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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. |
The valuses of x for which `1+x log_e (x+sqrt(x^2+1)) le sqrt(x^2 +1)` areA. `x le 0 `B. `0 le x le 1`C. `x ge 0 `D. none of these |
| Answer» Correct Answer - C | |
| 2. |
`(1+x)^n le 1+ x^n` whereA. `n gt 1`B. `0 le n le 1 and x gt 0 `C. `n lt 1 and x lt 0`D. `x lt 0 ` |
| Answer» Correct Answer - B | |
| 3. |
If the function `f(x)=cos|x|-2a x+b`increases along the entire number scale, then(a) `a=b`(b) `a=1/2b`(c) `alt=-1/2`(d) `a >-3/2`A. `a le b `B. `a=b/2`C. `a lt - 1/2`D. `a gt - 3/2` |
| Answer» Correct Answer - C | |
| 4. |
If `f(x)=unerset(x^2)overset(x^(2+1))e^(-t^2)` dt then f(x) increases onA. `(-2,2)`B. `(0,oo)`C. `(-oo,0)`D. none of these |
| Answer» Correct Answer - C | |
| 5. |
Find the value of `a`in order that `f(x)=sqrt(3)sinx-cosx-2a x+b`decreases for all real values of `xdot`A. `a lt 1`B. ` a le 1`C. ` a le sqrt(2)`D. `a lt sqrt(2)` |
| Answer» Correct Answer - B | |
| 6. |
The function` f(x)=tan^(-1)x-x` is decreasing on the setA. RB. `(0,oo)`C. R-[0]D. none of these |
| Answer» Correct Answer - A | |
| 7. |
For what value of a,`f(x)=-x^3+4ax^2+2x-5` decreasing for all x .A. (1,2)B. (3,4)C. RD. no value of a |
| Answer» Correct Answer - D | |
| 8. |
A function is matched below against an interval where it is supposed tobe increasing. Which of the following parts is incorrectly matched?Interval,Function[2, `oo)`, `2x^3-3x^2-12 x+6``(-oo,oo)`, `x^3=3x^2+3x+3``(-oo-4)`, `x^3+6x^2+6``(-oo,1/3)`, `3x^2-2x+1`A. `{:("","interval","Function"),((a),(-oo,-4],f(x)=x^3+6x^2+6):}`B. `{:("","interval","Function"),((a),(-oo,1//3],g(x)=3x^3-2x+1):}`C. `{:("","interval","Function"),((a),(2,oo],h(x)=2x^3-3x^2+12x+6):}`D. `{:("","interval","Function"),((a),(-oo,oo],q(x)=x^3-3x^2+3x+3):}` |
| Answer» Correct Answer - B | |
| 9. |
Let the function `f(x) = tan^(1-) (sin x + cos x)` be defined on `[ 0, 2 pi] Then f(x) isA. increasing on `[0,pi//4) cup [ 5 pi/4, 2 pi]`B. decreasing on `(Pi//4,2pi)`C. increasing on `(0,pi//4,) cup (3 pi//4,2pi)`D. decreasing on `[pi//4,7 pi//4]` |
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Answer» Correct Answer - A We have `f(x) = tan^(-1)""(sin x + cos x )` `rArr f(x)=(1)/(1+(sin x+cos x )^2)xx (cos x - sin x)` Now `f(x) gt 0 rArr cos x -sin x gt 0 tArr x in [0,pi//4) cup (5pi //4 ,4,2pi]` and `f(x) lt 0 rArr cos x sin x lt 0 rArr x in (pi//4,5 pi//4)` decreasing `(pi//4,5 pi//4)` |
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| 10. |
the function `f(x)=logx/x` is increasing in the intervalA. `(1,2e)`B. `(0,e)`C. (2,2e)D. (1/e,2e) |
| Answer» Correct Answer - B | |
| 11. |
`f(x)=x|log_ex|,x lt 0` is monotonically decreasig inA. `(e,oo)`B. `(0,1//e)`C. `(-oo,-1)cup [1,oo)`D. (1,e) |
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Answer» Correct Answer - C We have `f(x)=|x log_e x|={{:(-x log_ex"," 0 lt x lt 1),(x log_ex","x ge1):}` `f(x)={{:(-(log_ex+1)"," 0 lt x lt 1 ),((log_ex+1)","x ge1):}` For f(x) to be decreaing ,we must have `f(x) lt 0 ` `rArr {:{(-(log_ex+1)lt0 " for " 0 lt x lt 1 ),(" "(log_e x+1)lt 0 " for " x gt 1 ):}` `rArr {:{(log_ex+1gt0 " for " 0 lt x lt 1 ),(log_e x+1lt 0 " for " x gt 1 ):}` `rArr {:{(x gt1//e " for "0 lt x lt 1 ),(x lt 1//e" for "x gt1):}` `rArr x gt 1/e "for " 0 lt x lt 1 ` `rArr x in (1//e,1)` Hence f(x) is decreasing in [1/e,1] |
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| 12. |
the function `(|x-1|)/x^2` is monotonically decreasing at the pointA. `(2,oo)`B. `(0,1)`C. `(-oo ,1)`D. `(oo,oo)` |
| Answer» Correct Answer - C | |
| 13. |
The values of a for which the function `(a+2)x^3 - 3x^2 + 9ax-1` decreases monotonically throughout for all real x are :-A. `a lt -2 `B. `a gt -2 `C. `-3 lt a lt 0 `D. `-oo lt a le -3` |
| Answer» Correct Answer - D | |
| 14. |
If f(x)= kx-sin x is monotonically increasing thenA. `k gt 1`B. ` k gt -1`C. ` k lt 1`D. `k lt - 1` |
| Answer» Correct Answer - A | |
| 15. |
The function `f(x)=x^(1//x)` is increasing in the intervalA. `(e,oo)`B. `(-oo,e)`C. `(-e,e)`D. none of these |
| Answer» Correct Answer - D | |
| 16. |
Let `f(x)=2 tan^(-1)((1-x)/(1+x))`A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-5B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-5C. Statement-1 True statement -1 is True,Statement -2 is FalseD. Statement-1 is False ,Statement -2 is True |
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Answer» Correct Answer - A `f(x)=2tan^(-1)""(1-x)/(1+x)(tan^(-1)-tan^(-1)x)"-="pi/2-2tan^(-1)""x` `rArr f(x)=-(2)/(1+x^2)lt0 for all x in [0,1]` `rArr f(x)` decreases on [0,1] `rArr ` Range of `f=[f(1),f(0)]=[0,pi//2]` Hence both the statements are ture and statement are ture and statement -2 is a correct explanation of statement - 1 |
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| 17. |
If `a le 0 f(x) =e^(ax)+e^(-ax)` and S={x:f(x) is monotonically increasing then S equalsA. `f {x:x gt 0 } `B. `{x:x lt 0}`C. `{x:x lt 1}`D. `{x:x lt 1}` |
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Answer» Correct Answer - A We have `f(x)=e^(ax)+e^(-ax) rArr f(x) =a(e^(ax)-e^(-ax))` For f(x) to be increasing we must have `f(x) gt 0 ` `rArr a(e^ax-e^(-ax)) gt 0 ` `rArr a(e^(ax)-e^(-ax)) gt 0` `rArr e^(-bx)-e^(bx) lt 0 ` where a = -b and `b gt 0 ` `rArr e^(bx)-e^(-bx) gt 0 rArr x lt 0 ` Hence `S= {x: x gt 0 }` |
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| 18. |
If `a |
| Answer» Correct Answer - B | |
| 19. |
On which of the following intervals is the function f given by `f(x)=x^(100)+sinx-1`strictly decreasing ?(A) (0, 1) (B) `(pi/2,pi)` (C) `(0,pi/2)` (D) None of theseA. `(0,(pi)/2)`B. `(0,1)`C. `(pi//2,pi)`D. none of these |
| Answer» Correct Answer - D | |
| 20. |
Statement-1 `e^(x)+e^(-x) gt 2 +x^2` is an increasing function on R.A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-3B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-3C. Statement-1 True statement -1 is True,Statement -2 is FalseD. Statement-1 is False ,Statement -2 is True |
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Answer» Correct Answer - A We have `f(x)=e^x+e^(-x)-2-x^2` `rArr f(x)=e^x-e^(-x)-2x` `rArr f(x)= e^x+e^(-x)-2=((e^x-1)^2)/(e^x)gt 0 `for all `x ne 0` ltbr gt `rArr ` f(x) in increasing in R `rArr f(x) gt f(0) " for all " x in R , x ne 0 ` `f(x) gt 0 "for all " x (ne 0) in R` `f (x) gt f(0) " for all " x ne 0 ` `e^x+e^(-x)-2-x^2 lt 0 " for all " x ne 0 ` `rArr e^x+e^(-x) gt 2+ x^2 " for all " x ne 0` Hence both the statements are true statement-2 is a correct explanation of statment-1 |
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| 21. |
Let the function g:`(-oo,oo)rarr (-pi //2,pi//2)` be given by g(u) `= 2 tan^(-1) (e^u)-pi/2` Then g isA. even and is strictly increasing in `(0,oo)`B. odd and is strictly decreasing `(-oo,oo)`C. odd and is strictly increasing in `(-oo,oo)`D. neither even nor odd , but is stictly increasing in `(-oo,oo)` |
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Answer» Correct Answer - C we have ` g(u)=2 tan^(-1) (e^u)-pi/2` `rArr g(u)=(2e^u)/(1+ e^(2u))gt 0 "for all u " rarr (-oo ,oo)` `rarr` g is stictly increasing function in `(-oo,oo)` Now `g(u) = tan^(-1)(e^u)-pi/2` `rArr g(u) = tan^(-1)(e^u )-(pi/2-tan ^(-1)(e^u))` `rArr g(u)=tan^(-1)(e^u)-cot^(-1)(e^(u))` `rArr g(-u)=tan ^(-1)(e^u)-cot^(-1)(e^(-u))` `rArr g(-u) = tan^(-1)(e^(1//u))-cot^(-1)(e^(1//u))` `rArr g(-u)=cot^(-1)(e^u)-tan^(-1)(e^u)=-g(u)` Hence g(u) is odd and is strictly increasing `(-oo,oo)` |
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| 22. |
If the function f(x)=3 cos |x| -6 ax +b increases for all `x in R` then the range of value of a given byA. `(-1/2, oo)`B. `(-oo,-1//2)`C. `(-oo,-2)`D. `(-2 ,oo)` |
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Answer» Correct Answer - B We have f(x)=3 cos |x|-6ax +b `rArr f(x) =-3 sin x -6ax +b` `rArr f(x) =3 sin -6 ax + b` `[ because cos |x| =cos x for all x ]` `rArr f(x)=-3 sinx -6a` For f(x) to be increasing on R we must have `f(x) gt 0 "for all" n in R` `rArr -3 sin x-6 a gt 0 "for all"x in R` `rArr sin x+2a lt 0 "for all x in R` `rArr sin x lt -2a " for all " x in R` `rArr 1 lt - 2a " "[ because "`Max .value of sin x is 1] `rArr a lt -1/2` `rArr a in (-oo,-1 //2)` |
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| 23. |
The fucntion f(x)`=(sin x)/(x)` is decreasing in the intervalA. (-lt/2, 0 )B. `(0, pi//2)`C. `(0, pi)`D. none of these |
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Answer» Correct Answer - B::C we have ` f(x) =(sin x)/(x)` `rArr f(x)=g(x)/x^2` , where g(x)=x cos x - sin x . Now g(x)=- sin x Consider the interval `(-pi //2,0)` In this inetrval we obseve that `g(x) lt g( 0)` `rArr` g(x) is decresing on `(- pi //2,0)` `rArr g(x) gt 0 for all x in (-pi//2, 0)` `therefore f(x)=(g(x)/(2) gt 0 for all x in (-pi //2 , 0)` `rArr f(x) " in creasing on" (-pi //2 , 0)` Consider now the interval `(0,pi//2)` In this interval ,we have `g(x) lt 0 "for all " x in (0 pi//2)` `rAr g(x) "decresing on " (0,pi//2)` `f(x) lt 0 for x in (0,pi//2)` |
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| 24. |
Statement-1 `e^(pi) gt pi^( e)` Statement -2 The function `x^(1//x)( x gt 0)` is strictly decreasing in `[e ,oo)`A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-1B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-1C. Statement-1 True statement -1 is True,Statement -2 is FalseD. Statement-1 is False ,Statement -2 is True |
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Answer» Correct Answer - A Let `f(x)=x^(1//x)` Then , `f(x)=x^(1//x)(1/x^2 - (log x)/(x^2))=x^(1//x)((1-log x )/(x^2))`f `rArr f(x) lt 0` for all `x in (e,oo)` `rArr f(x) ` is strictly decreasing in `[e,oo)` `rArr (e ) gt f(pi)` `rArr e^(1//e) gt pi^(1//pi) rArr e^(pi) gt pi^(e)` Hence both the statements are ture and statement -2 is a correct explanation for statement -1 |
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| 25. |
Let f (x ) and g(x) be increasing and decreasing functions respectively from `[0,oo) "to" [ 0 , oo)` Let h (x) = fog (x) If h(0) =0 then h(x) isA. always 0B. always positiveC. always negativeD. strictly increasing |
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Answer» Correct Answer - A since composition of an increasing function and a decreasing function is always a decreasing fuction .Therefore h(x)` [ 0, oo) rarr [0,oo)`is a decreasing function . `h(x) le 0 for all x le 0` `rArr h(x) =0 for all x ge 0 " "[{:(because h(x)in [0","oo)),(rArr h(x) ge 0 for all x in [0","oo)):}]` |
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| 26. |
The function `f(x)=(ln(pi+x))/(ln(e+x))`isincreasing in `(0,oo)`decreasing in `(0,oo)`increasing in `(0,pi/e),`decreasing in `(pi/e ,oo)`decreasing in `(0,pi/e),`increasing in `(pi/e ,oo)`A. increasing function on `[0,oo)`B. decreases on [1/2,1]C. increasing on `[0,pi//e]` and increasing on `[pi//e, oo)`D. decreasing on`[0,pi//e)` and increasing on `[pi//e,oo)` |
| Answer» Correct Answer - B | |
| 27. |
`y={x(x-3)^2` increases for all values of x lying in the intervalA. `0 lt x lt 3/2`B. `0 lt x lt oo`C. `- oo lt x lt 0 `D. `1 lt x lt 3` |
| Answer» Correct Answer - A | |
| 28. |
The value of b for which the function f(x)=sin x-bx+c is decreasing in the interval `(-oo,oo)` is given byA. `b lt 1`B. `b ge 1`C. `b gt 1`D. `b le 1 ` |
| Answer» Correct Answer - C | |
| 29. |
The interval in which the function `x^3` increases less rapidly than `6x^2+15x+5`A. `(-oo,-1)`B. `(-5,1)`C. `(-1,5)`D. `(5,oo)` |
| Answer» Correct Answer - C | |
| 30. |
The interval in which the function `f(x)=x^(e^(2-x))` increases isA. `(-oo,0)`B. `(2,oo)`C. `(0,2)`D. none of these |
| Answer» Correct Answer - D | |
| 31. |
The function `f(x)=cot^(-1)x+x`increases in the interval(a) `(1, oo)`(b) `(-1, oo)`(c) `(-oo, oo)`(d) `(0, oo)`A. `(1,oo)`B. `(-1,oo)`C. `(-oo,oo)`D. `(0,oo)` |
| Answer» Correct Answer - A | |
| 32. |
If `f(x)=2 x cot ^(-1)x + log (sqrt(1+x^2)-x ` then f(x)A. increases on RB. decreases in `[0,oo)`C. neither increasing nor decreasing in `(0,oo)`D. none of these |
| Answer» Correct Answer - A | |
| 33. |
Leg `f(x)=x^3+a x^2+b x+5sin^2x`be an increasing function on the set `Rdot`Then find the condition on `aa n dbdot`A. `a^2-3b-15 gt 0`B. `a^2-3b+15 gt 0 `C. `a^2-3b+15 lt 0 `D. `a gt 0 and b lt 0 ` |
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Answer» Correct Answer - C `f(x)=x^3+ax^2+bx + 5 sin ^2x` increasing on R `rArr (x) lt for all in R` `rArr 3x^2a + (b-5)gt 0 for all x in R` `rArr 3x^2+2ax(b-5) lt 0 rArr a^2-3b+15 lt 0 ` |
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| 34. |
Consider the following statements in S and RS: Both `sinxa n dcosx`are decrerasing function in the interval `(pi/2,pi)`R: If a differentiable function decreases in an interval `(a , b),`then its derivative also decrease in `a , b)dot`Which of the following it true?Both S and R are wrong.Both S and R are correct, but R is not the correct explanation of S.S is correct and R is the correct explanation for S.S is correct and R is wrong.A. Both S and R are wrongB. Both S and R are correct but R is not correct explanation for SC. S is correct and R wrongD. d |
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Answer» Correct Answer - D It is evident that both sinx and cosx are decreasing function in the interval `(pi/2,pi)`So is correct , Statement R is not correct because sinx is decreasing in the interval `(pi/2,(3pi)/2)` but its derivative i.e. cosx is not so |
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