class 12

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JEE Advanced

For non-negative inger n, let $f(n)=k=0∑n sin_{2}(n+1k+1 π)k=∑n sin(n+1k+1 π)sin(n+1k+2 π) $ Assuming $cos_{−1}x$ takes values in $[0,π]$ which of the following options is/are correct?

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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 $f:RR$be a continuous odd function, which vanishes exactly at one point and $f(1)=21 ˙$Suppose that $F(x)=∫_{−1}f(t)dtforallx∈[−1,2]andG(x)=∫_{−1}t∣f(f(t))∣dtforallx∈[−1,2]I˙G(x)f(lim)_{x1}(F(x)) =141 ,$Then the value of $f(21 )$is

For $3×3$matrices $MandN,$which of the following statement (s) is (are) NOT correct ?$N_{T}MN$is symmetricor skew-symmetric, according as $m$is symmetric or skew-symmetric.$MN−NM$is skew-symmetric for all symmetric matrices $MandN˙$$MN$is symmetric for all symmetric matrices $MandN$$(adjM)(adjN)=adj(MN)$for all invertible matrices $MandN˙$

Let $a$ and $b$ be two unit vectors such that $a.b=0$ For some $x,y∈R$, let $c=xa+yb+(a×b$ If $(∣c∣=2$ and the vector $c$ is inclined at same angle $α$ to both $a$ and $b$ then the value of $8cos_{2}α$ is

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), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$For $a=2 ,if$a tangent is drawn to a suitable conic (Column 1) at the point of contact $(−1,1),$then which of the following options is the only CORRECT combination for obtaining its equation?(I) (ii) (Q) (b) (III) (i) (P)(II) (ii) (Q) (d) $(I)(i)(P)$

Let $O$be the origin and let PQR be an arbitrary triangle. The point S is such that$OPO˙Q+ORO˙S=ORO˙P+OQO˙S=OQ$.$OR+OPO˙S$Then the triangle PQ has S as its:circumcentre (b) orthocentre (c) incentre (d) centroid

Let $X$be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11,,¨ $and $Y$be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23,¨$. Then, the number of elements in the set $X∪Y$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. $1332$