Area of the region bounded by the curve \(y=\frac{1}{2} \cos x\) and x-axis between \(x=0\) and \(x=2 \pi\) is:

Which of the following aze correct about the linear programming problem (LPP)?
(A) The objective function is always linear.
(B) A corner point of a feasible region is a polut in the region which is the intersection of two boundary lines.
(C) The common region determined by all the linear constraints of a LPP is called the feasible region.
(D) If an LPP admits optimal solution at two points then its optimal values occurs at an infinite number of points.
Choose the correct answer from the options given below :

The maximum number of equivalence relations on the set \(A=\{3,5,7\}\) is :

A boat 10 m high floating at a uniform speed of 13 meters per minute \(( m / min )\) away from a lamp post 15 m high.
Then the rate at which the length of shadow of the boat increases is :

Consider the following data of expenses (in lakhs) of an organization year wise:

Year	2001	2002	2003	2004	2005
Expenses (Rs. lakh)	160	185	220	300	510

Then expected expenses trend for the year 2006 using method of least squares is:

The order of the differential equation \(\left[1+\left(\frac{d y}{d x}\right)^2\right]^2=\left(\frac{d^3 y}{d x^3}\right)\) is:

\(\int_{\sqrt{\log _e 2}}^{\sqrt{\log _e 4}} x e^{x^2} d x\) is equal to

Read the following passage carefully and answer the given questions
The separated DNA fragments can be visualized only after staining the DNA with ethidium bromide followed by exposure to UV radiation. Bright orange colored bands of DNA in ethidium bromide stained gel exposed to UV light can be seen. The separated bands of DNA are cut out from the agarose gel and extracted from the gel piece. This step is known as elution. The DNA fragments purified in this way are used in constructing recombinant DNA by joining them with a cloning vector.
The extraction of separated bands of DNA from the gel piece is called:

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{j}\).
Then a vector \(\vec{c}\) satisfying \(\vec{a} \times \vec{c}=\vec{b}\) and \(\vec{a} \cdot \vec{c}=8\) is equal to:

What is the coordination number of Fe in \(\left[ Fe ( CO )_5\right]\) and \(\left[ Fe \left( C _2 O _4\right)_3\right]^{3-}\) respectively?

The value of \(\int_{-5}^5|x+3|\) \(dx\) is

Arrange the stages of oogenesis in order of their occurance.
(A) Formation of ovum & IInd polar body
(B) Meiosis II begins but held at Metaphase II
(C) Meiosis I started but held at Prophase I
(D) Formation of primary oocyte from oogonia
(E) Completion of Meiosis-I and formation of secondary oocyte and first polar body
Choose the correct answer from the options given below :

Answer the questions on the basis of passage given below :
As aldehydes ketones possess the carbonyl functional group, they show similar reactions. They exhibit nucleophilic addition reactions, reduction, oxidations, reactions due to alpha-Hydrogen atom, and many other reactions, like Cannizzaro reaction to electrophilic substitution reactions (in case of aromatic aldehydes ketones).
These reactions help in the preparation of a number of important compounds, like acetic acid, vinyl acetate, polymers and in perfume and dye industries.
Nucleophilic addition reactions form cyanohydrins, acetals, alcohols and N -substituted derivatives like imine, oxime, semicarbazone, phenylhydrazone Schiff's base. Reactions due to alpha-hydrogen atoms help in formation of aldols and alpha, beta-unsaturated aldehydes ketones.
Which of the following combinations of Grignard's reagent and carbonyl compound can give \(C _6 H _5- C ( OH )\left( CH _3\right)- CH \left( CH _3\right)_2\)
A. \(C _6 H _5 CH _2 MgBr +\left( CH _3\right)_2 CH - CHO\)
B. \(C _6 H _5 MgBr + CH _3- CO - CH \left( CH _3\right)_2\)
C. \(CH _3 MgBr + C _6 H _5- CO - CH \left( CH _3\right)_2\)
D. \(C _6 H _5- CO - CH _3+\left( CH _3\right)_2 CHMgBr\)
E. \(C _6 H _5 CH ( MgBr ) CH _3+\left( CH _3\right)_2 CO\)
Choose the correct answer from the option given below :