class 12

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JEE Advanced

A sample of $._{19}K_{40}$ disintegrates into two nuclei Ca & Ar with decay constant $λ_{Ca}=4.5×10_{−10}S_{−1}$ and $λ_{Ar}=0.5×10_{−10}S_{−1}$ respectively. The time after which 99% of $._{19}K_{40}$ gets decayed is

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The common tangents to the circle $x_{2}+y_{2}=2$ and the parabola $y_{2}=8x$ touch the circle at $P,Q$ andthe parabola at $R,S$. Then area of quadrilateral $PQRS$ is

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover cards numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isa.$264$ b. $265$ c. $53$ d. $67$

Three boys and two girls stand in a queue. The probability, that the number of boys ahead is at least one more than the number of girls ahead of her, is (A) $21 $ (B) $31 $ (C) $32 $ (D) $43 $

Let $s,t,r$be non-zero complex numbers and $L$be the set of solutions $z=x+iy(x,y∈R,i=−1 )$of the equation $sz+tz+r=0$, where $z=x−iy$. Then, which of the following statement(s) is (are) TRUE?If $L$has exactly one element, then $∣s∣=∣t∣$(b) If $∣s∣=∣t∣$, then $L$has infinitely many elements(c) The number of elements in \displaystyle{\Ln{{n}}}{\left\lbrace{z}\right|}{z}-{1}+{i}{\mid}={5}{\rbrace}is at most 2(d) If $L$has more than one element, then $L$has infinitely many elements

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) $xy=4$

Let a,b ,c be positive integers such that $ab $ is an integer. If a,b,c are in GP and the arithmetic mean of a,b,c, is b+2 then the value of $a+1a_{2}+a−14 $ is

The coefficient of $x_{9}$ in the expansion of $(1+x)(1+x_{2})(1+x_{3})….(1+x_{100})$ is

Let $a,b,andc$ be three non coplanar unit vectors such that the angle between every pair of them is $3π $. If $a×b+b×x=pa+qb+rc$ where p,q,r are scalars then the value of $q_{2}p_{2}+2q_{2}+r_{2} $ is