A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains atleast 8 units of Vitamin A and 10 units of Vitamin C. Food 'I' contains 2 units/kg of Vitamin A and 1 unit/kg of Vitamin C while food 'II' contains 1 unit/kg of Vitamin A and 2 units/kg of Vitamin C. It costs ₹5 per kg to purchase food I and ₹7 per kg to purchase food II. If 'x' kg of food I is mixed with 'y' kg of food II, then the constraint pertaining to vitamin C contents will be:

The value(s) of \(x\) for which the matrix \(A=\left[\begin{array}{ccc}x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1\end{array}\right]\) is singular, is/are :

A pipe can fill a tank in 2 hours. Because of a leak in the tank, the pipe takes 2 hours 40 minutes to fill the tank. The time the leakage takes to empty the full tank is :

If \(A=\left[\begin{array}{cc}1 & 0 \\ \frac{1}{25} & 1\end{array}\right]\), then the value of \(A^{125}\) is :

The values of \(a\) for which the function \(f(x)=a^x\) is increasing on \(R\) are given by :

The integral \(\int e^x\left(\frac{x-1}{2 x^2}\right) d x\) is equal to:

Let \(e^y(x+1)=1\). Then which of the following are TRUE
(A) \(\frac{d^2 y}{d x^2}=-\frac{1}{(x+1)^2}\)
(B) \(\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^2\)
(C) \(\left.\frac{d^2 y}{d x^2}\right|_{x=0}=-1\)
(D) \(\left.\frac{d^2 y}{d x^2}\right|_{x=0}=1\)
(E) \(\left.\frac{d^2 y}{d x^2}\right|_{x=1}=\frac{1}{4}\)
Choose the correct answer from the options given below :

Read the passage carefully and answer the questions
Ethers with the general formula \(C _n H _{2 n+2} O\) and functional group C-O-C may be divided as symmetrical and unsymmetrical.
Lower symmetrical ether may be obtained by the dehydration of alcohols at low temperatures in the presence of protic acids,
such as \(H _2 SO _4\) and \(H _3 PO _4\) whereas at high temperature intramolecular dehydration takes place.
Formation of ether from primary alcohols is also accompanied by the formation of small amount of alkenes.
Due to the steric hinderance, secondary and tertiary alcohols undergo elimination reaction rather than substitution reaction yielding alkene as major product. Ethers can also be obtained by heating alkyl halides with sodium or potassium alkoxides.
Substituted Ethers (secondary & tertiary) can also be prepared using this method.
If primary halides are replaced with secondary and tertiary halides, alkenes are the major products.
The major product in the following reaction is \(ROH + HOR \xrightarrow{ H _2 SO _4, 140^{\circ} C }\)

The sides of an equilateral triangle are increasing at the rate of \(5 cm / sec\). The rate at which the area increases when the side is 20 cm , is

If A and B are square matrices of order n × n, then \((A-B)^2\) is equal to:
(A) \(A^2-A B-B A+B^2\)
(B) \(A^2-2 A B+B^2\)
(C) \(A^2-2 B A+B A\)
(D) \(A^2-B^2\)
(E) \(A^2+2 A B+B^2\)

In reference to Inferential Statistics, if \(\bar{x}\) is a sample mean of random data \(\left\{x_i\right\}_{i=1}^n\) and n is the sample size, then the formula \(\frac{1}{n-1} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2\) represents

If \(\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & -2 \\ 7 & 3\end{array}\right|\) then the value of x is:

A charging capacitor has