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JEE Advanced

$sec_{−1}∣41 k=0∑10 (sec(127π +2kπ )sec(127π +(k+1)(2π ))]$ will be

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The option(s) with the values of a and L that satisfy the following equation is (are) $∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt =L$

Football teams T1 and T2 have to play two games against each other. It is assumed that theoutcomes of the two games are independent. The probabilities of T1 winning, drawing andlosing a game against T2 are1/ 2,and1/6,1/3respectively. Each team gets 3 points for a win,1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total pointsscored by teams T1 and T2, respectively, after two gamesP $(X=Y)$ is

Let $f:(−2π ,2π )R$be given by $f(x)=(g(secx+tanx))_{3}$then$f(x)$is an odd function$f(x)$is a one-one function$f(x)$is an onto function$f(x)$is an even function

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

Let $b_{1}>1$ for $i=1,2,……,101.$ Suppose $g_{e}b_{1},g_{e}b_{10}$ are in Arithmetic progression $(A.P.)$ with the common difference $g_{e}2.$ suppose $a_{1},a_{2}……….a_{101}$ are in A.P. such $a_{1}=b_{1}anda_{51}=b_{51}.$ If $t=b_{1}+b_{2}+……+b_{51}ands=a_{1}+a_{2}+……+a_{51}$ then

Let w = ($3 +2ι )$ and $P={w_{n}:n=1,2,3,…..},$ Further $H_{1}={z∈C:Re(z)>21 }andH_{2}={z∈c:Re(z)<−21 }$ Where C is set of all complex numbers. If $z_{1}∈P∩H_{1},z_{2}∈P∩H_{2}$ and O represent the origin, then $∠Z_{1}OZ_{2}$ =

Let $XandY$be two arbitrary, $3×3$, non-zero, skew-symmetric matrices and $Z$be an arbitrary $3×3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?a.$Y_{3}Z_{4}Z_{4}Y_{3}$b. $x_{44}+Y_{44}$c. $X_{4}Z_{3}−Z_{3}X_{4}$d. $X_{23}+Y_{23}$

Consider two straight lines, each of which is tangent to both the circle $x_{2}+y_{2}=21 $and the parabola $y_{2}=4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $2 $, then which of the following statement(s) is (are) TRUE?For the ellipse, the eccentricity is $2 1 $and the length of the latus rectum is 1(b) For the ellipse, the eccentricity is $21 $and the length of the latus rectum is $21 $(c) The area of the region bounded by the ellipse between the lines $x=2 1 $and $x=1$is $42 1 (π−2)$(d) The area of the region bounded by the ellipse between the lines $x=2 1 $and $x=1$is $161 (π−2)$