class 12

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JEE Advanced

if (a,d) denotes an A.P with first term a and common different d. if the A.P formed by intersection of three A.P's given (1,3), (2,5),and (3,7) is a new A.P (A,D). Then the value of A+D is

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The function $f(x)=2∣x∣+∣x+2∣=∣∣x∣2∣−2∣x∣∣$has a local minimum or a local maximum at $x=$$−2$ (b) $−32 $ (c) 2 (d) $32 $

For how many values, of p, the circle $x_{2}+y_{2}+2x+4y−p=0$and the coordinate axes have exactly three common points?

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 n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of $nm $ is ____

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

For every twice differentiable function $f:R→[−2,2]$with $(f(0))_{2}+(f_{prime}(0))_{2}=85$, which of the following statement(s) is (are) TRUE?There exist $r,s∈R$where $r<s$, such that $f$is one-one on the open interval $(r,s)$(b) There exists $x_{0}∈(−4,0)$such that $∣∣ f_{prime}(x_{0})∣∣ ≤1$(c) $(lim)_{x→∞}f(x)=1$(d) There exists $α∈(−4,4)$such that $f(α)+f(α)=0$and $f_{prime}(α)=0$

Let $O$be the origin, and $OXxOY,OZ$be three unit vectors in the direction of the sides $QR$, $RP$, $PQ$, respectively of a triangle PQR.If the triangle PQR varies, then the minimum value of $cos(P+Q)+cos(Q+R)+cos(R+P)$is:$−23 $ (b) $35 $ (c) $23 $ (d) $−35 $

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60