Binomial Theorem Quiz

1.

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

A.

True
B.

False

Correct Answer
A. True

Explanation
The statement is true because when expanding the binomial (1-b)^n, the coefficients will have alternating positive and negative signs. This is because the binomial theorem states that the coefficients are given by the combination formula, which alternates between positive and negative values. Therefore, the binomial expansion of (1-b)^n will have the same coefficients as the expansion of (1+b)^n, but with alternating signs.

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5

2.

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

A.

13440
B.

3360
C.

-15360

Correct Answer
C. -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.

A.

True
B.

False

Correct Answer
B. 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

A.

150x^3
B.

140x^3
C.

91x^3

Correct Answer
B. 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.

A.

10
B.

8
C.

3

Correct Answer
B. 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.

A.

3360x^6
B.

3000x^6
C.

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.

A.

True
B.

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

A.

20412
B.

10512
C.

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

A.

435
B.

-882
C.

632

Correct Answer
B. -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.

A.

True
B.

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

A.

23
B.

40
C.

51

Correct Answer
B. 40