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IB Mathematics AI SL - Popular Quizzes

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Question 1

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easy

[Maximum mark: 6]

Let Q=(sin2x+b)(2sinx1)a24tanxQ = \dfrac{(\sin 2x + b)(2\sin x - 1)}{a^2 - 4\tan x}, where x=45°x = \ang{45}, a=18a = 18 and b=2b = \sqrt{2}.

  1. Find the exact value of QQ. [2]

  2. Give your answer to part (a) correct to

    1. three decimal places;

    2. three significant figures. [2]

  3. Calculate the percentage error if QQ is given to three decimal places. [2]

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Question 2

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easy

[Maximum mark: 6]

After solving a problem, John has an exact answer of z=0.1475z = 0.1475.

  1. Write down the exact value of zz in the form a×10ka\times10^k, where 1a<10,kZ1 \leq a < 10, k\in \mathbb{Z}.[2]

  2. State the value of zz given correct to 22 significant figures. [1]

  3. Calculate the percentage error if zz is given correct to 22 significant figures. [3]

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Question 3

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easy

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits 40004000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 66 %, compounded semi-annually.

  1. Find the amount of interest that Mia will earn over the next 2.52.5 years. [3]

Ella also deposits AUD into a bank account. Her bank pays a nominal annual interest\text{interest} rate of 44 %, compounded monthly. In 2.52.5 years, the total amount in Ella's account will be 40004000 AUD.

  1. Find the amount that Ella deposits in the bank account. [3]

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Question 4

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easy

[Maximum mark: 6]

The table shows the first four terms of three sequences: unu_n, vnv_n, and wnw_n.

c39694c1cf7513ffce115791e6b0f1c54c230963.svg

  1. State which sequence is

    1. arithmetic;

    2. geometric. [2]

  2. Find the sum of the first 5050 terms of the arithmetic sequence. [2]

  3. Find the exact value of the 1313th term of the geometric sequence. [2]

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Question 5

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easy

[Maximum mark: 6]

On the first day of September, 20192019, Gloria planted 55 flowers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowers she is going to plant in the last day of September is 6363.

  1. Find the common difference of the sequence. [2]

  2. Find the total number of flowers Gloria is going to plant during September.[2]

  3. Gloria estimated she would plant 10001000 flowers in the month of September. Calculate the percentage error in Gloria's estimate. [2]

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Question 6

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easy

[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

Benjamin spends € 3200032\hspace{0.15em}000 buying bitcoin mining hardware for his cryptocurrency mining\text{mining} business. The hardware depreciates by 1616 % each year.

  1. Find the value of the hardware after two years. [3]

  2. Find the number of years it will take for the hardware to be worth less than 8000\text{\euro\hspace{0.05em}\(8000\)}. [3]

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Question 7

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easy

[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

A population of goats on an island starts at 232232. The population is expected
to increase by 1515 % each year.

  1. Find the expected population size after:

    1. 1010 years;

    2. 2020 years. [4]

  2. Find the number of years it will take for the population to reach 1500015\hspace{0.15em}000. [2]

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Question 8

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[Maximum mark: 7]

Two college students, David and Lisa, decide to invest money they have saved from working part-time jobs. David's investment strategy results in an increase of his investment amount by $1000\$ 1\,000 each year. Lisa's investment strategy results in her investment amount increasing by 5%5 \% each year.

At the start of the second year of investing, David's total investment amount is $21000\$21\,000 and Lisa's is $11655\$11\,655.

  1. Calculate
    1. the original amount David invested.

    2. the original amount Lisa invested.[4]

During a certain year, nn, Lisa's investment amount becomes larger than David's amount for the first time.

  1. Find the value of nn. [3]

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Question 9

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medium

[Maximum mark: 6]

The fifth term, u5u_5, of an arithmetic sequence is 2525. The eleventh term, u11u_{11}, of the same sequence is 4949.

  1. Find dd, the common difference of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Find S100S_{100}, the sum of the first 100100 terms of the sequence. [2]

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Question 10

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medium

[Maximum mark: 6]

Greg has saved 20002000 British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of 8%\text{\(8\)\hspace{0.05em}\%}, compounded monthly, for two years.

  1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.

  1. Calculate the nominal annual interest rate the second bank offers. [3]

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Question 11

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medium

[Maximum mark: 15]

Charles has a New Years Resolution that he wants to be able to complete 100100 pushups in one go without a break. He sets out a training regime whereby he completes 2020 pushups on the first day, then adds 55 pushups each day thereafter.

  1. Write down the number of pushups Charles completes
    1. on the 44th training day;

    2. on the nnth training day. [3]

On the kkth training day Charles completes 100100 pushups for the first time.

  1. Find the value of kk. [2]

  2. Calculate the total number of pushups Charles completes on the first 1010 training days. [4]

Charles is also working on improving his long distance swimming in preparation for an Iron Man event in 1212 weeks time. He swims a total of 1000010\hspace{0.15em}000 metres in the first week, and plans to increase this by 1010 % each week up until the event.

  1. Find the distance Charles swims in the 66th week of training. [3]

  2. Calculate the total distance Charles swims until the event. [3]

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Question 12

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medium

[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

Michelle takes out a loan of $12000\$12\hspace{0.15em}000. The unpaid balance on the loan has an interest rate of 4.34.3 % per year, compounded annually.

  1. The loan is to be repaid in payments of $1500\$1500 made at the end of each year.

    1. Find the number of years it will take to repay the loan.

    2. Calculate the total amount that has been paid in amortising the loan.[3]

  2. The loan is to be amortised over 55 years.

    1. Find the annual payment made at the end of each year.

    2. Calculate the total amount that has been paid in amortising the loan.[3]

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Question 13

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medium

[Maximum mark: 6]

The first term of an arithmetic sequence is 2424 and the common difference is 1616.

  1. Find the value of the 6262 nd term of the sequence. [2]

The first term of a geometric sequence is 88. The 44th term of the geometric sequence is equal to the 1313th term of the arithmetic sequence given above.

  1. Write down an equation using this information. [2]

  2. Calculate the common ratio of the geometric sequence. [2]

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Question 14

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medium

[Maximum mark: 6]

Melinda has $300000\$300\hspace{0.15em}000 in a private foundation. Each year she donates 10%10\hspace{0.05em}\% of the money remaining in her private foundation to charity.

  1. Find the maximum number of years Melinda can donate to charity while keeping at least $100000\$100\hspace{0.15em}000 in the private foundation. [3]

Bill invests $400000\$400\hspace{0.15em}000 in a bank account that pays a nominal interest rate of 44 %, compounded quarterly, for ten years.

  1. Calculate the value of Bill's investment at the end of this time. Give your answer correct to the nearest dollar. [3]

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Question 15

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medium

[Maximum mark: 14]

Bruce goes into a car dealership to purchase a new vehicle. The one he wants to buy costs $16000\$16\hspace{0.15em}000, however he doesn't have that much money in his bank. The salesman offers him a financing option of a 3030 % deposit followed by 1212 monthly payments of $1150\$1150.

  1. Find the amount of the deposit. [1]

  2. Calculate the total cost of the loan under this financing option. [2]

Bruce's father generously offers him an interest free loan of $16000\$16\hspace{0.15em}000 to buy the car to avoid the expensive loan repayments. They agree that Bruce will repay the loan by paying his father $x\$\hspace{0.05em}x in the first month and $y\$\hspace{0.05em}y every following month until the $16000\$16\hspace{0.15em}000 is repaid.

The total amount Bruce's father receives after 1212 months is $5200\$5200. This can be expressed by the equation x+11y=5200x + 11y = 5200. The total amount that Bruce's father receives after 2424 months is $10600\$10\hspace{0.15em}600.

  1. Write down a second equation involving xx and yy. [1]

  2. Determine the value of xx and the value of yy. [2]

  3. Calculate the number of months it will take Bruce's father to receive
    the $16000\$16\hspace{0.15em}000. [3]

Bruce decides to buy a cheaper car for $12000\$12\hspace{0.15em}000 and invest the remaining $4000\$4000. He is considering two investment options over four years.

Option A: Compound interest at an annual rate of 6.56.5 %.

Option B: Compound interest at a nominal annual rate of 66 %, compounded monthly.

Express each answer in part (f) to the nearest dollar.

  1. Calculate the value of each investment option after four years.

    1. Option A.

    2. Option B. [5]

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Question 16

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hard

[Maximum mark: 7]

Jenni is conducting an experiment with a spring and has attached a mass so that it will oscillate up and down.

She is measuring the yy-coordinate of the centre of the mass.

At the start of the experiment the mass is at rest with its centre being at the point (0,0)(0, 0).

She gives the mass a nudge upwards in the positive yy-direction. She makes her first measurement of (0,37.5)(0, 37.5) when the centre of the mass is at the first maximum point (n=1n=1). The units of the yy-coordinate are in millimetres.

The mass then moves downwards passing the xx-axis and reaching its first minimum point (n=2n=2). Jenni makes her second measurement of the yy-coordinate of the centre of a the mass as (0,a)(0, a).

The mass then moves up past the xx-axis to the next maximum point (n=3n=3) and Jenni makes her third measurement of (0,24)(0, 24).

The diagram below shows how the mass moves up and down until Jenni makes her 33rd measurement.

springs

Jenni notices that the yy-coordinates of the three measurements 37.5,  a,  2437.5,\; a,\; 24 form a geometric sequence.

  1. Find aa. [2]

The spring continues to oscillate up and down with Jenni measuring the yy-coordinate in the same way as described.

The results continue to form a geometric sequence.

  1. Find the 66th term in the sequence. Give your answer to 3 decimal places. [2]

  2. Show that the total distance travelled in the yy-direction by the mass when the 66th measurement is made is 264.408264.408 mm. [3]

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Question 17

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hard

[Maximum mark: 7]

The half-life, TT, in days, of a radioactive isotope can be modelled by the function

T(k)=ln0.5ln(1k100),0<k<100,\begin{aligned} T(k) &= \dfrac{\ln 0.5}{\ln\left(1 - \frac{k}{100}\right)}, \hspace{0.5em} 0 < k < 100,\end{aligned}

where kk is the decay rate, in percent, per day of the isotope.

  1. The decay rate of Gold-196196 is 6.26.2 % per day. Find its half-life.[2]

The half-life of Phosphorus-3232 (P-3232) is 14.314.3 days. A sample containing 120120 grams of P-3232 is produced and stored in a biochemistry laboratory.

  1. Find the decay rate per day of P-3232. [2]

  2. Find the amount of P-3232 left in the sample after:

    1. 14.314.3 days;

    2. 5050 days. [3]

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Question 18

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hard

[Maximum mark: 19]

On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in 2525 seconds. Each lap Eddy cycles takes him 1.61.6 seconds longer than the previous lap.

  1. Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]

Eddy cycles his last lap in 55.455.4 seconds.

  1. Find how many laps he has cycled on Wednesday. [3]

  2. Find the total time, in minutes, cycled by Eddy on Wednesday. [4]

On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in 2525 seconds. Each lap Mario cycles takes him 1.051.05 times as long as his previous lap.

  1. Find the time, in seconds, Mario takes to cycle his fifth lap. [3]

  2. Find the total time, in minutes, Mario takes to cycle his first ten laps. [3]

Each lap Eddy cycles again takes him 1.61.6 seconds longer that his previous lap.
After a certain number of laps Eddy takes less time per lap than Mario.

  1. Find the number of the lap when this happens. [3]

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Question 19

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hard

[Maximum mark: 12]

Lily and Eva both receive 5000050\hspace{0.15em}000 Australian dollars (AUD) on their 1818th birthday. Lily deposits her 5000050\hspace{0.15em}000 AUD into a bank account. The bank pays an annual interest rate of 55 %, compounded yearly. Eva invests her 5000050\hspace{0.15em}000 AUD into a high-yield mutual fund that returns a fixed amount of 30003000 AUD per year.

  1. Calculate:

    1. the amount in Lily's bank account at the end of the first year;

    2. the total amount of Eva's funds at the end of the first year. [2]

  2. Write down an expression for:

    1. the amount in Lily's bank account at the end of the nnth year;

    2. the total amount of Eva's funds at the end of the nnth year. [4]

  3. Calculate the year in which the amount in Lily's bank account becomes
    greater than the amount in Eva's fund. [2]

  4. Calculate:

    1. the interest amount that Lily earns if invested for 1212 years, giving your answer correct to two decimal places;

    2. the amount of funds that Eva earns for her investment if invested for 1212 years. [4]

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Question 20

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hard

[Maximum mark: 15]

Consider the sequence u1,u2,u3,,un,u_1,\, u_2,\, u_3,\, \dots,\, u_n,\, \dots where

u1=860,u2=980,u3=1100,u4=1220.\begin{aligned} u_1 = 860,\hspace{0.3em} u_2 = 980,\hspace{0.3em} u_3 = 1100,\hspace{0.3em} u_4 = 1220.\end{aligned}

The sequence continues in the same manner.

  1. Find the value of u50u_{50}. [3]

  2. Find the sum of the first 1010 terms of the sequence. [3]

Now consider the sequence v1,v2,v3,,vn,v_1,\, v_2,\, v_3,\, \dots,\, v_n,\, \dots where

v1=2,v2=4,v3=8,v4=16.\begin{aligned} v_1 = 2,\hspace{0.3em} v_2 = 4,\hspace{0.3em} v_3 = 8,\hspace{0.3em} v_4 = 16.\end{aligned}

This sequence continues in the same manner.

  1. Find the exact value of v13v_{13}. [3]

  2. Find the sum of the first 1010 terms of this sequence. [3]

kk is the smallest value of nn for which vnv_n is greater than unu_n.

  1. Calculate the value of kk. [3]

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