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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.
| 51. |
`tan^(-1)` का `cos^(-1)((1-x^(2))/(1+x^(2)))` के सापेक्ष अवलन गुणांक ज्ञात कीजिये । |
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Answer» माना `y_(1)=tan^(-1)((2x)/(1-x^(2)))` `x=tan theta` रखने पर , `y_(1)=tan^(-1)((2tan theta)/(1-tan^(2)theta))` या `y_(1)=tan^(-1)(tan 2 theta)` या `y_(1)=2theta` या `y_(1)2 tan^(-1)x` या `(dy_(1))/(dx)=(2)/(1+x^(2)) ` ...(i) पुनः माना `y_(2)=cos^(-1)((1-x^(2))/(1+x^(2)))` `x=tan theta` रखने पर , `y_(2)=cos^(-1)((1-tan^(2)theta)/(1+tan^(2)theta))` `=cos^(-1)(cos 2 theta)` `=2 theta = 2 tan^(-1)x` `:. (dy_(2))/(dx)=(2)/(1+x^(2))` अब , `(dy_(1))/(dy_(2))=((dy_(1))/(dx))/((dy_(2))/(dx))=((2)/(1+x^(2)))/((2)/(1+x^(2)))=1` |
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| 52. |
`x` के सापेक्ष अवकलन कीजिये: `(5x)^(3 cos x 2x)` |
| Answer» Correct Answer - `(5x)^(3 cos 2x)[(3 cos 2x)/(x)- 6 sin2x log 5x]` | |
| 53. |
यदि `y=A cos nx+Bsinnx` तो `(d^(2)y)/(dx^(2))=`A. `n^(2)y`B. `-y`C. `-n^(2)y`D. इसमें से कोई नहीं |
| Answer» Correct Answer - C | |
| 54. |
यदि `y=cos(sin x^(2))`, तो `x=sqrt(x/2)` पर `(dx)/(dx)` का मान हैA. -2B. 2C. `-2sqrt(x/2)`D. 0 |
| Answer» Correct Answer - D | |
| 55. |
`e^(x)` sin x का अवलन गुणांक है -A. `e^(x) (sin x-cos x)`B. `e^(x) (sin x+cos x)`C. `e^(x) (cos x-sin x)`D. `e^(x)/sqrt2 (cos x+sin x)` |
| Answer» Correct Answer - B | |
| 56. |
यदि `y=x^(x)`, तब `(dx)/(dx)=`A. `x^(x)(1+log_(e)x)`B. `x^(x)(1+1/x)`C. `1+log x`D. `x^(x)log x` |
| Answer» Correct Answer - A | |
| 57. |
यदि `y=("log x)/x` तो `(dy)/(dx)` का मान हैA. `(1-log x)/(x^(2))`B. `(1+log x)/(x^(2))`C. `(log x-1)/(x^(2))`D. इसमें से कोई नहीं |
| Answer» Correct Answer - A | |
| 58. |
`x` के सापेक्ष अवकलन कीजिये: `(cos^(-1)(x/2))/sqrt(2x+7), -2 lt x lt 2` |
| Answer» Correct Answer - `-[(1)/(sqrt(4-x^(2))sqrt(2x+7))+(cos^(-1)""(x)/(2))/((2x+7)^((3)/(2)))]` | |
| 59. |
यदि `xy=c^(2)` तो `(dy)/(dx)=`A. -`c^(2)/x^(2)`B. `x-c^(2)/x^(2)`C. `c^(2)/y^(2)`D. `-c^(2)/y^(2)` |
| Answer» Correct Answer - B | |
| 60. |
यदि `(x^(2)+y^(2))^(2)=xy,` तो `(dy)/(dx)` का मान ज्ञात कीजिये । |
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Answer» `(x^(2) + y^(2) ) ^(2) = xy " "` …(1) दोनों पक्षों का x के सापेक्ष अवकलन करने पर , ` 2(x^(2) + y^(2)) ( 2x + 2y (dy)/(dx)) = x (dy)/(dx) + y ` ` rArr 4 ( x^(2) + y^(2)) ( x + y (dy)/(dx)) = x (dy)/(dx) + y ` ` rArr { 4 y ( x^(2) + y^(2) ) -x} (dy)/(dx) = x (dy)/(dx)=y- 4(x^(2) + y^(2) ) x` `rArr (dy)/(dx) = (y-4(x^(2) + y^(2) )x)/(4y (x^(2) + y^(2) - x ))` ` = (y-4sqrt(xy) x)/(4y sqrt (xy)- x) " "` [समीकरणों (1) से ] ` = ( sqrt(y) ( sqrt(y)-4x sqrt(x)))/(sqrt(x)(4y sqrt(y) - sqrt(x)))` |
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| 61. |
`sin^(2)y+cosxy=k` |
| Answer» Correct Answer - `(ysinxy)/(sin2y-xsinxy)` | |
| 62. |
`xy+y^(2)=tanx+y` |
| Answer» Correct Answer - `(sec^(2)x-y)/(x+2y-1)` | |
| 63. |
`xy=e^(x-y)` |
| Answer» Correct Answer - `(y(x-1))/(x(y+1))` | |
| 64. |
`x^(3)+x^(2)y+xy^(2)+y^(3)=81` |
| Answer» Correct Answer - `-(3x^(2)+2xy+y^(2))/(x^(2)+2xy+3y^(2))` | |
| 65. |
यदि फलन `f(x)={{:((x^(2)-4)/(x-2),xne2),("k,",x=2):}x=2` पर संतत है तो k का मान ज्ञात कीजिये । |
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Answer» `f(2) = k ` दायी सीमा ` = underset ( x to 2 + 0) lim f ( x ) ` ` = underset ( h to 0 ) lim f(2+h) ` ` = underset ( h to 0 ) lim ((2+h)^(2)-4)/(2+h-2) ` ` = underset ( h to 0 ) lim (4 + h ^(2) + 4h - 4 )/h` `underset ( h to 0 ) lim ( h + 4) = 4 ` बायीं सीमा ` = underset ( x to 2 - 0 ) lim f ( x) ` ` = underset ( h to 0 ) lim f ( 2 - h ) ` ` = underset (h to 0 ) lim (( 2-h)^(2) - 4 )/(2-h -2)` `= underset ( h to 0 ) lim ( 4+h^(2) - 4h - 4 )/(-h)` ` = underset ( h to 0 ) lim ( 4 - h ) = 4 ` यदि फलन `f(x),x = 2 ` पर संतत है तो दायी सीमा = बायीं सीमा = k ` " या " 4=4 = k " या " k=4 ` |
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| 66. |
सिद्ध कीजिये कि फलन `f(x)={{:(x^(2)sin(1/x), if ,xne0),(0,if,x=0):}` द्वारा परिभाषित एक संतत फलन है |
| Answer» Correct Answer - हाँ, प्रत्येक `x in R` के लिए f संतत है। | |
| 67. |
`f(x)={{:("2x, if " x lt0),("0, if " 0lexle1),("4x, if " xgt1):}` |
| Answer» Correct Answer - केवल `x=1` असंतायता का बिंदु है। | |
| 68. |
क्या `f(x)={{:("3, if " 0lexle1),("4, if " 1lt x lt3),("5, if " 3lexle10):}` |
| Answer» Correct Answer - `x=1` और `x=3` पर f संतत नहीं है। | |
| 69. |
क्या `f(x)={{:("x+1, if " xge1),(x^(2)+1", if " xlt1):}` द्वारा परिभाषित फलन, एक सतत फलन है? |
| Answer» Correct Answer - असंतत्यत का कोई बिंदु नहीं | |
| 70. |
क्या `f(x)={{:(x "," if xle 1),("5," if xgt1):}` द्वारा परिभाषित फलन f ` x=1` पर संतत है? |
| Answer» Correct Answer - f,`x=0` और `x=2` पर संतत है: परन्तु `x=1` पर संतत नहीं है। | |
| 71. |
`f(x)={{:(x^(10)-1", if " xle1),(x^(2)", if " xgt1):}` |
| Answer» Correct Answer - `x=1` पर f असंतत है। | |
| 72. |
क्या `f(x)={{:("x+5, if " xle1),("x-5, if " xgt1):}` द्वारा परिभाषित फलन, एक सतत फलन है? |
| Answer» Correct Answer - `x=1` पर f असंतत नहीं है। | |
| 73. |
`y=sin^(-1)((1-x^(2))/(1+x^(2))), 0 lt x lt 1` |
| Answer» Correct Answer - `-2/(1+x^(2))` | |
| 74. |
`y=cos^(-1)((1-x^(2))/(1+x^(2))), 0 lt x lt 1` |
| Answer» Correct Answer - `2/(1+x^(2))` | |
| 75. |
यदि `y=cos^(-1)2xsqrt(1-x^(2))` तो सिद्ध कीजिये कि `(dy)/(dx)=-2(1/(sqrt(1-x^(2))))`. |
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Answer» `y=cos^(-1)(2xsqrt(1-x^(2)))` `x=sin theta ` रखने पर , `y=cos^(-1)(2sin thetasqrt(1-sin^(2)theta))` या `y=cos^(-1)(2sin theta cos theta)` या `ycos^(-1)(sin 2 theta)` या `y=cos^(-1)[cos((pi)/(2)-2theta)]` या `y=(pi)/(2) - 2 theta` या `y=(pi)/(2) - 2 sin^(-1) x ` या `(dy)/(dx)=-2((1)/(sqrt(1-x^(2))))` ( इति सिध्दम ) |
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| 76. |
फलन `f(x)=sin^(-1)2xsqrt(1-x^(2))` का `sin^(-1)x` x के सापेक्ष अवलन गुणांक ज्ञात कीजिये । |
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Answer» `(d{f(x)})/(dsin^(-1)x)=((d)/(dx)f(x))/((d)/(dx)(sin^(-1)x))` `((d)/(dx){sin^(-1)(2xsqrt(1-x^(2)))})/((1)/(sqrt(1-x^(2))))`, जहाँ `x=sin theta` `sqrt(1-x^(2))(d)/(dx) {sin^(-1)(2 sin theta cos theta)}` `=sqrt(1-x^(2))(d)/(dx){sin^(-1)(sin 2 theta)}` `=sqrt(1-x^(2))(d)/(dx)(2 theta)` `=sqrt(1-x^(2))(d)/(dx) (2 sin^(-1)x)` `=sqrt(1-x^(2))(2)/(sqrt(1-x^(2)))=2`. |
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| 77. |
`y=sin^(-1)((2x)/(1+x^(2)))` |
| Answer» Correct Answer - `2/(1+x^(2))` | |
| 78. |
`y=cos^(-1)((2x)/(1+x^(2))), 0 lt x lt 1` |
| Answer» Correct Answer - `-2/(1+x^(2))` | |
| 79. |
`f(x)={{:(kx^(2)", if " xle2),("3, if " xgt2):}` , द्वारा परिभाषित फलन `x=2` पर |
| Answer» Correct Answer - `k=3/4` | |
| 80. |
`f(x)= {{:("kx+1, if " xlepi),("cos x, if " xgt pi):}` द्वारा परिभाषित फलन `x=pi` पर |
| Answer» Correct Answer - `k=-2/pi` | |
| 81. |
`y=sin^(-1)(2xsqrt(1-x^(2))), -1/sqrt(2) lt x lt 1/sqrt(2)` |
| Answer» Correct Answer - `2/sqrt(1-x^(2))` | |
| 82. |
`y=tan^(-1)[(3x-x^(3))/((1-3x^(2))]],(-1/sqrt(3) lt x lt 1/sqrt(3))` |
| Answer» Correct Answer - `3/(1+x^(2))` | |
| 83. |
`y=sec^(-1)(1/(2x^(2)-1)), 0 lt x lt 1/sqrt(2)` |
| Answer» Correct Answer - `-2/(sqrt(1-x^(2))` | |
| 84. |
`sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5))` |
| Answer» Correct Answer - `(1)/(2)sqrt(((x-1)(x-2))/((x-3)(x-4)(x-5)))(1)/(x-1)+(1)/(x-2)-(1)/(x-3)-(1)/(x-4)-(1)/(X-5)` | |
| 85. |
`dy/dx ` ज्ञात कीजिए` , y=(sinx-cosx)^((sinx-cosx)),(pi)/(4)ltx lt (3pi)/(4)` |
| Answer» Correct Answer - `(sinx -cosx)^(sinx -cosx)(cosx+ sinx)(1+ log(sinx-cosx)), sinx gt cosx ` | |
| 86. |
`(x cosx)^(x)+(x sinx)^(1/x)` |
| Answer» Correct Answer - `(x cosx)^(x)[1-x tan x+ log(x cosx)]+(x sinx)^((1)/(x))[(x cotx+1-log(x sinx))/(x^(2))]` | |
| 87. |
`cosx.cos 2x.cos3x` |
| Answer» Correct Answer - `- cosx cos 2x cos 3x [tan x +2 tan 2x + 3 tan 3x]` | |
| 88. |
`log(logx), x gt 1` |
| Answer» Correct Answer - `1/(xlogx), x gt 1` | |
| 89. |
`x=2at^(2),y=at^(4)` |
| Answer» Correct Answer - `t^(2)` | |
| 90. |
`(cosx)/(logx), x gt 0` |
| Answer» Correct Answer - `-(xsinx.logx+cosx)/(x(logx)^(2)), x gt 0` | |
| 91. |
`cos(logx+e^(x))` |
| Answer» Correct Answer - `-(1/x+e^(x))sin(logx+e^(x)), x gt 0` | |
| 92. |
`x=a costheta, y=b costheta` |
| Answer» Correct Answer - `(b)/(a)` | |
| 93. |
`x=sint, y=cos 2t` |
| Answer» Correct Answer - `-4 sint` | |