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Applications of Derivatives Test- 5 Class 12 Notes | EduRev

Class 12 : Applications of Derivatives Test- 5 Class 12 Notes | EduRev

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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
– 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]
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