Let \(X\) and \(Y\) be any two invertible square matrices of order \(n\). Then which of the statements are true?
(A) \((A B)^{-1}=A^{-1} B^{-1}\)
(B) \((A B)^{-1}=B^{-1} A^{-1}\)
(C) \((A B)^T=A^T B^T\), where \(A^T\) denotes transpose of A
(D) \((A B)^T=B^T A^T\), where \(A^T\) denotes transpose of A
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Particular solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) given that when x = 0 y = 0 is:

The solution of the differential equation \((\cot y) d x=x d y\) is:

The product of order and degree of the differential equation \(\left(\frac{d^3 y}{d x^3}\right)^2+x^2 y\left(\frac{d^2 y}{d x^3}\right)^3=2 x^5\) is:

Under which of the following conditions the Poisson distribution is the limiting case of, binomial distribution :
(A) The number of trials is indefinitely large.
(B) The probability of success for each trial is indefinitely small.
(C) The product of the number of trials and the probability of success for each trial is finite.
(D) The probability of success for each trial is indefinitely large.
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Match List I with List II

LIST I	LIST II
A. \(\frac{d}{dx} [x^{-3}(5+3x)]\)	I. \(-\frac{2}{(x+1)^2} + \frac{3}{(x-2)^2}\)
B. \(\frac{d}{dx} [(x-a)(x-b)]\)	II. \(-\frac{15}{x^2} - \frac{6}{x^3}\)
C. \(\frac{d}{dx} [\frac{2}{x+1} - \frac{3}{x-2}]\)	III. \(-\frac{2}{(x-1)^2}\)
D. \(\frac{d}{dx} [\frac{x+1}{x-1}]\)	IV. \(2x - a - b\)

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Mr. X wishes to purchase a flat for Rs. 44, 60, 800 with a down payment of Rs. 10,00,000 and the balance in equal monthly instalments (EMI) for 20 years. If the bank charges 7.5% per annum compounded monthly, then the EMI is :
[Given that \((1.00625)^{240}=4.4608\) ]

Which of the following causes electric conduction in a semiconductor?

In a diffraction pattern due to a narrow slit of width a, alternate dark and bright fringe pattern is obtained on a screen kept at distance D from the slit. Choose the correct option.
(A) The angular width of central maximum is given by 2D/a, linear width is given by 2D\lambda/a
(B) When the size of aperture, a is quite large, the diffraction effect is negligible,
(C) For light of smaller wavelength, angular width of central maximum increases, while linear width decreases,
(D) Angular width of central maximum does not decrease while linear width decreases on decreasing distance of screen from slit.
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Arrange the trophic levels in an order from lowest to highest in a terrestrial food chain:
(A) Wolf
(B) Grass
(C) Cow
(D) Lion
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Identify the correct statements.
(A) The energy (E) and momentum (p) of a photon are related by p = E/c.
(B) The electrons and photons of the same wavelength, possess the same momentum.
(C) One photon having energy more than work function can emit one photo electron.
(D) Photons do not carry momentum.
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Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three vectors such that \(|\vec{a}|=2,|\vec{b}|=3,|\vec{c}|=5\) and each one of three vectors is perpendicular to the sum of the other two, then value of \(|\vec{a}+\vec{b}+\vec{c}|\) is

The source of electrical energy on the Apollo moon flight was: