For the following questions answer them individually
If the solution curve $$y =f (x)$$ of the differential equation
$$(x^{2}-4)y^{'}-2xy+2x(4-x^{2})^{2}=0,x>2,$$
passes through the point (3, 15), then the local maximum value of $$f$$ is __________
If the image of the point $$P(a, 2, a)$$ in the line $$\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$$ is Q and the image of Q in the line $$\frac{x-2b}{2}=\frac{y-a}{1}=\frac{z+2b}{-5}$$ is P, then a + b is equal to _____.
Let S denote the set of 4-digit numbers $$abcd$$ such that $$a > b > c > d$$ and P denote the set of 5-digit numbers having product of its digits equal to 20. Then $$n(S) + n(P)$$ is equal to ______
The number of elements in the
set $$ S=\left\{ x:x\in [0,100] \text{ and } \int_{0}^{x} t^{2} \sin(x-t)dt=x^{2}\right\}$$ is _________
Let $$A = \begin{bmatrix}0 & 2 & -3 \\-2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}$$ and B be a matrix such that $$B(I- A)=I+A.$$ Then the sumof the diagonal elements of $$B^{T}B$$ is equal to _________
One mole of an ideal diatomic gas expands from volume $$V$$ to $$2 V$$ isothermally at a temperature $$27^{o}C$$ and does W joule of work. lf the gas undergoes same magnitude of expansion adiabatically from $$27^{o}C$$ doing the same amount of work $$W$$, then its final temperature will be (close to) ____ ^{o}C.
$$(\log_{e}2 = 0.693)$$
A small metallic sphere of diameter 2 mm and density $$10.5 g/cm ^{3}$$ is dropped in glycerine having viscosity 10 Poise and density $$1.5 g/cm^{3}$$ respectively. The terminal velocity attained by the sphere is __ $$cm/s$$.
$$(\pi=\frac{22}{7} \text { and } g=m/s^{2})$$
Suppose a long solenoid of 100 cm length, radius 2 cm having 500 turns per unit length, carries a current $$I= 10 \sin (\omega t)$$ A, where $$\omega$$ = 1000 rad.ls. A circular conducting loop (B) of radius 1 cm coaxially slided through the solenoid at a speed $$v = 1 cm/s$$. The r.m.s. current through the loop when the coil B is inserted 10 cm inside the solenoid is $$\alpha/\sqrt{2}\mu A$$. The value of $$\alpha$$ is ______.
[Resistance of the loop= 10$$\Omega$$]
A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio 2 : 2 : 3. The two pieces of equal masses fly off perpendicular to each other with a speed of 18 m/s each. The velocity of the heavier fragment is ______m/s.
To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm. The least count of scale is 1 cm. The percentage error in the ratio of EMFs is______
