HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

A particle moves along a line according to the law s(t) = 2t³ – 9t² + 12t – 4, where t ≥ 0. Find the particle’s acceleration each time the velocity is zero

If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units

If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = `sqrt(3x)` then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres

A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45° with the shore?

A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall. How fast is the top of the ladder moving down the wall?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall, at what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing?

A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?

Find the slope of the tangent to the following curves at the respective given points.

y = x⁴ + 2x² – x at x = 1

Find the slope of the tangent to the following curves at the respective given points.

x = a cos³t, y = b sin³t at t = `pi/2`

Find the point on the curve y = x² – 5x + 4 at which the tangent is parallel to the line 3x + y = 7

Find the points on curve y = x³ – 6x² + x + 3 where the normal is parallel to the line x + y = 1729

Find the points on the curve y² – 4xy = x² + 5 for which the tangent is horizontal

Find the tangent and normal to the following curves at the given points on the curve

y = x² – x⁴ at (1, 0)

Find the tangent and normal to the following curves at the given points on the curve

y = x⁴ + 2e^x at (0, 2)

Find the tangent and normal to the following curves at the given points on the curve

y = x sin x at `(pi/2, pi/2)`

Find the tangent and normal to the following curves at the given points on the curve

x = cos t, y = 2 sin²t at t = `pi/2`

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x + 12y = 12

Find the equations of the tangents to the curve y = `- (x + 1)/(x - 1)` which are parallel to the line x + 2y = 6