class 12

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JEE Advanced

Match the reactions in Column I with nature of the reactions/type of the products in Column II. Indicate your answer by darkening the appropriate bubbles of the $4×4$ matrix given in the ORS.

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Let $u^=u_{1}i^+u_{2}j^ +u_{3}k^$ be a unit vector in be a unit vector in $R_{3}andw^=6 1 (i^+j^ +2k^)$.Given that there exists vector $v^$ in $R_{3}$ such that $∣u^×v∣=1andw^.(u^×v)=1$. Which of the following statement(s) is(are) correct?

A solution curve of the differential equation $(x_{2}+xy+4x+2y+4)(dxdy )−y_{2}=0$ passes through the point $(1,3)$ Then the solution curve is

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression is

Let $[x]$ be the greatest integer less than or equal to $x˙$ Then, at which of the following point (s) function $f(x)=xcos(π(x+[x]))$ is discontinuous? (a)$x=1$ (b) $x=−1$ (c) $x=0$ (d) $x=2$

Let $F(x)=∫_{x}[2cos_{2}t.dt]$ for all $x∈R$ and $f:[0,21 ]→[0,∞)$ be a continuous function.For $a∈[0,21 ]$, if F'(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is

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. $X_{23}+Y_{23}$

If$α=∫_{0}(e_{9}x+3tan_{(−1)x})(1+x_{2}12+9x_{2} )dxwherηn_{−1}$takes only principal values, then the value of $((g)_{e}∣1+α∣−43π )is$

Let $f:R→R$be a differentiable function with $f(0)=1$and satisfying the equation $f(x+y)=f(x)f_{prime}(y)+f_{prime}(x)f(y)$for all $x,y∈R$. Then, the value of $(g)_{e}(f(4))$is _______