HSC Science (General) १२ वीं कक्षा - Maharashtra State Board Important Questions

Find the approximate value of log₁₀ (1016), given that log₁₀e = 0⋅4343.

 A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle. 

Find the equation of the tangent to the curve at the point on it.

y = x² + 2e^x + 2 at (0, 4)

Verify Lagrange’s mean value theorem for the following function:

f(x) = log x, on [1, e]

Divide the number 20 into two parts such that sum of their squares is minimum.

Choose the correct option from the given alternatives : 

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.

Choose the correct option from the given alternatives:

Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.

Choose the correct option from the given alternatives :

If x = –1 and x = 2 are the extreme points of y = αlogx + βx² + x`, then ______.

The approximate value of tan (44°30'), given that 1° = 0.0175^c, is ______.

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a². Show that the maximum volume of the box is `a^3/(6sqrt(3)`.

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

Solve the following: 

Find the maximum and minimum values of the function f(x) = cos²x + sinx.

The slope of the normal to the curve y = x² + 2e^x + 2 at (0, 4) is ______.

If the tangent at (1, 1) on y² = x(2 − x)² meets the curve again at P, then P is

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 the when foot of the ladder is 4 m away from the wall decreases at the rate of ______

Show that the function f(x) = x³ + 10x + 7 for x ∈ R is strictly increasing

Find points on the curve given by y = x³ − 6x² + x + 3, where the tangents are parallel to the line y = x + 5.

Find the local maximum and local minimum value of  f(x) = x³ − 3x² − 24x + 5

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

The maximum value of the function f(x) = `logx/x` is ______.