class 12

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JEE Advanced

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Let $f:R0,1 $ be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)? $e_{x}−∫_{0}f(t)sintdt$ (b) $f(x)+∫_{0}f(t)sintdt$(c)$x−∫_{0}f(t)costdt$ (d) $x_{9}−f(x)$

From a point $P(λ,λ,λ)$, perpendicular PQ and PR are drawn respectively on the lines $y=x,z=1$ and $y=−x,z=−1$.If P is such that $∠QPR$ is a right angle, then the possible value(s) of $λ$ is/(are)

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

For each positive integer $n$, let $y_{n}=n1 ((n+1)(n+2)n+n˙ )_{n1}$For $x∈R$let $[x]$be the greatest integer less than or equal to $x$. If $(lim)_{n→∞}y_{n}=L$, then the value of $[L]$is ______.

Let $f:[a,b]1,∞ $be a continuous function and let $g:RR$be defined as$g(x)={0ifxbThen$$g(x)$is continuous but not differentiable at a$g(x)$is differentiable on $R$$g(x)$is continuous but nut differentiable at $b$$g(x)$is continuous and differentiable at either $a$or $b$but not both.

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______

Three randomly chosen nonnegative integers $x,yandz$are found to satisfy the equation $x+y+z=10.$Then the probability that $z$is even, is:$125 $ (b) $21 $ (c) $116 $ (d) $5536 $

A solution curve of the differential equation $(x_{2}+xy+4x+2y+4)(dxdy )−y_{2}=0$ passes through the point $(1,3)$ Then the solution curve is