Mathematics - Differentiability and Differentiation

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Question - 1

Consider the function f(x)=|logex|,\(\forall\)x>0.Then,

  • A LHD does not exist at x=1
  • B RHD does not exist at x=1
  • C f is differentiable at x=1
  • D f is not differential at x=1

Question - 2

Consider the function f(x) defined by \(\\ f(x)=\begin{cases} xsin(ln{ x }^{ 2 }),x\neq 0 \\ 0,\quad \quad \quad \quad x=0 \end{cases}\).Then

  • A f is differential at x=0
  • B f is not differential at x=0
  • C LHD exists but RHD does not exists at x=0
  • D RHD exists but LHD does not exists at x=0

Question - 3

If \(\sqrt{1-x^2}-\sqrt{1-y^2}=a(x-y)\), then\(dx\over dy\)is equal to 

  • A \(\sqrt{1-x^2\over1-y^2}\)
  • B \(\sqrt{1-y^2\over1-x^2}\)
  • C \(\sqrt{x^2-1\over1-y^2}\)
  • D \(\sqrt{y^2-1\over1-x^2}\)

Question - 4

Let x=a(cost+logtan\(t\over2\))and y=a sint, then \(dy\over dx\)is

  • A cot t
  • B tan t
  • C -tan t
  • D None of the above

Question - 5

Let f:[-5,5]\(\rightarrow\)R be a differentiable function such that f'(x) does not vanish anywhere, then

  • A f(-5)>f(5)
  • B f(-5)
  • C f(-5)=f(5)
  • D f(-5)\(\neq\)f(5)

Question - 6

If F(x)=f(x).g(x0 and f'(x).g'(x)=c, then

  • A \(F'=c[{f\over f'}+{g\over g'}]\)
  • B \(F'=c[{f\over f'}-{g\over g'}]\)
  • C \({F''\over c}={f''\over f}+{g''\over g}+{2c\over fg}\)

Question - 7

Observe the following columns

Column I Column II

A.The function
   \(f(x)=\begin{cases} { x }^{ 2 }+3x+a;\quad x\le 1 \\ bx+2;\quad \quad \quad x>1 \end{cases}\)
differentiable,\(\forall\ x\epsilon R\) then

 P.a=3

B.The function
    \(f(x)=\begin{cases} \frac { 1 }{ |x| } ;\quad \quad\quad |x|\le 1 \\ ax^{ 2 }+b;\quad |x|>1 \end{cases}\)

differentiable everywhere, then

 Q.b=5
C.The function
   \(f(x)=\begin{cases} { ax }^{ 2 }-bx+2;\quad x<3 \\ bx^{ 2 }-3b;\quad \quad\quad x\ge 1 \end{cases}\)
differentiable everywhere, then

R.\(a={35\over9}\)
 

S.\(b={3\over2}\)
 

T.\(a=-{1\over2}\)
  • A A B C PQ ST R
  • B A B C R SR RP
  • C A B C T SP RP
  • D None of the above

Question - 8

Let f(x)=x2+xg'(1)+g''(2) and g(x)=x2+xf'(2)+f''(3), then

  • A f'(1)=4-f'(2)
  • B g'(2)=8-g'(1)
  • C g''(2)+f''(3)=4
  • D None of the above

Question - 9

Consider the function f(x) defined by\(f(x)=\begin{cases} \frac { { x(e }^{ -1/x }-{ e }^{ 1/x } }{ { e }^{ -1/x }+{ e }^{ 1/x } } ,x\neq 0 \\ o,\quad \quad \quad \quad x=0 \end{cases}\).Then,

  • A  f is continuous and derivable at x=0
  • B f is continuous but not derivable at x=0
  • C  f is not continuous at x=0
  • D None of the above

Question - 10

Consider the function f(x) defined by f(x)=x-2|+|x|+|x+2|.Then,

  • A f is derivable at x=0,2
  • B f is derivable at x=-2,0
  • C f is not derivable at x=-2,2
  • D f is not derivable at x=-2,0,2
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