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JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7511

For the natural numbers $m,\ n$, if $(1-y)^m(1+y)^n=1+a_1y+a_2y^2+\cdots+a_{m+n}y^{m+n}$ and $a_1=a_2=10$, then the value of $(m+n)$ is equal to :

Binomial Theorem · Coefficient of a Given Power / Term Independent of x

$88$$64$$100$$80$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7512

Consider the line $L$ given by the equation $\dfrac{x-3}{2}=\dfrac{y-1}{1}=\dfrac{z-2}{1}$. Let $Q$ be the mirror image of the point $(2,3,-1)$ with respect to $L$. Let a plane $P$ be such that it passes through $Q$, and the line $L$ is perpendicular to $P$. Then which of the following points is on the plane $P$?

3D Geometry

$(-1,1,2)$$(1,1,1)$$(1,1,2)$$(1,2,2)$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7513

Let $g(t)=\displaystyle\int_{-\pi/2}^{\pi/2}\cos\left(\dfrac{\pi}{4}t+f(x)\right)dx$, where $f(x)=\log_e\left(x+\sqrt{x^2+1}\right)$, $x\in\mathbf{R}$. Then which one of the following is correct?

Definite Integration

$g(1)=g(0)$$\sqrt{2}\,g(1)=g(0)$$g(1)=\sqrt{2}\,g(0)$$g(1)+g(0)=0$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7514

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin\theta$ is equal to :

Trigonometric Ratio & Identities

$\dfrac{\sqrt{5}+1}{4}$$\dfrac{\sqrt{5}-1}{2}$$\dfrac{\sqrt{2}-1}{2}$$\dfrac{\sqrt{5}-1}{4}$

JEE Main 2021 (Online) 20th July Evening ShiftQuestion 7515

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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