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JEE Advanced

A few electric field lines for a system of two charges $Q_{1}$ and $Q_{2}$ fixes at two different points on the x-axis are shown in the figure. These lines suggest that

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Let P and Q be distinct points on the parabola $y_{2}=2x$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $ΔOPQ$ is $32$ , then which of the following is (are) the coordinates of $P?$

Let $P$be a point in the first octant, whose image $Q$in the plane $x+y=3$(that is, the line segment $PQ$is perpendicular to the plane $x+y=3$and the mid-point of $PQ$lies in the plane $x+y=3)$lies on the z-axis. Let the distance of $P$from the x-axis be 5. If $R$is the image of $P$in the xy-plane, then the length of $PR$is _______.

Let m be the smallest positive integer such that the coefficient of $x_{2}$ in the expansion of $(1+x)_{2}+(1+x)_{3}+(1+x)_{4}+……..+(1+x)_{49}+(1+mx)_{50}$ is $(3n+1)._{51}C_{3}$ for some positive integer n. Then the value of n is

Let $S_{n}=k=1∑4n (−1)2k(k+1) k_{2}˙$Then $S_{n}$can take value (s)$1056$b. $1088$c. $1120$d. $1332$

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

The slope of the tangent to the curve $(y−x_{5})_{2}=x(1+x_{2})_{2}$at the point $(1,3)$is.

Let $f:R0,1 $ be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)? $e_{x}−∫_{0}f(t)sintdt$ (b) $f(x)+∫_{0}f(t)sintdt$(c)$x−∫_{0}f(t)costdt$ (d) $x_{9}−f(x)$

Let $f(x)=∣1−x∣1−x(1+∣1−x∣) cos(1−x1 )$ for $x=1.$ Then: (A)$(lim)_{n→1_{−}}f(x)$ does not exist (B)$(lim)_{n→1_{+}}f(x)$ does not exist (C)$(lim)_{n→1_{−}}f(x)=0$ (D)$(lim)_{n→1_{+}}f(x)=0$