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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.
| 51. |
`lim_(xrarr(pi//4))(sec^2x-2)/(tanx-1)` isA. 3B. 1C. 0D. 2 |
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Answer» Correct Answer - d Given, `lim_(xto(pi//4))(sec^(x)-2)/(tanx-1)` `=lim_(xto(pi//4))(1+tan^(2)x-2)/(tanx-1)=lim_(xto(pi//4))(tan^(2)x-1)/(tanx-1)` = `=lim_(xto(pi//4))(tanx+1)` =2 |
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| 52. |
If `f(x) = (tanx)/(x-pi)`, then `lim_(xrarr(pi))f(x)=`………………. |
| Answer» Given, `f(x) = (tanx)/(x-pi)= lim_(xto(pi))(tanx)/(x-pi) = lim_(pi-xto0)(-tan(pi-x))/(-pi-x)` `[therefore lim_(xto0)(tanx)/(x) =1]` | |
| 53. |
`lim_(xrarr-1) (x^(10)+x^(5)+1)/(x-1)` |
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Answer» `underset(xrarr-1)"lim"(x^(10)+x^(5)+1)/(x-1)` `=((-1)^(10)+(-1)^(5)+1)/(-1-1)` `=(1-1+1)/(-2)=-(1)/(2)` |
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| 54. |
`lim_(xrarr0)(x^(2)cosx)/(1-cosx)` is equal toA. 2B. `3/2`C. `-3/2`D. 1 |
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Answer» Correct Answer - a Given, `lim_(xto0)(x^(2)cosx)/(1-cosx)` `[therefore 1-cosx=2sin^(2)((x)/2)]` `=2lim_(xto0)(x/2)^(2)/(sin^(2)(x/2))lim_(xto0)cosx=2.1=2` |
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| 55. |
if `y=(sin(x+9))/(cosx)`, then `(dy)/(dx)` at x=0 is equal toA. cos9B. sin9C. 0D. 1 |
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Answer» Correct Answer - a Given, `y=(sin(x+9))/(cosx)` `therefore (dy)/(dx) = (cosxcos(x+9)-sin(x+9)(-sinx))/(cosx)^(2)` [by quotient rule] `=cosxcos(x+9)+sinxsin(x+9)/(cos^(2)x)` `therefore ((dy)/(dx))_(x=0) = (costheta)/(1) = costheta` |
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| 56. |
if `y=(sinx+cosx)/(sinx-cosx)`, then `(dy)/(dx)` at x=0 is equal toA. `-2`B. 0C. `1/2`D. Does not exist |
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Answer» Correct Answer - a Given, `y=(sinx+cosx)/(sinx-cosx)` `therefore (dy)/(dx) = ((sinx-cosx)(cosx-sinx)-(sinx + cosx)(cosx+sinx))/(sinx-cosx)^(2)` [by quotient rule] `=((-sinx-cosx)^(2)-(sinx+cosx)^(2))/(sinx-cosx)^(2)` `(-[(sinx-cosx)^(2)-(sinx+cosx)^(2)])/(sinx-cosx)^(2)` `=-([(sin^(2)x+cos^(2)x-2sinxcosx+sin^(2)x+cos^(2)x+2sinxcosx)])/(sinx-cosx)^(2)` `=-2/(sinx-cosx)^(2)` `therefore(dy)/(dx)_(x=0)=-2` |
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| 57. |
Find the derivative of `sin x. "log"_(e) x` with respect to x.A.B.C.D. |
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Answer» Let `y=e^(x) sin x "log"_(e)x` `rArr(dy)/(dx)=(d)/(dx)(e^(x)sin x "log"_(e)x)` `=e^(x)sin x(d)/(dx)"log"_(e)x+e^(x)"log"_(e)x(d)/(dx)sinx` ` +"log"_(e)x.sinx(d)/(dx)e^(x)` `=[ (e^(x)sinx)/(x)+e^(x)"log"_(e)x.cos x+"log"_(e)x.sin x.e^(x)]` |
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| 58. |
Find the differential coefficient of `(sin x -x cos x)/(x sin x+cos x)` with respect to x. |
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Answer» Let `y=(sin x-x cos x)/(x sin x+cos x)` `(dy)/(dx)=(d)/(dx)((sinx-x cos x)/(x sinx+cos x))` `(xsinx+cosx).(d)/(dx)(sinx-xcos x)` `=(-(sinx-xcos x)(d)/(dx)(xsin+cosx))/((x sin x+cos x)^(2))` `(xsinx+cos x)(cos x-cos x+x sinx)` `=(-(sinx-xcoss)(x cos x+sinx-sinx))/((x sinx+cos x)^(2))` `=((x sin x+cos x).x sin x-(sinx-x cos x)x cos x)/((x sin x+cos x)^(2))` `=(x(xsin^(2)x+sinxcos x-sinxcos x+x cos^(2)x))/((x sin x+cos x)^(2))` `=(x^(2)(sin^(2)x+cos^(2)x))/((x sin x+cos x)^(2))` `(x^(2))/((x sin x +cos x)^(2))` |
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| 59. |
Differentiate w.r.t to x ,`(ax^(2)+cotx)(p+qcosx)` |
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Answer» Let `y=(ax^(2)+ cotx)(p+qcosx)` `therefore (dy)/(dx) = (ax^(2)+cotx)d/(dy)(p+qcosx) + (p+qcosx)d/(dy)(ax^(2)+cotx)` [by product rule] `=(ax^(2)+cotx)(-qsinx)+(p+qcosx)(2ax-"cosec"^(2)x)` `=-qsinx(ax^(2)+cotx)+(p+qcosx)(2ax-"cosec"^(2)x)` |
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| 60. |
Differentiate the following function with respect of `x" ":1/(a x^2+b x^2+c)` |
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Answer» Let `y=1/(ax^(2)+bx+c) = (ax^(2)+bx+c)^(-1)` `therefore (dy)/(dx) = -(ax^(2)+bx+c)^(-2)(2ax + b)` [By chain rule] `=(-2ax+b)/(ax^(2)+bx+c)^(2)` |
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| 61. |
Differentiate the following function with respect of `x :(a x+b)//(c x+d)` |
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Answer» Let `f(x) = (ax + d)/(cx+d)` `f(x+h) = (a(x+h)+b)/(c(x+h)+d)` `therefore d/(dx)f(x)= underset(hto0)"lim"1/h[f(x+h)-f(x)]` `=underset(hto0)"lim"1/h[(a(x+h)+b)/(c(x+h)+d)-(ax+b)/(cx+d)]` `=underset(hto0)"lim"1/h[(ax+b+ah)/(c(x+h)+d)-(ax+b)/(cx+d)]` `=underset(hto0)"lim"1/h[((ax+ah+b)(cx+d)-(ax+b)(cx+ch+d))/((c(x+h)+d)(cx+d))]` `=underset(hto0)"lim"1/h[(acx^(2)+achx+bcx+adx+adh+bd-acx^(2)+achx+adx+bcx+bch+bd)/(c(x+h)+d(cx+d))]` `underset(hto0)"lim"1/h[(acx^(2)+achx+bcx+adx+adh+bd-acx^(2)-achx-adx-bcx-bch-bd)/(c(x+h)+d(cx+d))]` `underset(hto0)"lim"1/h[(adh-bch)/(c(x+h)+d(cx+d)]]` `=(ac-bd)/(c(x+h)+d(cx=d))` `=(ac-bd)/(cx+d)^(2)` |
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| 62. |
`(secx-1)(secx+1)` |
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Answer» Let `y=(secx-1)(secx+1)` `y=(sec^(2)-1) [therefore (a+b)(a-b)=a^(2)-b^(2)]` `=tan^(2)x` `therefore (dy)/(dx) 2tanx. d/(dx)tanx` `=2tanx.sec^(2)x` [By chain rule] |
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| 63. |
Differentiate with respect to x,`(x+1/x)^(3)` |
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Answer» Let `y=(x+1/x)^(3)` `therefore (dy)/(dx) = d/(dx)(x+1/x)^(3)= 3(x+1/x)^(3-1)d/(dx)(x+1/x)` [by chain rule] `=3(x+1/x)^(2)(1-1/x^(2))` `=3x^(2)-3/x^(2)-3/x^(4)+3` |
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| 64. |
For `xgt0, lim_(xto0) {(sinx)^(1//x)+((1)/(x))^sinx}`, is |
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Answer» `lim_(x->o^+)(sinx)^(1/x)+(1/x)^(sinx)` let`L=lim_(x->0^+)(1/x)^sinx` `lnL=lim_(x->0^+)sinx ln(1/x)` `=lim_(x->0)ln(1/x)/(cosecx)` LH hospital `lnL=(x(1/x^2))/cot^2x=1/(xcot^2x)=tan^2x/x` `lim_(x->0^+)tanx/x*tanx=0` `lnL=0` `L=e^0=1` |
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| 65. |
Evaluate: `lim_(xrarra) (sqrt(x)-sqrt(a))/(x-a)` |
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Answer» `underset(xrarra)"lim"(sqrt(x)-sqrt(a))/((sqrt(x)-sqrt(a))(sqrt(x)+sqrt(a)))` `=underset(xrarra)"lim"(1)/(sqrt(x)+sqrt(a))` `=(1)/(sqrt(a)+sqrt(a))=(1)/(2sqrt(a))` |
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| 66. |
Evaluate: `lim_(xrarr1) ((2)/(x^(2)-1)+(1)/(1-x))` |
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Answer» `underset(xrarr1)((2)/(x^(2)-1)+(1)/(1-x))` `=underset(xrarr1)("lim")((2)/(x^(2)-1)-(1)/(x-1))` `=underset(xrarr1)("lim")(2-(x+1))/(x^(2)-1)` `=underset(xrarr1)("lim")(1-x)/((x-1)(x+1))` `=underset(xrarr1)("lim")(-1)/(X+1)=(-1)/(1+1)=(-1)/(2)` |
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| 67. |
Evaluate: `lim_(xrarr1) (x-1)/(2x^(2)-7x+5)` |
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Answer» `underset(xrarr1)"lim"(x-1)/(2x^(2)-7x+5)` `=underset(xrarr1)"lim"(x-1)/((x-1)(2x-5))` `=underset(xrarr1)"lim"(1)/(2x-5)` `=(1)/(2-5)=-(1)/(3)`. |
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| 68. |
Evaluate: `lim_(xrarr1) (1-sqrt(X))/((cos^(-1)sqrt(x))^(2))` |
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Answer» `underset(xrarr1)"lim"(1-sqrt(x))/((cos^(-1)sqrt(x))^(2 ))` Let `x=cos^(2)theta` `=underset(thetararr0)"lim"(1-cos theta)/(theta^(2))rArr cos^(2)thetararr1` `=underset(thetararr0)"lim"(2sin^(2)(theta)/(2))/(theta^(2))` `=underset(thetararr0)"lim"2[(sin(theta)/(2))/(theta)]^(2)= underset(thetararr0)2[(sin(theta)/(2))/(2.(theta)/(2))]^(2)` `=2xx(1)/(4)=(1)/(2)` |
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| 69. |
`lim_(h->0) (e^((x+h)^2)-e^(x^2))/h` |
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Answer» `lim_(h->0)((e^(x+h)^2-e^x^2)/h)` `lim_(h->0)((e^(x+h)^2*2(x+h)-0)/1)` `(e^x^2*2x-0)/1` `2xe^x^2`. |
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| 70. |
Find the sum of an infinite geometric series whose first term is the limit of the function `f(x)=(tan x-sin x)/(sin^3x)` as `x->0` and whose common ratio is the limit of the function `g(x) =(1-sqrt(x))/(cos^(-1)x)^2` as x->1 |
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Answer» `a=lim_(x->0)(tanx-sinx)/(sin^3x)` `=lim_(x->0)(sinx(secx-1))/(sin^3x)` `=lim_(x->0)(seccx-1)/sin^2x*(secx+1)/(secx+1)` `=lim_(x->0)(sec^2x-1)/(sin^2x(secx-1))` `=lim_(x->0)(tan^2x)/(sin^2x(secx+1))` `=(sec^2 0)/(sec 0+1)=1/(1+1)=1/2` `a=1/2` `r=lim_(x->1)(1-sqrtx)/(cos^(-1)x)^2*(1+sqrtx)/(1+sqrtx)` `1/(1+sqrt1)lim_(x->1)(1-x)/(cos^(-1)x)^2` `1/2lim_(t->0)(1-cost)/t^2` `1/2lim_(t->0)(2sin^2(t/2))/t^2` `lim_(t->0)1/4((sin(t/2))/(t/2))((sin(t/2))/(t/2))` `=1/4``a=1/2,r=1/4` `s_oo=a/(1-r)=(1/2)/(1-1/4)=1/2*4/3=2/3.` |
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| 71. |
If `lim_(x->0) (a+bx sin x + c cosx)/x^4=2` then a=, b=, c= |
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Answer» `lim_(x->0) (a+bxsinx+c cosx)/x^4 = 2` Here, we will use, `sinx = x-x^3/(3!)+x^5/(5!)+...` `cosx = 1-x^2/(2!)+x^4/(4!)+...` So, given equation becomes, `lim_(x->0) (a+(bx^2-(bx^4)/6+...)+(c-(cx^2)/2+(cx^2)/24+...))/x^4 = 2` Now, terms of `x` with power less than `4` and more than `4` will be `0` . `:. a+c = 0=> a = -c->(1)` `b-c/2 = 0=>c = 2b->(2)` Only terms with `x^4` will have some value. ` :. -b/6+c/24 = 2` `=>-4b+c = 48` From (2), `=>-4b+2b = 48 => b = -24` `=>c = 2**(-24) = -48` `=a = -c = 48` |
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| 72. |
`lim_(x->(pi/4)) (sqrt(2) cosx-1)/(cot x-1)` |
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Answer» `cos x -> 1/sqrt2 ` `cot x -> 1` `lim_(x-> pi/4) (d/dx(sqrt2 cos x - 1))/(d/dx(cot x - 1))` `lim_(x->pi/4) ( sqrt2 (-sinx))/(cosec^2 x)` `= (sqrt2 xx -1/sqrt2)/-(sqrt2)^2` `= -1/-2 = 1/2` Answer |
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| 73. |
Find the value of n, if `lim_(xto2) (x^(n)-2^(n))/(x-2)=80`, `n in N`.A. `3`B. `5`C. `4`D. `6` |
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Answer» Correct Answer - B Given, `underset(xto2)"lim"(x^(n)-2^(n))/(x-2)=80` `rArr n(2)^(n-1)=80` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a)=na^(n-1)]` `rArr n(2)^(n-1)= 5 xx 16` `rArr n xx 2^(n-1)= 5 xx (2)^(4)` `rArr n xx 2^(n-1)= 5 xx (2)^(5-1)` `therefore n=5` |
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| 74. |
`lim_(xrarr4) (4x+3)/(x-2)`A.B.C.D. |
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Answer» `underset(xrarr4)"lim"(4x+3)/(X-2)` `=(4(1)+3)/(4-2)=(19)/(2)` |
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| 75. |
Evaluate: `lim_( x to 2) (x^(2)-4)/(x^(3)-4x^(2)+4x)` |
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Answer» `underset(xrarr2)"lim"(x^(2)-4)/(x^(3)-4x^(2)+4x)` `=underset(xrarr2)"lim"((X-2)(x+2))/(x(x^(2)-4x+4))` `=underset(xrarr2)"lim"((x-2)(x+2))/(x(x-2)^(2))` `=underset(xrarr2)"lim"(X+2)/(x(x-2))=(4)/(0)=oo`. |
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| 76. |
Evaluate `lim_(hto0)(sqrt(x+h)-sqrt(x))/(h)` |
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Answer» Given, `underset(hto0)"lim"(sqrt(x+h)-sqrt(x))/(h) = underset(hto0)"lim"((x+h)^(1//2)-(x)^(1//2))/(x+h-x)` `=underset(hto0)"lim"((x+h)^(1//2)-(x)^(1//2))/((x+h)-x)` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a) = na^(n-1)]` `=1/2x^(1/2-1) = 1/2x^(-1//2)` `[therefore h-0 rArr x+h to x]` `=1/(2sqrt(x))` |
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| 77. |
Evaluate `lim_(xto3) ((x^(2)-9))/((x-3))` |
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Answer» Given, `underset(xto3)"lim"(x^(2)-9)/(x-3) = underset(xto3)"lim"(x^(2)-(3)^(2))/(x-3)` `underset(xto3)"lim"((x+3)(x-3))/((x-3))= underset(xto3)"lim"(x+3)` `=3+3=6` |
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| 78. |
Find `lim_(xrarr5) f(x)`, where `f(x)=|x|-5`. |
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Answer» `f(x)=|x|-5` at x=5 `LHL=underset(Xrarr5^(-)"lim"f(x)` `=underset(hrarr0)"lim"f(5-h)` `=underset(hrarr0)"lim"|5-h|-5` `=5-5=0` `RHL=underset(xrarr5^(+))"lim"f(x)` `=underset(hrarr0)"lim"f(5+h)` `=underset(hrarr0)"lim"|5+h|-5` `=5-5=0` `because LHL=RHL` `therefore underset(Xrarr5)f(x)=0` Let 5-h=x `rARr 5-hrarr5` `rArr hrarr0` Let `5+h=x` `rArr 5+hrarr 5` `rArr hrarr0` |
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| 79. |
Find `lim_(xrarr0) f(x)` where `f(x){{:((x)/(|x|),xne0),(0,x=0):}` |
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Answer» `f(x){{:((x)/(|x|),xne0),(0,x=0):}` at x=0 LHL `underset(xrarr0^(-))"lim"f(x)` `=underset(hrarr0)"lim"f(0-h)` `=underset(hrarr0)(h)/(|-h|)` `=underset(hrarr0)"lim"(h)/(-h)=underset(hrarr0)"lim"(-1)=-1` Let `0-h=x` `rArr 0-hrarr0` `rArr hrarr0` RHL`=underset(Xrarr0^(+))"lim"f(x)` ` underset(hrarr0)"lim"f(0+h)` `=underset(hrarr0)"lim"(h)/(|h |)` ` =underset(hrarr0)"lim"(h)/(h)=underset(hrarr0)(1)=1` `because LHLneRHL` `therefore underset(xrarr0)"lim"` f(x) does not exist. Let `0+h=x` `rArr 0+hrarr0` `rArr hrarr0` |
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| 80. |
`lim_(x-pi/2) (cot x - cosx)/(pi-2x)` equals: (1) `1/8` (2) `1/4` (3) `1/24` (4) `1/16` |
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Answer» Let `x = pi/2 - h`, Then, our limit becomes, `Lim_(h->0) (cot(pi/2-h) - cos(pi/2-h))/(pi -(2(pi/2-h))^3 )` `=Lim_(h->0) (tanh - sinh)/(pi -2(pi/2-h))^3 ` `=Lim_(h->0) (sinh(1-cosh)/(cosh))/(8h^3) ` `=Lim_(h->0) (tanh(2sin^2(h/2)))/(8h^3) ` `=Lim_(h->0) (tanh(sin^2(h/2)))/(4h^3) ` `=Lim_(h->0) (tanh/h)(sin^2(h/2))/(4*4(h/2)^2) ` `=1*1/16*1 = 1/16` `:. ` Option `(4)` is the correct option. |
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| 81. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(x)/(sin^(n)x)` |
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Answer» Let `y=(x)/(sin^(n)x)` `rArr (dy)/(dx)=(d)/(dx)((x)/(sin^(n)x))` `=(sin^(n)xx(d)/(dx)x-x(d)/(dx)(sin^(n)x))/((sin^(n)x)^(2))` `=(sin^(n)x.1-x.n sin^(n-1)x.(d)/(Dx)sinx)/(sin^(2n)x)` `=(sin^(n-1)x[sinx-nxcosx))/(sin^(n+1)x)` |
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| 82. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(x+sec x)(x-tan x)` |
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Answer» Let `y=(x+sec x)(x-tanx)` `rArr (dy)/(dx)=(d)/(dx)[(x +sec x)(x-tanx)]` `=(x+sec x)(d)/(dx)(x-tan x)` `+(x-tan x)(d)/(dx)(x+sec x)` `=(x+secx)(1-sec^(2)x)` `+(x-tan x)(1+sec x tan x)` |
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| 83. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(x)/(1+tanx)` |
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Answer» Let `y=(x)/(1+tan x)` `rArr (dy)/(dx)=(d)/(dx)[(x)/(1+tanx)]` `=((1+tanx)(d)/(dx)x-x(d)/(dx)(1+tan x))/((1+tanx)^(2))` `=(1+tan x-x sec^(2)x)/((1+tan x)^(2))` |
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| 84. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(x+a)` |
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Answer» Let y=x+a `rArr(dy)/(dx)=(d)/(dx)(x+a)` `=(d)/(dx)x+(d)/(dx)a=1+0=1` |
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| 85. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(px+q)((r)/(x)+s)` |
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Answer» Let `y=(pix+q)((r)/(x)+s)` `=pr+psx+(qr)/(x)+qs` `rArr(dy)/(dx)=(d)/(dx)(pr+psx+(qr)/(x)+qs)` `=(d)/(dx)pr+ps(d)/(dx)x+qr(d)/(dx)x^(-1)+(d)/(dx)qs` `=0+ps(1)+qr(-1x^(-2))+0` `=ps-(qr)/(x^(2))` |
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| 86. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(ax+b)(cx+d)^(2)` |
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Answer» Let `y=(ax+b)(cx+d)^(2)` `=(ax+b)(c^(2)x^(2)+2cdx+d^(2))` `rArr(dy)/(dx)=(d)/(dx)(ax+b)(c^(2)x^(2)+2cdx+d^(2))` `=(ax+b)(d)/(dx)(c^(2)x^(2)+2cdx+d^(2))` `+(c^(2)x^(2)+2cdx+d^(2))(ax+b)` `=(ax+b)(2c^(2)x+2cd+0)+(cx+d)^(2).(a+0)` `=(ax+b)2c(cx+d)+a(cx+d)^(2)` `=2c(cx+d)(aax+b)+a(Cx+d)^(2)` |
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| 87. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(ax+b)/(cx+d)` |
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Answer» Let `y=(ax+b)/(cx+d)` `rArr(dy)/(dx)=(d)/(dx)((ax+b)/(cx+d))` `=((cx+d)(d)/(dx)(ax_b)-(ax+b)(d)/(dx)(cx+d))/((cx+d)^(2))` `=(acx+ad-acx-bc)/((cx+d)^(2))=(ad-bc)/((cx+d)^(2))` |
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| 88. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(1+(1)/(x))/(1-(1)/(x))` |
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Answer» Let `y=(1+(1)/(x))/(1-(1)/(x))=(x+1)/(x-1)` `rArr(dy)/(dx)=(d)/(dx)((x+1)/(x-1))` `=((x-1)(d)/(dx)(x+1)-(x+1)(d)/(dx)(x-1))/((x-1)^(2))` `=((x-1).1-(x+1).1)/((x-1)^(2))=-(2)/((x-1)^(2))`. |
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| 89. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(1)/(ax^(2)+bx+c)` |
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Answer» Let `y=(1)/(ax^(2)+bx+c)` `(dy)/(dx)=(d)/(dx)((1)/(ax^(2)+bx+c))` `=((ax^(2)+bx+c)(d)/(dx)(1)-1(d)/(dx)(ax^(2)+bx+c))/((ax^(2)+bx+c)^(2))` `=(0-(2ax+b))/((ax^(2)+bx+c)^(2))=-(2ax+b)/((ax^(2)+bx+c)^(2))` |
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| 90. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(ax+b)/(px^(2)+qx+r)` |
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Answer» Let `y=(ax+b)/(px^(2)+qx+r)` `rArr(dy)/(dx)=(d)/(dx)((ax+b)/(px^(2)+qx+r))` ltbr. `(px^(2)+qx+r)(d)/(dx)(ax+b)` `-(ax+b)(d)/(dx)(px^(2)+qx+r)` `(px^(2)+qx+r)^(2)` `=(px^(2)+qx+r)^(2)` `(apx^(2)+aqx+ar)` `=-(2apx^(2)+aqx+2bpx+bq)` `(px^(2)+qx+r^(2))^(2)` |
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| 91. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(px^(2)+qx+r)/(ax+b)` |
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Answer» Let `y=(px^(2)+qx+r)/(ax+b)` `rArr(dy)/(dx)=(d)/(dx((px^(2)+qx+r)/(ax+b))` `(ax+b)(d)/(dx)(px^(2)+qx+r)` `-(px^(2)+qx+r)(d)/(dx)(ax+b)` `(ax+b)^(2)` `=((ax+b)(2px+q)-(px^(2)+qx+r).a)/((ax+b)^(2))` `=((2apx^(2)+aqx+2bpx+bq)-(apx^(2)+aqx+ar))/((ax+b)^(2))` `=((apx^(2)+2bpx+bq-ar))/((ax+b)^(2))` |
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| 92. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(a)/(x^(4))-(b)/(x^(2))+cosx` |
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Answer» Let `y==(a)/(x^(4))-(b)/(x^(2))+cos x` `=a.x^(-4)-b.x^(-2)+cos x` `rArr(dy)/(dx)=(d)/(dx)(ax^(4)-bx^(-2)++cos x)` `=a(d)/(dx)x^(-4)-b(d)/(dx)x^(-2)+(d)/(dx)cos x` `=a(-4)x^(-5)-b(-2)x^(-3)+(-sin x)` `=(-(4a)/(x^(5))+(2b)/(x^(3))-sinx)` |
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| 93. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `4sqrt(x)-2` |
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Answer» Let `y=4sqrt(x)-2` `rArr(dy)/(dx)=(d)/(dx)(4sqrt(x)-2)` `=4(d)/(dx)x^(1//2)-(d)/(dx)(2)` `=4.(1)/(2)x^(-1//2)-0=(2)/sqrt(x)` |
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| 94. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(ax+b)^(n)` |
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Answer» Let `y=(ax+b)` `rArry^(n)=(ax+b)^(n)` `rArr (d)/(dx)(y^(n))=(d)/(dx)(ax+b)^(n)` `rArr(d)/(dx)(ax+b)^(n)=(d)/(dx)(y^(n))` `=ny^(n-1).(d)/(dx)(y)` `=n(ax+b)^(n-1)(d)/(dx)(ax+b)` `=n.a(ax+b)^(n-1)` |
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| 95. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(ax+b)^(n)(cx+d)^(m)` |
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Answer» Let `Y=(ax+b)^(n)(cx+d)^(m)` `rArr(dy)/(dx)=(d)/(dx)(ax+b)^(n)(cx+d)^(m)` `rArr =(ax+b)^(n)(d)/(dx)(cx+d)^(m)` `+(cx+d)^(m)(d)/(dx)(ax+b)^(n)` `=(ax+b)^(n).m(cx+d)^(m-1).(d)/(dx)(cx+d)` `+(cx+d)^(m).n(ax+b)^(n-1)(d)/(dx)(ax+b)` `=m(ax+b)^(n) (cx+d)^(m-1).c` `+n(cx+d)^(m)(Ax+b)^(n-1)` `=(ax+b)^(n-1)(cx+d)^(m-1)cm(ax+b)+an(cx+d)` |
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| 96. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `sin (x+a)` |
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Answer» Let `y=sin(x+a)` ,brgt `rArr(dy)/(dx)=(d)/(dx)sin(x+a)` `=cos(x+a)(d)/(dx)(x+a)` `=cos(x+a).1=cos(x+a)` |
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| 97. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `co sec x cot x` |
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Answer» Let `y=co sec x cot x` `rArr(dy)/(dx)=(d)/(dx)(cosec x cot x)` `=co sec x(d)/(dx)cot x+cotx(d)/(dx)co sec x` `=co secx(-co sec^(2)x)+cot x(-co sec x cot x)` `=-co sec^(3)x-co sec xcot^(2)x` |
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| 98. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(x^(2)cos((pi)/(4)))/(sin x)` |
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Answer» Let `y=(x^(2)cos((pi)/(4)))/(sin x)` `(dy)/(dx)=cos (pi)/(4).[(sin x (d)/(dx)x^(2)-x^(2)(d)/(dx)sinx)/((sin x)^(2))]` `=(cos (pi)/(4)(2x sin x-x^(2)cos x))/(sin^(2)x)` `=(xcos(pi)/(4)(2sin x-x cos x))/(sin^(2)x)` |
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| 99. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(cos x)/(1+sin x)` |
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Answer» Let `y=(cos x)/(1+sin x)` `rArr (dy)/(dx)=(d)/(dx)((cos x)/(1+sin x))` `=((1+sin x)(d)/(dx)cos x -cos x(d)/(dx)(1+sin x))/((1+sinx)^(2))` `=((1+sin x)(-sin x)-cos x.(cos x))/((1+sinx)^(2))` `=(-sinx-sin^(2)x-cos^(2)x)/((1+sinx)^(2))` `=(-sinx-(sin^(2)x+cos^(2)x))/(1+sinx)^(2))=(-sinx-1)/((1+sinx)^(2))` `=(-1(1+sinx))/((1+sinx)^(2))=(-1)/(1+sinx)` |
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| 100. |
Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(sinx+cosx)/(sin x-cos x)` |
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Answer» Let `y=(sin x+cos x)/(sin xj-cos x)` `rArr(dy)/(dx)=(d)/(dx)((sinx+cosx)/(sin x-cos x))` `(sin x-cos x)(d)/(dx)(sin x+cos s)` `=(-(sin x+cos x)(d)/(dx)(sin x-cos x))/((sin xj-cos x)^(2))` `(sin x-cos x)(cos x-sinx)` `(2sin x cos x-sin^(2)x-cos^(2)x)` `=(-(sin^(2)x+cos^(2)x+2sin x cos x))/((sin x-cos x)^(2))` `(-1-1)/((sin x-cos x)^(2))=(-2)/((xin x-cos x)^(2))` |
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