The general solution of the differential equation \(\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0\) is :

Which of the following are correct?
(A) The probability curve in t-distribution is symmetric about the line \(t =0\).
(B) t-axis is an asymptote of the curve.
(C) The variable t of t-distribution ranges from \(-\infty\) to \(\infty\).
(D) As the number of degrees of freedom increases, the t-distribution curve moves closer to the binomial distribution.
Choose the correct answer from the options given below :

The value of \(x\) for which the slope of the curve \(y=-x^3+3 x^2+x^2+9 x-27\) is maximum, is:

The probability of a shooter hitting the target in one shot is \(\frac{1}{4}\)
The minimum number of shots needed so that the probability of hitting the target at least once is greater than \(\frac{7}{16}\) is:

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

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?

The solution of \(y^{\prime}-y^{\prime \prime}=2 x\) is:
A. \(y=x^2+2 x+2\)
B. \(y=x^2+2 x+1\)
C. \(y=x+2\)
D. \(y=x^2-2 x+1\)
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 :

The endosperm is absent in the seed of :

The function \(f: R \rightarrow R, f(x)=x^2\) is :

Interference fringes may be observed due to the superposition of two light waves represented by :

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

The integral \(\int e^x\left(\frac{x-1}{2 x^2}\right) d x\) is equal to: