The area of the triangle whose vertices are \((3,8),(-4,2)\) and \((5,1)\) is

Let \(f(x)=|x|\) and \(g(x)=[x]\), then the value of \(f \circ g\left(-\frac{9}{2}\right)-g \circ f\left(-\frac{9}{2}\right)\) is :

In a game, a man wins Rs. 5 for getting a number greater than 4 and loses Rs. 1 otherwise, when a fair die is thrown.
The man decides to throw the die thrice but quits as soon as he gets a number greater than 4 .
The expected value of the amount (in Rs.) he wins or loses is :

Rahul invested ₹ 20000 in a mutual fund in year 2018. If the value of mutual fund increased to ₹ 32000 in year 2023, then the compound annual growth rate of his investment is : [given that \((1.6)^{\frac{1}{5}}=1.098\) ]

The value of \(\left|\begin{array}{lll}2^x & 1 & 6^x \\ 4^x & 1 & 3^x \\ 2^x & 1 & 6^x\end{array}\right|\), where \(x \neq 0 ;\) is:

The values of \(\lambda\) for which the system of equation \(x+2 y+z=14,-x+y+z=10, x+\lambda y+z=2\) has unique solution is :

Area (in sq. units) of the region bounded by the curve \(y^2=4 x, y\)-axis and the line \(y=3\) is

Osmotic pressure measurements are less accurate and difficult to carry out. Therefore, this method is not useful for measuring molecular weights. But it is used to find out molecular weights of polymers eg. plastics, proteins and starch having many monomer units. Knowing the value of osmotic pressure of a given solution, molecular weight of the solute can be calculated using Van't Hoff equation.
A solution which shows deviations from Raoult's law is called:

In a LPP:
If two corner points yield the same maximum value of the objective function of an LPP then :
(A) the LPP has unique solution
(B) the LPP has 2 solutions
(C) the LPP has 3 solutions
(D) the LPP has infinite many solutions
Choose the correct answer from the options given below :

Which of the following compound(s) on being warmed with \(I_2\) solution and NaOH , will give Iodoform?
A. \(CH _3 CH _2 OH\)
B. \(CH _3 COCH _3\)
C. \(H _3 C - CHOH - CH _3\)
D. \(CH _3 OH\)
Choose the correct option

The process of evolution of different species in a given geographical area starting from a point and dispersing to other areas is called as:

Read the passage carefully and answer the questions
The resistance ( \(R\) ) of an object is given by the expression \(R=\rho \frac{l}{A}\).
The inverse of \(R\) is called conductance and the inverse of \(\rho\) is called conductivity or specific conductance represented by \(\kappa\).
The conductivity depends upon the nature of the material (insulator, semiconductor or conductor) and the temperature of the substance.
Similarly, the conductance due to the ions present in the solution (ionic conductance) depends upon the nature of the electrolyte, size of ions, the nature of the solvent and the temperature.
Molar conductivity ( \(\Lambda_m\) ) of a solution at a given concentration is the conductance of volume \(V\) containing one mole of electrolyte kept between two electrodes of area \(A\) and unit distance apart. It is given by the expression \(\Lambda_m=\kappa V\).
Molar conductivity increases with decrease in concentration and when concentration approaches zero, it is called limiting molar conductivity.
The decrease is linear for a strong electrolyte. However, for a weak electrolyte the change is gradual at higher concentrations and sharp near zero concentration.
Using limiting molar conductivity values of strong electrolytes we can calculate the limiting molar conductivity of weak electrolytes using Kohlrausch law of independent migration of ions.
From molar conductivity and limiting molar conductivity we can determine the degree of dissociation \(\left(\frac{\Lambda_m}{\Lambda_m^0}\right)\) and the ionization constant of a weak acid.
Which of the following is an insulator?

The simple random sample consists of six observations: \(5, 8, 10, 7, 10, 14\). The point estimate of the population mean is: