1)Show that the function f given by is strictly decreasing function on ().
2)Find the intervals in which the following function is increasing :=– .
3)A man 160 cm tall; walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/s. How fast is the length of his shadow increasing when he is 1 m away from the pole?
4)A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of ‘c’ m/min?
5)The bottom of a rectangular swimming pool is 25 m by 40 m. Water is pumped out into the tank at the rate of 500 m3/min. Find the rate at which the level of the water in the tank rising.
6)An inverted cone has a depth of 40 cm and base of radius 5 cm. Water is poured into it at a rate of 1.5 cm3/ min. Find the rate at which the level of water in the cone is rising when the depth is 4 cm.
7)Water is dripping through a tiny whole at the vertex in the bottom of a conical funnel at a uniform rate of 4 cm3 / s. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 1200
8)Oil is leaking at the rate of 16 mL / s from a vertically kept cylindrical drum containing oil. It the radius of the drum is 7 cm and its height is 60 cm, find the rate at which the level of the oil is changing when the oil level is 18 cm
9)Determine the values of for which , is increasing or decreasing.
10)Find the intervals in which is increasing or decreasing.
11)Separate the interval into sub intervals in which is increasing or decreasing.
12)Find the interval in which where is increasing or decreasing.
13)Show that is a decreasing function on R
14)Show that the function is increasing if λ> 2
15)Find the interval in which where is increasing or decreasing.
16)
17)Find the condition that the line x cos + y sin = p may touch the curve + = 1
18)If the tangent at the point( p, q) to the curve x3 + y3 = k meets the curve again at the point ( a, b), prove that + = -1
19)Show that + = 1 and + = 1 intersect orthogonally
20)If ax + by = 1 is a normal to the parabola y2 = 4px then show that pa3 + 2pab3 = b2
21)If the straight line , touches the ellipse , prove that
22)If touches , prove that
23)Show that touches the curve at the point where the curve crosses the axis of
24)Find the angle between the parabolas and at their point of intersection other than the origin.
25)Show that the equation of the tangent to the curve = 0 at is + + = 0
26)Show that the equation of the tangent to the curve at is
27)Prove that curve + = 2 touches the straight line at for all values of at the point
28)Show that the curve and touch each other.
29)Show that the condition that the curves = 1 and = 1 should intersect orthogonally is that
30)If the curve and cut orthogonally at (1, 1), then find the value of
31)If the curves and cut orthogonally, then find the relation between and .
32)Show that the curves and cut each other orthogonally, where and are constants.
33)A window is in the form of a rectangle above which there is a semi-circle. If the perimeter of the window is cm, show that the window will allow the maximum possible light only when the radius of the semi-circle is cm
34)A Cylindrical container with a capacity of 20 cubic feet is to be produced. The top and bottom of the container are to be made of a material that costs Rs.6 per squares foot while the side of the container is made of material costing Rs.3 per squares foot. Find the dimension that will minimize the total cost.
35)Assuming that the petrol burnt in a driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of km per hour is km per hour
36)An open box, with a square base , is to be made out a given quantity of metal sheet of area . Show that the maximum volume of the box is
37)Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
38)If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is
39)Prove that the perimeter of a right –angled triangle of given hypotenuse is maximum when the triangle is isosceles.
40)The cost of fuel for running a bus is proportional to the square of the speed generated in km/hr. It costs Rs. 48 per hour when the bus is moving with a speed of 20 km/hr. What is the most economical speed if the fixed charges are Rs. 108 for one hour over and above the running charges?
41)An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.
42)Prove that the radius of the right circular cylinder of greatest curved surface which can be inscribed in a given cone is half of that of the cone.
43)Show that all the rectangle with a given perimeter, the square has the largest area.
44)A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle so that its area is maximum. Find also this area.
45)Show that the height of a right circular cylinder of maximum total surface area, including the two ends that can be inscribed in a sphere of radius is given by .
46)A given quantity of metal is to be cast into a half cylinder with a rectangular base and semi-circular ends. Show that in order that total surface area is minimum , the ratio of length of cylinder to the diameter of its semi-circular ends is 11: 18.
47)A square tank of capacity 250 cubic metres has to be dug out. The cost of land is Rs.50 per square metre. The cost of digging varies with the depth and for the whole tank is 400 (depth)2 rupees. Find the dimension of the tank for the least total cost.
SECOND part
1)A man , 2m tall walks at the rate of m/s towards a street light which is m above the ground. At what rate is the tip of his shadow moving and at what rate is the length of the shadow changing when he is m from the base of the height?
2)Find the condition of no extreme value of .
3)If the curve passes through the origin and the tangents drawn to it at and are parallel to X-axis, then find the values of and c
4)If the function where attains its maximum and minimum at and respectively, such that , then find the value of
5)If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
6)Find the equation of the normal to the curve at . Also, find the distance from origin to the line.
7)A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and L = 200(10-t)2. How fast is the water running out at the end of 5 seconds. Water is the average rate at which the water flows out during the first 5 seconds?
8)Find the condition that the curves such that and intersects orthogonally.
9)Tangent to the circle at any point on it in the first quadrant makes intercepts OA and OB on X and Y axes respectively, O being the centre of the circle, find the minimum value of OA+ OB.
10)AB is a diameter of a circle and C is any point on the circle. Show that the area of is maximum, when it is isosceles.
11)A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs. 5 per square centimeter and the material for the sides costs Rs. 2.50 per square centimeter. Find the least cost of the box.
12)Let be the length of one of the equal sides of an isosceles triangle and let be the angle between them. If is increasing at the rate of m/hr and is increasing at the rate of radius/hr, then the rate in m2/hr at which the area of the triangle is increasing when and
Prepared by Mr. AnirbanNayak
PGT Mathematics, DPS Jodhpur
Mobile- 9828353006, 9001120453