class 12

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JEE Advanced

The Fischer presentation of D-glucose is given below. The correct structure(s) of $β$-L- glucopyranose is (are)

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Let the curve C be the mirror image of the parabola $y_{2}=4x$ with respect to the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=−5$, then the distance between A and B is

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 $αandβ$ be nonzero real numbers such that $2(cosβ−cosα)+cosαcosβ=1$ . Then which of the following is/are true? (a) $3 tan(2α )+tan(2β )=0$ (b) $3 tan(2α )−tan(2β )=0$ (c) $tan(2α )+3 tan(2β )=0$ (d) $tan(2α )−3 tan(2β )=0$

Let $f:RR$be a continuous odd function, which vanishes exactly at one point and $f(1)=21 ˙$Suppose that $F(x)=∫_{−1}f(t)dtforallx∈[−1,2]andG(x)=∫_{−1}t∣f(f(t))∣dtforallx∈[−1,2]I˙G(x)f(lim)_{x1}(F(x)) =141 ,$Then the value of $f(21 )$is

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

Let `f(x)``=x+log_ex-xlog_ex ,x(0,oo)dot`Column 1 contains information about zeros of `f^(prime)(x)f^(prime)(x)a n df^(x)dot`Column 2 contains information about the limiting behaviour of `f^(prime)(x)f^(prime)(x)a n df^(x)`at infinity.Column 2 contains information about the increasing/decreasing nature of `f(x)a n df^(prime)(x)dot`Column I, Column 2, Column 3I, `f(x)=0forsom ex(l , e^2)`, (i), `("lim")_("x"vecoo"")f^(prime)(x)=0`, (P), `f`is increasing in (0,1)II, `f'(x)=0forsom ex(l , e)`, (ii), `("lim")_("x"vecoo"")f^(x)=-oo`, (Q), `f`is decreasing in `(e ,e^2)`III, `f'(x)=0forsom ex(0,1)`, , `("lim")_("x"vecoo"")f^(prime)(x)=-oo`, (R), `f`is increasing in (0,1)IV, `f^(' '(x))=0forsom ex(1, e)`, , `("lim")_("x"vecoo"")f^prime^'(x)=0`, (S), `f`is decreasing in (`e , e^2`)Which of the following options is the only CORRECT combination?(I) (ii) (P) (b) (IV) (iv) (S) (III) (iii) (R) (d) (II) (ii) (Q)Which of the following option is the only incorrect combination?(III) (i) (R) (b) (I) (iii) (P)(II) (iii) (P) (d) (II) (iv) (Q)Which of the following options is the only CORRECT combination?(I) (ii) (R) (b) (II) (iii) (S)(III) (iv) (P) (d) (IV) (i) (S)

A box $B_{1}$, contains 1 white ball, 3 red balls and 2 black balls. Another box $B_{2}$, contains 2 white balls, 3 red balls and 4 black balls. A third box $B_{3}$, contains 3 white balls, 4 red balls and 5 black balls.

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