JEE Advanced 2021 Paper 2 Online | JEE Advanced Year Wise Previous Years Questions

JEE Advanced 2021 Paper 2 Online

MCQ (More than One Correct Answer)

-2

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

$$\cos P \ge 1 - {{{p^2}} \over {2qr}}$$

$$\cos R \ge \left( {{{q - r} \over {p + q}}} \right)\cos P + \left( {{{p - r} \over {p + q}}} \right)\cos Q$$

$${{q + r} \over p} < 2{{\sqrt {\sin q\sin R} } \over {\sin P}}$$

If p < q and p < r, then $$\cos Q > {p \over r}$$ and $$\cos R > {p \over q}$$

JEE Advanced 2021 Paper 2 Online

MCQ (More than One Correct Answer)

-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?

The equation $$f(x) - 3\cos 3x = 0$$ has at least one solution in $$\left( {0,{\pi \over 3}} \right)$$

The equation $$f(x) - 3\sin 3x = - {6 \over \pi }$$ has at least one solution in $$\left( {0,{\pi \over 3}} \right)$$

$$\mathop {\lim }\limits_{x \to 0} {{x\int_0^x {f(t)dt} } \over {1 - {e^{{x^2}}}}} = - 1$$

$$\mathop {\lim }\limits_{x \to 0} {{\sin x\int_0^x {f(t)dt} } \over {{x^2}}} = - 1$$

JEE Advanced 2021 Paper 2 Online

MCQ (More than One Correct Answer)

-2

For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation $${{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1$$. Let $$S = \{ {y_{\alpha ,\beta }}(x):\alpha ,\beta \in R\} $$. Then which of the following functions belong(s) to the set S?

$$f(x) = {{{x^2}} \over 2}{e^{ - x}} + \left( {e - {1 \over 2}} \right){e^{ - x}}$$

$$f(x) = - {{{x^2}} \over 2}{e^{ - x}} + \left( {e + {1 \over 2}} \right){e^{ - x}}$$

$$f(x) = {{{e^x}} \over 2}\left( {x - {1 \over 2}} \right) + \left( {e - {{{e^2}} \over 4}} \right){e^{ - x}}$$

$$f(x) = {{{e^x}} \over 2}\left( {{1 \over 2} - x} \right) + \left( {e + {{{e^2}} \over 4}} \right){e^{ - x}}$$

JEE Advanced 2021 Paper 2 Online

MCQ (More than One Correct Answer)

-2

Let O be the origin and $$\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k$$ and $$\overrightarrow {OB} = \widehat i - 2\widehat j + 2\widehat k$$ and $$\overrightarrow {OC} = {1 \over 2}\left( {\overrightarrow {OB} - \lambda \overrightarrow {OA} } \right)$$ for some $$\lambda$$ > 0. If $$\left| {\overrightarrow {OB} \times \overrightarrow {OC} } \right| = {9 \over 2}$$, then which of the following statements is (are) TRUE?

Projection of $$\overrightarrow {OC} $$ on $$\overrightarrow {OA} $$ is $$ - {3 \over 2}$$

Area of the triangle OAB is $${9 \over 2}$$

Area of the triangle ABC is $${9 \over 2}$$

The acute angle between the diagonals of the parallelogram with adjacent sides $${\overrightarrow {OA} }$$ and $${\overrightarrow {OC} }$$ is $${\pi \over 3}$$

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