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.