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CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 11 - MCQExams.com

The value of limx0(sin3xtanx)4 is
  • 0
  • 81
  • 4
  • 1
The value of limxsinπ4xcosπ4x is
  • π/4
  • π/2
  • 0

Given that f(x)
is a differentiable function of x and that f(x) . f(y) =  f(x)+ f(y) + f(xy)2 and that
f(2)=5.

Then f(3) is equal to?

  • 6
  • 24
  • 15
  • 19
lf {f}'({x})={g}({x}) and {g}'({x})=-{f}({x}) for all x and {f}(2)=4= {g}(2), then {f}^{2}(24)+{g}^{2}(24) is
  • 32
  • 24
  • 64
  • 48
Assertion(A): Let { f }({ x }) be twice differentiable function such that f^{ '' }(x)=-{ f }({ x }) and f^{ ' }(x)={ g }({ x }). lf { h }({ x })=[{ f }({ x })]^{ 2 }+[{ g }({ x })]^{ 2 } and { h }(1)=8, then { h }(2)=8

Reason (R): Derivative of a constant function is zero.
  • Both A and R are true R is correct reason of A
  • Both A and R are true R is not correct reason of A
  • A is true but R is false
  • A is false but R is true
Assertion (A): lf f(x)=\cos^{2}x+\cos^{2}\left(x+\dfrac{\pi}3\right)- \cos x \cos \left(x+\dfrac{\pi}3\right) then f'(x)=0

Reason(R): Derivative of constant function is zero
  • Both A & R are true, R is correct explanation for A
  • Both A & R are true,R is not correct explanation for A
  • A is true but R is false
  • A is false but R is true

 Let \mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\mathrm{x}^{\mathrm{n}}\sin\frac{1}{\mathrm{x}},\quad  \mathrm{x}\neq 0\\0, \quad \mathrm{x}=0\end{array}\right. , then f(x) is continuous but not differentiable at x=0 if
  • \mathrm{n}\in(0, 1 ]
  • \mathrm{n}\in[1, \infty)
  • \mathrm{n}\in(-\infty,0)
  • \mathrm{n}=0
The graph of the function y = f (x) has a unique tangent at the point (e^{a} ,0) through which the graph passes then \displaystyle \lim_{x\rightarrow e^{a}}\frac{log_{e}\{1+7f(x)\}-sinf(x)}{3f(x)} is
  • 1
  • 2
  • 0
  • -1
f(x)=|x-1|+|x-3| then f^{'}(2)=
  • -2
  • 2
  • 0
  • 1
lf \displaystyle \mathrm{f}(\mathrm{x})=\mathrm{x}.\sin \frac{1}{x} for x \neq 0,\ \mathrm{f}(\mathrm{0})=0 then?
  • f is continuous at x=0
  • f is differentiable at x=0
  • {f}'(0^{-}) exists but {f}'(0^{+}) does not exist
  • f is discontinuous at x=0
lf \mathrm{f}(\mathrm{x}) is a quadratic expression which is positive for all real vaues of \mathrm{x} and \mathrm{g}(\mathrm{x})=\mathrm{f}(\mathrm{x})+\mathrm{f}'(\mathrm{x})+\mathrm{f}''(\mathrm{x}) then for any real value of \mathrm{x}
  • \mathrm{g}(\mathrm{x})<0
  • \mathrm{g}(\mathrm{x})>0
  • \mathrm{g}(\mathrm{x})=0
  • \mathrm{g}(\mathrm{x})\underline{>}0
Let f be a twice differentiable function such that f''\left( x \right) =-f\left( x \right) and f'(x)=g(x)..
If h'\left( x \right) ={ \left[ f\left( x \right) \right]  }^{ 2 }+{ \left[ g\left( x \right) \right]  }^{ 2 },h\left( 1 \right) =6 and h(0)=4 then h(4) is equal to?
  • 16
  • 12
  • 13
  • None of these
If y=e^{\displaystyle\sqrt{x}}+e^{\displaystyle -\sqrt{x}}, then \displaystyle\frac{dy}{dx} is equal to
  • \displaystyle\frac{e^{\displaystyle\sqrt{x}}-e^{\displaystyle -\sqrt{x}}}{2\sqrt{x}}
  • \displaystyle\frac{e^{\displaystyle\sqrt{x}}-e^{\displaystyle -\sqrt{x}}}{2x}
  • \displaystyle\frac{1}{2\sqrt{x}}\sqrt{y^2-4}
  • \displaystyle\frac{1}{2\sqrt{x}}\sqrt{y^2+4}
If for all x, y the function f is defined by; f(x)+f(y)+f(x)\cdot f(y)=1 and f(x) > 0.When f(x) is differentiable f'(x)= ,
  • -1
  • 1
  • 0
  • cannot be determined
The function f(x)=e^x+x being differentiable and one to one, has a differentiable inverse f^{-1}(x), then find \dfrac {d}{dx} (f^{-1}(x)) at the point f(log_e 2).
  • \dfrac {1}{3}
  • 1
  • 3
  • 0
Given, f(x)=-\displaystyle \frac {x^3}{3}+x^2 \sin 1.5 a-x \sin a\cdot \sin 2a-5 arc \sin (a^2-8a+17), then
  • f(x) is not defined at x=sin 8
  • f' (sin 8) > 0
  • f' (x) is not defined at x=sin 8
  • f'(sin 8) < 0
If \sqrt {y+x}+\sqrt {y-x}=c (where c\neq 0), then \displaystyle \frac {dy}{dx} has the value equal to
  • \displaystyle \frac {2x}{c^2}
  • \displaystyle \frac {x}{y+\sqrt {y^2-x^2}}
  • \displaystyle \frac {y-\sqrt {y^2-x^2}}{x}
  • \displaystyle \frac {c^2}{2y}
Let f be a twice differentiable such that f''(x)=-f(x) and f'(x)=g(x). If h(x)=\left \{f(x)\right \}^2+\left \{g(x)\right \}^2, where h(5)=11. Find h(10)
  • 1
  • 10
  • 11
  • 100
 Let \mathrm{f}(\mathrm{x})=\mathrm{x}+\tan^{-1}\mathrm{x}, \displaystyle \mathrm{g}(\mathrm{x})=\frac{\mathrm{x}}{1+\mathrm{x}^{2}}(\mathrm{x}>0) Then
  • \mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x}), \mathrm{x}>0
  • \mathrm{f}(\mathrm{x})>\mathrm{g}(\mathrm{x}), \mathrm{x}>0
  • \mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x}) in [1, \infty)
  • None of these
Find the derivative of |x|+a_0x^n+a_1x^{n-1}+a_2x^{n-2}+....+a_{n-1}x+a_n
  • \frac {x}{|x|}+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n-1}
  • 1+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n-1}
  • \frac {x}{|x|}+na_0x^{n-1}+(n-1)a_1x^{n-2}+(n-2)a_2x^{n-3}+....+a_{n}
  • None of these
If f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}, then the value of 10 f'(102^+) is
  • -1
  • 0
  • 1
  • Does not exist
Let f(x) be a polynomial function of second degree.If f(1)=f(-1) and  a,b,c are in A.P f'(a),f'(b),f'(c) are in 
  • G.P.
  • H.P.
  • A.G.P
  • A.P.
If \phi (x) =\displaystyle \lim_{n \rightarrow \infty} \frac{x^{2n} f(x) + g(x)}{1 + x^{2n}}, then
  • \phi (x) = g(x) for all x \in R
  • \phi (x) = f(x) for all x \in R
  • \left\{\begin{matrix}g(x) & for -1 < x < 1\\ f(x) & for |x| \geq 1\end{matrix}\right.
  • \left\{\begin{matrix}g(x) & for |x| < 1\\ f(x) & for |x| > 1 \\\displaystyle \frac{f(x) + g(x)}{2} & for |x| = 1\end{matrix}\right.
Let f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}}, 1 \le x \le 26 be a real valued function, then f'(x) for 1 < x < 26 is
  • 0
  • \displaystyle\frac{1}{\sqrt{x - 1}}
  • 2\sqrt{x - 1}
  • 1
If \displaystyle \frac { \cos ^{ 4 }{ \theta  }  }{ x } +\frac { \sin ^{ 4 }{ \theta  }  }{ y } =\frac { 1 }{ x+y } then \displaystyle \frac { dy }{ dx } =
  • xy
  • \tan ^{ 2 }{ \theta  }
  • 0
  • \left( { x }^{ 2 }+{ y }^{ 2 } \right) \sec ^{ 2 }{ \theta  }
Let f be a differentiable function satisfying f(x) + f(y) + f(z) + f(x)f(y)f(z) = 14 for all x,\space y,\space z \in R
Then,
  • f'(x) < 0 for all x \in R
  • f'(x) = 0 for all x \in R
  • f'(x) > 0 for all x \in R
  • none of these
Given  f(x)=-\displaystyle \frac{x^3}{3}+x^2\sin 1.5a-x\sin a.\sin 2a-5 \arcsin (a^2-8a+17) then :
  • f(x) is not defined at x=\sin 8
  • {f}'(\sin 8)>0
  • f'(x) is not defined at x=\sin 8
  • {f}'(\sin 8)<0
Suppose, A=\displaystyle \frac {dy}{dx} of x^2+y^2=4 at (\sqrt 2, \sqrt 2), B=\displaystyle \frac {dy}{dx} of sin y+sin x=sin x\cdot sin y at (\pi, \pi) and C=\displaystyle \frac {dy}{dx} of 2e^{xy}+e^xe^y-e^x=e^{xy+1} at (1, 1), then (A-B-C) has the value equal to .....
  • \displaystyle \frac { 1 }{ 2 }
  • \displaystyle \frac { 1 }{ 3 }
  • 1
  • 2
Suppose A=\displaystyle \frac{dy}{dx} when x^2+y^2=4 at (\sqrt{2},\sqrt{2}), B=\displaystyle \frac{dy}{dx} when \sin y+ \sin x=\sin x-\sin y at (\pi,\pi) and C=\displaystyle  \frac{dy}{dx} when 2e^{xy}+e^x e^y-e^x-e^y=e^{xy+1} at (1,1), then (A+B+C) has the value equal to 
  • -1
  • e
  • -3
  • 0
If \displaystyle\lim_{x\rightarrow a}{(f(x)+g(x))}=2 and \displaystyle\lim_{x\rightarrow a}{(f(x)-g(x))}=1
then the value of \displaystyle\lim_{x\rightarrow a}{f(x)g(x)} is?
  • Does not exist
  • Exists and is \displaystyle\frac{3}{4}
  • Exists and is \displaystyle-\frac{3}{4}
  • Exists and is \displaystyle\frac{4}{3}
0:0:1


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