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JEE Advanced

The total number of contributing structures showing hyperconjugation (involving C-H bonds) for the following carbocations is

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Let $f:[0,2]→R$ be a function which is continuous on [0,2] and is differentiable on (0,2) with $f(0)=1$$Let:F(x)=∫_{0}f(t )dtforx∈[0,2]I˙fF_{prime}(x)=f_{prime}(x)$ . for all $x∈(0,2),$ then $F(2)$ equals (a)$e_{2}−1$ (b) $e_{4}−1$(c)$e−1$ (d) $e_{4}$

The number of 5 digit numbers which are divisible by 4, with digits from the set ${1,2,3,4,5}$and the repetition of digits is allowed, is ________.

The total number of ways in which 5 balls of differert colours can be distributed among 3 persons so thai each person gets at least one ball is

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) $(0,2e−1 )$

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(21 ,21 ,21 )$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If $p =SP,q =SQ ,r=SR$ and $t=ST$ then the value of $∣(p ×q )×(r×(t)∣is$

For any positive integer $n$, define $f_{n}:(0,∞)→R$as $f_{n}(x)=j=1∑n tan_{−1}(1+(x+j)(x+j−1)1 )$for all $x∈(0,∞)$.Here, the inverse trigonometric function $tan_{−1}x$assumes values in $(−2π ,2π )˙$Then, which of the following statement(s) is (are) TRUE?$j=1∑5 tan_{2}(f_{j}(0))=55$(b) $j=1∑10 (1+fj_{′}(0))sec_{2}(f_{j}(0))=10$(c) For any fixed positive integer $n$, $(lim)_{x→∞}tan(f_{n}(x))=n1 $(d) For any fixed positive integer $n$, $(lim)_{x→∞}sec_{2}(f_{n}(x))=1$

For $x∈(0,π),$ the equation $sinx+2$sin$x−sin3x=3$ has (A)infinitely many solutions (B)three solutions (C)one solution (D)no solution

Let $n_{1},andn_{2}$, be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$,be the number one red and b of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probablity that this red ball was drawn from box II is $31 $ then the correct option(s) with the possible values of $n_{1},n_{2},n_{3},andn_{4}$, is(are)