The following system of equations:
x + y - z = 7
\(4 x+\lambda y-\lambda z=3\)
3x + 2y - 4x = 5
does not possess a solution if the value of \(\lambda\) is:

If matrix \(A=\left[\begin{array}{cc}p & -3 \\ -4 & p\end{array}\right]\) and \(\left|A^3\right|=64\), then the value of \(p\) is :

Distance of the point \((2,4,-1)\) from the line \(\frac{10+2 x}{2}=\frac{y+3}{4}=\frac{6-z}{9}\) is

The angle at which the line \(\frac{x-1}{0}=\frac{2-y}{-1}=\frac{2 z-3}{-2}\) is inclined with the positive direction of \(z\)-axis is

Let \(A=\left[a_{i j}\right]\) is given by \(A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 4 & -5 \\ 2 & -1 & 3\end{array}\right]\). Then the matrix \(B=\left[b_{i j}\right]\), where \(b_{i j}\) = Minor of \(a_{i j}\) is:

The angle between thee curves \(x y=\sqrt{2}\) and \(x^2-y^2=1\) at \((\sqrt{2}, 1)\) is:

If \(\vec{a}, \vec{b}\) and \(\sqrt{3} \vec{a}-\vec{b}\) are three unit vectors, then the angle between \(\vec{a}\) and \(\vec{b}\) is :

What is the total number of chiral carbons present in glucopyranose and fructofuranose respectively?

A long solenoid with 25 turns per cm has a small loop of area \( 2.0 \text{ cm}^2 \) placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 3.0 A to 5.0 A in 0.1 s. The induced emf in the loop due to change in the current is

For \(95 \%\) confidence interval for a population mean reported to be 132 to 142 with standard deviation \(\sigma=17.85\) then the sample size used in this case, is :
[Given that : \(Z_{.0125}=1.96\)]

The value of b for which the function \(f(x) = \sin x - bx + C\), where b and C are constants is decreasing for \(x \in R\) is given by

Identify the correct statement(s) among the following:
(A) If three electrical cells each of emf, E and internal resistance r are connected in series to a resistance R, the current drawn from them is 3E/(R+3r).
(B) If these three cells are connected in parallel combination, across resistance R then the current drawn will be E/(3R+r).
(C) If the two cells are connected in series and then one in parallel which are then connected across resistance R, then the current drawn will be 2E/(2R+2r).
(D) In a mixed grouping of cells, current will be maximum when total external resistance is equal to total equivalent internal resistance.
Choose the correct answer from the options given below :

Area bounded by \(x=\frac{y^2}{4}\) and its latus rectum is: