class 12

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JEE Advanced

$I=π2 −π/4∫π/4 (1+e_{sinx})(2−cos2x)dx $ then find $27I_{2}$

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A box $B_{1}$, contains 1 white ball, 3 red balls and 2 black balls. Another box $B_{2}$, contains 2 white balls, 3 red balls and 4 black balls. A third box $B_{3}$, contains 3 white balls, 4 red balls and 5 black balls.

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

Let $f:RR$be a differentiable function such that $f(0),f(2π )=3andf_{prime}(0)=1.$If $g(x)=∫_{x}[f_{prime}(t)cosect−cottcosectf(t)]dtforx(0,2π ],$then $(lim)_{x0}g(x)=$

Let $n_{1},andn_{2}$, be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$,be the number one red and b of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probablity that this red ball was drawn from box II is $31 $ then the correct option(s) with the possible values of $n_{1},n_{2},n_{3},andn_{4}$, is(are)

Let $f:R→R$and $g:R→R$be two non-constant differentiable functions. If $f_{prime}(x)=(e_{(f(x)−g(x))})g_{prime}(x)$for all $x∈R$, and $f(1)=g(2)=1$, then which of the following statement(s) is (are) TRUE?$f(2)<1−(g)_{e}2$(b) $f(2)>1−(g)_{e}2$(c) $g(1)>1−(g)_{e}2$(d) $g(1)<1−(g)_{e}2$

Let $f_{prime}(x)=2+sin_{4}πx192x_{3} forallx∈Rwithf(21 )=0.Ifm≤∫_{21}f(x)dx≤M,$ then the possible values of $mandM$ are (a)$m=13,M=24$ (b) $m=41 ,M=21 $(c)$m=−11,M=0$ (d) $m=1,M=12$

Coefficient of $x_{11}$ in the expansion of $(1+x_{2})_{4}(1+x_{3})_{7}(1+x_{4})_{12}$ is 1051 b. 1106 c. 1113 d. 1120