6: Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) . Find...
• 13M.2.sl.TZ1.9b: Solve \(f(x) = 95\) .
• 13M.2.sl.TZ2.7b: The area of the shaded region is \(25\) cm2 . Find the value of \(r\) .
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 14M.1.sl.TZ1.4b: Hence, solve \({\log _3}27 + {\log _8}\frac{1}{8} - {\log _{16}}4 = {\log _4}x\).
• 14M.1.sl.TZ2.8c: The graph of \(f\) has its vertex on the \(x\)-axis. Write down the solution of \(f(x) = 0\).
• 14M.2.sl.TZ2.8c: How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?
• 14M.2.sl.TZ2.8d: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 14M.2.sl.TZ2.8e: The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function...
• 13N.1.sl.TZ0.10d: The graph of \(g\) intersects the graph of \(f'\) when \(x = q\). Find the value of \(q\).
• 14N.1.sl.TZ0.4b: Hence or otherwise, solve \(3\ln 2 - \ln 4 =  - \ln x\).
• 14N.2.sl.TZ0.4b: Solve \(f(x) = 0\).
• 14N.2.sl.TZ0.9b: Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where...
• 14N.2.sl.TZ0.9c: Find the smallest value of \(n\) for which \({v_n} > {u_n}\).
• 15M.2.sl.TZ1.7b: The graph of \(f\) is translated to the graph of \(g\) by the vector...
• 15M.2.sl.TZ2.7: Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\). The graphs of \(f\) and \(g\) intersect at two...
• 15N.2.sl.TZ0.4c: Find the least value of \(n\) such that \({S_n} > 75\,000\).
• 15N.2.sl.TZ0.9e: During which year were the number of coyotes the same as the number of foxes?
• 16M.2.sl.TZ1.2a: Solve \(f(x) = g(x)\).
• 16M.2.sl.TZ1.2b: Find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 16M.2.sl.TZ1.7a: (i)     Find the value of \(k\). (ii)     Interpret the meaning of the value of \(k\).
• 16M.2.sl.TZ1.7b: Find the least number of whole years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).
• 16M.2.sl.TZ1.9a: Find the displacement of P from O after 5 seconds.
• 16M.2.sl.TZ1.9b: Find when P is first at rest.
• 16M.2.sl.TZ1.9c: Write down the number of times P changes direction.
• 16M.2.sl.TZ1.9d: Find the acceleration of P after 3 seconds.
• 16M.2.sl.TZ1.9e: Find the maximum speed of P.
• 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
• 16M.2.sl.TZ1.10b: Show that...
• 16M.2.sl.TZ1.10c: Find \(\theta \).
• 16M.2.sl.TZ1.10d: (i)     Show that...
• 16M.1.sl.TZ2.4: Three consecutive terms of a geometric sequence are \(x - 3\), 6 and \(x + 2\). Find the...
• 16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
• 16M.2.sl.TZ2.9b: Find \(f'(x)\).
• 16M.2.sl.TZ2.9c: Write down the value of \(b\).
• 16M.2.sl.TZ2.9d: Given that \(g'(1) =  - e\), find the value of \(a\).
• 16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\) have...
• 16N.2.sl.TZ0.1c: Solve \((g \circ f)(x) = 0\).
• 16N.2.sl.TZ0.4b: Hence, find the area of the region enclosed by the graphs of \(f\) and \(g\).
• 17M.2.sl.TZ1.10a.i: Write down the value of \(q\);
• 17M.2.sl.TZ1.10a.ii: Write down the value of \(h\);
• 17M.2.sl.TZ1.10a.iii: Write down the value of \(k\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17M.2.sl.TZ2.5: Consider a geometric sequence where the first term is 768 and the second term is 576. Find the...
• 17M.2.sl.TZ2.6a: Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).
• 17M.2.sl.TZ2.6b: On the following grid, sketch the graph of \((f \circ g)(x)\), for...
• 17M.2.sl.TZ2.6c: The equation \((f \circ g)(x) = k\) has exactly two solutions, for...
• 17M.2.sl.TZ2.10a.i: Find \(q\).
• 17M.2.sl.TZ2.10a.ii: Find \(p\).
• 17M.2.sl.TZ2.10b.i: Write down the probability of drawing three blue marbles.
• 17M.2.sl.TZ2.10b.ii: Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).
• 17M.2.sl.TZ2.10b.iii: The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).
• 17M.2.sl.TZ2.B10c: Jill plays the game nine times. Find the probability that she wins exactly two prizes.
• 17M.2.sl.TZ2.10d: Grant plays the game until he wins two prizes. Find the probability that he wins his second prize...
• 17N.1.sl.TZ0.8a: Show that \(f’(1) = 1\).
• 17N.1.sl.TZ0.8b: Find the equation of \(L\) in the form \(y = ax + b\).
• 17N.1.sl.TZ0.8d: Find the area of the region enclosed by the graph of \(f\) and the line \(L\).
• 17N.1.sl.TZ0.8c: Find the \(x\)-coordinate of Q.
• 17N.2.sl.TZ0.5a: Find the value of \(p\).
• 17N.2.sl.TZ0.5b: The following diagram shows part of the graph of \(f\). The region enclosed by the graph of...
• 18M.2.sl.TZ1.1a: Find f '(x).
• 18M.2.sl.TZ1.1b: Find f "(x).
• 18M.2.sl.TZ1.1c: Solve f '(x) = f "(x).
• 18M.2.sl.TZ1.7a: Given that xk + 1 = xk + a, find a.
• 18M.2.sl.TZ1.7b: Hence find the value of n such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

• 08M.2.sl.TZ1.4b: Solve the equation \(f(x) = 1\) .
• 08M.2.sl.TZ1.10c: There are two values of x for which the gradient of f is equal to the gradient of g. Find both...
• 08M.2.sl.TZ2.10b(i) and (ii): At the end of 2000 there were \(25600\) people in the city who used taxis. After n years the...
• 09M.2.sl.TZ2.3: Solve the equation \({{\rm{e}}^x} = 4\sin x\) , for \(0 \le x \le 2\pi \) .
• 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.TZ1.9c(i), (ii) and (iii): (i)     Using your value of k , find \(f'(x)\) . (ii)    Hence, explain why f is a decreasing...
• 10M.2.sl.TZ2.5a: On the diagram above, sketch the graph of g.
• 10M.2.sl.TZ2.5b: Solve \(f(x) = g(x)\) .
• 10M.2.sl.TZ2.5c: Write down the set of values of x such that \(f(x) > g(x)\) .
• 11M.2.sl.TZ1.6: Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) . Find...
• 11M.2.sl.TZ1.9a: Identify the two points of inflexion.
• 11M.2.sl.TZ1.9b(i) and (ii): (i)     Find \(f'(x)\) . (ii)    Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .
• 11M.2.sl.TZ1.9c: Find the x-coordinate of each point of inflexion.
• 11M.2.sl.TZ1.9d: Use the second derivative to show that one of these points is a point of inflexion.
• 13M.2.sl.TZ1.9b: Solve \(f(x) = 95\) .
• 13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the value of...
• 14N.2.sl.TZ0.9c: Find the smallest value of \(n\) for which \({v_n} > {u_n}\).
• 15N.2.sl.TZ0.9e: During which year were the number of coyotes the same as the number of foxes?
• 18M.2.sl.TZ1.7a: Given that xk + 1 = xk + a, find a.
• 18M.2.sl.TZ1.7b: Hence find the value of n such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).

Solving \(a{x^2} + bx + c = 0\) , \(a \ne 0\) .

• 10M.1.sl.TZ2.6: Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .
• 11N.2.sl.TZ0.8a(i) and (ii): Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) . (i)    ...
• 11N.2.sl.TZ0.8b(i) and (ii): Consider an arithmetic sequence with n terms, with first term (\( - 36\)) and eighth term...
• 11N.2.sl.TZ0.8c: The sum of the infinite geometric sequence is equal to twice the sum of the arithmetic sequence....
• 14N.1.sl.TZ0.1b: Solve \(f(x) = 0\).
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...

The quadratic formula.

• 16M.1.sl.TZ1.7: Let \(f(x) = 3{\tan ^4}x + 2k\) and \(g(x) =  - {\tan ^4}x + 8k{\tan ^2}x + k\), for...
• 16M.1.sl.TZ2.4: Three consecutive terms of a geometric sequence are \(x - 3\), 6 and \(x + 2\). Find the...
• 16M.2.sl.TZ2.7: A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after...
• 16N.2.sl.TZ0.1c: Solve \((g \circ f)(x) = 0\).
• 17M.1.sl.TZ1.9a: Find the value of \(p\).
• 17M.1.sl.TZ1.9b: Find the value of \(a\).
• 17M.1.sl.TZ1.9c: The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).
• 17M.1.sl.TZ1.10a: Show that \(\cos \theta  = \frac{3}{4}\).

The discriminant \(\Delta = {b^2} - 4ac\) and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

• 12N.1.sl.TZ0.7: The equation \({x^2} - 3x + {k^2} = 4\) has two distinct real roots. Find the possible values of k .
• 12M.1.sl.TZ2.6: Consider the equation \({x^2} + (k - 1)x + 1 = 0\) , where k is a real number. Find the values...
• 09M.2.sl.TZ2.7a: Find the possible values of k.
• 09M.2.sl.TZ2.7b: Write down the values of k for which \({x^2} + (k - 3)x + k = 0\) has two equal real roots.
• 11N.1.sl.TZ0.7a: Find the values of k such that \(f(x) = 0\) has two equal roots.
• 11N.1.sl.TZ0.7b: Each value of k is equally likely for \( - 5 \le k \le 5\) . Find the probability that...
• 11M.1.sl.TZ1.7a: Find the value of k .
• 11M.1.sl.TZ1.7b: The line \(y = p\) intersects the graph of f . Find all possible values of p .
• 14M.1.sl.TZ1.7b: Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).
• 14M.1.sl.TZ2.8a(i): Write down the value of the discriminant.
• 14M.1.sl.TZ2.8a(ii): Hence, show that \(p = 3\).
• 13N.1.sl.TZ0.7: The equation \({x^2} + (k + 2)x + 2k = 0\) has two distinct real roots. Find the possible values...
• 15M.1.sl.TZ1.6a: Show that the discriminant of \(f(x)\) is \(100 - 4{p^2}\).
• 15M.1.sl.TZ1.6b: Find the values of \(p\) so that \(f(x) = 0\) has two equal roots.
• 15M.2.sl.TZ2.7: Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\). The graphs of \(f\) and \(g\) intersect at two...
• 16M.1.sl.TZ1.7: Let \(f(x) = 3{\tan ^4}x + 2k\) and \(g(x) =  - {\tan ^4}x + 8k{\tan ^2}x + k\), for...
• 16N.1.sl.TZ0.7: Let \(f(x) = m - \frac{1}{x}\), for \(x \ne 0\). The line \(y = x - m\) intersects the graph of...
• 17M.1.sl.TZ1.9a: Find the value of \(p\).
• 17M.1.sl.TZ1.9b: Find the value of \(a\).
• 17M.1.sl.TZ1.9c: The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).
• 18M.1.sl.TZ2.6: Let \(f\left( x \right) = p{x^2} + qx - 4p\), where p ≠ 0. Find Find the number of roots for the...

Solving exponential equations.

• 10M.2.sl.TZ2.7a: Find the value of n when \(t = 0\) .
• 10M.2.sl.TZ2.7b: Find the rate at which n is increasing when \(t = 15\) .
• 10M.2.sl.TZ2.7c: After k minutes, the rate of increase in n is greater than \(10000\) bacteria per minute. Find...
• 14N.2.sl.TZ0.9b: Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where...
• 15M.1.sl.TZ1.3b: Hence or otherwise solve \({8^{2x + 1}} = {16^{2x - 3}}\).
• 16M.1.sl.TZ1.7: Let \(f(x) = 3{\tan ^4}x + 2k\) and \(g(x) =  - {\tan ^4}x + 8k{\tan ^2}x + k\), for...
• 16M.2.sl.TZ2.8a: (i)     Find the correlation coefficient. (ii)     Write down the value of \(a\) and of \(b\).
• 16M.2.sl.TZ2.8b: Use the regression equation to estimate the price of Lina’s car, giving your answer to the...
• 16M.2.sl.TZ2.8c: Calculate the price of Lina’s car after 6 years.
• 16M.2.sl.TZ2.8d: Find the year when Lina sells her car.