If \(x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t\), then value of \(\frac{d y}{d x}\) at \(t=\pi / 4\) is

Value of \(\int \frac{2}{(x-3) \sqrt{x+1}} d x\) is : (Here \(C\) is an arbitrary constant)

The total cost \(C(x)\) in Rupees associated with the production of \(x\) units of an item is given by
\(C(x)=0.007 x^3+26 x^2+15 x+400\) 
The marginal cost when 10 items are produced is:

Let \(y(x)=a(x+1) \log (x+1)+b x+5\) be the solution of the differential equation \(e^{\frac{d y}{d x}}=x+1, y(0)=5\), then the value of \((a+b)\) is :

The number of tangents to the curve \(x y-3 y+2=0\) having slope 2 is :

How many minimum number of times must a man toss a fair coin so that the probability of having at least one head is more than \(90 \%\) ?

Let \(A=\left[a_{i j}\right]\) be a square matrix, where \(a_{i j}=\left\{\begin{array}{ll}0, & \text { when } i=j \\ 1, & \text { otherwise }\end{array}\right.\)
If \(|\operatorname{adj} A|=|A|^2\), then which of the following statements are correct?
(A) A is a skew-symmetric matrix.
(B) A is a non-singular matrix.
(C) A is a square matrix of order 4.
(D) A is a symmetric matrix.
Choose the correct answer from the options given below :

\((\vec{a} \cdot \hat{i})^2+(\vec{a} \cdot \hat{j})^2+(\vec{a} \cdot \hat{k})^2\) is equal to:

Identify the correct statement among the following in the context of a nuclear reactor.

The work function of a photoelectric surface is 3.3 eV. The value of the threshold frequency for the surface is :

Neutral or weakly alkaline \(KMnO_4\) solution will oxidise iodide ion into :

The effective rate equivalent to a nominal rate of 12% compounded quarterly is:
(Given \((1.03)^4=1.1256\) )

Read the passage carefully and answer the questions based on the passage:
Alcohols are versatile compounds as they can act as both nucleophile and electrophile. They undergo reactions based on cleavage of \(\mathrm{O}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{O}\) bonds. The cleavage of \(\mathrm{O}-\mathrm{H}\) bond reflects the acidic character of alcohols. However they are weaker acids than water, phenol and carboxylic acids. Alkoxide ion formed by loss of proton from the OH group of alcohols reacts with water to form the corresponding conjugate acid - base pair. They act as Bronsted base due to presence of unshared electron pairs on oxygen.Both alcohols and phenol react with carboxylic acids, acid chlorides and anhydrides to form esters. The introduction of acetyl group in alcohols or phenol is known as acetylation. Alcohols react with hydrogen halides to form alkyl halides which form the basis of Lucas test. They also undergo acid catalyzed dehydration to form alkenes. Alcohols can be easily oxidized with variety of oxidizing agents. The nature of the product depends on the type of oxidizing agent and the nature of alcohol.
Lucas reagent is