The number of distinct solutions of the equation \(\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|\) in the interval \([0,2 \pi],\) is

For \(\alpha, \beta \in \mathrm{R}\) and a natural number \(\mathrm{n}\), let \(A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|\). Then \(2 A_{10}-A_8\)

If \(1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots\) \(+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k\), where \(m, n, k \in \mathbf{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :

If the range of the function \(f(x)=\frac{5-x}{x^2-3 x+2}\), \(x \neq 1,2\), is \((-\infty, \alpha] \cup[\beta, \infty)\), then \(\alpha^2+\beta^2\) is equal to:

The value of the integral \(\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} {{{\sin }^4}\,x\left( {1 + \log \left( {\frac{{2 + \sin \,x}}{{2 - \sin \,x}}} \right)} \right)\,dx} \) is

If \({\cos ^{ - 1}}\left( {\frac{2}{{3x}}} \right) + {\cos ^{ - 1}}\,\left( {\frac{3}{{4x}}} \right) = \frac{\pi }{2},\,x > \frac{3}{4}\) then \(x\) is equal to

If \(f(x)\) and \(g(x)\) are two polynomials such that the polynomial \(P ( x )=f\left( x ^{3}\right)+ xg \left( x ^{3}\right)\) is divisible by \(x^{2}+x+1,\) then \(P(1)\) is equal to ....... .

Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be the solution of the differential equation, \(\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}=-\cos x, y>0, y(0)=1,\) If \(y(\pi)=a\) and \(\frac{\mathrm{dy}}{\mathrm{dx}}\) at \(\mathrm{x}=\pi\) is \(b\), then the ordered pair \((a, b)\) is equal to :

In the expansion of \(\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}\), \(x>0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

If the orthocentre of the triangle formed by the lines \(y=x+1, y=4 x-8\) and \(y=m x+c\) is at \((3,-1)\), then \(\mathrm{m}-\mathrm{c}\) is :

An ellipse passes through the foci of the hyperbola, \(9x^2 - 4y^2 = 36\) and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is \(\frac {1}{2}\), then which of the following points does not lie on the ellipse?

The letters of the word \(OUGHT\) are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word \(TOUGH\) is :