Binomial Theorem Quiz

2.

In the expansion of (x^3-2/x^2)^10, find the coefficient of 1/x^5.

13440
3360
-15360

Correct Answer

A. -15360

Explanation

To find the coefficient of 1/x^5 in the expansion of (x^3-2/x^2)^10, we need to find the term that contains 1/x^5. This term can be obtained by choosing the x^3 term from the first factor (x^3)^10 and the (-2/x^2) term from the second factor (-2/x^2)^10. The exponent of x^3 is 10, and the exponent of (-2/x^2) is also 10. Multiplying these exponents gives us x^30 * (-2)^10 / x^20 = -2^10 * x^10 / x^20 = -1024 * x^10 / x^20 = -1024 / x^10. The coefficient of 1/x^5 is therefore -1024. However, since the question asks for the coefficient of 1/x^5, we need to multiply -1024 by x^5, resulting in -1024x^5. Therefore, the correct answer is -1024.

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8

3.

There is no difference between the power n and the expansion of the binomial theorem.

True
False

Correct Answer

A. False

Explanation

The given statement is false. There is a difference between the power n and the expansion of the binomial theorem. The binomial theorem is a formula used to expand expressions of the form (a + b)^n, where a and b are numbers and n is a positive integer. The expansion involves a series of terms with coefficients and powers of a and b. On the other hand, the power n refers to the exponent in an expression, which can be any number, not necessarily a positive integer. Therefore, the statement is incorrect.

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3

4.

Find the term x^3 in the expansion of (1+5)^2(1-2x)^6. 150x^3

150x^3
140x^3
91x^3

Correct Answer

A. 140x^3

Explanation

To find the term x^3 in the expansion of (1+5)^2(1-2x)^6, we need to use the binomial theorem. The binomial theorem states that the term with x^k in the expansion of (a+b)^n is given by the formula: (n choose k) * a^(n-k) * b^k. In this case, we have (1+5)^2(1-2x)^6, so a = 1+5 = 6, b = 1-2x, and n = 6. Plugging these values into the formula, we get: (6 choose 3) * 6^(6-3) * (1-2x)^3 = 20 * 216 * (1-2x)^3 = 4320 * (1-2x)^3. Therefore, the term x^3 is 4320 * (1-2x)^3, which is equal to 140x^3 after simplifying.

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1

5.

In the expansion of (2+3x)^n, the coefficients of x^3 and x^4 are in the ratio of 8:15. Find the value of n.

10
8
3

Correct Answer

A. 8

Explanation

The ratio of the coefficients of x^3 and x^4 in the expansion of (2+3x)^n is 8:15. This means that the coefficient of x^3 is 8 times smaller than the coefficient of x^4. In the expansion of (2+3x)^n, the coefficient of x^3 is given by the formula (n choose 3) * 2^(n-3) * (3^3), where (n choose 3) represents the binomial coefficient. Similarly, the coefficient of x^4 is given by (n choose 4) * 2^(n-4) * (3^4). By comparing these two expressions, we can set up the equation (n choose 3) * 2^(n-3) * (3^3) = 8 * (n choose 4) * 2^(n-4) * (3^4). Simplifying this equation, we can find the value of n, which is 8.

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1

6.

The special triangle with angles of 30°, 60°, and 90° is often attributed to ______, a Greek mathematician and philosopher who is considered the "father of geometry.

Correct Answer
Euclid, euclid

Explanation
Euclid's Elements, a comprehensive treatise on geometry, includes a detailed study of triangles, including the special 30-60-90 triangle. This triangle has unique properties and relationships between its sides and angles, making it a fundamental concept in geometry and trigonometry.

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4

7.

Find the 7th term of (2+x)^10.

3360x^6
3000x^6
4500x^7

Correct Answer

A. 3360x^6

Explanation

The 7th term of (2+x)^10 can be found using the binomial theorem. The general formula for finding the kth term of (a+b)^n is given by (n choose k) * a^(n-k) * b^k. In this case, a = 2, b = x, and n = 10. Plugging in these values and simplifying the expression, we get (10 choose 6) * 2^4 * x^6 = 210 * 16 * x^6 = 3360x^6. Therefore, the 7th term of (2+x)^10 is 3360x^6.

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8.

BINOMIAL , in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a+b, a-b.

True
False

Correct Answer

A. True

Explanation

The explanation for the given correct answer is not available.

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2

9.

Find the coefficient of x^6 in (1-3x)^8

20412
10512
23110

Correct Answer

A. 20412

Explanation

To find the coefficient of x^6 in (1-3x)^8, we can use the binomial theorem. The binomial theorem states that the coefficient of x^k in the expansion of (a+b)^n is given by the binomial coefficient C(n,k) multiplied by a^(n-k) multiplied by b^k.

In this case, a=1, b=-3x, n=8, and k=6. Plugging these values into the formula, we get C(8,6) * 1^(8-6) * (-3x)^6 = C(8,6) * 1 * (-3)^6 * x^6. Simplifying further, we get C(8,6) * 729 * x^6.

The binomial coefficient C(8,6) can be calculated as 8! / (6! * 2!) = 8 * 7 / 2 = 28.

Therefore, the coefficient of x^6 in (1-3x)^8 is 28 * 729 = 20412.

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1

10.

Find the coefficient of x^3 in the expansion of (3-4x)(2-x/2)^7

435
-882
632

Correct Answer

A. -882

Explanation

To find the coefficient of x^3 in the expansion of (3-4x)(2-x/2)^7, we can use the binomial theorem. The binomial theorem states that the coefficient of x^k in the expansion of (a+b)^n is given by the formula: C(n, k) * a^(n-k) * b^k. In this case, a = 3-4x, b = 2-x/2, n = 7, and k = 3. Plugging these values into the formula, we get: C(7, 3) * (3-4x)^(7-3) * (2-x/2)^3. Simplifying further, we get: C(7, 3) * (3-4x)^4 * (2-x/2)^3. The coefficient of x^3 is the coefficient of x in this expression, which is -882.

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1

11.

With the help of this binomial theorem for positive integral index indices , we can expand any power of x + y into a sum of terms forming a polynomial.

True
False

Correct Answer

A. True

Explanation

The binomial theorem states that for any positive integral index, we can expand any power of x + y into a sum of terms forming a polynomial. This means that we can express expressions like (x + y)^n as a polynomial with multiple terms. Therefore, the statement "With the help of this binomial theorem for positive integral index indices, we can expand any power of x + y into a sum of terms forming a polynomial" is true.

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1

12.

Find the coefficients of x^2 in the expansion of (1-2x)^5.23

23
40
51

Correct Answer

A. 40

Binomial Theorem Quiz

The binomial expansion of (1-b)^ is the same as that of (1+b)^n except that the coefficients have alternate + and - signs.

Quiz Preview

In the expansion of (x^3-2/x^2)^10, find the coefficient of 1/x^5.

There is no difference between the power n and the expansion of the binomial theorem.

Find the term x^3 in the expansion of (1+5)^2(1-2x)^6. 150x^3

In the expansion of (2+3x)^n, the coefficients of x^3 and x^4 are in the ratio of 8:15. Find the value of n.

The special triangle with angles of 30°, 60°, and 90° is often attributed to ______, a Greek mathematician and philosopher who is considered the "father of geometry.

Find the 7th term of (2+x)^10.

BINOMIAL , in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a+b, a-b.

Find the coefficient of x^6 in (1-3x)^8

Find the coefficient of x^3 in the expansion of (3-4x)(2-x/2)^7

With the help of this binomial theorem for positive integral index indices , we can expand any power of x + y into a sum of terms forming a polynomial.

Find the coefficients of x^2 in the expansion of (1-2x)^5.23