The window of the house is in the form of a rectangle surmounted by a semi-circular portion, having a perimeter of 10 m as shown in figure.

If \(x\) and \(y\) represent the length and breadth of rectangular region, then relation between \(x\) and \(y\) is:

The integral \(\int e^x\left(1+\frac{1}{x}+\log x\right) d x\) is equal to:

A cell of emf 1.1 V and internal resistance \(0.5 \Omega\) is connected to a \(0.5 \Omega\) wire. Another cell of same emf is added in series, but current remains unchanged. The second cell's internal resistance is :

For a linear programming problem, the feasible region is shown in the figure by shaded portion, then linear constraints are

Area bounded by the curve \(y=\log x, x\)-axis, \(x=1\) and \(x=2\) in square units is:

The function \(f(x)=x^2-x+1\) is

The solution of the differential equation \((x+1) \frac{d y}{d x}+1-2 e^{-y}=0 ; y(0)=0\) is

Read the passage carefully and answer the questions
Replacement of a hydrogen atom in a hydrocarbon by an alkoxy or carboxyl group yields a class of compounds known as ethers. Ethers are classified as symmetrical or unsymmetrical on the basis of groups attached to the oxygen atoms. Diethyl ether, a symmetrical ether, has been widely used as an inhalation anesthetic. Ethers can be prepared by acid catalyzed intermolecular dehydration of alcohols and Williamson's synthesis. Acid catalyzed dehydration of alcohols is not generally preferred as it gives a mixture of elimination and substitution products. In Williamson's synthesis, an alkyl halide is allowed to react with sodium alkoxide. Ethers containing substituted Alkyl groups may also be prepared by this method. The C-O bond in ether is weakly polar and is cleaved under drastic conditions with excess of hydrogen halides. In electrophilic substitution, the alkoxy group deactivates the aromatic ring and directs the incoming group to ortho and para positions.
When ethanol is dehydrated in the presence of H₂SO₄, at 443 K and 413 K respectively, the products formed are

The largest open interval in which the function \(f(x)=4 x^3-5 x^2-8 x+12\) increases, is:

The first clinical gene therapy was given in 1990 to a 4-year old girl suffering from:

Match List-l with List-ll

List-l (Techniques)	List-II (Features)
(A) Bioprocess engineering	(I) Use of bio resources without proper authorisation
(B) Somatic hybridisation	(II) Correction of gene defect
(C) Gene therapy	(III) Fusion of protoplasts from two different organisms
(D) Biopiracy	(IV) Maintenance of sterile ambiance for microbial growth

Choose the correct answer from the options given below:

An unchanged capacitor is connected to a battery. How much amount of energy supplied by the battery is lost as heat while charging the capacitor ?

A solenoid of length 30 cm and 3000 turns carries a current of 10 A. The value of the magnetic field (\( \vec{B} \)) at the centre of solenoid is: (Given \( \mu_0 = 4\pi \times 10^{-7} \text{ Wb/A-m} \))