Let the domain of the function \(f(x)=\log _{4}\left(\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)\right)\) be \((a, b)\). Then the value of the integral \(\int_{a}^{b} \frac{\sin ^{3} x}{\left(\sin ^{3} x+\sin ^{3}(a+b-x)\right)} d x\) is equal to \(.....\)

Let \(f : R \rightarrow R\) be continuous function satisfying \(f ( x )+ f ( x + k )= n\), for all \(x \in R\) where \(k >0\) and \(n\)is a positive integer. If \(I _{1}=\int\limits_{0}^{4 n k} f ( x ) dx\) and \(I _{2}=\int\limits_{- k }^{3 k } f ( x ) dx\), then

If the area of the region \(\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}\) is \(A\), then \(12 A\) is equal to

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

If \(\frac{tan(A-B)}{tan~A}+\frac{sin^{2}C}{sin^{2}A}=1,\) \(A, B, C\in(0,\frac{\pi}{2})\), then

If the system of equations:
\(x+y+z=5\)
\(x+2y+3z=9\)
\(x+3y+\lambda z=\mu\)
has infinitely many solutions, then the value of \(\lambda+\mu\) is:

The mean and standard deviation of \(10\) observations are \(20\) and \(84\) respectively. Later on, it was observed that one observation was recorded as \(50\) instead of \(40\). Then the correct variance is:

If  \(A\) and \(B\)  are any two events such that \(P\left( A \right)\, = \frac{2}{5}\) and \(P\left( {A \cap \,B} \right)\, = \frac{3}{{20}},\) then the conditional probability, \(P\left( {A\,|\,A'\, \cup \,B')} \right),\) where \(A'\) denotes the complement of \(A,\)  is equal to 

If \((a, b)\) be the orthocentre of the triangle whose vertices are \((1,2),(2,3)\) and \((3,1)\), and \(I_1=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}, \mathrm{I}_2=\int_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}\) , then \(36 \frac{\mathrm{I}_1}{\mathrm{I}_2}\) is equal to :

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \quad \vec{b}=-\hat{i}-8 \hat{j}+2 \hat{k} \quad\) and \(\overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) be three vectors such that \(\vec{b} \times \vec{a}=\vec{c} \times \vec{a}\). If the angle between the vector \(\vec{c}\) and the vector \(3 \hat{i}+4 \hat{j}+\hat{k}\) is \(\theta\), then the greatest integer less than or equal to \(\tan ^2 \theta\) is :

The coefficient of \(x^{18}\) in the expansion of \(\left(x^4-\frac{1}{x^3}\right)^{15}\) is \(...........\).

Let \(p, q\) and \(r\)  be real numbers \((p \ne q,r \ne 0),\) such that the roots of the equation \(\frac{1}{{x + p}} + \frac{1}{{x + q}} = \frac{1}{r}\) are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to .

Let \(S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots\) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is \(\sqrt{2026 \mathrm{~S}_{2025}}\), then the absolute difference betwen \(20^{\text {th }}\) and \(15^{\text {th }}\) terms of the A.P. is