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JEE Advanced

Let $fR→R$ be a function we say that f has property 1 if $h→0lim ∣h∣ f(h)−f(0) $ exist and is finite. Property 2 if $h→0lim h_{2}f(h)−f(0) $ exist and is finite. Then which of the following options is/are correct?

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Which of the following values of $α$satisfying the equation $∣∣ (1+α)_{2}(1+2α)_{2}(1+3α)_{2}(2+α)_{2}(2+2α)_{2}(2+3α)_{2}(3+α)_{2}(3+2α)_{2}(3+3α)_{2}∣∣ =−648α?$$−4$b. $9$c. $−9$d. $4$

Let $a,b,xandy$ be real numbers such that $a−b=1andy=0.$ If the complex number $z=x+iy$ satisfies $Im(z+1az+b )=y$ , then which of the following is (are) possible value9s) of x? (a)$−1−1−y_{2} $ (b) $1+1+y_{2} $(c)$−1+1−y_{2} $ (d) $−1−1+y_{2} $

Coefficient of $x_{11}$ in the expansion of $(1+x_{2})_{4}(1+x_{3})_{7}(1+x_{4})_{12}$ is 1051 b. 1106 c. 1113 d. 1120

Football teams T1 and T2 have to play two games against each other. It is assumed that theoutcomes of the two games are independent. The probabilities of T1 winning, drawing andlosing a game against T2 are1/ 2,and1/6,1/3respectively. Each team gets 3 points for a win,1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total pointsscored by teams T1 and T2, respectively, after two gamesP $(X=Y)$ is

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is

The value of $∫_{0}4x_{3}{dx_{2}d_{2} (1−x_{2})_{5}}dxis$

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______

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