Page 1 CBSE TEST PAPER-05 CLASS - XII MATHEMATICS (Calculus: Application of Derivatives) Topic: - application of derivatives 1. The two equal side of on isosceles ? with fixed base b are decreasing at the rate of 3cm/s. How fast is the area decreasing when the two equal sides are equal to the base? [4] 2. A men of height 2m walks at a uniform speed of 5km/h away from a lamp, past which is 6m high. Find the rate at which the lengths of his shadow increase. [4] 3. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is tan -1 (0.5) water is poured into it at a constant rate of 5cm 3 /hr. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4m. [4] 4. Find the interval in which the function given by 4 3 2 3 4 36 ( ) 3 11 10 5 5 f x x x x x = - - + + is (a) Strictly increasing (b) Strictly decreasing [4] 5. Show that 1 ( ) tan (sin cos ) f x x x - = + is always an increasing function in 0, 4 p ? ? ? ? ? ? [4] 6. For the curve y = 4x 3 â€“ 2x 5 , find all the point at which the tangent passes through the origin. [4] 7. Prove that the curves x = y 2 , and xy = K cut at right angles if 8k 2 = 1 [4] 8. Find the maximum area of an isosceles ? inscribed in the ellipse 2 2 2 2 1 x y a b + = with its vertex at one end of the major axis. [4] 9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is 8m 3 . If building of tank costs Rs 70 per sq. metres for the base and Rs 45 per sq. metres for sides what is the cost of least expansive tank? [6] 10. The sum of the perimeter of a circle and square is k, where K is some constant. Prove that the sum of their area is least when the side of square is double the radius of circle. [6] 11. A window is the form of a rectangle surmounted by a semi circular opening the total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening. [6] 12. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is ( ) 3 2 2 3 2 3 a b + . [6]Read More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!