Test: Limit (Competition Level) - 3


30 Questions MCQ Test Mathematics for JEE Mains | Test: Limit (Competition Level) - 3


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QUESTION: 3

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 limx→0(3sinx−sin3x)/(x3
=limx→0 [d/dx(3sinx−sin3x)][d/dx(x3)]
We have indeterminate form 0/0 L'Hospital's rule applies
=limx→0 (3cosx−3cos3x)/(3x2)
=limx→0 [d/dx(3cosx−3cos3x)]/[d/dx(3x2)]
We have indeterminate form 0/0 L'Hospital's rule applies
=limx→0 (−3sinx+9sin3x)/(6x)
=limx→0 d/dx(−3sinx+9sin3x)/[d/dx(6x)]
We have indeterminate form 0/0 L'Hospital's rule applies
=limx→0 (−3cosx+27cos3x)/(6)
=(−3+27)/6
= 4

QUESTION: 4

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QUESTION: 9

Let α and β be the roots of ax2 + bx + c = 0, then 

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QUESTION: 10

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Put x = tanθ. Then x → 0 ⇒ θ → 0


QUESTION: 11

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QUESTION: 13

Let : R → R be a positive increasing function with  

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As fis a positive increa sin g function, we have f (x) < f (2x)< f (3x)
Dividing by f 
 we have by squeez theorem or sandwich theorem, 

QUESTION: 14

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QUESTION: 15

The values of constants a and b so that 

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Since the lim it is '0', the deg ree of numeration is less than the deg ree of Deno min ator.
∴ Coefficient of x2 = 0, coefficient of x = 0

⇒ 1 - a = 0, a - b = 0

⇒ a = 1, b = -1.

QUESTION: 16

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QUESTION: 19

 +.......n terms (a > 0,d > 0) =

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QUESTION: 24

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QUESTION: 26

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Requried limit = 

QUESTION: 27

The value of 

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 [on putting x = 1/h as x → ∞, h → 0]

QUESTION: 28

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= 1/24

QUESTION: 29

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Given limit is 

QUESTION: 30

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