The following system of equations \(2 x-y+3 z=5,3 x+2 y-z=7,4 x+5 y-\lambda z=\mu\) is consistent. Then

If \(\vec{a}, \vec{b}, \vec{c}\) are three mutually perpendicular unit vectors, then \(|\vec{a}+\vec{b}+\vec{c}|\) is equal to:

\(A\) and \(B\) are two independent events. The probability that both events \(A\) and \(B\) occur is \(\frac{1}{6}\) and the probability that neither of them occur is \(\frac{1}{3}\). If \(P(A)=x, P(B)=y\) then the value of \(x+y\) is.

Consider the differential equation, \(x \frac{d y}{d x}=y\left(\log _e y-\log _e x+1\right)\), then which of the following are true?
(A) It is a linear differential equation
(B) It is a homogenous differential equation
(C) Its general solution is \(\log _e\left(\frac{y}{x}\right)=C x\), where \(C\) is constant of integration
(D) Its general solution is \(\log _e\left(\frac{x}{y}\right)=C_y\), where \(C\) is constant of integration
(E) If \(y(1)=1\), then its particular solution is \(y=x\)
Choose the correct answer from the options given below:

\(y=e^{x+e^{\infty+e^{x+\cdots \infty}}}\) then \(\frac{d y}{d x}=\) ______________

The maximum value of \(\left( \frac{1}{x} \right)^x\) is :

The amount to be deposited at the end of every 6 months to accumulate Rs. 50,000 in 8 years, if money is worth 6% p.a. compounded semiannually, is : (Given \((1.03)^{16}=1.6047\) )

The straight line \(\frac{x+3}{3}=\frac{y+2}{4}=\frac{z+1}{0}\) is

Read the following paragraph and answer the given questions.
Bio-fertilisers are organisms that enrich the nutrient quality of the soil like cyanobacteria, while there are organisms that help to control plant diseases and pests. The very familiar beetle, the ladybird and dragonflies are useful to get rid of aphids and mosquitoes, respectively. An example of microbial biocontrol agents that can be introduced to control butterfly caterpillars is certain bacteria, which are available in sachets in the form of dry spores. These spores are mixed with water and sprayed onto vulnerable plants. Trichoderma species that are very common in the root ecosystem, are very effective biocontrol agents of several plant pathogens.Baculoviruses attack insects and other arthropods. Usage of these viruses is desirable when an ecological sensitive area is being treated.
Find the odd one out of the given organisms?

If \( \vec{M} \) is the magnetic moment and \( \vec{B} \) is the magnetic field, then the torque is given by

The function \(f(x)=2 x^3-3 x^2-12 x+4\) has:
(A) two points of local maxima.
(B) two points of local minima.
(C) one local maxima and one local minima.
(D) no local maxima or local minima.

These structures represent which particular kind of halides :

Which of the following is a correct statement?