JEE Mathematics / Practice

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 1566

Let $A=\{a_{ij}\}$ be a $3\times3$ matrix, where $a_{ij}=\begin{cases}(-1)^{j-i} & \text{if } i<j,\\ 2 & \text{if } i=j,\\ (-1)^{i+j} & \text{if } i>j\end{cases}$ then $\det\left(3\,\mathrm{Adj}(2A^{-1})\right)$ is equal to $\_\_\_\_\_\_\_\_$ .

Matrices

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 1567

The number of solutions of the equation $\log_{(x+1)}(2x^2+7x+5)+\log_{(2x+5)}(x+1)^2-4=0$, $x>0$, is :

Quadratic Equation · Equations Reducible to Quadratic

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 1568

If the point on the curve $y^2=6x$, nearest to the point $\left(3,\dfrac{3}{2}\right)$ is $(\alpha,\beta)$, then $2(\alpha+\beta)$ is equal to $\_\_\_\_\_\_\_\_$ .

Parabola

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 1569

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence such that $a_1=1$, $a_2=1$ and $a_{n+2}=2a_{n+1}+a_n$ for all $n\ge1$. Then the value of $47\displaystyle\sum_{n=1}^{\infty}\dfrac{a_n}{2^{3n}}$ is equal to $\_\_\_\_\_\_\_\_$ .

Sequence and Series · Logarithmic and Exponential Series, and Recursive Sequences

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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 1570

For $k\in\mathbf{N}$, let $\dfrac{1}{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+20)}=\displaystyle\sum_{k=0}^{20}\dfrac{A_k}{\alpha+k}$, where $\alpha>0$. Then the value of $100\left(\dfrac{A_{14}+A_{15}}{A_{13}}\right)^2$ is equal to $\_\_\_\_\_\_\_\_$ .

Sequence and Series · Miscellaneous

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