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Let a,b ,c be positive integers such that $ab $ is an integer. If a,b,c are in GP and the arithmetic mean of a,b,c, is b+2 then the value of $a+1a_{2}+a−14 $ is

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 $P?$

A line L : y = mx + 3 meets y-axis at E (0, 3) and the arc of the parabola $y_{2}=16x$ $0≤y≤6$ at the point art $F(x_{0},y_{0})$. The tangent to the parabola at $F(X_{0},Y_{0})$ intersects the y-axis at $G(0,y)$. The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum P) m= Q) = Maximum area of $△EFG$ is (R) $y_{0}=$ (S) $y_{1}=$

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)$(lim)_{n→1_{+}}f(x)=0$

Let $f:[−21 ,2]→R$ and $g:[−21 ,2]→R$ be functions defined by $f(x)=[x_{2}−3]$ and $g(x)=∣x∣f(x)+∣4x−7∣f(x)$, where [y] denotes the greatest integer less than or equal to y for $y∈R$. Then,

Let $S={1,2,3,¨ 9}F˙ork=1,2,5,letN_{k}$be the number of subsets of S, each containing five elements out of which exactly $k$are odd. Then $N_{1}+N_{2}+N_{3}+N_{4}+N_{5}=?$210 (b) 252 (c) 125 (d) 126

Let $f:(0,∞)→R$ be a differentiable function such that $f_{′}(x)=2−xf(x) $ for all $x∈(0,∞)$ and $f(1)=1$, then

Let $f[0,1]→R$ (the set of all real numbers be a function.Suppose the function f is twice differentiable, $f(0)=f(1)=0$,and satisfies $f_{′}(x)–2f_{′}(x)+f(x)≤e_{x},x∈[0,1]$.Which of the following is true for $0<x<1?$