24) The points on the curve , where the normal to the curve makes equal intercepts with the axes are
23) The normal to the curve passing (1,2) is
(A) x + y = 3
(B) x – y = 3
(C) x + y = 1
(D) x – y = 1
22) The normal at the point (1,1) on the curve is
(A) x + y = 0
(B) x – y = 0
(C) x + y +1 = 0
(D) x – y = 1
20) The slope of the tangent to the curve at the point
(2,– 1) is
A ) 22/7
B ) 6/7
C ) 7/6
D ) -6 /7
19) A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314
cubic metre per hour. Then the depth of the wheat is increasing at the rate of
(A) 1 m/h
(B) 0.1 m/h
(C) 1.1 m/h
(D) 0.5 m/h
18) Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle a is one-third that of the cone and the greatest volume of cylinder is
17) Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is
. Also, find the maximum volume.
16) Let f be a function defined on [a, b] such that , for all. Then prove that f is an increasing function on (a, b).
15) Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3
13) Find the points at which the function f given byhas
(i) local maxima (ii) local minima (iii) point of inflexion
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
11) A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
10) The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
9) A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
8) Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis.