Q. If the Binomial Theorem is applied to (x + 2)^4, what is the term containing x^2?
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A.
12x^2
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B.
24x^2
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C.
36x^2
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D.
48x^2
Solution
The term containing x^2 is C(4,2) * x^2 * 2^2 = 6 * x^2 * 4 = 24x^2.
Correct Answer: B — 24x^2
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Q. If the coefficient of x^k in the expansion of (x + 1)^n is given by C(n,k), what does C(n,k) represent?
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A.
The number of ways to choose k items from n.
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B.
The total number of terms in the expansion.
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C.
The sum of the coefficients.
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D.
The product of the coefficients.
Solution
C(n,k) represents the number of ways to choose k items from n, which corresponds to the coefficient of x^k.
Correct Answer: A — The number of ways to choose k items from n.
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Q. If the expansion of (x + y)^n contains a term with x^4y^3, what can be inferred about n?
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A.
n must be 7.
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B.
n must be greater than 7.
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C.
n must be less than 7.
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D.
n can be any integer.
Solution
The sum of the exponents in the term x^4y^3 is 4 + 3 = 7, hence n must be 7.
Correct Answer: A — n must be 7.
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Q. In the context of the Binomial Theorem, which of the following statements is true?
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A.
The coefficients in the expansion are always positive.
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B.
The Binomial Theorem applies only to integers.
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C.
The expansion of (a + b)^n has n + 1 terms.
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D.
The theorem can only be applied when n is even.
Solution
The expansion of (a + b)^n indeed has n + 1 terms, regardless of whether n is even or odd.
Correct Answer: C — The expansion of (a + b)^n has n + 1 terms.
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Q. In the expansion of (2x - 3y)^5, what is the sign of the term containing x^3y^2?
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A.
Positive
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B.
Negative
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C.
Zero
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D.
Indeterminate
Solution
The term containing x^3y^2 will have a negative sign due to the -3y factor raised to an even power.
Correct Answer: B — Negative
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Q. In the expansion of (a + b)^6, which term will contain a^2b^4?
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A.
The 3rd term
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B.
The 4th term
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C.
The 5th term
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D.
The 6th term
Solution
The term containing a^2b^4 corresponds to C(6,2) * a^2 * b^4, which is the 4th term in the expansion.
Correct Answer: B — The 4th term
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Q. What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?
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A.
-6720
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B.
6720
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C.
13440
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D.
-13440
Solution
The coefficient is C(8,5) * (3^5) * (-2)^3 = 56 * 243 * (-8) = -6720.
Correct Answer: A — -6720
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Q. Which of the following best describes the Binomial Theorem?
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A.
A method for solving quadratic equations.
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B.
A formula for expanding powers of binomials.
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C.
A technique for finding limits.
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D.
A principle in calculus.
Solution
The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n.
Correct Answer: B — A formula for expanding powers of binomials.
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Q. Which of the following expressions represents the coefficient of x^3 in the expansion of (2x + 3)^5?
Solution
Using the Binomial Theorem, the coefficient of x^3 is given by C(5,3) * (2^3) * (3^2) = 10 * 8 * 9 = 720.
Correct Answer: C — 90
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Q. Which of the following is a correct application of the Binomial Theorem?
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A.
Finding the roots of a polynomial.
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B.
Calculating the area under a curve.
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C.
Expanding (x + 1)^n for any integer n.
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D.
Solving differential equations.
Solution
The Binomial Theorem is specifically used for expanding expressions of the form (x + y)^n.
Correct Answer: C — Expanding (x + 1)^n for any integer n.
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