The slope of normal to the curve \(y=3 x^2-6 x\) at \(x=0\) is:

The area (in sq. units) of the region \(\left\{(x, y): 3 x^2 \leq y \leq|x|\right\}\) is equal to :

The value of \(\tan ^{-1} 5+\tan ^{-1} 3-\cot ^{-1} \frac{4}{7}\) is :

If \(y=\sin ^{-1} \sqrt{\frac{x}{x+1}}+\sec ^{-1} \sqrt{\frac{x+1}{x}}\), then \(\frac{d y}{d x}\) is

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 :

If \(\left|\begin{array}{ccc}-a^2 & a b & a c \\ b a & -b^2 & b c \\ a c & b c & -c^2\end{array}\right|=4 x\), then \(x=\)

The area of the smaller region enclosed by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\) and the ne straight straight line \(\frac{x}{4}+\frac{y}{3}=1\) is \(a \pi+b\). The value of a + b is:

Read the passage carefully and answer the questions
Replacement of a hydrogen atom in a hydrocarbon by an alkoxy or carboxyl group yields a class of compounds known as ethers. Ethers are classified as symmetrical or unsymmetrical on the basis of groups attached to the oxygen atoms. Diethyl ether, a symmetrical ether, has been widely used as an inhalation anesthetic. Ethers can be prepared by acid catalyzed intermolecular dehydration of alcohols and Williamson's synthesis. Acid catalyzed dehydration of alcohols is not generally preferred as it gives a mixture of elimination and substitution products. In Williamson's synthesis, an alkyl halide is allowed to react with sodium alkoxide. Ethers containing substituted Alkyl groups may also be prepared by this method. The C-O bond in ether is weakly polar and is cleaved under drastic conditions with excess of hydrogen halides. In electrophilic substitution, the alkoxy group deactivates the aromatic ring and directs the incoming group to ortho and para positions.
When ethanol is dehydrated in the presence of H₂SO₄, at 443 K and 413 K respectively, the products formed are

Two numbers are selected without replacement at random, one at a time from the first six positive integers. Let x denotes the larger of the two numbers. Match List-I with List-II

List-I	List-II
(A) \(P(x=2)\)	(I) \(\frac{4}{15}\)
(B) \(P(x=3)\)	(II) \(\frac{1}{15}\)
(C) \(P(x=4)\)	(III) \(\frac{2}{15}\)
(D) \(P(x=5)\)	(IV) \(\frac{1}{5}\)

Choose the correct answer from the options given below:

Which of the following statements is true?

Which traditional drink is made by fermenting palm sap?

Identify the statements which are correct with respect to fertilization.
(A) Sperm comes in contact with zona pellucida layer of the ovum
(B) Acrosome secretions help the sperm to pass through the cells of corona radiata
(C) Entry of sperm into the cytoplasm of ovum induces the completion of the meiotic division of the secondary oocyte
(D) Fertilization can occur if the ovum and sperms are transported simultaneously to the ampullary region
Choose the correct answer from the options given below:

1.02 g of the following substances are dissolved in 100 mL of water. Arrange them in increasing order of their molarity:
(A) NaCl
(B) KCI
(C)\(MgSO _4\)
(D)NaOH
Choose the correct answer from the options given below