Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively. Both pipes A and B are opened together for some time and then pipe B is turned off. If the tank is filled in 15 minutes, then time for which the pipe B works is :

Let \(R\) be a relation on the set of integers \(Z\) such that \(R=\left\{(a, b), a=2^k b, a, b, k \in Z\right\}\), then \(R\) is :

Central Limit Theorem implies that as the size of the sample increases, the distribution approaches:

The function \(f(x) = x^2 - 2x\) has a critical point at:

Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)	I. Linear Differential Equation of type \(\frac{d y}{d x}+P y=Q\)
B. \(\frac{d y}{d x}-2 y=\cos 3 x\)	II. Homogeneous Differential Equation
C. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	III. Linear Differential Equation of type \(\frac{d x}{d y}+P x=Q\)
D. \(\left(x+2 y^3\right) \frac{d y}{d x}=y\)	IV. Variable separable

Choose the correct answer from the options given below:

On 1st April 2024, person ' \(X\) ' purchased a machinery costing ₹ 65000 and spent ₹ 10000 on its installation. The estimated effective life of the machinery is 5 years with a scrap value of ₹ 10000 . The annual depreciation using the straight-line method with the accounting year ending on 31st March 2025 is :

\(\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x\) equals :

Answer the question on the basis of passage given below :
Adsorption is a surface phenomenon which can be expressed by means of an empirical relationship know as Freundlich Adsorption isotherm. The relationship given by \(\frac{x}{m} = Kp^{\frac{1}{n}}\ (n > 1)\)
x = Mass of the gas adsorbed
m = Mass of adsorbent
p = Pressure at which adsorption takes place
K and n are constant
The curve of above relationship indicate that at a fixed pressure there is a decrease in physical adsorption with increase in temperature. At high pressure, curves become parallel to pressure axis, indicating saturation of the surface of adsorbent.
Adsorption of a gas follow Freundlich isotherm shown below. \(\frac{x}{m}\) is proportional to

Due to which of the following, the irregular variations in a time series are caused:
(A) Floods
(B) Rise in prices before festivals
(C) A fire in a factory
(D) Epidemics
Choose the correct answer from the options given below:

A takes a loan of ₹ \(5,00,000\) at an interest rate for 5 years. The EMI paid by him is ₹ 10,000 . The effective interest rate for A is : (Under flat rate system)

In a survey question for a sample of 250 individuals, 120 persons gave response 'yes', 80 persons gave response 'no' and 50 gave 'no response'. The point estimate of the proportion in the population who responded 'yes' is:

Pick the correct structure for carbolic acid among the following compounds:

What is the ratio of the frequencies of the long wavelength limits of Lyman and Balmer series of hydrogen spectrum ?