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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
The smallest positive inteter n for which `n!lt((n-1)^(n))/(2)` holds , is __________A. 4B. 3C. 2D. 1 |
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Answer» Correct Answer - A Among the options, identify the value that satisfies the given indquqlity . |
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| 2. |
Find the coefficient of `x^(11)` in the expansion of `(1+2x+x^(2))^(6)`A. 1B. 2C. 6D. 12 |
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Answer» Correct Answer - D `(1+2x+x^(2))^(6)=[(1+x)^(2)]^(6)=(1+x)^(12)` Coefficient of `x^(11)"is" ""^(12)C_(11)=12` |
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| 3. |
`(x+y)^(3)`=_____________ |
| Answer» Correct Answer - `x^(3)+3x^(2)y+3xy^(2)+y^(3)` | |
| 4. |
For all `n in N,41^(n)-40n-1` is divisible by _________A. 41B. 40C. 300D. 500 |
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Answer» Correct Answer - B Substituting `n=1,2,3,4,......` in the given expression and verify from the options. |
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| 5. |
`(n-r)!`= ___________ |
| Answer» Correct Answer - `1*2*3....(n-r-1)*(n-r)` | |
| 6. |
In `(n+y)^(n)-(x-y)^(n)`, if the number of term is 5, then find n.A. 6B. 5C. 10D. 9 |
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Answer» Correct Answer - D `(x+y)^(n)-(x-y)^(n)` has `(n)/(2)` terms when n is even and `(n+1)/(2)`n terms when is odd. |
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| 7. |
The third term form the end in the expansion of `((4x)/(3y)-(3y)/(2x))^(9)` is _____________ |
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Answer» Correct Answer - B The nth term from the end in the expansion `(x+y)^(n)"is" (n-r+2)` the term from the beginning . |
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| 8. |
If m and n are the coeficients of `x^(a^(2)) and x^(b^(2))` respectively in `(1+x)^(a^(2)+b^(2))`, then ____________A. `n=2m`B. `m+n=0`C. `2n=m`D. `m=n` |
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Answer» Correct Answer - D (i) `"Use"""^(n)C_(r)=""^(n)C_(n-r)` (ii) `m=(a^(2)+b^(2))c_(a^(2))x^(a^(2)) and=(a^(2)+b^(2))c_(b^(2))x^(b^(2))` (ii) Find. |
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| 9. |
If `""^(n)C_(3)=""^(n)C_(15)`, then `""^(20)C_(n)` is __________ |
| Answer» Correct Answer - 190 | |
| 10. |
The sum of the elements in the sixth row of Pascal triangle is ___________A. 32B. 63C. 128D. 64 |
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Answer» Correct Answer - D The sum of the elements in the nth row of Pascal triangle is `2^(n)` |
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| 11. |
The number of terms in the expansion of `[(2x+3y)^(4)(4x-6y)^(4)]^(9)` is ____________A. 36B. 37C. 10D. 40 |
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Answer» Correct Answer - B The number of terms in the expansion of `(x+y)^(n)` is n+1 |
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| 12. |
The elmennts in the fifth row of Pascal triangle is ____________ |
| Answer» Correct Answer - `1,5,10,10,5,1` | |
| 13. |
In the exapansion `(6+9x)^(5)` the coefficient of `x^(3)` is __________A. `2^(2)xx3^(8)`B. `2^(4)xx3^(7)`C. `2^(3)xx3^(8)xx5`D. `2^(4)xx3^(7)5` |
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Answer» Correct Answer - C Given expression is `(6+9x)^(5)` `T_(3+1)=""^(5)C_(3)(6)^(3)(9x)^(3)=2xx5xx(2xx3)^(2)xx3^(6)xx^(3)` `:.` The required coefficient `=2^(3)xx3^(8)xx5` |
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| 14. |
The sum of the first n even natural numbers is ___________ |
| Answer» Correct Answer - `n(n+1)` | |
| 15. |
Find the value of `(98)^(4)` by using the binomial theorem.A. 92236846B. 92239816C. 9233886D. 92236806 |
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Answer» Correct Answer - B `(98)^(4)=(100-2)^(4)` . Expand using binomial theroem. |
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| 16. |
Constant term in the expansion of `(x+3)^(16)` is ____________ |
| Answer» Correct Answer - `3^(16)` | |
| 17. |
In Pascal, triangle, each row of coefficiets is bounded on both sides by ____________ |
| Answer» Correct Answer - 1 | |
| 18. |
The sum of the first n odd natural number is |
| Answer» Correct Answer - `n^(2)` | |
| 19. |
Number of non-zero terms in the exapansion of `(5sqrt(5)x+sqrt(7))^(6)+(5sqrt(5)x-sqrt(7))^(6)` is __________A. 4B. 10C. 12D. 14 |
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Answer» Correct Answer - A The number of terms in the expansion of `(x+y)^(n)+(x-y)^(n)"is" (n)/(2)+1` when n is even and `(n+1)/(2)` when n is odd. |
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| 20. |
`n^(2)-n+1` is an odd number for all ____________A. `ngt1`B. `ngt2`C. `nge1`D. `nge5` |
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Answer» Correct Answer - C For each, choice, check if the given expression is divisible by it for the given vlaues of n. |
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| 21. |
find the middle term of terms or the expanison of `(x+(1)/(x))^(6)` |
| Answer» Correct Answer - `""^(6)C_(3)` | |
| 22. |
Find the sum of the coefficients in the expansion of `(5x^6-4/(x^9))^(10)`A. `5^(10)`B. 1C. `4^(10)`D. 0 |
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Answer» Correct Answer - B The given expression is `(5x^(6)-(4)/(x^(9)))^(10)` Put x=1, so we get the sum of the coefficeints, `i.e.,(5-4)^(10)=1^(10)=1` |
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| 23. |
Find the middle the terms in the expansion of `(5x-7y)^(7)` |
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Answer» Since n is an odd number, the expansion contains two middle terms `((7+1)/(2))th and ((7+3)/(2))` th terms are the two middle terms in the expansion of `(5x-7y)^(7)` `t_(4)=T_(3+1)=(-1)^(3)""^(7)C_(3)(5x)^(7-3)(7y)^(3) =-""^(7)C_(3)(5x)^(4)(7y)^(3)` =`T_(5)=T_(4+1)=(-1)^(4)*""^(7)C_(4)(5x)^(7-4)*(7y)^(4)=""^(7)C_(4)(5x)^(3)(7y)^(4)` |
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| 24. |
Evalute: ` (sqrt(3)+1)^(5) -(sqrt(3)-1)^(5)` |
| Answer» Correct Answer - 152 | |
| 25. |
In the 10th term of `(x+y)^(n)`, the exponent of x is 3, then the exponent of x in the 7th term is __________ |
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Answer» Correct Answer - B The given exapnsion is `(x+y)^(n)` `T_(10)-T_(9+1)=""^(n)C_(9)(x)^(n-9)(y)^(9)` Given `n-9=3` `n=12` `T_(7)=T_(6+1)=""^(12)C_(6)(x)^(6)(y)^(6)` `:.` The exponet of x is 6 |
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| 26. |
`2^(n)-1` gives the set of all odd ntural numbers for all `n in N`. Comment on he given statement .A. True for alll values of nB.C. True for only odd values of nD. True for only prime values of n |
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Answer» Correct Answer - B Subsitute n=,1,2,3,4.... In the given expression. |
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| 27. |
Find the 7th term in the expansion of `(5x-(1)/(7y))^(9)` |
| Answer» Correct Answer - `""^(9)C_(6)(5x)^(3)((1)/(7y))^(6)` | |
| 28. |
In the 5th of `(x_y)^(n)`, the exaponent of y is 4 then the exponent of y in the 8th term is ___________A. 1B. 7C. 5D. 9 |
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Answer» Correct Answer - B As we go from left to right the eponent of y in the epansion of `(x+y)^(n)`n increases . |
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| 29. |
If `x=-""^(n)C_(1)+""^(n)C_(2)(2)-""^(n)C_(3)(2)^(2)+....` (where n is odd), then x=___________A. 1B. `-1`C. 0D. 12 |
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Answer» Correct Answer - B Given `x=-""^(n)C_(1)+""^(n)C_(2)(2)....` `2x=-""^(n)C_(1)(2)+""^(n)C_(2)(2)^(2)-""^(n)C_(3)(2)^(3)+.....` `1+2x=1-""^(n)C_(1)(2)+""^(n)C_(2)(2)^(2)-""^(n)C_(3)(2)^(3)+.....` `1+2x=(1-2)^(n)` `1+2x=(-1)^(n)` `1+2x=-1` (given n is odd) `2x=-2` `x=-1` |
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| 30. |
If the expansion `(x^(3)+(1)/(x^(2)))^(n)` contains a term independent of x, then the value of n, can be __________A. 18B. 20C. 24D. 22 |
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Answer» Correct Answer - B Use of formula `r=(np)/(p+q)` to find the independet term in the expression of `(ax^(p)+(b)/(x^(q)))^(n)` |
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| 31. |
The coefficient of x in the expansion of `(2x+3)^(5)` is __________ |
| Answer» Correct Answer - 810 | |
| 32. |
Find the independent term in the expansion of `(5x^(2)-(1)/(x^(4)))^(6)`A. 8250B. 8560C. 9250D. 9375 |
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Answer» Correct Answer - D In the expanssion `(ax^(p)+(b)/(x^(q)))^(n)` , the independent terms is `T_(r+1)`, where `r=(np)/(p+q)` Here, `r=(6xx2)/(6)=2` The independent term is `T_(3)` `T_(3)=T_(2+1)=""^(6)C_(2)(5x^(2))^(4)((-1)/(x^(4)))^(2)` `5^(4)xx15=9375` |
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| 33. |
If the expansion `(x^(2)+(1)/(x^(3)))^(n)` is to contain and independent term, then what should be the vlaue of m ? |
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Answer» General term `T_(r+1)=""^(n)C_(r)*x^(n-r)y^(r),"for"(x+y)^(n)` `rArr` general term of `(x^(2)+(1)/(x^(3)))"is" ""^(n)C_(r)*x^(2n-2r)*(1)/(x^(3r))=""^(n)C_(r)*x^(2n-5r)` For a term to be indep3edent of `x,2n-5r` should be equal to zero. That is ,` 2n-5r=0` `rArrr=(2)/(5)n`, since r can take only integer values , n has to be a multiple of 5 |
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| 34. |
The ratio of the coefficients of `x(4)` to that of the term independent of x in the expansion of `(x^(2)+(9)/(x^(2)))^(18)` is ____________A. `1:6`B. `3:8`C. `1:10`D. `1:8` |
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Answer» Correct Answer - C In the expansion of `(ax^(p)+(b)/(x^(q)))^(n)` to find (i) the independent term, use the formula, `r=(np)/(p+q)` and (ii) the coefficient of `x^(k)`, use the formula `r=(np-k)/(p+q)` |
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| 35. |
The number of irrational term s in the expansion of `(x^(2//3)+y^(1//4))^(81)` is ___________A. 70B. 12C. 75D. 13 |
| Answer» Correct Answer - C | |
| 36. |
`sum_(r=1)^(30) r (""^(30)C_(r))/(""^(30)C_(r-1))=` ___________A. 930B. 465C. 310D. 630 |
| Answer» Correct Answer - B | |
| 37. |
If three consective coefficient in the expansion of `(1+x)^(n)`, where n is a natural number are 36,84 and 126 respectively, then n is __________A. 8B. 9C. 10D. 36 |
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Answer» Correct Answer - B Let the three consecutive in the expansion `(1+x)^(n)"be" ""^(n)C_(r) ""^(n)C_(r+2)` |
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| 38. |
Find the independent term in the expansion of `(x^(4)+(3)/(8x^(3)sqrt(x)))^(15)`A. `""^(15)C_(4)((3)/(8))^(16)`B. `""^(15)C_(12)((3)/(8))^(8)`C. `""^(15)C_(8)((3)/(8))^(8)`D. `""^(15)C_(7)((3)/(8))^(8)` |
| Answer» Correct Answer - C | |
| 39. |
Find the term independent of x in `(2x^(5)+(1)/(3x^(2)))^(21)` |
| Answer» Correct Answer - `""^(21)C_(15)(2^(6))/(3^(15))` | |
| 40. |
Find the term independent of x in `(6x^(2)-(1)/(7x^(3)))^(10)` |
| Answer» Correct Answer - `""^(10)C_(4)(6^(6))/(7^(4))` | |
| 41. |
Find the coefficient of the independent term in the expansion of `(x^(1//2)+7x^(-1//3))^(10)`A. `""^(10)C_(4)7^(4)`B. `""^(10)C_(6)7^(6)`C. `""^(10)C_(6)7^(5)`D. `""^(10)C_(4)7^(7)` |
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Answer» Correct Answer - B To find the independent, in the expansion of `(ax^(p)+(b)/(x^(q)))^(n)`. Use of formula `r=(np)/(p+q)` |
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| 42. |
Find the coefficient of `x^(7)` in the expansion of `(7x+(2)/(x^(2)))^(13)`A. `78xx8^(8)xx4`B. `78xx7^(6)xx4^(2)`C. `78xx7^(11)xx4`D. `78xx7^(11)xx4^(2)` |
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Answer» Correct Answer - C To find the coefficient of `x^(k)` in the expansion `(ax^(p)+(b)/(x^(q)))^(n)`, use `r=(np-k)/(p+q)` |
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| 43. |
The term independent of x in the exapansion of `(x^(3)-(1)/(x^(2)))^(10)` is ___________A. `""^(10)C_(6)`B. `""^(10)C_(7)`C. `""^(10)C_(9)`D. `-""^(10)C_(6)` |
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Answer» Correct Answer - A To find the independent term in the expression of `(ax^(p)+(b)/(x^(q)))^(n)` use the formula , `r=(np)/(p+q)` |
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| 44. |
Find the value of k for which the term independent of x in `(x^(2)+(k)/(x))^(12)` is 7920A. `(1)/(sqrt(2))`B. `(1)/(2)`C. `sqrt(2)`D. 2 |
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Answer» Correct Answer - C To find the independent term in the expansion `(ax^(p)+(b)/(x^(q)))^(n)` , use `r=(np)/(p+q)` |
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