The triangle formed by the lines \(2 x^2+x y-6 y^2=0\) and \(x+y-1=0\) is

If \(x_1, x_2, \ldots, x_n\) are \(n\) observations such that \(\sum_{i=1}^n x_1^2=400\) and \(\sum_{i=1}^n x_1=80\), then the least value of \(n\) is

The value of \(\sqrt{42+\sqrt{42+\sqrt{42+\ldots \ldots .}}}\) is equal

Consider the following statements.
I. In \(\triangle A B C\), if \(c=6\) and \(\cos C=\frac{-11}{25}\), then
\[
R=\frac{25}{2 \sqrt{14}}
\]
II. In \(\triangle A B C\), if \(a=3, b=4, c=6\), then \(A B C\) is acute angled triangle.

Which of the above statements is/are true?

If \(\omega\) is the cube root of unity \(\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2}=\)

\(\sin \frac{2 \pi}{5}+\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}+\sin \frac{8 \pi}{5}\) is equal to

\(\lim _{x \rightarrow 1}\left(\lim _{7 \rightarrow \infty} y\left((e x)^{1 / y}-1\right)\right)=\)

Two towers \(A\) and \(B\), each of height \(20 \mathrm{~m}\) are situated a distance \(200 \mathrm{~m}\) apart. A body thrown horizontally from the top of the tower \(A\) with a velocity \(20 \mathrm{~ms}^{-1}\) towards the tower \(B\) hits the ground at point \(P\) and another body thrown horizontally from the top of tower \(B\) with a velocity \(30 \mathrm{~ms}^{-1}\) towards the tower \(A\) hits the ground at point \(Q\). If a car starting from rest from \(P\) reaches \(Q\) in 10 seconds, then the acceleration of the car is \(\left(\right.\) acceleration due to gravity \(\left.=10 \mathrm{~ms}^{-2}\right)\)

If $\cos \theta =-\frac{\sqrt{3}}{2}$ and $\sin \alpha =-\frac{3}{5}$, where $\theta$ does not lie in the third quadrant, then $\frac{2\tan \alpha +\sqrt{3}\tan \theta }{{\cot }^2\theta +\cos \alpha }$ is equal to

Equation \(x^5-5 x^3+5 x^2-1=0\) has equal roots

The incorrect statement from the following is

If the lines \(x^2+k x y+y^2=0\) and \(x+y=1\) make up the sides of an equilateral triangle, then the value of \(k^2\) equals

Let \(Q\) be the set of all rational numbers in \([0,1]\) and \(f:[0,1] \rightarrow[0,1]\) be defined by
\[
f(x)=\left\{\begin{array}{r}
x \text { for } x \in Q \\
1-x \text { for } x \notin Q
\end{array}\right.
\]
Then, the set \(S=\{x \in[0,21]:(f o f)(x)\}\) is equal to