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 :

By using the determinants the area of the triangle with vertices \((0,0),(0,1)\) and \((1,0)\) is :

The equation of the normal to the curve \(y=\sin x\) at \((0,0)\) is :

Area of the region bounded by \(y=x^2\) and the line \(y=16\) is

If A and B are invertible matrices of order \(3,|A|=2\) and \(\left|(A B)^{-1}\right|=-\frac{1}{0}\), then the value of \(|B|\) is :

The solution set of the inequality \(2x + 3y < 4\) is :

\(\int_1^2 \frac{1}{x(x+1)} d x, x>0\) equals

Antiseptics and disinfectants either kill or prevent growth of micro-organism. Identify which of the following statements is not true?
(A) Disinfectants harm the living tissues.
(B) A 0.2% solution of phenol is an antiseptic while 1% solution acts as a disinfectant.
(C) Chlorine and Iodine are used as strong disinfectants.
(D) Dilute solution of boric acid and hydrogen peroxide are antiseptics
Choose the correct answer from the options given below :

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
Hind II always cut DNA molecules at a particular point by identifying a specific sequence of six base pairs. This specific base sequence is called _______ for Hind II.

The general solution of the differential equation \(x\left(1+y^2\right) d x+y\left(1+x^2\right) d y=0\) is

Answer the question on the basis of passage given below :
There are some deposits of nitrates and phosphates in the earth’s crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under laboratory conditions, but microbes do it easily. Ammonia forms a large number of complexes with transition metals. Hybridisation easily explains the ease of sigma donation capability of NH₃ and PH₃. Phosphine is an inflammable gas, and it is prepared from white phosphorus.
Which of the following is the correct statement?

In a Young's double-slit experiment, the slits are separated by 0.56 mm and the screen is placed 2.8 m away from the slits. The distance between the central bright fringe and the fourth bright fringe is measured to be 1.2 cm. Determine the wavelength of light used in the experiment.

If under pure competition demand and supply functions are given by \(p=\sqrt{10-x}\) and \(p=\frac{1}{2}(x-2)\) respectively, where \(p\) is price per unit and \(x\) is quantity, then the consumer surplus is :