Matematika Wajib Kelas Xi

1. Diketahui S(n) adalah rumus dari 1 + 5 + 9 + . . . + (4n - 3) = 2n² - n Andaikan S(n) benar untuk n=k, maka ...

1 + 5 + 9 + . . . + (4k - 3) = 2n2 - n

(4k - 3) = 2k2 - k

(4k - 3) = 2n2 - n

(4n - 3) = 2k2 - k

1 + 5 + 9 + . . . + (4k - 3) = 2k2 - k

The given answer states that 1 + 5 + 9 + ... + (4k - 3) is equal to 2k^2 - k. This can be derived from the equation (4k - 3) = 2n^2 - n, where n represents any positive integer. By substituting k for n in the equation, we get (4k - 3) = 2k^2 - k. Therefore, the given answer is correct.

Explanation

The given answer states that 1 + 5 + 9 + ... + (4k - 3) is equal to 2k^2 - k. This can be derived from the equation (4k - 3) = 2n^2 - n, where n represents any positive integer. By substituting k for n in the equation, we get (4k - 3) = 2k^2 - k. Therefore, the given answer is correct.

2.

2 X 1

2 X 2

2 X 3

3 X 2

3 X 3

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Explanation

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3. Dengan induksi matematika, n(n+1) dengan n bilangan asli akan habis dibagi ...

2

3

4

5

6

The given expression n(n+1) is divisible by 2 for all natural numbers. This is because when n is even, both n and n+1 are even, and when n is odd, one of n and n+1 is even. In either case, the product n(n+1) is divisible by 2.

Explanation

The given expression n(n+1) is divisible by 2 for all natural numbers. This is because when n is even, both n and n+1 are even, and when n is odd, one of n and n+1 is even. In either case, the product n(n+1) is divisible by 2.

4. Boneka jenis A yang harga belinya Rp. 35.000,00 dijual dengan harga Rp. 59.000,00 per buah, sedangkan boneka jenis B yang harga belinya Rp. 70.000,00 dijual dengan harga Rp. 95.000,00 per buah. Seorang pedagang boneka mempunyai modal Rp. 3.500.000,00 dan kiosnya dapat menampung paling banyak 80 buah boneka akan mendapat keuntungan maksimum jika ia membeli ...

20 buah boneka jenis A dan 60 buah boneka jenis B

60 buah boneka jenis A dan 20 buah boneka jenis B

30 buah boneka jenis A dan 50 buah boneka jenis B

80 buah boneka jenis A saja

80 buah boneka jenis B saja

Based on the given information, the trader has a capital of Rp. 3,500,000. The profit obtained from selling each type of doll can be calculated by subtracting the buying price from the selling price. The profit per doll for type A is Rp. 59,000 - Rp. 35,000 = Rp. 24,000, and for type B is Rp. 95,000 - Rp. 70,000 = Rp. 25,000. To maximize the profit, the trader should buy the type of doll with the highest profit per doll. Since the profit per doll for type B is higher, the trader should buy 20 dolls of type A and 60 dolls of type B.

Explanation

Based on the given information, the trader has a capital of Rp. 3,500,000. The profit obtained from selling each type of doll can be calculated by subtracting the buying price from the selling price. The profit per doll for type A is Rp. 59,000 - Rp. 35,000 = Rp. 24,000, and for type B is Rp. 95,000 - Rp. 70,000 = Rp. 25,000. To maximize the profit, the trader should buy the type of doll with the highest profit per doll. Since the profit per doll for type B is higher, the trader should buy 20 dolls of type A and 60 dolls of type B.

5. Diketahui S(n) adalah rumus dari 6 + 12 + 18 + 24 + . . . +6n = 3(n² +n). Langkah pertama dalam pembuktian pernyataan di atas dengan induksi matematika adalah ...

S(n) benar untuk n = 0

S(n) benar untuk n = 1

S(n) benar untuk n bilangan bulat

S(n) benar untuk n bilangan rasional

S(n) benar untuk n bilangan real

The given statement is a mathematical formula that represents the sum of a series. The formula states that the sum of the series up to the nth term is equal to 3 times the sum of the squares of n and n squared. In order to prove this statement using mathematical induction, the first step is to show that the formula holds true for the base case, which is n = 0. Since the formula is valid for n = 1, it can be concluded that it is also true for n = 1.

Explanation

The given statement is a mathematical formula that represents the sum of a series. The formula states that the sum of the series up to the nth term is equal to 3 times the sum of the squares of n and n squared. In order to prove this statement using mathematical induction, the first step is to show that the formula holds true for the base case, which is n = 0. Since the formula is valid for n = 1, it can be concluded that it is also true for n = 1.

6.

(0,0)

(1,2)

(2,2)

(7,3)

(8,3)

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Explanation

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7.

1

2

3

4

5

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Explanation

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8.

I

II

III

IV

V

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9. Sebuah perusahaan pelayaran hendak mengangkut 420 mobil sedan dan 120 bus, dengan 2 kapal feri, yaitu feri jenis A dan feri jenis B. Feri A dapat mengangkut hingga 30 bus dan 30 sedan, sedangkan feri B dapat emngangkut 10 bus dan 70 sedan dalam sekali angkut. Jika biaya menggunakan sebuah feri A dan sebuah feri B masing-masing adalah Rp. 500.000,00 dan Rp. 300.000,00, biaya minimum yang dikeluarkan untuk mengangkut semua kenderaan tersebut menggunakan kedua jenis feri adalah ...

Rp. 2.750.000,00

Rp. 3.000.000,00

Rp. 3.250.000,00

Rp. 3.500.000,00

Rp. 3.750.000,00

The minimum cost can be calculated by determining the number of trips needed for each ferry type to transport all the vehicles. Since the ferry A can carry 30 buses and 30 sedans, it will require 4 trips to transport all the buses and sedans. On the other hand, ferry B can carry 10 buses and 70 sedans, so it will require 12 trips to transport all the buses and sedans. Therefore, the total cost will be 4 trips x Rp. 500,000.00 (ferry A) + 12 trips x Rp. 300,000.00 (ferry B) = Rp. 3,000,000.00.

Explanation

The minimum cost can be calculated by determining the number of trips needed for each ferry type to transport all the vehicles. Since the ferry A can carry 30 buses and 30 sedans, it will require 4 trips to transport all the buses and sedans. On the other hand, ferry B can carry 10 buses and 70 sedans, so it will require 12 trips to transport all the buses and sedans. Therefore, the total cost will be 4 trips x Rp. 500,000.00 (ferry A) + 12 trips x Rp. 300,000.00 (ferry B) = Rp. 3,000,000.00.

10.

A

B

C

D

E

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11.

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12.

(6,6) dan 2

(2,1) dan 3

(5,7) dan 3

(6,6) dan - 2

(5,7) dan - 3

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Explanation

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13.

12

14

16

18

20

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Explanation

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14. Dengan induksi matematika bentuk 4ⁿ⁺¹ - 4 habis dibagi ...

4

8

12

16

18

The given sequence consists of numbers in the form 4n+1, where n is a positive integer. We need to find a number in the sequence that is divisible by 4. By substituting different values of n, we can see that 12 is the only number in the sequence that satisfies this condition. Therefore, 12 is the correct answer.

Explanation

The given sequence consists of numbers in the form 4n+1, where n is a positive integer. We need to find a number in the sequence that is divisible by 4. By substituting different values of n, we can see that 12 is the only number in the sequence that satisfies this condition. Therefore, 12 is the correct answer.

15.

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16. Dengan induksi matematika, angka terbesar yang habis membagi n(n+1)(n+2) dengan n bilangan asli adalah ...

2

3

4

5

6

The expression n(n+1)(n+2) represents the product of three consecutive numbers. To find the largest number that divides this expression, we need to find the largest prime factor of n(n+1)(n+2). Since n is a natural number, the largest prime factor of n(n+1)(n+2) will be the largest prime number less than or equal to n+2. Therefore, the largest number that divides n(n+1)(n+2) is 6.

Explanation

The expression n(n+1)(n+2) represents the product of three consecutive numbers. To find the largest number that divides this expression, we need to find the largest prime factor of n(n+1)(n+2). Since n is a natural number, the largest prime factor of n(n+1)(n+2) will be the largest prime number less than or equal to n+2. Therefore, the largest number that divides n(n+1)(n+2) is 6.

17.

Option 4

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Explanation

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18. Diketahui 2 + 4 + 6+ 8 + . . . + 2n. Dengan induksi matematika rumus deret tersebut adalah ...

n

n2

n(n+1)

The given series is an arithmetic progression with a common difference of 2. The sum of an arithmetic progression can be found using the formula Sn = (n/2)(a + l), where Sn is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term. In this case, the first term is 2 and the last term is 2n. Substituting these values into the formula, we get Sn = (n/2)(2 + 2n) = n(n+1), which matches the given answer.

Explanation

The given series is an arithmetic progression with a common difference of 2. The sum of an arithmetic progression can be found using the formula Sn = (n/2)(a + l), where Sn is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term. In this case, the first term is 2 and the last term is 2n. Substituting these values into the formula, we get Sn = (n/2)(2 + 2n) = n(n+1), which matches the given answer.

19.

- 6

- 2

2

3

6

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20. Diketahui 1 + 3 + 5+ 7 + . . . + (2n – 1). Dengan induksi matematika rumus deret tersebut adalah ...

n

n2

The given series is an arithmetic series with a common difference of 2. The first term is 1 and the last term is (2n-1). The sum of an arithmetic series can be calculated using the formula Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term. Plugging in the given values, we get Sn = (n/2)(1 + 2n - 1) = (n/2)(2n) = n^2. Therefore, the formula for the sum of the given series is n^2.

Explanation

The given series is an arithmetic series with a common difference of 2. The first term is 1 and the last term is (2n-1). The sum of an arithmetic series can be calculated using the formula Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term. Plugging in the given values, we get Sn = (n/2)(1 + 2n - 1) = (n/2)(2n) = n^2. Therefore, the formula for the sum of the given series is n^2.

21.

2

3

4

5

6

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22. Dengan induksi matematika, rumus 1² + 2² + 3²+ . . . + n² adalah ...

Rumus yang dimaksud adalah rumus penjumlahan deret kuadrat. Dalam rumus tersebut, setiap suku deret adalah hasil dari mengkuadratkan bilangan bulat dari 1 hingga n, dan kemudian semua suku tersebut dijumlahkan.

Explanation

Rumus yang dimaksud adalah rumus penjumlahan deret kuadrat. Dalam rumus tersebut, setiap suku deret adalah hasil dari mengkuadratkan bilangan bulat dari 1 hingga n, dan kemudian semua suku tersebut dijumlahkan.

23. Diketahui 1 + 2 + 3 + . . . + n. Dengan induksi matematika, rumus deret tersebut adalah . . . Which one do you like?

n

n2

The correct answer is n(n+1)/2. This can be derived using the formula for the sum of an arithmetic series. The sum of the series 1 + 2 + 3 + ... + n can be expressed as n(n+1)/2. This formula can be proven using mathematical induction, which is a method of mathematical proof used to establish that a statement is true for all natural numbers.

Explanation

The correct answer is n(n+1)/2. This can be derived using the formula for the sum of an arithmetic series. The sum of the series 1 + 2 + 3 + ... + n can be expressed as n(n+1)/2. This formula can be proven using mathematical induction, which is a method of mathematical proof used to establish that a statement is true for all natural numbers.

24. Diketahui empat pernyataan berikut :

Nilai paling besar bagi 4x – 3y adalah 60
Nilai terkecil bagi 3y +2x adalah 60
Jumlah 2y dan 3x tidak boleh melebihi 90
Nilai bagi 3y – x lebih dari 15

Model matematika dalam bentuk sistem pertidaksamaan dari keempat pernyataan tersebut ...

The given statements can be translated into a system of inequalities as follows:

1. 4x - 3y ≤ 60 (maximum value)
2. 3y + 2x ≥ 60 (minimum value)
3. 2y + 3x ≤ 90
4. 3y - x > 15

These inequalities represent the mathematical model for the given statements.

Explanation

The given statements can be translated into a system of inequalities as follows:

1. 4x - 3y ≤ 60 (maximum value)
2. 3y + 2x ≥ 60 (minimum value)
3. 2y + 3x ≤ 90
4. 3y - x > 15

These inequalities represent the mathematical model for the given statements.

25. Seorang pedagang sepatu menjual sepatu merek A dan merek B. Harga beli sebuah sepatu merek A adalah Rp. 180.000,00 dan sebuah sepatu merek B adalah Rp. 240.000,00. Pedagang tersebut mempunyai modal sebesar Rp. 12.000.000,00 dan mempunyai tempat yang dapat menampung 60 sepatu. Misalkan x adalah banyak sepatu merek A dan y adalah banyaknya sepatu merek B. Model matematika yang sesuai adalah ...

The appropriate mathematical model for this situation is 180,000x + 240,000y ≤ 12,000,000, where x represents the number of brand A shoes and y represents the number of brand B shoes. This equation represents the constraint that the total cost of the shoes purchased must be less than or equal to the available capital of Rp. 12,000,000.

Explanation

The appropriate mathematical model for this situation is 180,000x + 240,000y ≤ 12,000,000, where x represents the number of brand A shoes and y represents the number of brand B shoes. This equation represents the constraint that the total cost of the shoes purchased must be less than or equal to the available capital of Rp. 12,000,000.