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JEE Advanced

Image of an object approaching a convex mirror of radius of curvature 20 m along its optical axis is observed to move from $325 $ to $750 $ m in 30 seconds. What is the speed of the object in km per hour ?

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Let $f:[0,∞)→R$be a continuous function such that $f(x)=1−2x+∫_{0}e_{x−t}f(t)dt$for all $x∈[0,∞)$. Then, which of the following statement(s) is (are) TRUE?The curve $y=f(x)$passes through the point $(1,2)$(b) The curve $y=f(x)$passes through the point $(2,−1)$(c) The area of the region ${(x,y)∈[0,1]×R:f(x)≤y≤1−x_{2} }$is $4π−2 $(d) The area of the region ${(x,y)∈[0,1]×R:f(x)≤y≤1−x_{2} }$is $4π−1 $

For a real number $α,$ if the system $⎣⎡ 1αα_{2} α1α α_{2}α1 ⎦⎤ ⎣⎡ xyz ⎦⎤ =⎣⎡ 1−11 ⎦⎤ $ of linear equations, has infinitely many solutions, then $1+α+α_{2}=$

Let m and n be two positive integers greater than 1.If $α→0lim α_{m}e_{cosα_{n}}−e =−(2e )$ then the value of $nm $ 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

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

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(21 ,21 ,21 )$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If $p =SP,q =SQ ,r=SR$ and $t=ST$ then the value of $∣(p ×q )×(r×(t)∣is$

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) $e_{4}$

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