Let $F:R\to R$ be a function. We say that $f$ has:
 PROPERTY $1$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{\sqrt{\left|h\right|}}$ exists and is finite, and 
PROPERTY $2$  if $\underset{h\to 0}{\lim }\frac{f\left(h\right)-f\left(0\right)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?

For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

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Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).


Question:

The probability that \(x_{1}+x_{2}+x_{3}\) is odd, is

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

If \(0 < x < 1\), then \(\sqrt{1+x^2}\left[\left\{x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1\right]^{\frac{1}{2}}\) is equal to

Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

Let \(\omega=e^{i \pi / 3}\) and \(a, b, c, x, y, z\) be non-zero complex numbers such that \(a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z\).
Then, the value of \(\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}\) is

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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
The equation of circle \(C\) is

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The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Precipitate \(\mathbf{X}\) is

A vertical line passing through the point $\left(\text{h},\text{0}\right)$ intersects the ellipse $\frac{{\text{x}}^2}{4}+\frac{{\text{y}}^2}{3}=1$ at the points $\text{P}$ and $\text{Q}$. Let the tangents to the ellipse at $\text{P}$ and $\text{Q}$ meet at the point $\text{R}$. If $\Delta \left(\mathrm{h}\right)=$ area of triangle $\mathrm{PQR},{\Delta }_1=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\max }\Delta \left(\mathrm{h}\right)$ and ${\mathrm{\Delta }}_2=\underset{\frac{1}{2}\le \mathrm{h}\le 1}{\min }\Delta \left(\mathrm{h}\right)$, then $\frac{8}{\sqrt{5}}{\Delta }_1-8{\Delta }_2=$

Let \(\left(x_0, y_0\right)\) be the solution of the following equations \((2 x)^{\log 2}=(3 y)^{\log 3}\), \(3^{\log x}=2^{\log y}\), then \(x_0\) is equal to

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

Question:
If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point \(B\), immediately after it strikes the second incline is

Consider an elliptically shaped rail PQ in the vertical plane with $\mathrm{O}\mathrm{P}\mathrm{}=\mathrm{}3\mathrm{}\mathrm{m}$ and $\mathrm{O}\mathrm{Q}\mathrm{}=\mathrm{}4\mathrm{}\mathrm{m}$ . A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $(\mathrm{n}\times 10)$ Joules. The value of n is (take acceleration due to gravity = $10\mathrm{}\mathrm{m}{\mathrm{s}}^{–2}$ )