If \(a, b, c\) are non-zero real numbers and if the system of equations \((a - 1 )x = y + z,\)  \((b - 1 )y = z + x ,\) \((c - 1 )z= x + y,\) has a non-trivial solution, then \(ab + bc + ca\) equals

A hyperbola has its centre at the origin, passes through the point \((4, 2)\) and has transverse axis of length \(4\) along the \(x -\) axis. Then the eccentricity of the hyperbola is

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

The number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}\) is equal to

If the foot of the perpendicular from the point \(A (-1,4,3)\) on the plane \(P : 2 x + my + nz =4\), is \(\left(-2, \frac{7}{2}, \frac{3}{2}\right)\), then the distance of the point \(A\) from the plane \(P\), measured parallel to a line with direction ratios \(3,-1,-4\), is equal to.

Let \(\alpha = 3\sin^{-1}\left(\dfrac{6}{11}\right)\) and \(\beta = 3\cos^{-1}\left(\dfrac{4}{9}\right)\), where inverse trigonometric functions take only the principal values.
Given below are two statements:
Statement I: \(\cos(\alpha+\beta) > 0\).
Statement II: \(\cos(\alpha) < 0\).
In the light of the above statements, choose the correct answer from the options given below:

If the system of linear equations \(2 x+y-z=3\) \(x-y-z=\alpha\) \(3 x+3 y+\beta z=3\) has infinitely many solution, then \(\alpha+\beta-\alpha \beta\) is equal to .... .

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

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

If the length of the latus rectum of an ellipse is \(4\,units\) and the distance between a focus and its nearest vertex on the major axis is \(\frac {3}{2}\,units\) , then its eccentricity is?

The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is :

The total number of \(4 -\)digit numbers whose greatest common divisor with \(18\) is \(3,\) is .... .

Let \(n\) be a positive integer. Let  \(A =\sum_{ k =0}^{ n }(-1)^{ k } n _{ C _{ k }}\left[\left(\frac{1}{2}\right)^{ k }+\left(\frac{3}{4}\right)^{ k }+\left(\frac{7}{8}\right)^{ k }+\left(\frac{15}{16}\right)^{ k }+\left(\frac{31}{32}\right)^{ k }\right]\) . If \(63 A =1-\frac{1}{2^{30}},\) then \(n\) is equal to ...... .