If \(\lim _{x \rightarrow 0} \frac{2 a \sin x-\sin 2 x}{\tan ^{3} x}\) exists and is equal to 1 then the value of \(a\) is

\(\lim _{x \rightarrow 0}(\sin x)^{2 \tan x}\) is equal to

If \(P(x)=a x^2+b x+c\) and \(Q(x)=-a x^2+d x+c\) where \(a c \neq 0\), then \(P(x) \cdot Q(x)=0\) has \((a, b, c, d\) are real)

Let \(f: R \rightarrow R\) be given by \(f(x)=\left|x^{2}-1\right|, x \in R\). then

If \(\mathrm{I}_1=\int_0^{\pi / 4} \sin ^2 \mathrm{x} d \mathrm{x}\) and \(\mathrm{I}_2=\int_0^{\pi / 4} \cos ^2 \mathrm{x} d \mathrm{x}\), then,

\(\text {If } \log _{0.2}(x-1)>\log _{0.04}(x+5), \text { then }\)

The number of ways in which the letters of the word 'VERTICAL' can be arranged without changing the order of the vowels is

In aluminium extraction by the Bayer process, alumina is extracted from bauxite by sodium hydroxide at high temperature and pressures :
\(\mathrm{Al}_2 \mathrm{O}_3(\mathrm{~s})+2 \mathrm{OH}^{-}(\mathrm{aq}) \rightarrow 2 \mathrm{Al}_2 \mathrm{O}_2(\mathrm{aq})+\mathrm{H}_2 \mathrm{O}(1)\)

Solid impurities such as \(\mathrm{Fe}_2 \mathrm{O}_3\) and \(\mathrm{SiO}_2\) are removed and then \(\mathrm{Al}(\mathrm{OH})_4^{-}\)is reprecipitated : \(2 \mathrm{Al}(\mathrm{OH})_4^{-} \rightarrow \mathrm{Al}_2 \mathrm{O}_3 \cdot 3 \mathrm{H}_2 \mathrm{O}(\mathrm{s})+2 \mathrm{OH}^{-}\)(aq) . In the industrial world :

Which of the following oxides is paramagnetic?

The correct dimensional formula for impulse is given by

If the kinetic energies of an electron, an alpha particle and a proton having same de-Broglie wavelength are \(\varepsilon_1, \varepsilon_2\) and \(\varepsilon_3\) respectively, then

\(A B C\) is an isosceles triangle with an inscribed circle with centre \(O\). Let \(P\) be the midpoint of \(B C\). If \(A B=A C=15\) and \(B C=10\), then \(\mathrm{OP}\) equals

We define a binary relation - on the set of all \(3 \times 3\) real matrices as \(A \sim B,\) if and only if there exist invertible matrices \(P\) and \(Q\) such that \(B=P A Q^{-1} .\) The binary relation \(\sim\) is