Avail KVPY Study Material On Discount (Quiz 5)

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:

3

4

6

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 =

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:

Amylose

Teflon

Bakelite

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.

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

3

6

8

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.

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. What is the Maximum number of hydrogen bonds possible involving a water molecule:

0

1

2

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.

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.

5. 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:

25 J

50 J

75 J

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.

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.