class 12

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JEE Advanced

Let $f:R→R$ be given by $f(x)=⎩⎨⎧ x_{5}+5x_{4}+10x_{3}+3x+1x_{2}−x+1(2/3)x_{3}−4x_{2}+7x−(8/3)(x−2)ln(x−2)−x+(10/3) x<00≤x<11≤x<3x≥3 $ Then which of the following options is/are correct?

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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) $m=1,M=12$

Let $u^=u_{1}i^+u_{2}j^ +u_{3}k^$ be a unit vector in be a unit vector in $R_{3}andw^=6 1 (i^+j^ +2k^)$.Given that there exists vector $v^$ in $R_{3}$ such that $∣u^×v∣=1andw^.(u^×v)=1$. Which of the following statement(s) is(are) correct?

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60

Q. The value of is equal $k=1∑13 (sin(4π +(k−1)6π )sin(4π +k6π )1 $ is equal

If $Ik=1∑98 ∫_{k}x(x+1)k+1 dx,then:$$I<5049 $ (b) $I>(g)_{e}99$$I>5049 $ (d) $I<(g)_{e}99$

Let $F_{1}(x_{1},0)$ and $F_{2}(x_{2},0)$, for $x_{1}<0$ and $x_{2}>0$, be the foci of the ellipse $9x_{2} +8y_{2} =1$ Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral $MF_{1}NF_{2}$ is

If$α=∫_{0}(e_{9}x+3tan_{(−1)x})(1+x_{2}12+9x_{2} )dxwherηn_{−1}$takes only principal values, then the value of $((g)_{e}∣1+α∣−43π )is$

The area enclosed by the curves$y=sinx+cosxandy=∣cosx−sinx∣$ over the interval $[0,2π ]$