The image of the line \(\frac{{x - 1}}{3} = \frac{{y - 3}}{1} = \frac{{z - 4}}{{ - 5}}\) in the plane \(2x - y + z + 3 = 0\)  is the line :

The mean and the standard deviation (s.d.) of \(10\) observations are \(20\) and \(2\) resepectively. Each of these \(10\) observations is multiplied by \(\mathrm{p}\) and then reduced by \(\mathrm{q}\), where \(\mathrm{p} \neq 0\) and \(\mathrm{q} \neq 0 .\) If the new mean and new s.d. become half of their original values, then \(q\) is equal to

Let \(X=\mathbf{R} \times \mathbf{R}\). Define a relation \(R\) on \(X\) as :
\(\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2\)

Statement I : \(\quad \mathrm{R}\) is an equivalence relation.
Statement II : For some \((a, b) \in X\), the set \(S=\{(x, y) \in X:(x, y) R(a, b)\}\) represents a line parallel to \(y=x\).
In the light of the above statements, choose the correct answer from the options given below :

If \(x, y, z\) are in arithmetic progression with common difference \(d , x \neq 3 d ,\) and the
determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(k ^{2}\) is ..... .

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

The area (in sq. units) of the part of the circle \(x ^{2}+ y ^{2}=36\) which is outside the parabola \(y ^{2}=9 x ,\) is

Consider the following three statements for the function \( f:(0,\infty)\rightarrow\mathbb{R} \) defined by
\( f(x)=|log_{e}x|-|x-1| \):
(I) f is differentiable at all \( x>0 \).
(II) f is increasing in (0, 1).
(III) f is decreasing in (1, ∞).
Then:

Let \(\mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\mathrm{L}_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}\) be two lines. Then which of the following points lies on the line of the shortest distance between \(L_1\) and \(L_2\) ?

The tangent to the parabola \(y^2 = 4x\) at the point where it intersects the circle \(x^2 + y^2 = 5\) in the first quadrant, passes through the point

Area (in sq. units) of the region outside \(\frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1\) and inside the ellipse \(\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{9}=1\) is

The function \(f : R \rightarrow R\) defined by \(f(x)=\lim _{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is continuous for all \(x\) in.

The value of \(\lambda\) and \(\mu\) such that the system of equations \(x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu\) has no solution, are :

Let \(\quad f:(-\infty, \infty)-\{0\} \rightarrow R\) be a differentiable function such that \(f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)\). Then \(\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a\) is equal to