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Definite Integration question

2021 · Shift 2 · Q22

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PYQ Library
JEE Advanced
Mathematics
Definite Integration
2021 · Shift 2 · Q22

Definite Integration question

2021 · Shift 2 · Q22

JEE AdvancedMathematicsDefinite IntegrationMultiple correct+4 / −2

Let

f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R

be a continuous function such that

f(0) = 1

and

\int_0^{{\pi \over 3}} {f(t)dt = 0}

. Then which of the following statements is(are) TRUE?

A
The equation $f(x) - 3\cos 3x = 0$ has at least one solution in $\left( {0,{\pi \over 3}} \right)$
B
The equation $f(x) - 3\sin 3x = - {6 \over \pi }$ has at least one solution in $\left( {0,{\pi \over 3}} \right)$
C
$\mathop {\lim }\limits_{x \to 0} {{x\int_0^x {f(t)dt} } \over {1 - {e^{{x^2}}}}} = - 1$
D
$\mathop {\lim }\limits_{x \to 0} {{\sin x\int_0^x {f(t)dt} } \over {{x^2}}} = - 1$

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