Let \(A B C D\) be a quadrilateral with area 18 , with side \(A B\) parallel to the side \(C D\) and \(A B=2 C D\). Let \(A D\) be perpendicular to \(A B\) and \(C D\). If a circle is drawn inside the quadrilateral \(A B C D\) touching all the sides, then its radius is

Consider \(L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0\) where \(p\) is a real number and \(C: x^2+y^2+6 x-10 y+30=0\).
Statement 1 If line \(L_1\) is a chord of circle \(C\), then line \(L_2\) is not always a diameter of circle \(C\).
Statement 2 If line \(L_1\) is a diameter of circle \(C\), then line \(L_2\) is not a chord of circle \(C\).

Let \(a_1, a_2, a_3, \ldots, a_{11}\) be real numbers satisfying \(a_1=15,27-2 a_2>0\) and \(a_k=2 a_{k-1}-a_{k-2}\) for \(k=3,4, \ldots, 11\).
If \(\frac{a_1^2+a_2^2+\ldots+a_{11}^2}{11}=90\), then the value of \(\frac{a_1+a_2+\ldots+a_{11}}{11}\) is equal to

The total number of distinct $x\in R$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

The value of \(\int_{\sqrt{\log 2}}^{\sqrt{\log 3}} \frac{x \sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)} d x\) is

The area of the region $\left\{\left(\mathrm{x},\mathrm{y}\right):\mathrm{x}\mathrm{y}\le 8,\mathrm{ }1\le \mathrm{y}\le {\mathrm{x}}^2\right\}$ is

Let \(X\) be a random variable, and let \(P(X=x)\) denote the probability that \(X\) takes the value \(x\). Suppose that the points \((x, P(X=x)), x=0,1,2,3,4\), lie on a fixed straight line in the \(x y\)-plane, and \(P(X=x)=0\) for all \(x \in \mathbb{R}-\{0,1,2,3,4\}\). If the mean of \(X\) is \(\frac{5}{2}\), and the variance of \(X\) is \(\alpha\), then the value of \(24 \alpha\) is ___

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
The area of the region bounded by the curve \(y=f(x)\), the \(X\)-axis and the lines \(x=a\) and \(x=b\), where \(-\infty < a < b < -2\), is

If the straight lines \(\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}\) and \(\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}\) are coplanar, then the plane (s) containing these two lines is (are)

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The allowed energy for the particle for a particular value of \(n\) is proportional to

Let \(a\) and \(b\) be two nonzero real numbers. If the coefficient of \(x^5\) in the expansion of \(\left(a x^2+\frac{70}{27 b x}\right)^4\) is equal to the coefficient of \(x^{-5}\) in the expansion of \(\left(a x-\frac{1}{b x^2}\right)^7\), then the value of \(2 b\) is

Let the straight line \(x=b\) divide the area enclosed by \(y=(1-x)^2, y=0\) and \(x=0\) into two parts \(R_1(0 \leq x \leq b)\) and \(R_2(b \leq x \leq 1)\) such that \(R_1-R_2=\frac{1}{4}\). Then, \(b\) equals to