Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}, \quad A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}, \quad A\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 2\end{pmatrix}.$ If $X = (x_1, x_2, x_3)^T$ and $I$ is an identity matrix of order $3$, then the system $(A - 2I)X = \begin{pmatrix}4 \\ 1 \\ 1\end{pmatrix}$ has:
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Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the val…
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The value of $\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} …
Topic: JEE Main 2022 (25 June Morning Shift)
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Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$…
Topic: JEE Main 2022 (25 June Morning Shift)
If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \ove…
Topic: JEE Main 2022 (25 June Morning Shift)
Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$. Then, the value of $\mathop {\lim }\l…
Topic: JEE Main 2022 (25 June Morning Shift)
Let E1 and E2 be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$, $P({E_2}|{E_1}) = {…
Topic: JEE Main 2022 (25 June Morning Shift)
Let $A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$. If M and N are two matrices given by $M = \sum…
Topic: JEE Main 2022 (25 June Morning Shift)
Let $g:(0,\infty ) \to R$ be a differentiable function such that $\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}…
Topic: JEE Main 2022 (25 June Morning Shift)
Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = …
Topic: JEE Main 2022 (25 June Morning Shift)
Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} > 0$, $i = 1,2,3$ be a vector whi…
Topic: JEE Main 2022 (25 June Morning Shift)
Let $y = y(x)$ be the solution of the differential equation $(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$, with $y(0) = {1 \ove…
Topic: JEE Main 2022 (25 June Morning Shift)
If the solution curve $y = y(x)$ of the differential equation ${y^2}dx + ({x^2} - xy + {y^2})dy = 0$, which passes thro…
Topic: JEE Main 2022 (25 June Morning Shift)
