Let $a,b$ and $c$ be three positive real numbers such that the sum of any two of them is greater than the third. All the values of $\lambda$ such that the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real, are given by

If \(\frac{\tan (\alpha+\beta-\gamma)}{\tan (\alpha-\beta+\gamma)}=\frac{\tan \gamma}{\tan \beta}\) and \(\beta \neq \gamma\), then the value of \(\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma\) is

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola \(x^2-y^2=a^2\) is

If \(2 x^2+3 x-2=0\) and \(3 x^2+a x-2=0\) have one common root then the sum of all possible values of \(a\) is

If \(A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]\) is expressed as a sun of a symmetric matrix \(\mathrm{P}\) and skew symmetric matrix \(\mathrm{Q}\), then \(\mathrm{P}^{\mathrm{T}}-\mathrm{Q}^{\mathrm{T}}=\)

If \(f_n(x)=\log \log \log \ldots \log x \quad(\log\) is repeated \(n\)-times), then
\(\int\left(x f_1(x) f_2(x) \ldots f_n(x)\right)^{-1} d x\) is equal to

Let a, b and c are 3 non zero vectors such that no 2 of these are collinear. If vector a + 2b is collinear with c and b + 3c is collinear with a (l being some non zero scalar) then
a + 2b + 6c equals

If \(f(x)=(\cos x)(\cos 2 x) \ldots(\cos n x)\) then \(f^{\prime}(x)+\sum_{r=1}^n(r \tan r x) f(x)\) is equal to

\(\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\frac{1}{4 \cdot 9}+\ldots\) is equal to

A body of mass \(1 \mathrm{~g}\) and carrying a charge \(10^{-8} \mathrm{C}\) passes from two points \(P\) and \(Q . P\) and \(Q\) are at electric potentials. \(600 \mathrm{~V}\) and \(0 \mathrm{~V}\), respectively. The velocity of the body at \(Q\) is \(20 \mathrm{cms}^{-1}\). It velocity in \(\mathrm{ms}^{-1}\) at \(P\) is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

If the slope of one of the pair of lines represented by \(2 x^2+3 x y+K y^2=0\) is 2 , then the angle between the pair of lines is

If \(\alpha\) is a repeated root of multiplicity 2 of the equation \(18 x^3-33 x^2+20 x-4=0\), then