For the system of linear equations
x + y + z = 5000
6x + 7y + 8z = 35800
6x + 7y - 8z = 7000
the values of x, y and z are:

The relation \(R=\left\{(a, b): a \leq b^2\right\}\) on the set of real numbers is:

Match List-I with List-II

List-I	List-II
(A) \(f ( x )=| x |\)	(I) Not differentiable at \(x=-2\) only
(B) \(f(x)=|x+2|\)	(II) Not differentiable at \(x=0\) only
(C) \(f(x)=\left|x^2-4\right|\)	(III) Not differentiable at \(x=2\) only
(D) \(f(x)=|x-2|\)	(IV) Not differentiable at \(x=2,-2\) only

Choose the correct answer from the options given below:

\(\int e^x \sec x(1+\tan x) d x\) equals:

In a game, a person is paid Rs. \(2\) if he gets all heads or all tails when three coins are tossed and he will pay Rs. \(2\) if either one or two heads show. What can he expect to win on an average per game?

There are 4 bulbs out of which 2 are defective. Each bulb is tested in random order till both the defective bulbs are identified. The probability that only two tests are required to identify the defective bulbs is:

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

Consider the region bounded by the lines \(y-1=x, x=-2, x=3\) and \(x\)-axis. Then,
(A) The area of the bounded region is given by \(\int_{-2}^3(x+1) d x\)
(B) The numerical value of the area is \(\frac{15}{2}\) sq. units
(C) The numerical value of the area is 8 sq . units
(D) The numerical value of the area is \(\frac{17}{2}\) sq. units
Choose the correct answer from the options given below:

The height of transmitting antenna and receiving antenna are 45 m and 80 m. If the radius of earth is \( 6.4 \times 10^6 \text{ m} \), then the maximum distance between the two antennas for satisfactory communication in LOS mode is:

Read the Passage carefully and answer the Questions
Kinetic studies not only help us to determine the rate of a chemical reaction but also describe the conditions by which the reaction rates can be altered. The factors such as concentration, temperature, pressure and catalyst affect the rate of a reaction.
The speed of a reaction or the rate of a reaction can be defined as the change in concentration of a reactant or product in unit time.
Rate \(=k[A]^x[B]^y\)
Here \(k\) is the rate constant
\(x+y\) gives the overall order of a reaction, whereas \(x\) and \(y\) represent the order with respect to the reactants \(A\) and \(B\), respectively.
Which of the following statements are correct?
(A) Catalysts increase the rate of a chemical reaction.
(B) Catalysts can catalyze both spontaneous and non-spontaneous reactions.
(C) Higher the activation energy for a chemical reaction, lower is the rate of the reaction.
(D) Catalyst increases the potential energy barrier.
Choose the correct answer from the options given below:

\(\int \frac{\left(x^4-x\right)^{1 / 4}}{x^5} d x\) is equal to

Observe the figure and answer the following questions

Match List I with List II

LIST I (Name of the structure)	LIST II (LABEL)
A. Graafian Follicle	I. B
B. Primary Follicle	II. D
C. Corpus Luteum	III. A
D. Ovum	IV. E

Choose the correct answer from the options given below:

Read the following passage given below and answer the question.
Aldehydes and ketones undergo nucleophilic addition reactions. A nucleophile attacks the electrophilic carbon atom of polar carbonyl group from perpendicular to plane of \(sp^2\) hybridised orbitals of carbonyl carbon. The hybridisation of carbon changes from \(sp^2\) to \(sp^3\) and a tetrahedral alkoxide intermediate is formed. This captures \(H^+\) to give neutral product. In carboxylic acids, carbonyl carbon is less electrophilic than carbonyl carbon in aldehydes and ketones.
An organic compound \(C_3H_6O\) does not give a precipitate with 2,4-dinitrophenylhydrazine and also does not react with metallic sodium. It could be :