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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7501

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 7502

The value of $k\in\mathbf{R}$, for which the following system of linear equations $3x-y+4z=3$, $x+2y-3z=-2$, $6x+5y+kz=-3$, has infinitely many solutions, is :

Determinant

$3$$-5$$5$$-3$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7503

In a triangle $ABC$, if $|\overrightarrow{BC}|=3$, $|\overrightarrow{CA}|=5$ and $|\overrightarrow{BA}|=7$, then the projection of the vector $\overrightarrow{BA}$ on $\overrightarrow{BC}$ is equal to :

Vector Algebra

$\dfrac{19}{2}$$\dfrac{13}{2}$$\dfrac{11}{2}$$\dfrac{15}{2}$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7504

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 7505

Let $y=y(x)$ satisfy the equation $\dfrac{dy}{dx}-|A|=0$, for all $x>0$, where $A=\begin{bmatrix}y&\sin x&1\\0&-1&1\\2&0&\dfrac{1}{x}\end{bmatrix}$. If $y(\pi)=\pi+2$, then the value of $y\left(\dfrac{\pi}{2}\right)$ is :

Differential Equation

$\dfrac{\pi}{2}+\dfrac{4}{\pi}$$\dfrac{\pi}{2}-\dfrac{1}{\pi}$$\dfrac{3\pi}{2}-\dfrac{1}{\pi}$$\dfrac{\pi}{2}-\dfrac{4}{\pi}$