If the sum of the first n terms of an A.P. is given by 3n²+5n, find the common difference of the A.P.

A man on the deck of a ship is 10m above water level.He observes that the angle of elevation of the top of a hill is 60⁰ and the angle of depression of the base of the hill is 30⁰.Calculate the distance of the hill from the ship and the height of the hill.

A bird is sitting on the top of a tree which is 60m high.The angle of elevation of the bird from a point on the ground is 45⁰.the bird flies away from the point o observation horizontally and remains at a constant height.After 2seconds, the angle of elevation of the bird from the point of observation becomes 30⁰ .Find the speed of flying of bird.

In the given figure, AB is a diameter of the circle, with centre O and AT is a  tangent. Calculate the numerical value of x.

Draw an equilateral triangle of altitude 4 cm. Construct another triangle similar to it such that its sides are \(\frac { 2 }{ 3 } \) of the given triangle.

Draw a circle of diameter 8 cm. From a point P, 7 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangent segments.

Find the area of the shaded region of the figure along side.

ABCD is a field in the shape of a trapezium.AB||DC and \(\angle\)ABC=60⁰,\(\angle\)DAB=90⁰.Four sector are formed with centres A, B, C and D.THe radius of each sector is 17.5m.Find:
(i)the total area of the four sectors
(ii)the area of remaining portion, given that AB=75m and CD=50m.

A vessel is in form of an inverted cone.Its height is 8cm and the radius of its top, which is open, is 5cm.It is filled with water up to the brim.When lead shots, each of which is a sphere of radius 0.5cm are dropped into the vessel, one-fourth of the water flows out.Find the number of lead shots dropped in the vessel.

In a bag-A, there are four cards numbered 1,3,5 and 7 respectively.In another bag-B, there are three cards numbered 2, 4 and 6 respectively.A card is drawn at random from each bag.
(i)Write the possible outcomes i.e., sample  space
(ii)Find the probability that the sum of these two cards drawn is:
(a)7
(b)even
(c)odd
(d)more than 7