CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. Anil Kumar Tondak
CBSE CLASS XII
Application of Derivatives
Q. 1. The volume of a cube is increasing aa constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.
Q. 2. Use differentials to find the approximate value of 
Q. 3. It is given that for the function f(x) = x3 – 6x2 + ax + b on [1, 3], Rolle’s theorem holds with
c = 2+ 
. Find the values of a and b if f(1)= f(3) = 0
Q. 4. Find a point on the curve y = (x – 3)2, where the tangent is parallel to the line joining (4, 1) and (3, 0).
Q. 5. Find the intervals in which the function f(x) = x4 – 8x3 + 22x2 – 24x + 21 is decreasing or increasing.
Q. 6. Find the local maximum or local minimum of the function. 
Q. 7. Find the point on the curve y2 = 4x which is nearest to the point (2, 1).
Q. 8. A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.
Q. 9. A balloon which always remain spherical has a variable diameter 
. Find the rate of change of its volume with respect to x.
Q. 10. Find the intervals in which f(x) = (x+1)3 (x – 3)3 is strictly increasing or decreasing.
Q. 11. Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1
Q. 12. Using differentials, find the approximate value of (26.57)1/3
Q. 13. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Q. 14. Find the equation of the tangent and normal to the hyperbola 
at the point (x0,y0)
Q. 15. Find the intervals of the function 
is strictly increasing or strictly decreasing.
Q. 16. An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.
Q. 17. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 
of the volume of the sphere.
Q. 18. Show that the right circular cylinder of given surface area and maximum volume is such that its height is equal to the diameter of the base.
Q. 19. 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.
Q. 20. A window is in the form of a rectangle surmounded 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.
Q. 21. Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Q. 22. For the curve y = 4 x3– 2 x5 , find all the points at which the tangent passes through the origin.
Q. 23. An Apache helicopter of enemy is flying along the curve given by y = x2+ 7. A soldier, placed at (3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.
Q. 24. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?
Q. 25. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Q. 26. 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
Paper By Mr. Anil Kumar Tondak
Email Id : [email protected]
