The area of the region
$\left\{(x,y):0\le x\le \frac{9}{4}, 0\le y\le 1, x\ge 3y, x+y\ge 2\right\}$ is

Let the vectors \(\mathbf{P Q}, \mathbf{Q R}, \mathbf{R S}, \mathbf{S T}, \mathbf{T U}\) and UP represent the sides of a regular hexagon.
Statement I PQ \(\times(\mathbf{R S}+\mathbf{S T}) \neq \mathbf{0}\)
Statement II \(\mathbf{P Q} \times \mathbf{R S}=\mathbf{0}\) and \(\mathbf{P Q} \times \mathbf{S Q} \neq \mathbf{0}\)

Let $f:R\to R$ be given by $f\left(x\right)$ $=\left\{\begin{array}{cc}{\mathrm{x}}^5+5{\mathrm{x}}^4+10{\mathrm{x}}^3+10{\mathrm{x}}^2+3\mathrm{x}+1,& \mathrm{x}<0;\\ {\mathrm{x}}^2-\mathrm{x}+1,& 0\le \mathrm{x}<1;\\ \frac{2}{3}{\mathrm{x}}^3-4{\mathrm{x}}^2+7\mathrm{x}-\frac{8}{3},& 1\le \mathrm{x}<3;\\ \left(\mathrm{x}-2\right){\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{e}}\left(\mathrm{x}-2\right)-\mathrm{x}+\frac{10}{3},& \mathrm{x}\ge 3\end{array}\right.$
Then which of the following options is/are correct?

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The probability that \(X \geq 3\) equals

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) is

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\)\(\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\). Find a least odd natural number \(n_0\), so that \(B_n>A_n, \forall n \geq n_0\).

The maximum value of the function \(f(x)=2 x^3-15 x^2+36 x-48\) on the set \(A=\left\{x \mid x^2+20 \leq 9 x\right\}\) is

In plotting stress versus strain curves for two materials \(P\) and \(Q\), a student by mistake puts strain on the \(y\)-axis and stress on the \(x\)-axis as shown in the figure. Then the correct statement(s) is(are)

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The reaction of compound \(\mathbf{P}\) with \(\mathrm{CH}_{3} \mathrm{MgBr}\) (excess) in \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O}\) followed by addition of \(\mathrm{H}_{2} \mathrm{O}\) gives \(\mathbf{Q}\). The compound \(\mathbf{Q}\) on treatment with \(\mathrm{H}_{2} \mathrm{SO}_{4}\) at \(0^{\circ} \mathrm{C}\) gives \(\mathbf{R}\). The reaction of \(\mathbf{R}\) with \(\mathrm{CH}_{3} \mathrm{COCl}\) in the presence of anhydrous \(\mathrm{AlCl}_{3}\) in \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) followed by treatment with \(\mathrm{H}_{2} \mathrm{O}\) produces compound \(\mathbf{S}\). [Et in compound \(\mathbf{P}\) is ethyl group]


Question :
The reactions, \(\mathbf{Q}\) to \(\mathbf{R}\) and \(\mathbf{R}\) to \(\mathbf{S}\), are

If \(x^2-10 a x-11 b=0\) have roots \(c\) and \(d\) and \(x^2-10 c x-11 d=0\) have roots \(a\) and \(b\), then, find \(a+b+c+d\).

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
1
Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?

When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is :
\(\mathrm{CHO}-\mathrm{CH}_{2}-\mathrm{CHOH}-\mathrm{CHOH}-\mathrm{CHOH}\) \(-~\mathrm{CH}_{2} \mathrm{OH}\)