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JEE Advanced

Extraction of metal from the ore cassiterite involves

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Which of the following is (are) NOT the square of a $3×3$ matrix with real entries? (a)$⎣⎡ 100 010 00−1 ⎦⎤ $ (b) $⎣⎡ −100 0−10 00−1 ⎦⎤ $ (c)$⎣⎡ 100 010 001 ⎦⎤ $ (d) $⎣⎡ 100 0−10 00−1 ⎦⎤ $

Let $n_{1},andn_{2}$, be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$,be the number one red and b of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probablity that this red ball was drawn from box II is $31 $ then the correct option(s) with the possible values of $n_{1},n_{2},n_{3},andn_{4}$, is(are)

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of $nm $ is ____

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $27 $ on y-axis is (are)

A box $B_{1}$, contains 1 white ball, 3 red balls and 2 black balls. Another box $B_{2}$, contains 2 white balls, 3 red balls and 4 black balls. A third box $B_{3}$, contains 3 white balls, 4 red balls and 5 black balls.

Let $M$be a $2×2$symmetric matrix with integer entries. Then $M$is invertible ifThe first column of $M$is the transpose of the second row of $M$The second row of $M$is the transpose of the first column of $M$$M$is a diagonal matrix with non-zero entries in the main diagonalThe product of entries in the main diagonal of $M$is not the square of an integer

Let $−61 <θ<−12π $ Suppose $α_{1}andβ_{1}$, are the roots of the equation $x_{2}−2xsecθ+1=0$ and $α_{2}andβ_{2}$ are the roots of the equation $x_{2}+2xtanθ−1=0$. If $α_{1}>β_{1}$ and $α_{2}>β_{2}$, then $α_{1}+β_{2}$ equals

If $f:RR$ is a twice differentiable function such that $f(x)>0forallxR,and$ $f(21 )=21 ,f(1)=1$ then: $f_{prime}(1)>1$ (b) $f_{prime}(1)≤0$(c)$21 <f_{prime}(1)<1$ (d)="" $0<f_{prime}(1)≤21 $