class 12

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JEE Advanced

For the given compound X, the total number of optically active stereoisomers is ____.

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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 ____

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) $m=1,M=12$

The coefficients of three consecutive terms of $(1+x)_{n+5}$are in the ratio 5:10:14. Then $n=$___________.

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)

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

Three randomly chosen nonnegative integers $x,yandz$are found to satisfy the equation $x+y+z=10.$Then the probability that $z$is even, is:$125 $ (b) $21 $ (c) $116 $ (d) $5536 $

Let $f:[0,2]→R$ be a function which is continuous on [0,2] and is differentiable on (0,2) with $f(0)=1$$Let:F(x)=∫_{0}f(t )dtforx∈[0,2]I˙fF_{prime}(x)=f_{prime}(x)$ . for all $x∈(0,2),$ then $F(2)$ equals (a)$e_{2}−1$ (b) $e_{4}−1$(c)$e−1$ (d) $e_{4}$