HSC Science (General) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

In ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0

In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0

In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0

In ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`

If y = sec (tan⁻¹x), then `dy/dx` at x = 1 is ______.

If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______

If sin⁻¹(x³ + y³) = a then `("d"y)/("d"x)` = ______

If x = cos⁻¹(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

If y = cos⁻¹ [sin (4^x)], find `("d"y)/("d"x)`

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin²x

Suppose y = f(x) is a differentiable function of x on an interval I and y is one – one, onto and `("d"y)/("d"x)` ≠ 0 on I. Also if f^–1(y) is differentiable on f(I), then `("d"x)/("d"y) = 1/(("d"y)/("d"x)), ("d"y)/("d"x)` ≠ 0

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. The time when the acceleration is 14 ft/sec², is 

The edge of a cube is decreasing at the rate of 0.6 cm/sec then the rate at which its volume is decreasing when the edge of the cube is 2 cm, is

A particle moves along the curve y = 4x² + 2, then the point on the curve at which y – coordinate is changing 8 times as fast as the x – coordinate is

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. Find the velocity when 𝑡 = 2 sec

A car is moving in such a way that the distance it covers, is given by the equation s = 4t² + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

Water is being poured at the rate of 36 m³/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising

A ladder 10 meter 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 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

The volume of the spherical ball is increasing at the rate of 4π cc/sec. Find the rate at which the radius and the surface area are changing when the volume is 288 π cc.