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JEE Advanced

The total number of cyclic isomers possible for a hydrocarbon with the molecular formula $C_{4}H_{6}$ is

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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 $0<x<1?$

The coefficients of three consecutive terms of $(1+x)_{n+5}$are in the ratio 5:10:14. Then $n=$___________.

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

If$α=∫_{0}(e_{9}x+3tan_{(−1)x})(1+x_{2}12+9x_{2} )dxwherηn_{−1}$takes only principal values, then the value of $((g)_{e}∣1+α∣−43π )is$

The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x))=0$ has A. only purely imaginary roots B. all real roots C. two real and purely imaginary roots D. neither real nor purely imaginary roots

For $a∈R$ (the set of all real numbers), $a=−1),$$(lim)_{n→∞}((n+1)_{a−1}[(na+1)+(na+2)+……(na+n)]1_{a}+2_{a}++n_{a} =60.1 $Then $a=$(a)$5$ (b) 7 (c) $2−15 $ (d) $2−17 $

Let $f:RR$be a differentiable function such that $f(0),f(2π )=3andf_{prime}(0)=1.$If $g(x)=∫_{x}[f_{prime}(t)cosect−cottcosectf(t)]dtforx(0,2π ],$then $(lim)_{x0}g(x)=$

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 of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is