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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
prove that `7 + 77 + 777 +...... + 777........._(n-digits) 7 = 7/81 (10^(n+1) - 9n - 10)` for all `n in N`A. a=1,b=9,c=9B. a=1,b=9,c=10C. a=1,b=-9,c=-10D. a=1,b=-9,c=-10 |
| Answer» Correct Answer - B | |
| 2. |
Statement-1: `cos10^(@)+cos20^(@)+…..+cos170^(@)=0` Statement-2: `cos alpha+cos(alpha+beta)+....+cos(alpha+(n-1)beta)=(cos(alpha+((n-1)beta)/(2))sin((nbeta)/(2)))/(sin((beta)/(2))), beta ne 2npi.` |
| Answer» Correct Answer - A | |
| 3. |
Statement-1: For every natural number `n ge 2, (1)/(sqrt1)+(1)/(sqrt2)+…..(1)/(sqrtn) gt sqrtn` Statement-2: For every natural number `n ge 2, sqrt(n(n+1) lt n+1` |
| Answer» Correct Answer - A | |
| 4. |
Statement-1: `sin10^(@)+sin120^(@)+....+sin350^(@)=0` Statement-2: `sin alpha+sin (alpha+beta)+....+sin(alpha+(n-1)beta)=(sin(alpha+((n-1)beta)/(2))sin((nbeta)/(2)))/(sin((beta)/(2))), beta ne 2npi.` |
| Answer» Correct Answer - A | |
| 5. |
Statement-1: The sum of n terms of the series `a+(a+d)+(a+2d)+...(a+(n-1)d)=(n)/(2)[2n+(n-1)d]` Statement-2:- Mathematical induction is valid only for natural numbers. |
| Answer» Correct Answer - A | |
| 6. |
Most of the formulae are verified on the basis of induction method by putting n=1,2,…. `27^(n)-8^(n)` is divisible by `("for "n in N)`.A. `3^(n)-2^(n)`B. `3^(n)+2^(n)`C. `4^(n)-3^(n)`D. `4^(n)+3^(n)` |
| Answer» Correct Answer - A | |
| 7. |
Statement-1: sin`(pi+theta)=-sin theta` Statement-2: `sin(npi+theta)=(-1)^(n) sin theta`, `n in N`. |
| Answer» Correct Answer - A | |
| 8. |
Statement-1: `n(n+1)(n+2)` is always divisible by 6 for all `n in N`. Statement-2: The product of any two consecutive natural number is divisible by 2. |
| Answer» Correct Answer - B | |
| 9. |
Statement-1: `7^(n)-3^(n)` is divisible by 4. Statement-2: `7^(n)=(4+3)^(n)`. |
| Answer» Correct Answer - A | |
| 10. |
Statement-1: `1^(2)+2^(2)+....+n^(2)=(n(n+1)(2n+1))/(6)"for all "n in N` Statement-2: `1+2+3....+n=(n(n+1))/(2),"for all"n in N` |
| Answer» Correct Answer - A | |
| 11. |
`5^(n)-1` is always divisible by `(n in N)`A. 3B. 4C. 5D. 6 |
| Answer» Correct Answer - B | |
| 12. |
For all `n( gt 1) in N`, by using mathematical induction or otherwise `1+(1)/(2)+(1)/(3)+....+(1)/(n)` in its lowest form isA. Odd integerB. Even integerC. `("Odd integer")/("Even integer")`D. `("Even integer")/("Odd integer")` |
| Answer» Correct Answer - C | |
| 13. |
Mathematical induction is a tool or technique which is use to prove a proposiiton about allA. NumberB. IntegersC. Whole numberD. None of these |
| Answer» Correct Answer - D | |
| 14. |
`2^(n) lt n!` holds forA. All nB. `n gt 1`C. `n gt 3`D. `n gt 4` |
| Answer» Correct Answer - D | |
| 15. |
Most of the formulae are verified on the basis of induction method by putting n=1,2,…. Which of the following is true?A. `(1)/(1.2)+(1)/(2.3)+...+(1)/(n(n_1))=(2n)/(n+1)`B. `(1)/(1.2)+(1)/(2.3)+....+(1)/(n(n+1))=(3n)/(n+1)`C. `(1)/(1.2)+(1)/(2.3)+....+(1)/(n(n+1))=(4n)/(n+1)`D. `(1)/(1.2)+(1)/(2.3)+....+(1)/(n(n+1))=(n)/(n+1)` |
| Answer» Correct Answer - D | |
| 16. |
Choose the propagation among the following that is true for all `n in N`A. `tan^(-1)((1)/(3))+tan^(-1)((1)/(7))+...+tan^(_1)((1)/(n^(2)+n+1))=tan^(-1)((3n-1)/(3n+3))`B. `tan^(-1)((1)/(3))+tan^(-1)((1)/(7))+....+"tan"^(-1)((1)/(n^(2)+n+1))=tan^(-1)((3n-1)/(4n+2))`C. `tan^(-1)((1)/(3))+tan^(-1)((1)/(7))+....+"tan"^(-1)((1)/(n^(2)+n+1))=tan^(-1)((n)/(n+2))`D. `tan^(-1)((1)/(3))+tan^(-1)((1)/(7))+....+"tan"^(-1)((1)/(n^(2)+n+1))=tan^(-1)((2n-1)/(3n))` |
| Answer» Correct Answer - C | |
| 17. |
Statement-1: 1 is a natual number. Statement-2: `(n^(5))/(5)+(n^(3))/(3)+(6n)/(15)` is a natural number for `n in N`. |
| Answer» Correct Answer - C | |
| 18. |
`2^(n) gt 2n+1` for all natual number n is for `n>2`A. TrueB. FalseC. Cannot be determinedD. Depends on n. |
| Answer» Correct Answer - B | |
| 19. |
`3^(2n)-1` is divisible by 8, for all natural numbers n.A. 3B. 5C. 6D. 8 |
| Answer» Correct Answer - D | |
| 20. |
Prove by using the principle of mathematical induction that for all `n in N, 10^(n)+3.4^(n+2)+5` is divisible by 9 |
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Answer» Let f(n)=`10^(n)+3. 4^(n+2)+5` Let P(n): f(n) is divisible by 9. f(1)=`10^(1)+3.4^(1+2)+5` =10+192+5=207, which is divisible by 9. `therefore P(1)` is true. (a) `Rightarrow f(k)=10^(k)+3.4^(k+2)+5=9m, k in N` [where m is any integer] To prove P(k+1) is true, i.e. f(k+1) is divisible by 9. Now, f(k+1)=`10^(k+1)+3.4^(k+1)+2=5` `=10^(k).10+3.4^(k+3)+5` `=90m-18.4^(k+2)-5)` which is divisible by 9. Hence P(k+1) is true whenever P(k) is true. .....(b) From (a) and (b), by the principle of mathematical induction it follows that P(n) is true for all `n in NN`. |
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| 21. |
Show that P(n): `3^(n)` is less than 15 is true for `n ne 2` |
| Answer» From P(n) `Rightarrow P(n+1)` we cannot say that P(n) is true for any. For the principleof mathematical induction it should have been give that `P(n) Rightarrow P(n+1)` and P(n) is true for some fixed +ve integer say n=m | |
| 22. |
Which of the following is true for `n in N`?A. `(1)/(2)+(1)/(4)+(1)/(8)....+(1)/(2^(n))lt 1`B. `(1)/(n+1)+(1)/(n+2)+....+(1)/(3n+1)lt 1`C. `n^(4) gt 10^(n)`D. `.^(2n)C_(n)gt (4^(n))/(n+1)` |
| Answer» Correct Answer - A | |
| 23. |
Statement-1: `2^(33)-1` is divisible by 7. Statement-2: `x^(n)-a^(n)` is divisibel by x-a, for all `n in N` and `x ne a`. |
| Answer» Correct Answer - A | |
| 24. |
Use the principle of mathematical induction to prove that for all `n in N` `sqrt(2+sqrt(2+sqrt2+...+...+sqrt2))=2cos ((pi)/(2^(n+1)))` When the LHS contains n radical signs. |
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Answer» Let P(n) =`sqrt(2+sqrt(2+sqrt2+...+...+sqrt2))=2cos ((pi)/(2^(n+1))).....(i)` Step-1 For n=1 `"LHS of "(i)=sqrt2 and "RHS of (i) "=2cos((pi)/(2^(2)))` `2cos((pi)/(4))` `2.(1)/(sqrt2)` `=sqrt2` Therefore, P(1) is true. Step II. Assume it is true for n=k, `P(k)=underset("k radical sign")(sqrt(2+sqrt(2+sqrt2+...+...+sqrt2)))=2cos ((pi)/(2^(n+1)))` `=sqrt({2+P(k)})` `sqrt(2+2cos((pi)/(2^(k+1))))" "("By assumption step")` `sqrt(2(1+cos((pi)/(2^(k+1)))-1))` `sqrt(4cos((pi)/(2^(k+2))))` `2cos((pi)/(2^(k+2)))` This shows that the result is true for n=k+1. Hence by the principle of mathematical, induction, the result is true for all `n in N`. |
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| 25. |
For each `n in N, 3.(5^(2n+1))+2^(3n+1)` is divisible byA. 17B. 19C. 21D. 23 |
| Answer» Correct Answer - A | |
| 26. |
Choose the proposition that is not true for `n gt 1 (n in N)`.A. `1+(1)/(4)+(1)/(9)+...+(1)/(n^(2))lt 2 -(1)/(n)`B. `1+(1)/(sqrt2)+(1)/(sqrt3)+...+(1)/(sqrtn)gt sqrtn`C. `(1)/(n+1)+(1)/(n+2)+....+(1)/(2n)gt (13)/(24)`D. `(1)/(2).(3)/(4).(5)/(6)...+(2n-1)/(2n)gt(1)/(sqrt(3n+1))` |
| Answer» Correct Answer - D | |
| 27. |
The statement `n! gt 2^(n-1), n in N` is true forA. `n gt 1`B. `n gt 2`C. All nD. No n |
| Answer» Correct Answer - B | |
| 28. |
Give an example of a statement P(n) such that P(3) is true, but P(4) is not true. |
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Answer» Consider the statement P(n), `2^(n)` less than 12. Here P(3) is `2^(3)` is less than 12, which is true. `(therefore 8 lt 12)` But P(4) i.e. `2^(4)` is less than 12 is not true `(therefore 16 gt 12)` |
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| 29. |
`If P(n) stands for the statement n(n+1) (n+2) is divisible by 6, then what is p(3)? |
| Answer» P(3) is 3(3+1) (3+2) is divisible by 6, i.e., "60 is divisible by 6". | |
| 30. |
The statement `x^(n)-y^(n)` is divisible by (x-y) where n is a positive integer isA. Always trueB. Only true for `n lt 10`C. Always falseD. Only true for `n ge 10` |
| Answer» Correct Answer - A | |
| 31. |
`3.6+6.9+9.12+....+3n(3n+3)=`A. `b=n,c=n`B. b=n+1, c=n+2C. b=n, c=n+1D. b=n+1, c=n+1 |
| Answer» Correct Answer - B | |
| 32. |
For `a gt 01`, prove thjat `(1+a)^(n) ge (1+an)` for all natural numbers n. |
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Answer» When n=1 LHS=(1+a) and RHS =(1+a) Hence `(1+a)^(n) ge (1+na)` is true, when n=1 `therefore P(1)` is true. (a) Let P(k) be true `Rightarrow (1+a)^(k) ge (1+ka)`…..(i) To prove P(k+1) is true, i.e. `(1+a)^(k+1) ge 1+(k+1)a=x("say")` ......(ii) `"from "(i), (1+a)^(k+1) ge (1+ka)(1+a)y(say)` Now, (y-x)=`(1+ka) +a+ka^(2)-1-ka-a)` `=1+ka+a+ka^(2)-1-1ka-a` `=ka^(2) gt 0 therefore y gt x` From (iii) and (iv), we get `(1+a)^(k+1) ge 1,+(k+1)a` Hence, P(k+1) is true whenever P(k) is true......(b) From (a) and (b) by the principle of mathematical induction, it follows that P(n) is true for all natural number n. |
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| 33. |
Find the value of `1 xx 1!+2 xx 2!+3 xx 3!+........+n xx n!`A. True for no nB. True for all `n gt 1`C. True for all `n in N`D. None of these |
| Answer» Correct Answer - C | |
| 34. |
Prove that `1+(1)/(sqrt2)+(1)/(sqrt3)+....+(1)/(sqrtn) ge sqrtn, AA n in N` |
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Answer» For n=1, so LHS=RHS ….(i) Assume the result for n=k `i.e. 1+(1)/(sqrt2)+(1)/(sqrt3)+.....+(1)/(sqrtk) ge sqrtk ....(ii)` For n=k+1 LHS=`1+(1)/(sqrt2)+..+(1)/(sqrtk)+(1)/(sqrt(k+1))` `ge sqrtk+(1)/(sqrt(k+1))" "["using (ii)"]` `gt sqrtk+(1)/(sqrt(k+1)+sqrtk)"Note"` `=sqrtk+sqrt((k+1))-sqrtk=sqrt((k+1))` i.e. the result is true for n=k+1 Hence, by induction, the result is true `AA n in N`. |
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| 35. |
Shwo that `n^3+(n+1)^3+(n+2)^3` is divisible 9 for everynatural number n. |
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Answer» Let f(n)=`n^(3)+(n+1)^(3)+(n+2)^(3)` Let P(n): `f(n)i.e. n^(3)+(n+1)^(3)+(n+2)^(3)` is divisible by 9. Now, f(1)=`1^(3)+2^(3)+3^(2)=36` is divisible by 9. `therefore P(1) ` is true …….(A) Let P(m) be true `Rightarrow f(m)=m^(3)+(m+1)^(3)+(m+2)^(3)` is divisible by 9. `Rightarrow f(m)=m^(3)+(m+1)^(3)+(m+2)^(3)=9k` where k is an integer ...(i) To prove P(m+1) is true i.e. f(m+1) is divisible by 9. Now, `f(m+1)=m(m+1)^(3)+(m+2)^(3)+(m+3)^(3)` `=(m+1)^(3)+(m+2)^(3)+m^(3)+9m^(2)+27m+27` `=[m^(3)+(m+1)^(3)+(m+2)^(3)]+9m^(2)+27m+27` `=9k+9(m^(2)+3m+3)` which is divisible by 9 Hence P(m+1) is true whenever P(m) is true. .........(B) From (A) and (B). It follows that P(n) is true for all natural number n. |
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| 36. |
Show by using the principle of mathematical induction that for all natural number `n gt 2, 2^(n) gt 2n+1` |
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Answer» Let `P(n): 2^(n) gt 2n+1`, where `n gt 2` . When n=3, LHS=`2^(3)=8 and RHS=2xx3+1=7` clearly `8 gt 7` therefore `P(3)` is true. (a) Let P(k) be true In (i), multiplying both sides by 2, we get `2^(k+1) gt 4k+2…… (iii)` Now, (4k+2)-(2k+3)=`2k-1 gt 0` `Rightarrow 4k+2 gt 2k+3 ....(iv)` From (iii) and (iv), we get `2^(k+1) gt 2k+3` Hence P(k+1) is true whenever P(k) is true, From (a) and (b), it follows that P(n) is true for all natual number n. |
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| 37. |
For each `n in N, 3^(2n)-1` is divisible byA. 8B. 16C. 32D. 10 |
| Answer» Correct Answer - A | |
| 38. |
Prove by the principle of mathematical induction that `(n^5)/5+(n^3)/3+(7n)/(15)`is a natural number for all `n in Ndot` |
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Answer» Let P(n): `(n^(5))/(5)+(n^(3))/(3)+(7n)/(15)` is a natural number. When n=1, `(n^(5))/(5)+(n^(3))/(3)+(7n)/(15)=(1^(5))/(5)+(1^(3))/(3)+(7)/(15)=(1)/(5)+(1)/(3)+(7)/(15)=1`, which is a natual number Hence, P(1) is true……(A) Let P(m) be true `Rightarrow (m^(5))/(5)+(m^(3))/(3)+(7m)/(15)` is a natural number .....(i) To prove P(m+1) is true i.e, `(m+1)^(5)/(5)+(m+1)^(3)/(3)+(7(m+1))/(15)` is natural number .....(ii) Expanding (ii), we get `(1)/(5)(m^(5)+5m^4+10m^(3)+10m^(2)+5m+1)+(1)/(3)(m^(3)+m^(2)+3m+1)+(7)/(15)(m+1)` `=(m^(5))/(5)+(m^(3))/(3)+(7m)/(15)+(m^(4)+2m^(3)+3m^(2)+2m)+(1)/(5)+(1)/(3)+(7)/(15)` `=(m^(5))/(5)+(m^(3))/(3)+(7m)/(15)+(m^(4)+2m^(3)+2m+1)` =a natural number `[therefore m^(5)/(5)+(m^(3))/(3)+(7m)/(15)"is a natural number from (i) "]` Hence P(m+1) is true whenever P(m) is true .....(B) From (A) and (B) it follows that P(n) is true for all natural number n. |
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| 39. |
If P(n) stands for the statement n(n+1)(n+2) is divisible by 3, then what is P(4). |
| Answer» `therefore (n+3)^(2) le 2^(n+3)` is true for all `n in N`. | |
| 40. |
if `P(n)` be the statement `10n+3` is a prime number", then prove that P(1) and P(2) are true but P(3) is false.A. `n gt 2`B. `n gt 3`C. `n lt 4`D. `n ge 2` |
| Answer» Correct Answer - D | |
| 41. |
If `x!= y`, then for every natural number n, `x^n - y^n` is divisible byA. n is an odd integerB. n is an even integerC. n is a prime number onlyD. None of these |
| Answer» Correct Answer - B | |
| 42. |
Prove by the principle of mathematical induction that for all `n in N`:`1+4+7++(3n-2)=1/2n(3n-1)` |
| Answer» Here `P(n) Rightarrow P(n+1) and P(4)` is true, therefore, by the principle of mathematical indcution, P(n) is true for all `n ge 4`. | |
| 43. |
`9^(n)-8^(n)-1` is divisible by 64 isA. Always trueB. Always falseC. Always true for rational values of nD. Always true for irrational values of n. |
| Answer» Correct Answer - B | |
| 44. |
Consider the statement P(n): `n^(2) ge 100`. Here, `P(n) Rightarrow P(n+1)` for all n. Does it means thatA. P(n) is true for all nB. P(n) is true for all `n ge 2`C. P(n) is true for all `n ge 3`D. None of these |
| Answer» Correct Answer - D | |
| 45. |
Let P(n): `n^(2)+n` is odd, then `P(n) Rightarrow P(n+1)` for all n. and P(1) is not true. From here, we can conclude thatA. P(n) is true for all `n ge NN`B. P(n) is true for all `n ge 2`C. P(n) is false for all `n in NN`D. P(n) is true for all `n ge 3` |
| Answer» Correct Answer - C | |
| 46. |
Let P(n) be a statement such that `P(n) Rightarrow P(n+1)` for all `n in NN`. Also, if P(k) is true, `k in N`, then we can conclude that.-A. P(n) is true for all nB. P(n) is true for all n `n ge k`C. P(n) is true for all n `n gt k`D. None of these |
| Answer» Correct Answer - D | |
| 47. |
The statement i.e. `(n+3)^(2) gt 2^(n+3)` is true.A. For all nB. For all `n ge 3`C. For all `n ge 2`D. No, `n in N` |
| Answer» Correct Answer - D | |
| 48. |
A student was asked to prove a statement P(n) by using the principle of mathematical induction. He proved that `P(n) Rightarrow P(n+1)` for all `n in N` and also that P(4) is true: On the basis of the above he can conclude that P(n) is true.A. For all `n in N`B. For all `n gt 4`C. For all `n ge 4`D. For all `n lt 4`. |
| Answer» Correct Answer - C | |
| 49. |
Let P(n): `n^(2)-n+41` is a prime number, thenA. P(1) is not trueB. P(5) is not trueC. P(g) is not trueD. P(41) is not true |
| Answer» Correct Answer - D | |
| 50. |
Let P(n) be a statement and let P(n) `Rightarrow` P(n+1) for all natural number n, then P(n) is true.A. For all `n in N`B. For all `n ge m`, m being a fixed positive integerC. For all `n ge 1`D. Nothing canbe said. |
| Answer» Correct Answer - D | |