HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

`(3/2, (3√3)/2)`.

In ΔABC, if cot A, cot B, cot C are in A.P. then show that a², b², c² are also in A.P.

Solve the following equation by the method of inversion:

2x - y = - 2, 3x + 4y = 3

Solve the following equations by the method of inversion:

x + y+ z = 1, 2x + 3y + 2z = 2,
ax + ay + 2az = 4, a ≠ 0.

Solve the following equation by the method of inversion:

5x − y + 4z = 5, 2x + 3y + 5z = 2 and 5x − 2y + 6z = −1

Solve the following equations by the method of inversion:

x + y + z = - 1, y + z = 2, x + y - z = 3

Express the following equations in matrix form and solve them by the method of reduction:

x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2

Express the following equations in matrix form and solve them by the method of reduction:

`x + y = 1, y + z = 5/3, z + x 4/33`.

Express the following equations in matrix form and solve them by the method of reduction:

2x - y + z = 1, x + 2y + 3z = 8, 3x + y - 4z = 1.

Express the following equations in matrix form and solve them by the method of reduction:

x + 2y + z = 8, 2x + 3y – z = 11, 3x – y – 2z = 5.

The cost of 4 pencils, 3 pens, and 2 books is ₹ 150. The cost of 1 pencil, 2 pens, and 3 books is ₹ 125. The cost of 6 pencils, 2 pens, and 3 books is ₹ 175. Find the cost of each item by using matrices.

An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from the first two investments is ₹ 70 more than the income from the third, find the amount of each investment using matrix method.

Solve the following equations by the method of inversion:

2x + 3y = - 5, 3x + y = 3

Express the following equations in matrix form and solve them by the method of reduction:

x + 3y + 2z = 6,

3x − 2y + 5z = 5,

2x − 3y + 6z = 7

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?

The volume of a sphere increases at the rate of 20 cm³/sec. Find the rate of change of its surface area, when its radius is 5 cm

A man of height 2 metres walks at a uniform speed of 6 km/hr away from a lamp post of 6 metres high. Find the rate at which the length of the shadow is increasing.

A man of height 1.5 meters walks towards a lamp post of height 4.5 meters, at the rate of `(3/4)` meter/sec. Find the rate at which (i) his shadow is shortening (ii) the tip of shadow is moving.

A ladder 10 metres long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 metres per second, find how fast the top of the ladder is sliding down the wall, when the bottom is 6 metres away from the wall.

Choose the correct option from the given alternatives :

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of `(1.5 "m")/sec`. The length of the higher point of ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of