The sum of all the real values of \(x\) satisfying the equation \(\left(x^2-7 x+11\right)^{x^2-6 x-7}=1\) is

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

\(\int \frac{d x}{4+3 \cot x}=\)

A hyperbola passing through a focus of the ellipse \(\frac{x^2}{169}+\frac{y^2}{25}=1\). Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of eccentricities is 1 . Then, the equation of the hyperbola is

The straight lines \(x+3 y-4=0, x+y-4=0\) and \(3 x+y-4=0\)

Let \(S \equiv x^2+y^2-6 x-6 y+4=0\) and \(S^{\prime} \equiv x^2+y^2-2 x-4 y+3=0\) be two circles. The centre of a circle of radius \(\sqrt{14}\) and having same radical axes with either of \(S=0\) or \(S^{\prime}=0\) is

\(\int_0^{\frac{\pi}{2}} \frac{\sin ^2 x}{\sin x+\cos x} d x=\)

If \(A\) and \(B\) are acute angles satisfying \(3 \cos ^2 A+2 \cos ^2 B=4\) and \(\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}\), then \(A+2 B=\)

Identify all the products formed when \(\mathrm{XeF}_4\) is completely hydrolysed.

\(\int \frac{x+1}{x\left(1+x e^x\right)} d x=\)

If \(\alpha, \beta, \gamma\) are the roots of \(x^3-2 x^2+3 x-4=0\), then the value of \(\alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2\) is

Assertion : Aldehydes are more reactive than ketones towards nucleophilic addition reactions. Reason (R) : In aldehydes, carbonyl carbon is less electrophilic compared to ketones. The correct answer is

Two adjacent sides of a triangle are represented by the vectors \(2 \bar{i}+\bar{j}-2 \bar{k}\) and \(2 \sqrt{3} \bar{i}-2 \sqrt{3} \bar{j}+\sqrt{3} \bar{k}\). Then the least angle of the triangle and perimeter of the triangle are respectively