class 12

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JEE Advanced

In the reaction shown below, the major product(s) formed is/are .

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A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Let $f_{prime}(x)=2+sin_{4}πx192x_{3} forallx∈Rwithf(21 )=0.Ifm≤∫_{21}f(x)dx≤M,$ then the possible values of $mandM$ are (a)$m=13,M=24$ (b) $m=41 ,M=21 $(c)$m=−11,M=0$ (d) $m=1,M=12$

Let P and Q be distinct points on the parabola $y_{2}=2x$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $ΔOPQ$ is $32$ , then which of the following is (are) the coordinates of $P?$

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 PQ be a focal chord of the parabola $y_{2}=4ax$ The tangents to the parabola at P and Q meet at a point lying on the line $y=2x+a,a>0$. Length of chord PQ is

In a triangle PQR, P is the largest angle and $cosP=31 $. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

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 $P$be a point on the circle $S$with both coordinates being positive. Let the tangent to $S$at $P$intersect the coordinate axes at the points $M$and $N$. Then, the mid-point of the line segment $MN$must lie on the curve$(x+y)_{2}=3xy$(b) $x_{2/3}+y_{2/3}=2_{4/3}$(c) $x_{2}+y_{2}=2xy$(d) $x_{2}+y_{2}=x_{2}y_{2}$

Let $f:(0,∞)→R$ be a differentiable function such that $f_{′}(x)=2−xf(x) $ for all $x∈(0,∞)$ and $f(1)=1$, then