Mathematics Induction and Binomial Theorem MCQS Online Preparation Sample Paper Questions with Answer
Mathematical induction and the binomial theorem are very important topics in algebra and play an essential part in solving lots of mathematical questions. Concepts from these topics are often covered in the ECAT (Engineering College Admission Test), so students need to master them. It contains a complete quantity of Mathematics Induction and Binomial Theorem MCQs Tests and Provides proof techniques with binomial expansions and their applications. Nokryan.com Take a look at the site, it is a one-stop destination for great stuff, such as sample papers, practice questions, and thorough solutions designed by professionals to assist students.
Key Concepts in Mathematical Induction and Binomial Theorem
- Mathematical Induction: A proof technique used to establish the truth of an infinite number of statements by proving a base case and an inductive step.
- Base Case: Proving the statement is true for an initial value (usually n=1).
- Inductive Step: Assuming the statement holds for 𝑛 = 𝑘 (inductive hypothesis) and proving it also holds for n=k+1.
- Binomial Theorem: A formula that provides a systematic method for expanding expressions raised to a power, such as (a+b) n = n ∑ k=0 ( k n ) a n−k b k
- Binomial Coefficients: The coefficients in the binomial expansion, represented as.
- Summation Formulas: Various summation formulas are used in mathematical induction to prove numerical series properties.
Sample MCQs on Mathematics Induction and Binomial Theorem
1. (1-3y) ⁴ =
- 1+y+y²+y³+y⁴
- 1-4y+6y²-4y³+y⁴
- 1-12y+54y²-108y³+81y⁴ correct
- 1+4y+6y²+4y³+y⁴
2. (a² -b²)³ =
- a³-3a²b+3ab²-b³
- A⁶-3a⁴b²+3a²b⁴-b⁶ correct
- a³+3a²b+3ab²+b³
- a⁶+3a⁴b²+3a²b⁴+b⁶
3. (x-1) ³ =
- x³- 3x² + 3x – 1 correct
- 1 – 3x + 3x² – x³
- x³+ x² + x +1
- x³+ 3x² + 3x + 1
4. (x+1/x) ⁴ =
- 1+x⁴+4x² + 4/x²+1/x⁴
- 1-x⁴ +4x² + 4/x² + 1/x⁴
- x⁴+4x² +6+ 4/x²+1/x⁴ correct
- x⁴-4x² -6- 4/x² – 1/x⁴
5. If a statement P(n) is true for n=1 and the truth of P(n) for n = k implies the truth of P(n) for n = k + 1, then P(n) is true for all
- Positive integers n correct
- Positive real numbers n
- Real numbers n
- Integers n
6. (1+√2) ³ =
- 7-5√2
- 5+7√2
- 5-7√2
- 7+5√2 correct
7. If n is any positive integer, then 1³ + 2³ + 3³ + … + n =
- n(n+1)/2
- n²(n+1)²/4 correct
- n(n+1)²/4
- n²(n+1)/4
8. (1-x) ³=
- 1+3x+3x²+x³
- 1-x+ x²-x³
- 1+x+x²+x³
- 1-3x+ 3x²-x³ correct
9. In the expansion of (a – 2b)³ the coefficient of b² is
- -8a
- -2a²
- -4a
- 12a correct
10. If n is any positive integer, then 1 + 2 + 3 +…+ n =
- n/n+2
- n/n+1
- n!
- n(n+1)/2 correct
11. If n is any positive integer, then 3 + 6 + 9 + …. + 3n =
- 3n(n+1)
- 2n(n+1)/3
- 3n(n+1)/4
- 3n(n+1)/2 correct
12. If n is any positive integer, then n ! 3ⁿ¯¹ is true for all
- n > 5
- n > 3
- n ≥ 5 correct
- n ≥ 3
13. If n is any positive integer, then 1/1.2+1/2.3+1/3.4 +…+1/n(n+1)=
- n!
- n/n+1 correct
- n/2(n+1)
- n/n+2
14. (1+x)⁷ =
- 1+x+x²+x³+x⁴+x⁵+x⁶+x⁷
- 1-7x+21x²+35x³+35x⁴-21x⁵+7x⁶-x⁷
- 7+7x+21x²+35x³+35x⁴+21x⁵+7x⁶+x⁷
- 1+7x+21x²+35x³+35x⁴21x⁵+7x⁶+x⁷ correct
15. IF n is any positive integer, then 2ⁿ > 2 (n+1) is true for all
- n>3 correct
- n≤3
- n≥3
- n<3
16. If a statement P(n) is true for n = m, where m is some given natural number, and the truth of P(n) for n = k > m implies the truth of P(n) for n = k + 1, then P(n) is true for all positive integers
- n ≥ m correct
- m ≥ n
- n > m
- m > n
17. In the expansion of (a + b)⁷, the 2nd term is
- a⁷
- 7a⁶b correct
- None of these
- 7ab⁶
18. If n is any positive integer, then 1/3 + 1/9 + …..+ 1/3ⁿ =
- ½(1- 1/2ⁿ)
- ½ (1-1/3ⁿ) correct
- 1/3 (1-1/3ⁿ)
- 1/3(1-1/2ⁿ)
19. IF n is any positive integer, then 1/1.3+1/3.5+1/5.7 + …+ 1/(2n – 1)(2n + 1) =
- n/2(n+1)
- n/2n+1 correct
- 2n/n+1
- n/n+2
20. If n is any positive integer, then 2¹ + 2²+ 2³ + … + 2ⁿ =
- 2(2ⁿ¯¹ -1)
- 2(2 ⁿ⁺¹ – 1)
- 2(2 ⁿ – 1) correct
- 2(3 ⁿ – 1)
21. If n is any positive integer, then 1 + 3 + 5 + … + (2n – 1) =
- n+1
- 2n+1
- n
- N² correct
22. IF n is any positive integer, then 4ⁿ > 3ⁿ + 4 is true for all
- n<2
- N≥2 correct
- n>2
- n≤2
23. (x-1/x) ³ =
- x³+x + 1/x + 1/x³
- x³-3x + 3/x – 1/x³ correct
- x³+3x+3/x + 1/x³
- none of these
24. If n is any positive integer, then 1² + 2² + 3² + ……+ n² =
- n(n+1)(2n+1)/2
- n(n+1)(2n+1)/6 correct
- n(n+1)(2n+1)/3
- (n+1)(2n+1)/6
25. (1+2x)⁴ =
- 1-4x+6x²-4x³+ x⁴
- 1+4x+6x²+4x³+x⁴
- 1-8x+24x²-32x³+16x⁴
- 1+8x+24x²+32x³+16x⁴ correct
Conclusion
Mastering mathematical induction and binomial theorem is essential to score high in the ECAT Mathematics portion. Practicing these MCQs helps in building the problem-solving skills of the students and makes them a bit more confident. Nokryan.com provides structured study notes, detailed solutions, and hundreds of practice questions are available to make sure that effective preparation is done. Keep practicing regularly, and remain focused and you will excel in your ECAT exam with flying colors!