\(y=x(x-3)^2\) decreases for the value of \(x\) given by

If \(a, b, c\) are positive real numbers, then the least value of \((a+b+c)(a b+b c+c a)\) is :

The maximum value of \(Z\) for the linear Programming problem maximum \(Z=2 x+y\) subject to the constraints \(x+y \leq 2, \quad x \geq 0, \quad y \geq 0\) is :

Linear inequalities corresponding to the shaded feasible region \(O A B C O\) in the given figure are

Let \(A=\left[a_{i j}\right]_{3 \times 3}\) be a matrix, defined by \(a_{i j}=\left\{\begin{array}{ll}2 i+3 j & , i < j \\ 6 & , i=j \\ 3 i-2 j & , i > j\end{array}\right.\). The number of elements in A. which are greater than 6, is

Match List-I with List-II

LIST-I (Function \(f(x)\))	LIST-II (Interval)
(A) \(f(x)=x|x|\)	(I) Decreases on (0, \(\infty\) )
(B) \(f(x)=x^2+2 x-5\)	(II) Increases on ( \(3, \infty\) )
(C) \(f(x)=x^2-6 x+9\)	(III) Decreases on \((-\infty,-1)\)
(D) \(f(x)=-x^2\)	(iv) Increases on \((-\infty, \infty)\)

Choose the correct answer from the options given below:

A bag contains 4 red and 6 green balls. A ball is drawn at random, its colour is noted and is returned to the bag.
One additional ball of the colour drawn is put in the bag. Again a ball is then drawn from the bag. The probability of this ball to be of green colour is

Two sources of equal emf are connected in series. This combination is connected to an external resistance R.
The internal resistance of two sources are \( r_1 \) and \( r_2 \) (\( r_2 > r_1 \)).
If the potential difference across the source of internal resistance \( r_2 \) is zero, then R equals to:

Read the following passage carefully and answer the given questions.
Restriction endonuclease on finding its specific recognition sequence, bind to the DNA and cut each of the two strands of the double helix at specific points in their sugar -phosphate backbones. Each restriction endonuclease recognises a specific palindromic nucleotide sequences in the DNA. Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA along with other macromolecules such as RNA, proteins, polysaccharides and also lipids. This can be achieved by treating the bacterial cells/plant or animal tissue with enzymes such as lysozyme (bacteria), cellulase (plant cells), chitinase (fungus). Now a days the most commonly used matrix is agarose which is a natural polymer extracted from sea weeds. The DNA fragments separate (resolve) according to their size through sieving effect provided by the agarose gel. The first restriction endonuclease-Hind II, whose functioning depended on a specific DNA nucleotide sequence was isolated and characterised five years later. It was found that Hind II always cut DNA molecules at a particular point by recognising a specific sequence of six base pairs. In addition to 'ori', the vector requires a selectable marker, which helps in identifying and eliminating non-transformants and selectively permitting the growth of the transformants. Transformation is a procedure through which a piece of DNA is introduced in a host bacterium. Normally, the genes encoding resistance to antibiotics such as ampicillin, chloramphenicol, tetracycline or kanamycin, etc., are considered useful selectable markers for E. coli.
A natural polymer extracted from sea weeds used in the field of biotechnology is:

Answer the question on the basis of passage given below:
In the periodic table, the d-block contains the elements of group \(3\) to \(12\). The d-orbitals are progressively filled in each of the four long periods. The elements of d block referred as transition metals have partly filled d orbitals and exhibit certain characteristic properties such as variety of oxidation states, formation of coloured ions, act as catalyst and show paramagnetic behaviour.
The two inner transition metals series \(4f\) and \(5f\) are known as Lanthanoids and Actinoids respectively. The lanthanoids resemble one another more closely as compared to ordinary transition elements in any series.
Highest oxidation state of manganese in fluoride is \(+4\left( MnF _4\right)\) but highest oxidation state in oxides is \(+7\left( Mn _2 O _7\right)\) because:

Match List I with List II

LIst - I	List - II
(A) The present value of an immediate perpetuity of ₹R payable at end of any year forever at rate of i per period is:	(I) \(\frac{R}{i}(1+i)^m\)
(B) Present value of deferred perpetuity is	(II) \(\frac{A i}{(1+i)^n-1}\)
(C) Periodic payment R in a sinking fund is given by formula (for Amount A)	(III) \(A(1+i)^{-n}\)
(D) The present value of redemption price A of a bond is given by	(IV) \(\frac{R}{i}\)

\(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}\) is equal to:

For the following reaction, which is correct statement \(2 H ^{+}( aq )+2 e^{-} \rightarrow H _2(g)\)
under non-standard conditions.