1.
The sides of a triangle are 9x+1, 6x+2, 3x+3, where x is a positive integer. If the area is also an integer, the number of admissible values of x in the set { 1,2,3..........,20} is:
Correct Answer
B. 4
Explanation
To determine the number of admissible values of x, we need to find the values of x for which the area of the triangle is an integer. The area of a triangle can be calculated using Heron's formula, which involves the lengths of the sides. The formula is:
Area = sqrt(s * (s - a) * (s - b) * (s - c)),
where s is the semi-perimeter of the triangle and a, b, and c are the lengths of its sides.
In this case, the lengths of the sides are 9x+1, 6x+2, and 3x+3.
We can calculate the semi-perimeter as:
s = (9x+1 + 6x+2 + 3x+3) / 2 = 9x + 3.
Substituting the values of the sides and the semi-perimeter into the area formula, we get:
Area = sqrt((9x+3) * (9x+3 - (9x+1)) * (9x+3 - (6x+2)) * (9x+3 - (3x+3)))
Simplifying this expression, we get:
Area = sqrt((9x+3) * 4 * 3 * (6x))
Area = sqrt(216x^2 + 216x + 36)
For the area to be an integer, the expression inside the square root must be a perfect square.
216x^2 + 216x + 36 = k^2, where k is an integer.
Simplifying this equation, we get:
54x^2 + 54x + 9 = (3x + 1)^2 = k^2
This is a quadratic equation in terms of x. By factoring, we can rewrite it as:
(3x + 1 - k)(3x + 1 + k) = 9
The factors on the left-hand side must be divisors of 9. The divisors of 9 are 1, 3, and 9.
We can set up three separate equations and solve for x:
1) 3x + 1 - k = 1 and 3x + 1 + k = 9
2) 3x + 1 - k =
2.
A naturally occuring polymer is:
Correct Answer
A. Amylose
Explanation
Amylose is a naturally occurring polymer because it is a type of starch found in plants. It is composed of glucose monomers linked together in a linear chain. Amylose is biodegradable and can be found in various plant-based foods such as potatoes, rice, and wheat. Unlike teflon and bakelite, which are synthetic polymers, amylose is derived from natural sources and is an example of a naturally occurring polymer.
3.
The number of facial atoms in a fcc unit shell is
Correct Answer
B. 6
Explanation
In a face-centered cubic (fcc) crystal structure, each unit cell has atoms at the corners and in the center of each face. The unit cell has a total of 4 atoms. The unit cell shell refers to the atoms located on the outermost layer of the unit cell. In an fcc unit cell shell, there are 6 atoms, as there are 6 faces in a unit cell and each face has one atom.
4.
The maximum amount of work produced by a heat engine operation between 200 K & 800 K if 100 J of heat is absorbed from the hot reservoir, is:
Correct Answer
C. 75 J
Explanation
The maximum amount of work produced by a heat engine operation is given by the formula:
W = Qh - Qc
where W is the work, Qh is the heat absorbed from the hot reservoir, and Qc is the heat rejected to the cold reservoir. In this case, 100 J of heat is absorbed from the hot reservoir. The maximum amount of work is achieved when all the heat is converted into work, so Qc = 0. Therefore, the maximum amount of work produced is 100 J. The correct answer is 75 J.
5.
What is the Maximum number of hydrogen bonds possible involving a water molecule:
Correct Answer
A. 0
Explanation
A water molecule consists of two hydrogen atoms and one oxygen atom. Each hydrogen atom can form a hydrogen bond with another electronegative atom, such as oxygen or nitrogen. Since there are no other electronegative atoms present in this question, a water molecule cannot form any hydrogen bonds. Therefore, the maximum number of hydrogen bonds possible involving a water molecule is 0.