class 12

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JEE Advanced

Which one of the following reagents is used in the above reaction ?

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Consider the set of eight vector $V={ai^+bj^ +ck^;a,bc∈{−1,1}}˙$Three non-coplanar vectors can be chosen from $V$is $2_{p}$ways. Then $p$is_______.

If $g(x)=∫_{sinx}sin_{−1}(t)dt,then:$ (a)$g_{prime}(2π )=−2π$ (b) $g_{prime}(−2π )=−2π$ (c)$g_{prime}(−2π )=2π$ (d) $g_{prime}(2π )=2π$

For a real number $α,$ if the system $⎣⎡ 1αα_{2} α1α α_{2}α1 ⎦⎤ ⎣⎡ xyz ⎦⎤ =⎣⎡ 1−11 ⎦⎤ $ of linear equations, has infinitely many solutions, then $1+α+α_{2}=$

Consider the hyperbola $H:x_{2}−y_{2}=1$ and a circle S with centre $N(x_{2},0)$ Suppose that H and S touch each other at a point $(P(x_{1},y_{1})$ with $x_{1}>1andy_{1}>0$ The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle $ΔPMN$ then the correct expression is (A) $dx_{1}dl =1−3x_{1}1 $ for $x_{1}>1$ (B) $dx_{1}dm =3(x _{1}−1)x_{!} )forx_{1}>1$ (C) $dx_{1}dl =1+3x_{1}1 forx_{1}>1$ (D) $dy_{1}dm =31 fory_{1}>0$

Let RS be the diameter of the circle $x_{2}+y_{2}=1,$ where S is the point $(1,0)$ Let P be a variable apoint (other than $RandS$) on the circle and tangents to the circle at $SandP$ meet at the point Q.The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. then the locus of E passes through the point(s)- (A) $(31 ,3 1 )$ (B) $(41 ,21 )$ (C) $(31 ,−3 1 )$ (D) $(41 ,−21 )$

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 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

Let $S_{n}=k=1∑4n (−1)2k(k+1) k_{2}˙$Then $S_{n}$can take value (s)$1056$b. $1088$c. $1120$d. $1332$