class 12

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JEE Advanced

The correct statement(s) for orthoboric acid is/are

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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 complex numbers $αandα1 $ lies on circle $(x−x_{0})_{2}(y−y_{0})_{2}=r_{2}and(x−x_{0})_{2}+(y−y_{0})_{2}=4r_{2}$ respectively. If $z_{0}=x_{0}+iy_{0}$ satisfies the equation $2∣z_{0}∣_{2}=r_{2}+2$ then $∣α∣$ is equal to (a) $2 1 $ (b) $21 $ (c) $7 1 $ (d) $31 $

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 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

Let $g:R→R$ be a differentiable function with $g(0)=0,g_{′}(1)=0,g_{′}(1)=0$.Let $f(x)={∣x∣x g(x),0=0and0,x=0$ and $h(x)=e_{∣x∣}$ for all $x∈R$. Let $(foh)(x)$ denote $f(h(x))and(hof)(x)$ denote $h(f(x))$. Then which of thx!=0 and e following is (are) true?

Let $XandY$be two events that $P(X)=31 ,P(X|Y)=21 andP(Y|X)=52 $then:$P(Y)=154 $ (b) $P(X∪Y)=52 $$P(X_{prime}|Y)=21 $ (d) $P(X∩Y)=51 $

Let $y(x)$ be a solution of the differential equation $(1+e_{x})y_{prime}+ye_{x}=1.$ If $y(0)=2$ , then which of the following statements is (are) true? (a)$y(−4)=0$ (b)$y(−2)=0$ (c)$y(x)$ has a critical point in the interval $(−1,0)$ (d)$y(x)$ has no critical point in the interval$(−1,0)$

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