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Answer: p = 1.0658; A ≈ 24 800 (shown); A = value of the car on 1 Jan 2001; p = yearly multiplier (about +6.6 % a year); During 2022คำตอบ: p = 1.0658; A ≈ 24 800 (shown); A = value of the car on 1 Jan 2001; p = yearly multiplier (about +6.6 % a year); During 2022
1(a) Find $p$ and $A$1(a) หา $p$ กับ $A$
A vintage car was first valued on 1 January 2001. At $t$ years after that date, its value in GBP is modelled by $V=Ap^{t}$, with constants $A$ and $p$. The car was worth GBP 32 000 on 1 January 2005 and GBP 50 000 on 1 January 2012.
(a) (i) Find $p$ to 4 decimal places. (ii) Show that $A$ is about 24 800. [4]
1(b) What $A$ and $p$ mean1(b) $A$ กับ $p$ หมายถึงอะไร
A vintage car was first valued on 1 January 2001. At $t$ years after that date, its value in GBP is modelled by $V=Ap^{t}$, with constants $A$ and $p$. The car was worth GBP 32 000 on 1 January 2005 and GBP 50 000 on 1 January 2012.
(b) Say what each constant tells you about the car: (i) $A$, (ii) $p$. [2]
1(c) When does $V$ pass GBP 100 000?1(c) $V$ เกิน GBP 100 000 เมื่อไร
A vintage car was first valued on 1 January 2001. At $t$ years after that date, its value in GBP is modelled by $V=Ap^{t}$, with constants $A$ and $p$. The car was worth GBP 32 000 on 1 January 2005 and GBP 50 000 on 1 January 2012.
(c) Use the model with $A=24\,800$ and $p=1.0658$ to find the year in which the value of the car first goes above GBP 100 000. [4]
Answer: 265 thousand; rate 18.1 ≈ 18 thousand a year; T = 12.08คำตอบ: 265 thousand; rate 18.1 ≈ 18 thousand a year; T = 12.08
2(a)(b) Start count, then growth rate2(a)(b) จำนวนตอนเริ่ม แล้วอัตราการเพิ่ม
A scientist studies bees and wasps on an island, with $t$ measured in years after the study began. The number of bees, in thousands, is modelled by $N_b=45+220e^{0.05t}$. The number of wasps, in thousands, is modelled by $N_w=10+800e^{-0.05t}$.
(a)(b) Find how many bees there were when the study began (1 mark). Then show that at precisely $t=10$ the bee population was rising by about 18 thousand a year (3 marks). [4]
2(c) When bees equal wasps2(c) เมื่อผึ้งเท่ากับต่อ
A scientist studies bees and wasps on an island, with $t$ measured in years after the study began. The number of bees, in thousands, is modelled by $N_b=45+220e^{0.05t}$. The number of wasps, in thousands, is modelled by $N_w=10+800e^{-0.05t}$.
(c) At $t=T$ the two models give equal numbers of bees and wasps. Find $T$ to 2 decimal places. [4]
Answer: Initial value £1000; V = 1000 × 0.953^t; Model £315 is close to the real £320, so the model is reliableคำตอบ: Initial value £1000; V = 1000 × 0.953^t; Model £315 is close to the real £320, so the model is reliable
3(a)(b) Initial value, then the equation3(a)(b) ราคาเริ่มต้น แล้วสมการ
A mobile phone is worth $V$ pounds at $t$ months after purchase, and the model is $V=ab^{t}$, with constants $a$ and $b$. The graph of $\log_{10}V$ against $t$ is a straight line through the points $(0, 3)$ and $(10, 2.79)$.
(a)(b) Use the two points to work out what the phone was worth at the start (2 marks). Then write the full equation for $V$ in terms of $t$, with $a$ exact and $b$ to 3 s.f. (3 marks). [5]
3(c) Is the model reliable?3(c) โมเดลเชื่อถือได้ไหม
A mobile phone is worth $V$ pounds at $t$ months after purchase, and the model is $V=ab^{t}$, with constants $a$ and $b$. The graph of $\log_{10}V$ against $t$ is a straight line through the points $(0, 3)$ and $(10, 2.79)$.
(c) Take the model $V=1000\times0.953^{t}$. At precisely 2 years after purchase the phone was worth GBP 320. Say how reliable the model is. [2]
Answer: a = 4.06, b = 9.97คำตอบ: a = 4.06, b = 9.97
4 Find $a$ and $b$4 หา $a$ กับ $b$
The variables $x$ and $y$ satisfy $y=ab^{x}$, with constants $a$ and $b$. When $\ln y$ is plotted against $x$ for two pairs of values, the points are $(1, 3.7)$ and $(2.2, 6.46)$.
Use the two points to find $a$ and $b$. [4]
Answer: n = 0.8, C = 14.41คำตอบ: n = 0.8, C = 14.41
5 Find $n$ and $C$5 หา $n$ กับ $C$
The variables $x$ and $y$ satisfy $x^{n}y^{2}=C$, with constants $n$ and $C$. The graph of $\ln y$ against $\ln x$ is a straight line through the points $(0.31, 1.21)$ and $(1.06, 0.91)$.
Find $n$, and find $C$ to 2 decimal places. [5]
Answer: k = 1/20; a = 20; 14 hoursคำตอบ: k = 1/20; a = 20; 14 hours
6(a) Gradient and intercept6(a) ความชันและจุดตัดแกน
The number of bacteria, $P$, at time $t$ hours is modelled by $P=ae^{kt}$, with constants $a$ and $k$. The graph of $\ln P$ against $t$ is a straight line with gradient $\tfrac{1}{20}$ that meets the vertical axis at $(0, 3)$.
(a) Write down $k$, and find $a$ to 2 significant figures. [3]
6(b) Time to double6(b) เวลาที่เพิ่มเป็นสองเท่า
The number of bacteria, $P$, at time $t$ hours is modelled by $P=ae^{kt}$, with constants $a$ and $k$. The graph of $\ln P$ against $t$ is a straight line with gradient $\tfrac{1}{20}$ that meets the vertical axis at $(0, 3)$.
(b) Find how many hours $P$ takes to double, to the nearest hour. [2]
Answer: y = 2ˣ through (0, 1); y = 4ˣ − 6 through (0, −5) and (log₄ 6, 0); P = (log₂ 3, 3)คำตอบ: y = 2ˣ through (0, 1); y = 4ˣ − 6 through (0, −5) and (log₄ 6, 0); P = (log₂ 3, 3)
7(a) Sketch both curves7(a) ร่างสองกราฟ
The curve with equation $y=4^{x}$ is already drawn on a pair of axes.
(a) On the same axes, sketch (i) $y=2^{x}$ and (ii) $y=4^{x}-6$. Mark the coordinates of every point where a curve meets a coordinate axis. [4]
7(b) Find the point $P$7(b) หาจุด $P$
The curve with equation $y=4^{x}$ is already drawn on a pair of axes.
(b) The curves $y=2^{x}$ and $y=4^{x}-6$ meet at the point $P$. Use algebra to find the exact coordinates of $P$. [4]
Answer: a = 2^(1/10) = 1.07177… = 1.072 (shown); t = 4(1.072)^w − 5; t = 6.35คำตอบ: a = 2^(1/10) = 1.07177… = 1.072 (shown); t = 4(1.072)^w − 5; t = 6.35
8(a) Find the base $a$8(a) หาฐาน $a$
A baby mammal is weighed over a 16-month period. Its weight, $w$ kg, is $w=\log_{a}(t+5)-\log_{a}4$ for $2\le t\le18$, where $t$ is its age in months and $a$ is a constant. Its weight was 10 kg when $t=3$.
(a) Show that $a=1.072$ to 3 decimal places. [3]
8(b) Make $t$ the subject8(b) จัดรูปให้ $t$ เป็นตัวตั้ง
A baby mammal is weighed over a 16-month period. Its weight, $w$ kg, is $w=\log_{a}(t+5)-\log_{a}4$ for $2\le t\le18$, where $t$ is its age in months and $a$ is a constant. Its weight was 10 kg when $t=3$.
(b) Take $a=1.072$. Write an equation for $t$ in terms of $w$. [3]
8(c) Find $t$ when $w=15$8(c) หา $t$ เมื่อ $w=15$
A baby mammal is weighed over a 16-month period. Its weight, $w$ kg, is $w=\log_{a}(t+5)-\log_{a}4$ for $2\le t\le18$, where $t$ is its age in months and $a$ is a constant. Its weight was 10 kg when $t=3$.
(c) Use $t=4(1.072)^{w}-5$ from part (b) to find $t$ when $w=15$. Answer to 3 significant figures. [2]
Answer: Falling curve from N = 1 towards 0, no turning points; λ = (1/2)^(1/5700) = 0.999878 (shown); 10.1 grams; 2700 yearsคำตอบ: Falling curve from N = 1 towards 0, no turning points; λ = (1/2)^(1/5700) = 0.999878 (shown); 10.1 grams; 2700 years
9(a) Sketch the decay graph9(a) ร่างกราฟการสลายตัว
A scientist dates wooden items by carbon-14. The amount of carbon-14 in an item is $N=k\lambda^{t}$, where $N$ grams is the amount now, $k$ grams was the amount when the item was made, $t$ is the number of years since it was made, and $\lambda$ is a constant with $0\lt\lambda\lt1$.
(a) Sketch the graph of $N$ against $t$ when $k=1$. [2]
9(b) Show $\lambda$ is 0.9998789(b) แสดงว่า $\lambda$ คือ 0.999878
A scientist dates wooden items by carbon-14. The amount of carbon-14 in an item is $N=k\lambda^{t}$, where $N$ grams is the amount now, $k$ grams was the amount when the item was made, $t$ is the number of years since it was made, and $\lambda$ is a constant with $0\lt\lambda\lt1$.
(b) It takes 5700 years for the amount of carbon-14 to fall to half of its starting value. Show that $\lambda=0.999878$ to 6 decimal places. [2]
9(c) Carbon-14 left in Item A9(c) คาร์บอน-14 ที่เหลือในวัตถุ A
A scientist dates wooden items by carbon-14. The amount of carbon-14 in an item is $N=k\lambda^{t}$, where $N$ grams is the amount now, $k$ grams was the amount when the item was made, $t$ is the number of years since it was made, and $\lambda$ is a constant with $0\lt\lambda\lt1$.
(c) Item A started with 15 grams of carbon-14 and is thought to be 3250 years old. Use $\lambda=0.999878$ to find, to 3 significant figures, how much carbon-14 the equation gives for it now. [2]
9(d) Age of Item B9(d) อายุของวัตถุ B
A scientist dates wooden items by carbon-14. The amount of carbon-14 in an item is $N=k\lambda^{t}$, where $N$ grams is the amount now, $k$ grams was the amount when the item was made, $t$ is the number of years since it was made, and $\lambda$ is a constant with $0\lt\lambda\lt1$.
(d) Item B started with 25 grams of carbon-14 and now holds 18 grams. Use algebra to find the age of Item B, to the nearest 100 years. [3]
Answer: 2390; N = 4000(0.98)^t; T = 13.9คำตอบ: 2390; N = 4000(0.98)^t; T = 13.9
10(a) Dormice after 6 years10(a) จำนวน dormice หลัง 6 ปี
Show every stage of your working: a solution that relies only on a calculator will not be accepted.
Dormice and voles on an island are being counted. At the start there are 2000 dormice. A model says the number of dormice goes up by 3% each year, so the year-end totals form a geometric sequence.
(a) According to the model, how many dormice are there 6 years after counting began? Answer to 3 significant figures. [2]
10(b) The model for the voles10(b) โมเดลของ voles
Show every stage of your working: a solution that relies only on a calculator will not be accepted.
Dormice and voles on an island are being counted. At the start there are 2000 dormice. A model says the number of dormice goes up by 3% each year, so the year-end totals form a geometric sequence.
(b) The voles are counted over the same period. After 4 years there were 3690 voles, and after 7 years there were 3470. The year-end totals of voles form a geometric sequence. Find the model in the form $N=ab^{t}$, where $N$ is the number of voles and $t$ is the time in years since counting began. Give $a$ and $b$ to 2 significant figures. [3]
10(c) When the numbers match10(c) เมื่อจำนวนเท่ากัน
Show every stage of your working: a solution that relies only on a calculator will not be accepted.
Dormice and voles on an island are being counted. At the start there are 2000 dormice. A model says the number of dormice goes up by 3% each year, so the year-end totals form a geometric sequence.
(c) Use the dormice model $2000\times1.03^{t}$ and the voles model $N=4000(0.98)^{t}$. At $t=T$ the two numbers are equal. Find $T$ to 1 decimal place. [3]
Same method, new numbers. Write what you know first.วิธีเดิม ตัวเลขใหม่ · จดสิ่งที่รู้ก่อน
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