IB Questionbank

User interface language: English | Español

Date	May 2018	Marks available	4	Reference code	18M.2.sl.TZ1.7
Level	SL only	Paper	2	Time zone	TZ1
Command term	Find	Question number	7	Adapted from	N/A

Question

Let $f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}$ , for x > 0.

The k th maximum point on the graph of f has x-coordinate x_k where $k \in {\mathbb{Z}^ + }$ .

Given that x_k_{+ 1} = x_k + a, find a.

[4]

Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .

[4]

Markscheme

valid approach to find maxima (M1)

eg one correct value of x_k, sketch of f

any two correct consecutive values of x_k (A1)(A1)

eg x₁ = 1, x₂ = 5

a = 4 A1 N3

[4 marks]

recognizing the sequence x_1, x₂_, x_{3, …,} x_n is arithmetic (M1)

eg d = 4

correct expression for sum (A1)

eg $\frac{n}{2}\left( {2\left( 1 \right) + 4\left( {n - 1} \right)} \right)$

valid attempt to solve for n (M1)

eg graph, 2n² − n − 861 = 0

n = 21 A1 N2

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2 - Functions and equations » 2.7 » Solving equations, both graphically and analytically.

Show 78 related questions

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

Hide related questions

Question

Markscheme

Examiners report

Syllabus sections

View options