Mathematics DPP NO.
# 07 JEE (Main-Advanced) 2021
Time : 45 Minutes Maximum Marks-45
Special DPP on Methods of Differentiation
1. Differentiate the following functions w.r.t. x:-
5 æp xö
(i) x 2/3 - + 7 tan x (ii) x 2 .lnx.e (iii) ln tan ç + ÷
x
x è 4 2ø
sin x - x cos x 1 + x2 + x4
(iv) (v) 2ln( x - 2) - x 2 + 4 x + 1 (vi)
x sin x + cos x 1 + x + x2
2. Differentiate the following functions w.r.t. x:-
lnlnx
(i) ( lnx ) (iii) ( xlnx )
cos x
(ii) x - 2
x sin x
3. If f(x) = logx(lnx), then f’(x) at x=e is equal to
(A)1/e (B) e (C) 1 (D) zero
d2y
4. If x=at2 and y=2at, then is equal to
dx 2
1 1 1 1
(A) - (B) (C) - (D) -
t2 2at 2 t3 2at 3
dy
5. If x 1 + y + y 1 + x = 0 ,then is equal to
dx
1 1 1 1
(A) (B) - (C) (D) -
(1 + x )
2
(1 + x )
2
(1 + x 2 ) (1 + x2 )
x10
6. Let ɡ(x) is the inverse function of f(x) and f’(x) = . If ɡ(2)= a, then ɡ’(2) is equal to
1 + x2
a 1+ a2 1 + a10 a10
(A) (B) (C) (D)
210 a10 a2 1+ a2
dy
7. If xm.yn = (x+y)m+n, then is equal to
dx
(A) (B) xy (C) (D)
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DPP - 07 Mathematics
dy
8. If y = sec(tan-1x), then at x=1 is equal to
dx
1 1
(A) (B) (C) 1 (D) 2
2 2
dy
9. If 2x + 2y = 2x+y , then is equal to
dx
2y 1 2 x (1 - 2 y )
(A) - (C) 1 - 2
y
(B) (D)
2x 1 - 2x 2 y (2 x - 1)
dy
10. Find if:
dx
æ 1 tö
(i) x = a ç cos t + ln tan 2 ÷ and y = a sin t
è 2 2ø
(ii) x = sin t cos 2t and y = cos t cos 2t
11. Differentiate the following functions w.r.t. x:-
æ -1 1 - cos x ö
(i) y = tan ç tan
ç ÷,0<x< p
è 1 + cos x ÷ø
4x 2 + 3x
(ii) y = tan -1 + tan -1 , 0<x<1
1 + 5x 2
3 - 2x
æ 1 - x2 ö - 1+ x
(iii) y = sin -1 ç 2 ÷
, 0 < x <1 (iv) y = sin 1 , -1 < x < 1
è 1 + x ø 2
12. Let y be an implicit function of x defined by x2x – 2xx cot y – 1 = 0, then y’(1) is equal to
(A) 1 (B) log 2 (C) – log 2 (D) – 1
d2x
13. =
dy 2
-1 -1 -3 -2 -3
æ d2y ö æ d 2 y ö æ dy ö æ d 2 y ö æ dy ö æ d 2 y ö æ dy ö
(A) ç 2 ÷ (B) - ç 2 ÷ ç ÷ (C) ç 2 ÷ç ÷ (D) - ç 2 ÷ç ÷
è dx ø è dx ø è dx ø è dx ø è dx ø è dx ø è dx ø
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DPP - 07 Mathematics
d2y dy
14. If y = (sin -1 x ) 2 + (cos -1 x) 2 , then (1 - x 2 ) 2
-x is equal to
dx dx
(A) 4 (B) 3 (C) 1 (D) 0
t 1
15. The differential coefficient of sin -1 w.r.t. cos -1 is
1+ t 2
1+ t2
(A) 1 " t > 0 (B) – 1 " t < 0 (C) 1 " t ϵ R (D) – 1 " t ϵ R
Answer key
2 5
1. (i) 1/3
+ 2 + 7 sec 2 x (ii) ex (2lnx + 1+ x lnx) (iii) sec x
3x x
x2 2
(iv) (v) - 2x + 4 (vi) 2x – 1
( x sin x + cos x ) x-2
2
æ cos x ö
2. (i) ( lnx ) - sin xln ( lnx ) ÷ (ii) x (1 + lnx ) - ln 2.2
cos x x sin x
ç .cos x
è xlnx ø
lnlnx 1æ æ 2 öö
(iii) ( xlnx ) . ç1 + ln ( lnx ) ç1 + ÷÷
xè è lnx ø ø
3. (A) 4. (D) 5. (B) 6. (B) 7. (D) 8. (A)
9. (A,B,C,D) 10. (i) tan t (ii) – tan3t
1 x 5 2 1
11. (i) sec 2 (ii) (iii) - (iv)
2 2 1 + 25x 2 1 + x2 2 1 - x2
12. (D) 13. (D) 14. (A) 15. (A,B)
.
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