If \(\left|\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right|\), where \(a, b\) and \(c\) are non zero constants, then the value of \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\) is

In the following differential equation \(\frac{d^2 y}{d x^2}+x\left(\frac{d y}{d x}\right)^2=2 x^2 \log \left(\frac{d^2 y}{d x^2}\right)\) order and degree is:

The region represented by the system of inequalities \(x, y \geq 0 ;-2 x+y \leq 4 ; x+y \geq 3\) and \(x-2 y \leq 2\) is

Which of the following function is increasing for all values of x in the domain \((0, \infty)\)?

The differential equation of the family of parabolas \(y^2=4 a x\) where \(a\) is an arbitrary constant is :

Which of the following statements are true?
(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\) is \(\vec{r}=(5 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-8 \hat{k})\)
(B) Vector form of the equation of line \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is \(\vec{r}=(5 \hat{i}-4 \hat{j}+6 \hat{k})+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})\)
(C) The direction cosines of z-axis are (1, 1, 0).
(D) If a line has direction ratios 2, -1, -2, then its direction cosines are \(-2 / 3,-1 / 3,-2 / 3\).
Choose the correct answer from the options given below:

Let A be a non-singular matrix of order \(n \times n\)
Then \(|\operatorname{adj}(3 A)|\) is equal to:

Which one of the following statement is incorrect about ecological pyramids?

The value of \(\cot \left(\cos ^{-1} \frac{7}{25}\right)\) is

The rate constant for the first order reaction is \(60\) \(s^{-1}\). The time taken to reduce the initial concentration of the reactant to its \(1/26^{th}\) value is:
[Given \(\log 26 = 1.4150\)]

Arrange the following ions in increasing order of number of 3d-electrons
(A) Cr²⁺
(B) Cu⁺
(C) Ti³⁺
(D) Mn⁺
Choose the correct answer from the options given below :

Electric field intensity at a point varies as \( \frac{1}{r^3} \) due to:

The value of \(\frac{3}{2}(\hat{i} \cdot \hat{i})+\frac{1}{2}(\hat{j} \cdot \hat{j})+\frac{|\hat{i} \times \hat{i}|}{(\hat{k} \cdot \hat{k})}+\frac{(\hat{k} \cdot \hat{i})}{|\hat{i} \times \hat{j}|}\) is: