COMPACT QUESTIONS (DIFFERENTIATION) -2

EXERCISE-A

Find dy/dx:

1) sin⁻¹x+ sin⁻¹√(1- x²).                          0

2) sin⁻¹x w.r.t. cos⁻¹√(1- x²).                 1

3) tan⁻¹{2x/(1- x²)} w.r.t sin⁻¹{2x/(1+ x²)}.   1

4) x¹⁰ w.r.t. log₁₀x.                  10logₑx. x¹⁰

5) tan⁻¹{sinx/(1- cosx)}             -1/2

6) The differential coefficient of log(tanx) is
a) Secx b) cotx c) 2sec2x d) 2 cosec2x

7) tan⁻¹{cosx/(1+ sinx)}.
a) -1/2 b) -1 c) 1 d) 1/2

8) sec⁻¹ {(x+1)/(x-1)} + sin⁻¹{(x -1)/(x+1)}
a) 1 b) (x-1)/(x+1) c) 0 d) (x+1)/(x-1)

9) sin(cos⁻¹x) w.r.t. cos⁻¹x is
a) cosx b) - x c) x d) sin x

10) xˣ w.r.t. x logx
a) x logx b) xˣ c)  x⁻ˣ d) none 

11) sinx° at x=60
a) 0 b) 1/2 c) √3/2 d) none

12) x+ x²/2 + x³/3 + x⁴/4+ ......∞, 
a) y b) eʸ c) e⁻ʸ  d) none 

13) The function y= cos⁻¹(sinx) is not differentiable at
a) x=π b) x=2π c) x=π/2 d) none 

14) f(logx) w.r.t.x where f(x)= logx
a) x/logx b) logx/x c) 1/(x logx) d) none 

15) tan⁻¹{(cosx - sinx)/(sinx + cosx)
a) -1 b) 0 c) 1  d) none

16) x|x|
a) 2x b)  -2x  c) 2|x|  d) none

17) If f(x) be an even function, then f'(0) is
a) -1 b) 0 c) 1 d) none

18) If f(x)= logₑx, g(x)= x², h(x)= eˣ and F(x)= f{g(x)}, then dF/dx is equal to
a) 2 b) 2x  c) x² d) 0

19) 2x tan⁻¹x - logₑ(1+ x²).         2tan⁻¹x

20) tan²y = (1+ cos2x)/(1- cos2x).      ±1

21) tan⁻¹[{√(1+ x²) -1}/x] w.r.t. tan⁻¹x.   1/2

22) log √(x²+ y²)= tan⁻¹(y/x).     (x+ y)/(x - y)

23) xʸ + yˣ = aᵇ.           -(yxʸ⁻¹ + yˣlogy)/(xʸ logx + xyˣ⁻¹)

24) xˢᶦⁿˣ + log sinxˣ.        xˢᶦⁿˣ[sinx/x + cosx log x]+ cot(xˣ). xˣ(1+ logx)

25) sin⁻¹{x². √(1- x) - √x. √(2- x⁴)}.    2x/√(1- x⁴)  - 1/2√(x - x²).

26) tan⁻¹[(tanx + Secx -1)/(tanx - Secx +1)].                              1/2

27) y= cot⁻¹{√(1+ t²) - t}; x= tan⁻¹{(1+t)/(1- t)}.                         -1/2

28) y= (sinx)ᶜᵒˢˣ + (cosx)ˢᶦⁿˣ. (sinx)ᶜᵒˢˣ(cosx  + cot x - sinx log sinx)+ (cosx)ˢᶦⁿˣ(cosx log cosx - sinx tanx)

29) ₓsin⁻¹x w.r.t. sin⁻¹x.      ₓsin⁻¹x{logx + √(1- x²)/x . sin⁻¹x}

30) ₑxˣ + √(sin√x).        ₑxˣ .xˣ(1+ logx) + cos√x/4√(x sun√x).

31) xʸ. yˣ = x+ y.       y/x [x -y(x+ y)- x(x+ y)logy]/[y(x+ y)logx + x(x + y)- y]

32) y= ₓyˣ.         {y logy(1+ x logx logy}/{x logx(1- x logy)}.

33) y= b tan⁻¹{(x/a) + tan⁻¹(y/x)}.       b/a. {(x²+ y²- ay)/((x²+y²)sec²(y/b) - bx)}.

EXERCISE-B

1) If y= log₁₀x, find the value of x dy/dx.   log₁₀e

2) If y=f{f(x)}, f(0)=0 and f'(0)=2, find y'(0).   4

3) Given y= x³- 8x+7 and x= f(t). If x=3, when t=0 and df/dt=2, find dy/dx when t=0.               38

4) Find points on the curve y= x + 1/x where dy/dx=0.               (1,2) and (-1,-2)

5) Find sin²x at x=π/4.                         1

6) If f(x)= (x -1)/(2x²-7x +5), when x≠1,
=1/3, when x=1,
Then find the value of f'(1).             2/9

EXERCISE-C

Prove:
1) If x³+ y³- 3axy=0, show dy/dx= (ay - x²)/(y²- ax).

2) Show that the derivative of an even function is always an odd function.

3) If log(1+ sin2x)+ 2log sec(π/4- x) then dy/dx=0

4) If x²+ y²= t - 1/t and x⁴+ y⁴= t²+ 1/t² then x³y dy/dx = 1.

5) If xʸ = eˣ⁻ʸ, then dy/dx= logx/(1+ logx)² = logx/(log ex)²

6) If ₑsin⁻¹x and z= ₑ- cos⁻¹x then dy/dz = constant.

7) y= tan⁻¹{(a cosx- b Sinx)/(bcosx+ a Sinx)} show dy/dx=-1

8) y= cos⁻¹{(2 cosx+ 3sinx)/√13} show dy/dx= 1.

9) f(x)= {(a+ x)/(b+ x)}ᵃ⁺ᵇ⁺²ˣ, show f'(0)= [log(a²b²) + (b²- a²)/ab] (a/b)ᵃ⁺ᵇ.

10) x= tan(y/2) + log tan(y/2) - 2 log(1+ tan(y/2)), show dy/dx= 1/2  siny(1+ cosy + siny)

11) y = cos⁻¹(8x⁴- 8x²+1), show dy/dx + 4/√(1- x²)= 0.

12) If y= m cos2t(1- cos2t) and x= K sin2t(1+ cos2t), show cot t dy/dx = m/K

13) If siny = x sin(a+ y), show dy/dx = (sin²(a+ y))/sina= sina/(1- 2x cosa + x²).

14) If x - √(a²- y²)= a log{y/(a+√(a²- y²)) , show dy/dx= y/√(a²- y²).

15) If y= log{(1+ x)/(1- x)} + 1/2  log{(1+ x+ x²)/(1- x + x²)} + √3 tan⁻¹{x√3/(1- x²), then show dy/dx= 6/(1- x⁶).

16) If y= x²/2 + x√(x²+1)/2 + log{√x + √(x²+1)}, show 2y= x dy/dx + log dy/dx.

17) If √(1- x²)+ √(1- y²)= a(x - y), show dy/dx= √{(1-y²)/(1- x²)}.

18) {√(1+ x²) - √(1- x²)}/{√(1+ x²)+ √(1- x²)} w.r.t. √(1- x⁴) is show {√(1- x⁴) -1}/x⁶

19) If logx = tan⁻¹{(y -x²)/x²}, show dy/dx= 2x(1+ tan(logx))( x sec²(logx)).

20) If y = log{a+ b tan(x/2)}/{a - b tan(x/2) & z= 1/{a² cos²(x/2) - b²sin²(x/2)}, show dy/dx = ab/(a²+ b²) . (a² cot(x/2) - b² tan(x/2)).

21) If y= 1+ a/(x - a) + bx/{(x-a)(x - b)} + cx²/{(x -a)(x - b)(x - c), show dy/dx= y/x {a/(a - x) + b/(b - x) + c/(c - x)}.

22) If 2 {(a²- b²)/(a²+ b²)} . x{pᵖ√x/(p+1) + qᑫ√x/(q+1), then show dy/dx= {(a+ b)/(a- b)}⁽ᑫ⁺ᵖ⁾/⁽ᑫ⁻ᵖ⁾, when x= {(a+ b)/(a- b)}²ᵖᑫ/⁽ᑫ⁻ᵖ⁾

23) If (m - n cosy)(m+ n cosx)= m² - n², show dy/dx= √(m²- n²)/(m+ n cosx).

 

Comments

Post a Comment

Popular posts from this blog

SHORT QUESTIONS- X (All chapter)

Image
GOODS AND SERVICES TAX Multiple Choice Questions 1) IGST is charged on: a) interstate transaction b) intrastate transaction c) both a, b d) none 2) GST payable is equals to a)  ITC --output GST b) output GST + ITC c) output GST - ITC d) output GST 3) A dealer in Mumbai sold a washing machine to a consumer in Mumbai for Rs18000. If the rate of GST is 18% and GST, then SGST is: a) Rs1620 b) Rs3240 c) nil d) none 4) In a transaction from Delhi to Lucknow, MRP = Rs10000, discount= 10%, GST= 28%. Here IGST is: a)  Rs2520 b) Rs5040 c) nil  d) none 5) A refrigerator was sold for Rs15000 under intrastate transaction from station A to station B and the GST rate is 18%. CGST is: a) Rs1450 b) Rs 1350 c) Rs1300 d) Rs2700  6) A dealer purchased goods worth Rs 15000 and sold them for Rs21000 within the state. If the rate of GST is 12%, then the net SGST paid by the dealer is: a) Rs360 b) Rs720 c) Rs800 d) Rs850  7) A dealer in Delhi sold a water heater whose marked is Rs22000...
Read more

MULTIPLE CHOICE QUESTIONS -XII

MAXIMUM AND MINIMUM 1) The minimum value of f(x)= x²+ 250/x is a) 55  b) 25 c) 50 d) 75  2) If  f(x)=  1/(4x²+ 2x +1) , then its maximum value is  a) 2/3 b) 4/3  c) 3/4 d) 1  3) The function  f(x)=  2x³- 3x²- 12x +4 has a) no maximum and minimum b) one maximum and one minimum  c) two maximum d)  two minimum  4) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a is  a) 3a/2  b) √2a/3 c) 2 a/√3 d)  a/√3 5) Maximum value of (logx)/x in [0,∞) is a) (log2)/2 b) 0 c) 1/e d) e 6) let the function f: R--> R be defined by f(x)= 2x + cosx; then f(x).. a) has maximum value at x=0. b) has minimum value at x=π c) is a decreasing function d) is an increasing function  7) The maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is a) a- b b) a+ b c) √(a²+ b²) d) √(a²- b²) 8) Let x and y be two variables and x...
Read more

LAST REVISED MIXED QUESTION - XII

1) If A= 3 &  B= (1  -5  7),                1              -2 Verify that (AB)' = B'A'. 2) Find a 2 x 2 Matrix B such that BA = A where A= 1    -2  = 6     0                        4       2       1     4     0     6                        -1      1 3)  Find the equation of the tangent to the curve √x + √y = a, at the point (a²/4 , a²/4).      2y + 2x - a²=0 4) Discuss the continuity of the function f(x) at x=0, where  f(x)=   x, if x >0            1, if x=0           -x, if x <0                 discontinue  5 ...
Read more