IIT JEE & NEET Question Bank with Solution: 09/11/21

Differentiation

NCERT Continuity and Differentiability
This MCQs are based on the Mathematics Part I Textbook of Maharashtra (click here for reference).
Answers are in Bold, Italic and underline form of option name.

Q.1. If $y equals tan to the power of negative 1 end exponent open parentheses square root of fraction numerator a minus x over denominator a plus x end fraction end root close parentheses comma space w h e r e space minus a less than x less than a space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator x over denominator square root of a squared minus x squared end root end fraction$

(B) $fraction numerator a over denominator square root of a squared minus x squared end root end fraction$

(C) $negative fraction numerator 1 over denominator 2 square root of a squared minus x squared end root end fraction$

(D) $fraction numerator 1 over denominator 2 square root of a squared minus x squared end root end fraction$

Q.2. If $s e c open parentheses fraction numerator x plus y over denominator x minus y end fraction close parentheses equals a squared comma space t h e n space fraction numerator d squared y over denominator d x squared end fraction equals$ _______.

(A) y

(B) x

(D) 0

Q.3. If $y equals cos e c to the power of negative 1 end exponent open parentheses fraction numerator x squared plus 1 over denominator x squared minus 1 end fraction close parentheses space plus space cos to the power of negative 1 end exponent open parentheses fraction numerator x squared minus 1 over denominator x squared plus 1 end fraction close parentheses comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator negative 2 over denominator square root of x to the power of 4 minus 1 end root end fraction$

(B) 0

(C) $fraction numerator 2 over denominator square root of x to the power of 4 minus 1 end root end fraction plus fraction numerator 2 over denominator square root of x to the power of 4 plus 1 end root end fraction$

(D) $fraction numerator 2 over denominator square root of x to the power of 4 minus 1 end root end fraction$

Q.4. If $x to the power of y equals y to the power of x comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator x left parenthesis x space log space y minus y right parenthesis over denominator y left parenthesis y space log space x minus x right parenthesis end fraction$

(B) $fraction numerator y left parenthesis y space log space x minus x right parenthesis over denominator x left parenthesis x space log space y minus y right parenthesis end fraction$

(C) $fraction numerator y squared left parenthesis 1 minus log space x right parenthesis over denominator x squared left parenthesis 1 minus log space y right parenthesis end fraction$

(D) $fraction numerator y left parenthesis 1 minus log space x right parenthesis over denominator x left parenthesis 1 minus log space y right parenthesis end fraction$

Q.5. If $x equals a left parenthesis cos space theta plus theta space sin space theta right parenthesis comma space y equals a space left parenthesis sin space theta space minus space theta space cos space theta right parenthesis space t h e n space open square brackets fraction numerator d squared y over denominator d x squared end fraction close square brackets subscript theta equals straight pi over 4 end subscript equals$

(A) $fraction numerator 8 square root of 2 over denominator a straight pi end fraction$

(B) $negative fraction numerator 8 square root of 2 over denominator a straight pi end fraction$

(D) $fraction numerator 4 square root of 2 over denominator a straight pi end fraction$

Q.6. If $y equals tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator 1 plus square root of 1 minus x squared end root end fraction close parentheses space plus space sin open square brackets 2 tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 minus x over denominator 1 plus x end fraction end root close parentheses close square brackets comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator x over denominator square root of 1 minus x squared end root end fraction$

(B) $fraction numerator 1 minus 2 x over denominator square root of 1 minus x squared end root end fraction$

(C) $fraction numerator 1 minus 2 x over denominator 2 square root of 1 minus x squared end root end fraction$

(D) $fraction numerator 1 minus 2 x squared over denominator square root of 1 minus x squared end root end fraction$

Q.7. If g is the inverse of a function f and f '(x) = $fraction numerator 1 over denominator 1 plus x to the power of 7 end fraction$ , then the value of g'(x) is equal to

(A) $1 plus x to the power of 7$

(B) $fraction numerator 1 over denominator 1 plus open square brackets g left parenthesis x right parenthesis close square brackets to the power of 7 end fraction$

(C) $1 plus open square brackets g left parenthesis x right parenthesis close square brackets to the power of 7$

(D) $7 x to the power of 6$

Q.8. If $x square root of y plus 1 end root plus y square root of x plus 1 end root equals 0 space a n d space x not equal to y space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $1 over open parentheses 1 plus x close parentheses squared$

(B) $negative 1 over open parentheses 1 plus x close parentheses squared$

(D) $negative fraction numerator x over denominator x plus 1 end fraction$

Q.9. If $y equals sin space left parenthesis 2 space sin to the power of negative 1 end exponent x right parenthesis comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator 2 minus 4 x squared over denominator square root of 1 minus x squared end root end fraction$

(B) $fraction numerator 2 plus 4 x squared over denominator square root of 1 minus x squared end root end fraction$

(D) $fraction numerator 1 minus 2 x squared over denominator square root of 1 minus x squared end root end fraction$

Q.10. If $s e c open parentheses fraction numerator x plus y over denominator x minus y end fraction close parentheses equals a squared comma space t h e n space fraction numerator d squared y over denominator d x squared end fraction equals$ _______.

(A) y

(B) x

(D) 0

Q.11. Let $f left parenthesis 1 right parenthesis equals 3 comma space f apostrophe left parenthesis 1 right parenthesis equals negative 1 third comma space g left parenthesis 1 right parenthesis equals negative 4 space a n d space g apostrophe left parenthesis 1 right parenthesis equals negative 8 over 3.$ The derivative of $square root of open square brackets f left parenthesis x right parenthesis close square brackets squared plus open square brackets g left parenthesis x right parenthesis close square brackets squared end root$ w.r.t. x at x = 1 is

(A) $negative 29 over 15$

(B) $7 over 3$

(D) $29 over 15$

Q.12. If y is a function of x and log (x + y) = 2xy, then the value of y'(0) =

(A) 2

(B) 0

(D) 1

Q.13. If $f left parenthesis x right parenthesis equals sin to the power of negative 1 end exponent open parentheses fraction numerator 4 to the power of x plus begin display style 1 half end style end exponent over denominator 1 plus 2 to the power of 4 x end exponent end fraction close parentheses$ which of the following is not the derivative of f(x)

(A) $fraction numerator 2.4 to the power of x log 4 over denominator 1 plus 4 to the power of 2 x end exponent end fraction$

(B) $fraction numerator 4 to the power of x plus 1 end exponent log 2 over denominator 1 plus 4 to the power of 2 x end exponent end fraction$

(C) $fraction numerator 4 to the power of x plus 1 end exponent log 4 over denominator 1 plus 4 to the power of 4 x end exponent end fraction$

(D) $fraction numerator 2 to the power of 2 left parenthesis x plus 1 right parenthesis end exponent log space 2 over denominator 1 plus 2 to the power of 4 x end exponent end fraction$

Q.14. If $x equals t space log space t comma space y equals t to the power of t comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $e to the power of t$

(B) $1 space plus space log space t$

(D) $e to the power of x$

Q.15. If $y equals s e c left parenthesis tan to the power of negative 1 end exponent x right parenthesis space t h e n space fraction numerator d y over denominator d x end fraction space a t space x equals 1 comma space i s space e q u a l space t o colon$

(A) $1 half$

(B) 1

(C) $fraction numerator 1 over denominator square root of 2 end fraction$

(D) $square root of 2$

Q.16. If $y equals 1 minus cos space theta comma space x equals 1 minus sin space theta comma space t h e n space fraction numerator d y over denominator d x end fraction space a t space theta equals straight pi over 4 space i s$

(A) -1

(B) 1

(D) $fraction numerator 1 over denominator square root of 2 end fraction$

Q.17. If y = a cos (log x) and $A fraction numerator d squared y over denominator d x squared end fraction plus B fraction numerator d y over denominator d x end fraction plus C equals 0 comma$ then the values of A, B, C are

(A) $x squared comma negative x comma negative y$

(B) $x squared comma space x comma space y$

(D) $x squared comma negative x comma space y$

Q.18. If $x to the power of y equals e to the power of x minus y end exponent comma space t h e n space fraction numerator d y over denominator d x end fraction equals$

(A) $fraction numerator 1 plus x over denominator 1 plus log space x end fraction$

(B) $fraction numerator log space x over denominator left parenthesis 1 plus log right parenthesis space x squared end fraction$

(D) $fraction numerator 1 minus x over denominator 1 plus log space x end fraction$

Q.19. Derivatives of $tan cubed theta$ with respect to $s e c cubed theta space a t space theta equals straight pi over 3$ is _______.

(A) $3 over 2$

(B) $fraction numerator square root of 3 over denominator 2 end fraction$

(D) $negative fraction numerator square root of 3 over denominator 2 end fraction$

Q.20. If $y equals s e c to the power of negative 1 end exponent open parentheses fraction numerator square root of x minus 1 over denominator x plus square root of x end fraction close parentheses space plus space sin to the power of negative 1 end exponent open parentheses fraction numerator x plus square root of x over denominator square root of x minus 1 end fraction close parentheses comma space t h e n space fraction numerator d y over denominator d x end fraction equals$ .......

(A) x

(B) $1 over x$

(D) 0

Q.21. If $u equals tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 minus x squared end root minus 1 over denominator x end fraction close parentheses space a n d space v equals tan to the power of negative 1 end exponent open parentheses fraction numerator 2 x square root of 1 minus x squared end root over denominator 1 minus 2 x squared end fraction close parentheses comma space t h e n space fraction numerator d u over denominator d v end fraction space a t space x equals 0$ is

(A) 1

(B) $fraction numerator negative 1 over denominator 8 end fraction$

(C) $1 fourth$

(D) $1 over 8$

Q.22. If $2 y equals open parentheses c o t to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 cos space x space plus space sin space x over denominator cos space x space minus space square root of 3 sin space x end fraction close parentheses close parentheses squared comma space x element of space open parentheses 0 comma straight pi over 2 close parentheses space t h e n space fraction numerator d y over denominator d x end fraction$ is equal to

(A) $straight pi over 6 minus x$

(B) $x minus straight pi over 6$

(D) $2 x minus straight pi over 3$

Q.23. If $y left parenthesis alpha right parenthesis equals square root of 2 open parentheses fraction numerator tan space alpha plus c o t space alpha over denominator 1 plus space tan squared space alpha end fraction close parentheses plus fraction numerator 1 over denominator sin squared alpha end fraction end root comma space alpha element of open parentheses fraction numerator 3 straight pi over denominator 4 end fraction comma straight pi close parentheses comma space t h e n space fraction numerator d y over denominator d alpha end fraction space a t space alpha equals fraction numerator 5 straight pi over denominator 6 end fraction$ is

(A) 4

(B) $4 over 3$

(D) -4

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Saturday, September 11, 2021

HSC Differentiation All Solved MCQs: Maharashtra Board