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JEE Advanced

If in a hypothetical system if the angular momentum and mass are dimensionless. Then which of the following is true.

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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 ________.

Let $[x]$ be the greatest integer less than or equal to $x˙$ Then, at which of the following point (s) function $f(x)=xcos(π(x+[x]))$ is discontinuous? (a)$x=1$ (b) $x=−1$ (c) $x=0$ (d) $x=2$

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 $S={xϵ(−π,π):x=0,+2π }$The sum of all distinct solutions of the equation $3 secx+cosecx+2(tanx−cotx)=0$ in the set S is equal to

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$The tangent to a suitable conic (Column 1) at $(3 ,21 )$is found to be $3 x+2y=4,$then which of the following options is the only CORRECT combination?(IV) (iii) (S) (b) (II) (iii) (R)(II) (iv) (R) (d) (IV) (iv) (S)

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 $∣(p ×q )×(r×(t)∣is$

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

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. $X_{23}+Y_{23}$