Binomial Theorem Quiz

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| By Yuheng222
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Yuheng222
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Quizzes Created: 1 | Total Attempts: 2,534
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  • 1/12 Questions

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

    • True
    • False
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About This Quiz

The Binomial Theorem is a quick way of expanding a binomial expression that has been raised to some power. Most students have issues when it comes to working out these problems, having some practice helps. Take up the Binomial theorem quiz below to sharpen your math skills. All the best!

Binomial Theorem Quiz - Quiz

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  • 2. 

    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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  • 3. 

    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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  • 4. 

    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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  • 5. 

    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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  • 6. 

    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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  • 7. 

    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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  • 8. 

    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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  • 9. 

    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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  • 10. 

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

    • 23

    • 40

    • 51

    Correct Answer
    A. 40
  • 11. 

    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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  • 12. 

    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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  • Jan 26, 2025
    Quiz Edited by
    ProProfs Editorial Team
  • Feb 11, 2011
    Quiz Created by
    Yuheng222
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