For a triangle formed by \((0,0),(4,0)\) and \((3,4)\), the orthocenter is

The point of contact of the circles
\(x^2+y^2+2 x+2 y+1=0\)
and \(\quad x^2+y^2-2 x+2 y+1=0\) is

In a \(\triangle A B C, a=1, b=\sqrt{3}\) and \(\angle C=\pi / 6\). Then the measure of the third side \(c=\)

The equation of a circle which touches the $x$-axis and whose centre is $\left(1,2\right)$ is

\(\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{i}}=\overrightarrow{\mathbf{a}} \cdot(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})=\overrightarrow{\mathbf{a}}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})=1\), then \(\overrightarrow{\mathbf{a}}\) is equal to :

Sum of the coefficients of \(x^4\) and \(x^6\) in the expansion of \(\left(1+x-x^2\right)^6\) 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?

Which among the following pairs is not an acidic buffer?

When soap is dissolved in hard water, its cleaning ability comes down. This is due to the formation of:

If \(\cos (A-B)=3 / 5\) and \(\tan A \tan B=2\), then which one of the following is true?

If \((2-5 x)^{\frac{-1}{5}}=a_0+a_1 x+a_2 x^2+\ldots\), then \(\frac{a_1}{a_2}=\)

If \(t \in \mathbb{R}-\{-1\}\), then the locus of the point \(\left(\frac{3 a t}{1+t^3}, \frac{3 \mathrm{at}^2}{1+t^3}\right)\) is

The area of the region (in sq.units) bounded by the curves \(x^2+y^2=16\) and \(\mathrm{y}^2=6 \mathrm{x}\) is