INDUCTION-QUIZ-2

MATHEMATICS INDUCTION QUIZ-2

Dear Readers,

As per analysis for previous years, it has been observed that students preparing for JEE MAINS find Mathematics out of all the sections to be complex to handle and the majority of them are not able to comprehend the reason behind it. This problem arises especially because these aspirants appearing for the examination are more inclined to have a keen interest in Mathematics due to their ENGINEERING background. 

Furthermore, sections such as Mathematics are dominantly based on theories, laws, numerical in comparison to a section of Engineering which is more of fact-based, Physics, and includes substantial explanations. By using the table given below, you easily and directly access to the topics and respective links of MCQs. Moreover, to make learning smooth and efficient, all the questions come with their supportive solutions to make utilization of time even more productive. Students will be covered for all their studies as the topics are available from basics to even the most advanced. 

Q1. 3+13+29+51+79+⋯ to n terms =

•  2n²+7n³
•  n²+5n³
•  n³+2n²
•  None of these

Solution
Clearly, n³+2n² gives the sum of the series for n=1,2,3 etc.

Q2.

•  An integer
•  A natural number
•  A positive fraction
•  None of these

Q3.  The nth terms of the series 3+7+13+21+⋯ is ∑y=50,∑xy=220,∑x² =200,∑y² =262,n=10 is

•  4n-1
•  n²+2n
•  n²+n+1
•  n²+2

Solution
Putting n=1,2,3…, we observe that 4n-1 is the nth term

Q4. 10ⁿ+3(4ⁿ⁺² )+5 is divisible by (n∈N)

•  7
•  5
•  9
•  17

Solution
For n=1,10ⁿ+3∙4ⁿ⁺²+5 =10+3∙4³+5=207which is divisible by 9. ∴ By induction, the result is divisible by 9.

Q5.If n∈N , then 11ⁿ⁺²+12²ⁿ⁺¹ is divisible by

•  113
•  123
•  133
•  None of these

 On putting n=1 in 11ⁿ⁺²+12²ⁿ⁺¹, we get 11¹⁺²+12^2×1+1=11³+12³=3059 Which is divisible by 133

Q6. If P(n):3ⁿ< n!,n∈N, then P(n) is true

•  For n≥6
•  For n≥7
•  For n≥3
•  For all n

Solution

Q7.Matrix A is such that A²=2A-I where I is the identity matrix, then for n≥2,Aⁿ is equal to

• n A-(n-1)I
•  n A-I
•  2^n-1 A-(n-1)I
•  2^n-1 A-I

Solution
As we have A²=2A-I 

 ⟹ A² A=(2A-I)A=2A²-IA

 ⟹ A³=2(2A-I)-IA=3A-2I 

 Similarly, A⁴=4A-3I 

 A⁵=5A-AI 

 Aⁿ=nA-(n-1)I

Q8.If n∈N, then n(n²-1) is divisible by

•  6
•  16
•  36
•  24

Solution
We have, n(n²-1)=(n-1)(n+1), which is product of three consecutive natural numbers and hence divisible by 6

Q9.For each n∈N, 3²ⁿ-1 is divisible by

•  8
•  16
•  32
•  None of these

Solution

Q10. Let S(k)=1+3+5….+(2k-1)=3+k². Then, which of the following is true?

•  S(1) is correct
•  S(k)⟹S(k+1)
•  S(k)⇏S(k+1)
•  Principle of mathematical induction can be used to prove the formula

Solution