A pair of perpendicular straight lines passes through the origin and also through the point of intersection of the curve \(x^2+y^2=4\) with \(x+y=a\). The set containing the value of ' \(a\) ' is

The local maximum of \(y=x^3-3 x^2+5\) is attained at

The values of c such that the line \(y=4 x+c\) touches the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) is

\(2 \hat{i}-3 \hat{j}+\hat{k}\) and \(\hat{i}+2 \hat{j}-3 \hat{k}\) are the position vectors of two points \(A\) and \(B\) respectively and \(C\) divides \(A B\) in the ratio \(3: 2\). If \(3 \hat{i}-\hat{j}+2 \hat{k}\) is the position vector of a point \(D\), then the unit vector in the direction of \(\overrightarrow{C D}\) is

The equation of the plane passing through the points with position vectors \(A(2 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\), \(B(-3 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}-9 \hat{\mathbf{k}})\) and \(C(-5 \hat{\mathbf{i}}-6 \hat{\mathbf{k}})\) is

If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^4-4 x^3+3 x^2+2 x-2=0\) such that \(\alpha\) and \(\beta\) are integers and \(\gamma, \delta\) are irrational numbers, then \(\alpha+2 \beta+\gamma^2+\delta^2=\)

If \[ \int \frac{x-\sin x}{1+\cos x} d x=x \tan \left(\frac{x}{2}\right)+p \log \left|\sec \left(\frac{x}{2}\right)\right|+C, \] then \(p\) is equal to

\(A(2,3, k), B(-1, k,-1), C(4,-3,2)\) are the vertices of \(\triangle A B C\). If \(A B=A C\) and \(k\gt0\), then \(A B C\) is

The correct option for axial distances and axial angles for hexagonal crystal system is

The disperse phase, dispersion medium and nature of colloidal solution (lyophilic or lyophobic) of 'gold sol' respectively are :

If \(z=3+5 i\), then \(z^3+\bar{z}+198\) is equal to

An electrically charged particle enters into a uniform magnetic induction field in a direction perpendicular to the field with a velocity \(v\). Then, it travels

Consider the following statements: Assertion (A): The direction ratios of a line \(\mathrm{L}_1\) are \(2,5,7\) and the direction ratios of another line \(\mathrm{L}_2\) are \(\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}\). Then the lines \(\mathrm{L}_1, \mathrm{~L}_2\) are parallel Reason (R): If the direction ratios of a line \(L_1\) are \(a_1, b_1, c_1\), the direction ratios of a line \(\mathrm{L}_2\) are \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2\) and \(\mathrm{a}_1 \mathrm{a}_2+\mathrm{b}_1 \mathrm{~b}_2\) \(+c_1 c_2=0\), then the lines of \(L_1, L_2\) are parallel Which one of the following is True?