class 12

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JEE Advanced

The number of structural isomers for $C_{6}H_{14}$ is

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PARAGRAPH $X$Let $S$be the circle in the $xy$-plane defined by the equation $x_{2}+y_{2}=4.$(For Ques. No 15 and 16)Let $E_{1}E_{2}$and $F_{1}F_{2}$be the chords of $S$passing through the point $P_{0}(1,1)$and parallel to the x-axis and the y-axis, respectively. Let $G_{1}G_{2}$be the chord of $S$passing through $P_{0}$and having slope $−1$. Let the tangents to $S$at $E_{1}$and $E_{2}$meet at $E_{3}$, the tangents to $S$at $F_{1}$and $F_{2}$meet at $F_{3}$, and the tangents to $S$at $G_{1}$and $G_{2}$meet at $G_{3}$. Then, the points $E_{3},F_{3}$and $G_{3}$lie on the curve$x+y=4$(b) $(x−4)_{2}+(y−4)_{2}=16$(c) $(x−4)(y−4)=4$(d) $xy=4$

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$If a tangent to a suitable conic (Column 1) is found to be $y=x+8$and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)

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.

Let $f:R→R$be a differentiable function with $f(0)=1$and satisfying the equation $f(x+y)=f(x)f_{prime}(y)+f_{prime}(x)f(y)$for all $x,y∈R$. Then, the value of $(g)_{e}(f(4))$is _______

Consider the circle $x_{2}+y_{2}=9$ and the parabola $y_{2}=8x$. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.

PARAGRAPH AThere are five students $S_{1},S_{2},S_{3},S_{4}$and $S_{5}$in a music class and for them there are five seats $R_{1},R_{2},R_{3},R_{4}$and $R_{5}$arranged in a row, where initially the seat $R_{i}$is allotted to the student $S_{i},i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted five seats. For $i=1,2,3,4,$let $T_{i}$denote the event that the students $S_{i}$and $S_{i+1}$do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_{1}∩T_{2}∩T_{3}∩T_{4}$is$151 $(b) $101 $(c) $607 $(d) $51 $

A line $l$ passing through the origin is perpendicular to the lines $l_{1}:(3+t)i^+(−1+2t)j^ +(4+2t)k^,∞<t<∞,l_{2}:(3+s)i^+(3+2s)j^ +(2+s)k^,∞<t<∞$ then the coordinates of the point on $l_{2}$ at a distance of $17 $ from the point of intersection of \displaystyle{l}&{l}_{{1}} is/are:

Let $S$be the set of all column matrices $[b_{1}b_{2}b_{3}]$such that $b_{1},b_{2},b_{3}∈R$and the system of equations (in real variable)$−x+2y+5z=b_{1}$$2x−4y+3z=b_{2}$$x−2y+2z=b_{3}$has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $[b_{1}b_{2}b_{3}]∈S$?(a) $x+2y+3z=b_{1},4y+5z=b_{2}$and $x+2y+6z=b_{3}$(b) $x+y+3z=b_{1},5x+2y+6z=b_{2}$and $−2x−y−3z=b_{3}$(c) $−x+2y−5z=b_{1},2x−4y+10z=b_{2}$and $x−2y+5z=b_{3}$(d) $x+2y+5z=b_{1},2x+3z=b_{2}$and $x+4y−5z=b_{3}$