The area (in sq. units) of the region bounded by the line \(y=x+2, x=0, x=1\) and \(y=0\) is:

The area bounded by the curve \(y=4+3 x-x^2\) and the \(x\)-axis is equal to

The function f defined by \((x)=\left\{\begin{array}{ll}a x+1, & x \leq 3 \\ b x+3, & x>3\end{array}\right.\) is continuous at x = 3 if

If \(x=a \sec ^3 \theta, y=a \tan ^3 \theta\), then \(\frac{d y}{d x}\) at \(\theta=\frac{\pi}{3}\) is

Let \(A=\left[a_{i j}\right]\) be a square matrix of order 2 with elements either 0 or 1 . Then the difference between the possible number of singular and non-singular matrices is

For the differential equation \((x+y) d y+(x-y) d x=0\), which of the following is/are correct?
(A) Differential equation is homogeneous
(B) Order of differential equation is 1
(C) Integrating factor of differential equation is \(e ^{ x }\)
(D) Degree of the equation is not defined
Choose the correct answer from the options given below:

The equation of the plane which passes through the point (3, 2, -1) and is perpendicular to the line with direction ratios 2, 1, 2 is:

Read the following passage carefully and answer the given questions.
Insulin consists of two short polypeptide chains- chain A and chain B, that are linked together by disulphide bridges. Some strains of Bacillus thuringiensis produce proteins that kill certain insects such as lepidopterans, coleopterans and dipterans. Bt toxin protein is converted into an active form of toxin due to the alkaline pH of the gut which solubilise the crystals. Transgenic animals that produce useful biological products can be created by the introduction of the portion of DNA (or genes) which codes for a particular product such as human protein (alpha-1-antitrypsin) used to treat emphysema. 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. (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.
Transgenic animals that produce useful biological products such as human protein (alpha-1-antitrypsin) used to treat the disorder:

Allyl chloride on dehydrochlorination gives:

Arrange the following oxides in decreasing order of oxidation state of the metal ion.
(A) Cu\(_2\)\(\mathrm{O}\)
(B) CrO\(_3\)
(C) Mn\(_2\)\(\mathrm{O}_7\)
(D) V\(_2\)\(\mathrm{O}_5\)
Choose the correct answer from the options given below:

Let \(f(x)=\left\{\begin{array}{ll}2 x-1, & x<1 \\ 1, & x=1 \\ x^2, & x>1\end{array}\right.\) then at \(x=1\)

The ratio of the de-Broglie wavelengths associated with protons accelerated through a potential of 128 V and alpha particles accelerated through a potential of 64 V will be

In which region of the fallopian tube, fertilization of sperm and ovum takes place?