Limits PYQs Set 5 (PDF)

Set 5 of 5 · Questions 121-130 of 130

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Limits · Set 5 (Q121-130)
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121. $If the function f (x) = {\frac{l o g _{e} ( 1 - x + x ^{2} ) + l o g _{e} ( 1 + x + x ^{2} )}{s e c x - c o s x} k, x \in (- \frac{π}{2}, \frac{π}{2}) - {0}, x = 0 is continuous at x = 0, then k is equal to :$ (1) $1$ (2) $-1$ (3) $e$ (4) $0$ 122. $If α, β are the distinct roots of x^{2} + bx+ c = 0, then lim_{x \to β} \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}} is equal to:$ (1) $b^{2} + 4c$ (2) $2(b^{2} + 4c)$ (3) $2(b^{2} - 4c)$ (4) $b^{2} - 4c$ 123. $Let f :R \to R be a continuous function. Then lim_{x \to \frac{π}{4}} \frac{\frac{π}{4} \int _{2}^{s e c^{2} x} f ( x ) d x}{x ^{2} - \frac{π ^{2}}{16}} is equal to :$ (1) $f(2)$ (2) $2f(2)$ (3) $2 f (2)$ (4) $4f(2)$ 124. $lim_{x \to 2} (\sum_{n = 1}^{9} \frac{x}{n ( n + 1 ) x ^{2} + 2 ( 2 n + 1 ) x + 4}) is equal to :$ (1) $\frac{9}{44}$ (2) $\frac{5}{24}$ (3) $\frac{1}{5}$ (4) $\frac{7}{36}$ 125. $The value of lim_{x \to 1} \frac{( x ^{2} - 1 ) s i n ^{2} ( π x )}{x ^{4} - 2 x ^{3} + 2 x - 1} is equal to:$ (1) $\frac{π ^{2}}{6}$ (2) $\frac{π ^{2}}{3}$ (3) $\frac{π ^{2}}{2}$ (4) $π^{2}$ 126. $, is equal to$ (1) $\frac{1}{12}$ (2) $- \frac{1}{18}$ (3) $- \frac{1}{12}$ (4) $- \frac{1}{6}$ 127. $The value of lim_{x \to 0} (\frac{x}{8 1 - s i n x - 8 1 + s i n x}) is equal to:$ (1) $0$ (2) $4$ (3) $-4$ (4) $-1$ 128. $lim_{x \to 0} \frac{s i n ^{2} ( π c o s ^{4} x )}{x ^{4}} is equal to :$ (1) $π^{2}$ (2) $2 π^{2}$ (3) $4 π^{2}$ (4) $4π$ 129. $Let [t] denote the greatest integer < t and {t} denote the fractional part of t. Then integral value of a for which the left hand limit of the function at x = 0 is equal to is____$ 130. $lim_{x \to 0} (\frac{( x + 2 c o s x ) ^{3} + 2 ( x + 2 c o s x ) ^{2} + 3 s i n ( x + 2 c o s x )}{( x + 2 ) ^{3} + 2 ( x + 2 ) ^{2} + 3 s i n x ( x + 2 )})^{\frac{100}{x}} is equal to___________.$