Limits and Derivatives Test

Result

Limits and Derivatives Test - 58

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Weekly Quiz Competition

Question 1
1 / -0

Let $$\displaystyle x^{cos\,y} + y^{cox\,x} = 5 $$ , Then

A
$$ at x = 0 , y =0 ,{y}' =0 $$

B
$$ at x = 0 , y = 1 , {y}' = 0 $$

C
$$ at x = y , y = 1 , {y}' = -1 $$

D
$$ at x = 1 , y = 0 , {y}' = 1 $$

Solution
Question 2
1 / -0

Value of $$L=\displaystyle\lim_{n\rightarrow \infty n}\dfrac{1}{4}\left[1.\left(\displaystyle\sum_{k=1}^{n}k\right)+2.\left(\sum_{k=1}^{n-1}k\right)+3.\left(\sum_{k=1}^{n-2}k\right)+...+n.1\right]$$ is

A
$$1/24$$

B
$$1/12$$

C
$$1/6$$

D
$$1/3$$

Solution
Question 3
1 / -0

Solution of the equation $$ \dfrac {dy }{dx} + \dfrac {1}{x} \tan y = \dfrac {1}{x^2} \tan y \sin y$$ is

A
$$ 2x = sin y ( x ) $$

B
$$ 2x =sin y ( 1 +cx^2) $$

C
$$ 2x +sin y ( 1 +cx^2 ) $$

D
None of these

Solution
Question 4
1 / -0

$$\displaystyle \lim _{x \rightarrow 1}\left(\dfrac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\dfrac{1-\cos (x+1)}{(x+1)^{2}}} $$ is equal to:

A
1

B
$$(2 / 3)^{1 / 2} $$

C
$$ (3 / 2)^{1 / 2} $$

D
$$ e^{1 / 2} $$

Solution
Question 5
1 / -0

The value of $$\displaystyle \lim_{n\infty}\dfrac{1}{n^2}\left\{ sin^3\dfrac{\pi}{4n}+2sin^3\dfrac{2\pi}{4n} + ... + nsin^3\dfrac{n\pi}{4n}\right\}$$ is equal to

A
$$\dfrac{\sqrt{2}}{9\pi^2}(52-15 \pi )$$

B
$$\dfrac{2}{9\pi^2}(52-15n)$$

C
$$\dfrac{1}{9\pi^2}(15n-15)$$

D
None of these

Solution
Question 6
1 / -0

If $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x} $$ is non-zero finite, then $$ n $$ must be equal

A
4

B
1

C
2

D
3

Solution
Question 7
1 / -0

If $$ L=\displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}}=\infty $$

Equation $$ a x^{2}+b x+c=0 $$ has

A
real and equal roots

B
complex roots

C
unequal positive real roots

D
unequal roots

Solution
Question 8
1 / -0

$$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}} $$ is equal to

A
0

B
1

C
-1

D
Does not exists

Solution
Question 9
1 / -0

$$\displaystyle \lim _{x \rightarrow \pi / 2} \dfrac{\sin (x \cos x)}{\cos (x \sin x)} $$ is equal to

A
0

B
p/2

C
p

D
2p

Solution
Question 10
1 / -0

The value of $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos 2 x)}}{x} $$ is

A
1

B
-1

C
0

D
None of these

Solution