Algebra MCQ & Objective Questions
Algebra is a fundamental branch of mathematics that plays a crucial role in various exams, including school assessments and competitive tests. Mastering algebraic concepts not only enhances problem-solving skills but also boosts confidence in tackling objective questions. Practicing MCQs and important questions in algebra is essential for effective exam preparation, helping students identify their strengths and weaknesses.
What You Will Practise Here
Basic algebraic operations and properties
Linear equations and inequalities
Quadratic equations and their solutions
Polynomials and factorization techniques
Functions and their graphs
Exponents and logarithms
Word problems involving algebraic expressions
Exam Relevance
Algebra is a significant topic in various examinations such as CBSE, State Boards, NEET, and JEE. Students can expect questions related to algebraic expressions, equations, and functions. Common question patterns include solving equations, simplifying expressions, and applying algebraic concepts to real-life scenarios. Understanding these patterns is vital for scoring well in both school and competitive exams.
Common Mistakes Students Make
Misinterpreting word problems and failing to set up equations correctly
Overlooking signs while simplifying expressions
Confusing the properties of exponents and logarithms
Neglecting to check solutions for extraneous roots in equations
FAQs
Question: What are some effective ways to prepare for algebra MCQs?Answer: Regular practice with objective questions, reviewing key concepts, and solving previous years' papers can significantly improve your preparation.
Question: How can I identify important algebra questions for exams?Answer: Focus on frequently tested topics in your syllabus and practice questions that cover those areas thoroughly.
Start your journey towards mastering algebra today! Solve practice MCQs to test your understanding and enhance your skills. Remember, consistent practice is the key to success in exams!
Q. Determine the coefficient of x^5 in the expansion of (3x - 4)^7.
A.
252
B.
336
C.
672
D.
840
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Solution
The coefficient of x^5 in (3x - 4)^7 is C(7, 5) * (3)^5 * (-4)^2 = 21 * 243 * 16 = 68016.
Correct Answer:
A
— 252
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Q. Find the coefficient of x^2 in the expansion of (2x - 3)^4.
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Solution
Using the binomial theorem, the coefficient of x^2 in (2x - 3)^4 is given by 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.
Correct Answer:
C
— 54
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Q. Find the coefficient of x^2 in the expansion of (3x - 2)^5.
A.
-60
B.
-90
C.
90
D.
60
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Solution
The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.
Correct Answer:
B
— -90
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^6.
A.
96
B.
144
C.
216
D.
256
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Solution
The coefficient of x^2 is given by C(6, 2)(4)^4 = 15 * 256 = 3840.
Correct Answer:
A
— 96
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Q. Find the coefficient of x^3 in the expansion of (3x - 4)^5.
A.
-540
B.
-720
C.
720
D.
540
Show solution
Solution
The coefficient of x^3 in (3x - 4)^5 is given by 5C3 * (3)^3 * (-4)^2 = 10 * 27 * 16 = -720.
Correct Answer:
B
— -720
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Q. Find the coefficient of x^3 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^3 in (x + 1)^8 is given by 8C3 = 56.
Correct Answer:
C
— 84
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Q. Find the coefficient of x^4 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^4 is C(8, 4) = 70.
Correct Answer:
A
— 70
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Q. Find the coefficient of x^5 in the expansion of (3x + 2)^6.
A.
486
B.
729
C.
729
D.
486
Show solution
Solution
The coefficient of x^5 in (3x + 2)^6 is C(6, 5)(3)^5(2)^1 = 6 * 243 * 2 = 2916.
Correct Answer:
A
— 486
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Q. Find the conjugate of the complex number z = 2 - 5i.
A.
2 + 5i
B.
2 - 5i
C.
-2 + 5i
D.
-2 - 5i
Show solution
Solution
The conjugate of z = 2 - 5i is z* = 2 + 5i.
Correct Answer:
A
— 2 + 5i
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Q. Find the value of (1 + i)².
A.
2i
B.
2
C.
0
D.
1 + 2i
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Solution
(1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.
Correct Answer:
B
— 2
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Q. Find the value of (1 + x)^6 when x = 2.
A.
64
B.
128
C.
256
D.
512
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Solution
Using the binomial theorem, (1 + 2)^6 = 3^6 = 729.
Correct Answer:
C
— 256
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Q. Find the value of (a + b)^4 when a = 2 and b = 3.
A.
81
B.
125
C.
625
D.
256
Show solution
Solution
Using the binomial theorem, (a + b)^4 = C(4, 0)a^4b^0 + C(4, 1)a^3b^1 + C(4, 2)a^2b^2 + C(4, 3)a^1b^3 + C(4, 4)a^0b^4. Substituting a = 2 and b = 3 gives 16 + 4*6 + 6*9 + 4*27 + 81 = 81.
Correct Answer:
A
— 81
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Q. Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no real roots. (2020)
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Solution
The discriminant must be less than zero: k^2 - 4*1*16 < 0 leads to k < -8.
Correct Answer:
A
— -8
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Q. Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6.
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Solution
The coefficient of x^2 in (x + k)^4 is C(4, 2) * k^2 = 6. Thus, 6k^2 = 6, giving k^2 = 1, so k = 1 or -1.
Correct Answer:
B
— 2
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Q. Find the value of k in the expansion of (x + 2)^6 such that the term containing x^4 is 240.
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Solution
The term containing x^4 is C(6,4) * (2)^2 * x^4 = 15 * 4 * x^4 = 60x^4. Setting 60 = 240 gives k = 4.
Correct Answer:
A
— 4
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Q. Find the value of the binomial coefficient C(7, 4).
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Solution
C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7*6*5)/(3*2*1) = 35.
Correct Answer:
B
— 35
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Q. Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
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Solution
Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.
Correct Answer:
C
— 30
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Q. For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is a root?
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Solution
By substituting x = 1 into the equation, we find that it satisfies the equation, hence 1 is a root.
Correct Answer:
C
— 1
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Q. For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is true about its roots?
A.
All roots are real
B.
All roots are complex
C.
One root is real and two are complex
D.
Two roots are real and one is complex
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Solution
The roots can be found using the Rational Root Theorem and synthetic division, confirming that all roots are real.
Correct Answer:
A
— All roots are real
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Q. For the equation x^2 + 2x + 1 = 0, what is the nature of the roots?
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
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Solution
The discriminant is 2^2 - 4(1)(1) = 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. For the equation x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2023)
A.
k >= 0
B.
k <= 0
C.
k >= 16
D.
k <= 16
Show solution
Solution
The discriminant must be non-negative: 4^2 - 4*1*k >= 0 leads to 16 - 4k >= 0, thus k <= 4.
Correct Answer:
C
— k >= 16
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Q. For the equation x^2 + 6x + k = 0 to have no real roots, what must be the condition on k?
A.
k < 0
B.
k > 0
C.
k = 0
D.
k ≤ 0
Show solution
Solution
The condition for no real roots is that the discriminant must be less than zero: 6^2 - 4*1*k < 0 => 36 < 4k => k > 9.
Correct Answer:
D
— k ≤ 0
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Q. For the equation x^3 - 3x^2 + 3x - 1 = 0, how many real roots does it have?
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Solution
The equation can be factored as (x-1)^3 = 0, which has one real root (x = 1) with multiplicity 3.
Correct Answer:
A
— 1
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Q. For the equation x^3 - 4x^2 + 5x - 2 = 0, which of the following is a root? (2023)
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Solution
By substituting x = 2 into the equation, we find that it satisfies the equation, thus x = 2 is a root.
Correct Answer:
B
— 2
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Q. For the equation x^3 - 6x^2 + 11x - 6 = 0, what is the product of the roots? (2019)
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Solution
The product of the roots of the cubic equation ax^3 + bx^2 + cx + d = 0 is given by -d/a. Here, d = -6 and a = 1, so the product is -(-6)/1 = 6.
Correct Answer:
A
— 6
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Q. For the equation x^3 - 6x^2 + 11x - 6 = 0, which of the following is a root?
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Solution
By substituting x = 2 into the equation, we find that 2 is a root since 2^3 - 6(2^2) + 11(2) - 6 = 0.
Correct Answer:
B
— 2
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)
A.
All real and distinct
B.
All real and equal
C.
One real and two complex
D.
All complex
Show solution
Solution
The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.
Correct Answer:
B
— All real and equal
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, what is the value of the sum of the roots? (2019)
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Solution
The sum of the roots is given by -b/a = 3/1 = 3.
Correct Answer:
B
— 3
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Q. For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?
A.
All roots are real
B.
All roots are complex
C.
One root is real
D.
Two roots are real
Show solution
Solution
The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.
Correct Answer:
A
— All roots are real
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Q. For the quadratic equation 2x^2 + 4x + 2 = 0, what is the value of the discriminant? (2020)
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Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.
Correct Answer:
A
— 0
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