The inverse of the matrix \(\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]\) is:

If \(\int e^x\left(\frac{x-1}{(x+1)^3}\right) d x=\frac{A e^x}{(x+1)^B}+C\), where \(C\) is constant of integration, then which of the following are correct?
(A) \(A=-1\)
(B) \(A=1\)
(C) \(B=3\)
(D) \(B=2\)
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Rahul can run 34.4 m in the given time as Amit runs 50 m. By how much distance Rahul is away from Amit at the winning point, in a two km race?

Let \(A\) be a \(3 \times 7\) matrix, then each column of \(A\) contains :

If the lines \(\frac{1-x}{3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{6-z}{5}\) are perpendicular, then \(\lambda\) is equal to

The value of \(\tan ^{-1} 1+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)\) is

Consider the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\), then which of the following statements are correct?
(A) It is homogeneous
(B) It has \(\sec x\) as its integrating factor
(C) Its general solution is \(y \sec x=\tan x+c\), where \(c\) is an arbitrary constant.
(D) Its degree is not defined
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If A and B square matrices of order 3 such that |A|= -1, |B|= 5, then the value of |2AB| is:

A light ray is incident normal to one face of a right-angled isosceles prism and is totally internally reflected at the glass-air interface. If the angle of reflection is \(\frac{π}{4}\), what can be concluded about the refractive index?

Read the passage carefully and answer the Questions
Carboxylic acids are the earliest organic compounds isolated from nature. These are also known as organic acids but are weaker acids as compared to mineral acids. However, they are more acidic than alcohols and phenols because of greater resonance stabilization of the parent acid and its conjugate base. The acidic strength is determined by the \(K _a\), which is called the acidity constant.
The greater the value of \(K _a\), the greater the tendency of the acid to ionize and hence the stronger the acid. The acidic strength of the saturated aliphatic carboxylic acid depends upon the inductive effect and its position with respect to - COOH groups. Electron -donating substituents tend to decrease whereas electron-withdrawing substituents tend to increase the acidic strength. The acidic strength of the aromatic carboxylic acid, on the other hand depends upon both the inductive effect and the resonance effect of the substituents.
Arrange the following acids in decreasing order of acidity
(A) \(CH _3 CH _2 OH\)
(B) \(CH _3 COOH\)
(C) \(BrCH _2 COOH\)
(D) \(C _6 H _5 CH _2 COOH\)
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Choose an incorrect statement from the following about Kirchhoff's rules of electric circuits:

The unit of conductance(G) is

Which of the following statements are correct?
(A) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle, then its area is \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
(B) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a parallelogram, then its area is \(|\vec{a} \times \vec{b}|\)
(C) \(|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \cos \theta\)
(D) If \(\vec{a}\) and \(\vec{b}\) represent the 'diagonals' of a parallelogram, then its area is \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
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