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JEE Advanced

The thermal dissociation equilibrium of $CaCO_{3}(s)$ is studied under different conditions $CaCO_{3}(s)⇔CaO(s)+CO_{2}(g)$ For this equilibrium , the correct statement(s) is(are)

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Let $f:[21 ,1]→R$ (the set of all real numbers) be a positive, non-constant, and differentiable function such that $f_{prime}(x)<2f(x)andf(21 )=1$ . Then the value of $∫_{21}f(x)dx$ lies in the interval (a)$(2e−1,2e)$ (b) $(3−1,2e−1)$(c)$(2e−1 ,e−1)$ (d) $(0,2e−1 )$

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 $

If $g(x)=∫_{sinx}sin_{−1}(t)dt,then:$ (a)$g_{prime}(2π )=−2π$ (b) $g_{prime}(−2π )=−2π$ (c)$g_{prime}(−2π )=2π$ (d) $g_{prime}(2π )=2π$

In a triangle PQR, P is the largest angle and $cosP=31 $. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

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

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 ____

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

Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $