Which of the following statements is true for \(f(x)=(x-1)^3(x-2)^2\)?
(A) f(x) decreases on \(\left(-\infty, \frac{8}{5}\right)\)
(B) f(x) increases on \(\left[\frac{8}{5}, 2\right]\)
(C) f(x) has point of inflection at x = 1
(D) f(x) decreases on \((2, \infty)\)
Choose the correct answer from the options given below:

Consider the following test:
\(H _0: \mu \leq 12\)
\(H _{ a }: \mu>12\)
A sample of 36 provided a sample mean \(\bar{x}\) = 16 and a sample standard deviation S = 4.2 Then the value of the t-test statistic is:

The area (in sq. units) of the region bounded by the curve \(y=x^5\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is equal to:

If \(A\) and \(B\) are non singular square matrices of the same order,
then which of the statements are true ?
(A) \((\operatorname{adj} A)(\operatorname{adj} B)=(\operatorname{adj} B)(\operatorname{adj} A)\)
(B) \((A B)^T=A^T B^T\), if \(A^T\) denotes transpose of \(A\)
(C) \((A B)^{-1}=B^{-1} A^{-1}\)
(D) Multiplication of matrices is not commutative

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

If \(A=\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -3 & 5\end{array}\right]\) and \(B=\left[\begin{array}{cc}-2 & 3 \\ 4 & -5 \\ 1 & 2\end{array}\right]\), then which of the following statements are TRUE?
(A) AB is defined
(B) AB and BA both are defined and \(A B=I\), where I is an identity matrix of order 2
(C) BA is defined
(D) AB and BA both are defined and \(A B=B A\)
Choose the text correct answer from the options given below :

Let \(X\) denote the number of hours a student studies on a selected day. The probability distribution of \(X\) is given by (where \(k\) is some unknown constant)
\(P\left(X=x_i\right)=\left\{\begin{array}{ll}0.5, & \text { if } x_i=0 \\k x_i, & \text { if } x_i=1 \\k\left(4-x_i\right), & \text { if } x_i=2 \text { or } 3 \\0, & \text { otherwise }\end{array}\right.\)
Then the value of \(k\) is

Read the Passage carefully and answer the questions
When a ligand is bound to a metal ion through a single donor atom, it is said to be a unidentate ligand but when it bind through two donor atoms, then it is said to be didentate ligand and when several donor atoms are present in a single ligand, it is said to be a polydentate ligand. When a di- or polydentate ligand uses its two or more donor atoms simultaneously to bind a single metal ion, it is said to be a chelate ligand. The number of such ligating groups is called the denticity of the ligand. ligand which has two different donor atoms and either of the two ligetes in the complex is called ambidentate ligands. The number of ligand donor atoms to which the metal is directly bonded, is called coordination number of the metal ion in the complex. Complexes in which a metal is bound to only one kind of donor groups are known as homoleptic and in which a metal is bound to more than one kind of donor groups are known as heteroleptic complexes.
To make an octahesral complex of \(Mg ^{2+}\) ion, how many molecules of \(EDTA\) are used?

If x is real, the minimum value of \(x^2-8 x+17\) is:

Read the passage carefully and answer the questions based on the passage:
Pest Resistance Plants
Various nematodes parasitize a wide variety of plants and animals, including human beings. A nematode, Meloidogyne incognitia, infects the roots of tobacco plants and causes a great reduction in yield. A noval strategy was adopted to prevent this infestation which was based on the process of RNA interference (RNAi). RNAi takes place in all eukaryotic organisms as a method of cellular defense. This method involves silencing of a specific mRNA due to a complementary dsRNA molecule that binds to and prevents translation of the mRNA (silencing). Plants, bacteria, fungi and animals whose genes have been altered by manipulation are called Genetically Modified Organisms (GMO). GM plants have been useful in many ways. Genetic modification has: (i) made crops more tolerant to abiotic stresses (cold, drought, salt, heat). (ii) reduced reliance on chemical pesticides (pest-resistant crops). (iii) helped to reduce post harvest losses. (iv) increased efficiency of mineral usage by plants (this prevents early exhaustion of fertility of soil). (v) enhanced nutritional value of food, e.g., golden rice, i.e., Vitamin 'A' enriched rice. Genetic modification of organisms can have unpredicatable results when such organisms are introduced into the ecosystem. Therefore, the Indian Government has set up organisations such as GEAC (Genetic Engineering Approval Committee), which will make decisions regarding the validity of GM research and the safety of introducing GM-organisms for public services. Some strains of Bacillus thuringiensis produce proteins that kill certain insects such as lepidopterans (tobacco budworm, armyworm), coleopterans (beetles) and dipterans (flies, mosquitoes).
Which one infects the roots of tobacco plants and causes a great reduction in yield?

Irregular trends in a time series are caused by
(A) Epidemics
(B) Strikes
(C) Festival season
(D) Floods
Choose the correct answer from the options given below :

The correct IUPAC name for the following structure is:

In a laboratory population of 80 fruit flies, 8 died in a week, then the death rate of fruit flies is: