1  (IB/sl/2019/May/paper1tz1/q10)
[Maximum mark: 17] Consider $f(x)=\sqrt{x} \sin \left(\frac{\pi}{4} x\right)$ and $g(x)=\sqrt{x}$ for $x \geq 0$. The first time the graphs of $f$ and $g$ intersect is at $x=0$.
The set of all nonzero values that satisfy $f(x)=g(x)$ can be described as an arithmetic sequence, $u_{v}=a+b n$ where $n \geq 1$.
The following diagram shows part of the graph of $g$ reflected in the $x$axis. It also shows part of the graph of $f$ and the point $\mathrm{P}$.

2  (IB/sl/2019/May/paper1tz2/q10)
[Maximum mark: 14] Let $y=\left(x^{3}+x\right)^{\frac{3}{2}}$
Consider the functions $f(x)=\sqrt{x^{3}+x}$ and $g(x)=63 x^{3} \sqrt{x^{3}+x}$, for $x \geq 0$. The graphs of $f$ and $g$ are shown in the following diagram. The shaded region $R$ is enclosed by the graphs of $f, g$, the $y$axis and $x=1$.

3  (IB/sl/2019/May/paper2tz2/q2)
[Maximum mark: 5] Let $f(x)=42 \mathrm{e}^{\top}$. The following diagram shows part of the graph of $f$.

6  (IB/s1/2018/May/paper1tz2/q2)
[Maximum mark: 6] Let $f(x)=6 x^{2}3 x$. The graph of $f$ is shown in the following diagram.

7  (IB/sl/2018/May/paper2tz2/q3)
[Maximum mark: 5] Let $f(x)=\sin \left(\mathrm{e}^{t}\right)$ for $0 \leq x \leq 1.5$. The following diagram shows the graph of $f$.

8  (IB/s1/2017/November/Paper2/q5)
[Maximum mark: 5] Let $f(x)=6\ln \left(x^{2}+2\right)$, for $x \in \mathbb{R}$. The graph of $f$ passes through the point $(p, 4)$. where $p>0$.
The region enclosed by the graph of $f$, the $x$axis and the lines $x=p$ and $x=p$ is rotated $360^{\circ}$ about the $x$axis. Find the volume of the solid formed, 
9  (IB/s1/2017/May/paper1tz2/q10)
[Maximum mark: 17] Let $f(x)=x^{2}$. The following diagram shows part of the graph of $f$. The line $L$ is the tangent to the graph of $f$ at the point $\mathrm{A}\left(k, k^{2}\right)$, and intersects the $x$axis at point B. The point C is $(k, 0)$
The region $R$ is enclosed by $L$, the graph of $f$, and the $x$axis. This is shown in the following diagram.

10  (IB/sl/2017/May/paper2tz1/q10)
[Maximum mark: 15] Let $f(x)=\ln x$ and $g(x)=3+\ln \left(\frac{x}{2}\right)$, for $x>0$ The graph of $g$ can be obtained from the graph of $f$ by two transformations: a horizontal stretch of scale factor $q$ followed by a translation of $\left(\begin{array}{l}h \\ k\end{array}\right)$.
Let $h(x)=g(x) \times \cos (0.1 x)$, for $0 < x < 4$. The following diagram shows the graph of $h$ and the line $y=x$. The graph of $h$ intersects the graph of $h^{1}$ at two points. These points have $x$ coordinates $0.111$ and $3.31$, correct to three significant figures.

11  (IB/sl/2016/November/Paper2/q6)
[Maximum mark: 6] All lengths in this question are in metres. Let $f(x)=0.8 x^{2}+0.5$, for $0.5 \leq x \leq 0.5$. Mark uses $f(x)$ as a model to create a barrel. The region enclosed by the graph of $f$, the $x$axis, the line $x=0.5$ and the line $x=0.5$ is rotated $360^{\circ}$ about the $x$axis. This is shown in the following diagram.

12  (IB/s1/2016/May/paper1tz2/q10)
[Maximum mark: 16] The following diagram shows the graph of $f(x)=2 x \sqrt{a^{2}x^{2}}$, for $1 \leq x \leq a$, where $a>1$. The line $L$ is the tangent to the graph of $f$ at the origin, $O$. The point $P(a, b)$ lies on $L$.
The point $Q(a, 0)$ lies on the graph of $f .$ Let $R$ be the region enclosed by the graph of $f$ and the $x$axis. This information is shown in the following diagram. Let $A_{R}$ be the area of the region $R$.

13  (IB/s1/2019/May/paper1tz1/q5)
[Maximum mark: 5] The derivative of a function $f$ is given by $f^{\prime}(x)=2 \mathrm{e}^{3 x}$. The graph of $f$ passes through $\left(\frac{1}{3}, 5\right)$. Find $f(x)$. 
14  (IB/s1/2017/May/paper1tz1/q5)
[Maximum mark: 7]

15  (IB/s1/2017/May/paper1tz2/q5)
[Maximum mark: 6] Let $f^{\prime}(x)=\frac{3 x^{2}}{\left(x^{3}+1\right)^{5}}$, Given that $f(0)=1$, find $f(x)$. 
16  (IB/s1/2016/November/Paper1/q6)
[Maximum mark: 7] Let $f^{\prime}(x)=\sin ^{3}(2 x) \cos (2 x)$. Find $f(x)$, given that $f\left(\frac{\pi}{4}\right)=1$. 
17  (IB/s1/2019/November/Paper2/q10)
[Maximum mark: 14] A rocket is travelling in a straight line, with an initial velocity of $140 \mathrm{~m} \mathrm{~s}^{1}$. It accelerates to a new velocity of $500 \mathrm{~m} \mathrm{~s}^{1}$ in two stages. During the first stage its acceleration, $a \mathrm{~ms}^{2}$, after $t$ seconds is given by $a(t)=240 \sin (2 t)$. where $0 \leq t \leq k$.
The first stage continues for $k$ seconds until the velocity of the rocket reaches $375 \mathrm{~ms}^{1}$.
During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

18  (IB/s1/2019/May/paper1tz1/q7)
[Maximum mark: 7] A particle $\mathrm{P}$ starts from point $\mathrm{O}$ and moves along a straight line. The graph of its velocity, $\mathrm{vms}^{11}$ after $t$ seconds, for $0 \leq t \leq 6$, is shown in the following diagram. The graph of $v$ has $t$intercepts when $t=0,2$ and 4 . The function $s(t)$ represents the displacement of $\mathrm{P}$ from $\mathrm{O}$ after $t$ seconds. It is known that $\mathrm{P}$ travels a distance of 15 metres in the first 2 seconds. It is also known that $s(2)=s(5)$ and $\int_{2}^{4} v \mathrm{~d} t=9$

19  (IB/s1/2019/May/paper2tz2/q8)
[Maximum mark: 16] In this question distance is in centimetres and time is in seconds. Particle $\mathrm{A}$ is moving along a straight line such that its displacement from a point $\mathrm{P}$, after $t$ seconds, is given by $s_{A}=15t6 t^{3} \mathrm{e}^{0.1 .5 t}, 0 \leq t \leq 25$. This is shown in the following diagram.
Another particle, $\mathrm{B}$, moves along the same line, starting at the same time as particle $\mathrm{A}$. The velocity of particle $\mathrm{B}$ is given by $\mathrm{V}_{\mathrm{A}}=82 t, 0 \leq t \leq 25$.

20  (IB/s1/2018/November/Paper2/q4)
[Maximum mark: 7] A particle moves along a straight line so that its velocity, $v \mathrm{~m} \mathrm{~s}^{1}$, after $t$ seconds is given by $v(t)=1.4^{\prime}2.7$, for $0 \leq t \leq 5$.

21  (IB/s1/2018/May/paper2tz2/q9)
[Maximum mark: 15] A particle P moves along a straight line. The velocity $v \mathrm{~ms}^{1}$ of P after $t$ seconds is given by $v(t)=7 \cos t5 t^{\text {tow }}$, for $0 \leq t \leq 7$. The following diagram shows the graph of $v$.

22  (IB/s1/2017/November/Paper2/q9)
[Maximum mark: 14] Note: In this question, distance is in metres and time is in seconds. A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a=3 \vec{r}14 t+8$, for $0 \leq t \leq 5$,
When $t=0$, the velocity of $\mathrm{P}$ is $3 \mathrm{~ms}^{1}$.

23  (IB/s1/2017/May/paper2tz2/q7)
[Maximum mark: 6] Note: In this question, distance is in metres and time is in seconds. A particle moves along a horizontal line starting at a fixed point $\mathrm{A}$. The velocity $v$ of the particle, at time $t$, is given by $v(t)=\frac{2 t^{2}4 t}{t^{2}2 t+2}$, for $0 \leq t \leq 5$. The following diagram shows the graph of $v$ There are $t$intercepts at $(0,0)$ and $(2,0)$. Find the maximum distance of the particle from A during the time $0 \leq t \leq 5$ and justify your answer. 
24  (IB/s1/2016/November/Paper2/q9)
[Maximum mark: 14] A particle $\mathrm{P}$ starts from a point $\mathrm{A}$ and moves along a horizontal straight line, Its velocity $v \mathrm{~cm} \mathrm{~s}^{1}$ after $t$ seconds is given by $$v(t)=\left\{\begin{array}{c}2 t+2, \text { for } 0 \leq t \leq 1 \\3 \sqrt{t}+\frac{4}{t^{2}}7, \text { for } 1 \leq t \leq 12 \end{array}\right.$$ The following diagram shows the graph of $v$.
$\mathrm{P}$ is at rest when $t=1$ and $t=p$.
When $t=q$, the acceleration of $\mathrm{P}$ is zero.

25  (IB/s1/2016/May/paper2tz2/q7)
[Maximum mark: 7] A particle moves in a straight line. Its velocity $v \mathrm{~ms}^{1}$ after $t$ seconds is given by $$v=6 t6, \text { for } 0 \leq t \leq 2 $$ After $p$ seconds, the particle is $2 \mathrm{~m}$ from its initial position. Find the possible values of $p$. 
26  (IB/sl/2015/November/Paper1/q10)
[Maximum mark 15] Let $y=f(x)$, for $0.5 \leq x \leq 6.5$. The following diagram shows the graph of $f^{\prime}$, the derivative of $f .$ The graph of $f^{\prime}$ has a local maximum when $x=2$, a local minimum when $x=4$, and it crosses the $x$axis at the point $(5,0)$.
The following diagram shows the shaded regions $A, B$ and $C$. The regions are enclosed by the graph of $f^{\prime}$, the $x$axis, the $y$axis, and the line $x=6$. The area of region $A$ is 12 , the area of region $B$ is $6.75$ and the area of region $C$ is $6.75$.

28  (IB/sl/2015/November/Paper1/q3)
[Maximum mark: 6] Let $f^{\prime}(x)=6 x^{2}5$. Given that $f(2)=3$, find $f(x)$. 
29  (IB/s1/2015/May/paper1tz1/q7)
[Maximum mark: 8] Let $f(x)=\cos x$, for $0 \leq x \leq 2 \pi$. The following diagram shows the graph of $f$. There are $x$intercepts at $x=\frac{\pi}{2}, \frac{3 \pi}{2}$, The shaded region $R$ is enclosed by the graph of $f$, the line $x=b$, where $b>\frac{3 \pi}{2}$, and the $x$axis. The area of $R$ is $\left(1\frac{\sqrt{3}}{2}\right)$. Find the value of $b$. 
30  (IB/s1/2015/May/paper1tz2/q10)
[Maximum mark: 14] Consider a function $f$ with domain $3 < x < 3$. The following diagram shows the graph of $f^{\prime}$, the derivative of $f$. The graph of $f$ ' has $x$intercepts at $x=a, x=0$, and $x=d$. There is a local maximum at $x=b$ and local minima at $x=a$ and at $x=c$.

31  (IB/s1/2015/May/paper2tz1/q10)
[Maximum mark: 16] Consider a function $f$, for $0 \leq x \leq 10 .$ The following diagram shows the graph of $f^{\prime}$. the derivative of $f$. The graph of $f^{\prime}$ passes through $(2,2)$ and $(5,1)$, and has $x$intercepts at 0,4 and 6
Let $g(x)=\ln (f(x))$ and $f(2)=3$
The shaded region $A$ is enclosed by the curve, the $x$axis and the line $x=2$, and has area $0.66$ units $^{2}$. The shaded region $B$ is enclosed by the curve, the $x$axis and the line $x=5$, and has area $0.21$ units $^{2}$. Find $g(5)$ [4] 
32  (IB/sl/2015/May/paper2tz1/q5)
[Maximum mark: 6] Let $G(x)=95 \mathrm{e}^{0.02 x}+40$, for $20 \leq x \leq 200$.
Calculate the total cost for 45 guests. [3] 
33  (IB/s1/2015/November/Paper2/q6)
[Maximum mark: 6] The velocity $v \mathrm{~m} \mathrm{~s}^{1}$ of a particle after $t$ seconds is given by $$\mathrm{v}(t)=(0.3 t+0.1)^{\prime}4, \text { for } 0 \leq t \leq 5$$ The following diagram shows the graph of $v$.

Answer
[1](a) 2,10 (b) $a=6,b=8$ (c) $(154,\sqrt{154})$ (d) $$\int_0^{158}\left(\sqrt{x}\sin\left(\frac{\pi}{4}\right)x+\sqrt{x}\right) dx$$
[2](a) $\frac{3}{2}\sqrt{x^3+x}(3x^2+1)$ (b) $\frac{2}{3}(x^3+x)^{\frac{3}{2}}$ (c) $\int_0^1(63x^2\sqrt{x^3+x}\sqrt{x^3+x})\sqrt{x^3+x}dx$ (d) 6$\frac{4\sqrt{2}}{3}$
[3] (a) $x=\ln 2$ (b) volume $=3.43$
[4] area $=2$
[5] (a) $\frac{1}{2} \ln (2 x1)+C$ (b) $\frac{\pi}{2} \ln 17$
[6] (a) $2 x^{3}\frac{3 x^{2}}{2}+c$ (b) $\frac{19}{2}$
[7] (a) $x=\ln \pi$ (b) Volume $=2.50$
[8](a) $\quad p=\sqrt{e^{2}2}$ (b) Volume $=332$
[9](a)(i) $f^{1}(x)=2 x$ (ii) $2 k$ (b) Show (c) $\frac{k^{3}}{4}$ (d) $p=3$
[10](a)(i) $q=2$ (ii) $h=0$ (iii) $k=3$ (b) (i) $2.72$ (ii) $5.45$ (c) $x=0.974, y=2.27$
[11](a) $V=0.601$ (b) $\quad t=4.71$
[12](a)(i) $ y=2 a x2 a^{2}+b$ (ii) $b=2 a^{2}$ (b) Show (c) $k=\frac{3}{2}$
[13] $y=\frac{2}{3}e^{3x}+5+\frac{2}{3}e^{1}$
[14](a) $\frac{1}{2} e^{x^{2}1}+c$ (b) $f(x)=\frac{1}{2} e^{x^{2}1}+2.5$
[15] $f(x)=\frac{1}{4}\left(x^{3}+1\right)^{4}+\frac{5}{4}$
[16] $\quad f(x)=\frac{1}{8} \sin ^{4}(2 x)+\frac{7}{8}$
[17](a) $v=120\cos(2t)+260$ (b) 354 (c) 2.23s
[18](a) 9 (b) 33
[19](a) 15 (b) $t=2.47$ (c) $t=4.08$ (d) $17.7(e)(i) S_{B}(t)=8 tt^{2}+15$ (ii) $9.30$
[20](a) $\quad t=2.95$ (b) $a(2)=0.659$ (c) $5.35$
[21](a) $7$ (b) $24.7$ (c) 3 (d) $a=9.25$ (e) $63.9$
[22](a) $t=\frac{2}{3}$ (b) $\frac{2}{3}<t<4$ (c) $v=t^{3}7 t^{2}+8 t+3$ (d) $14.2$
[23] $2.28$
[24](a) $2$ (b) $p=5.22$ (c) (i) $q=1.95$ (ii) $1.76$ (d)(i) 4.45 (ii) $3.45$
[25] $p=0.423$ or $p=1.58$
[26](a) $f^{\prime}(5)=0, f^{\prime \prime}(5)>0$ (b) $\quad 2<x<4$ (c) $f(6)=2$ (d) $y=64 x380$
[28] $ f(x)=2 x^{3}5 x9$
[29] $b=\frac{5 \pi}{3}$
[30](a) $0 \leqslant x \leqslant d$ (b) (i) $x=d$ (ii) sign change (c) $8.5$
[31](a) $p=6$ (b) $f^{\prime}(2)=2$ (c) $g^{\prime}(2)=\frac{2}{3}$ (d) Verify (e) $g(5)=\ln 30.45$
[32](a) Graph (b) 3540
[33](a) $ b=4.28$ (b) $t=1.19$
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