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JEE Advanced

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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

Late $a∈R$and let $f:R$be given by $f(x)=x_{5}−5x+a,$then$f(x)$has three real roots if $a>4$$f(x)$has only one real roots if $a>4$$f(x)$has three real roots if $a<−4$$f(x)$has three real roots if $−4<a<4$

Let $XandY$be two arbitrary, $3×3$, non-zero, skew-symmetric matrices and $Z$be an arbitrary $3×3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?a.$Y_{3}Z_{4}Z_{4}Y_{3}$b. $x_{44}+Y_{44}$c. $X_{4}Z_{3}−Z_{3}X_{4}$d. $X_{23}+Y_{23}$

Let $S$be the set of all non-zero real numbers such that the quadratic equation $αx_{2}−x+α=0$has two distinct real roots $x_{1}andx_{2}$satisfying the inequality $∣x_{1}−x_{2}∣<1.$Which of the following intervals is (are) a subset (s) of $S?$$(21 ,5 1 )$b. $(5 1 ,0)$c. $(0,5 1 )$d. $(5 1 ,21 )$

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover cards numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isa.$264$ b. $265$ c. $53$ d. $67$

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) $(0,2e−1 )$

Let $f:(0,π)→R$be a twice differentiable function such that $(lim)_{t→x}t−xf(x)sint−f(x)sinx =sin_{2}x$for all $x∈(0,π)$. If $f(6π )=−12π $, then which of the following statement(s) is (are) TRUE?$f(4π )=42 π $(b) $f(x)<6x_{4} −x_{2}$for all $x∈(0,π)$(c) There exists $α∈(0,π)$such that $f_{prime}(α)=0$(d) $f(2π )+f(2π )=0$