## Topics : Continuity, Differentiation and Applications of Derivatives

TIME – 1:30 Hr                                                                                                                  M.M. 50

Very short answer type question (1 Mark)

1. Find the point of discontinuity, if any, for the function $f\left(x\right)=\frac{1}{x-5}$;
2.Verify LMV theorem for the function $f\left(x\right)=|x|\text{\hspace{0.17em}}in\text{\hspace{0.17em}}\left[-1,1\right]$

Short answers type questions. (2 marks)

3.Find two positive numbers whose sum is 15 and sum of whose squares is minimum.
4.Find the equation of the tangent to the curve $y=\sqrt{4x-3}$ , if slope of the tangent is 2/3.
5.Verify Rolle’s theorem $f\left(x\right)=\mathrm{sin}x+\mathrm{cos}x\text{\hspace{0.17em}}in\text{\hspace{0.17em}}\left[0,\frac{\Pi }{2}\right]$
6.For what value of k, is the following function continuous at x = 0? $f( x )={ k, x=0 sin5x 3x , x≠0$
7.A Particle moves along the curve $y=\frac{2}{3}{x}^{3}+1$.Find the points on the curve at which the y coordinates changes twice as fast as x coordinates.

Long answer type questions – I( 4 marks)

8.Prove that function $f\left(x\right)={\left\{}_{k,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=0}^{\frac{x}{|x|+2{x}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ne 0}$is discontinuous at x = 0.
9. If $x=2\mathrm{cos}\theta -\mathrm{cos}2\theta \text{\hspace{0.17em}}and\text{\hspace{0.17em}}y=2\mathrm{sin}\theta -\mathrm{sin}2\theta ,\text{\hspace{0.17em}}find\text{\hspace{0.17em}}{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}_{\theta =\frac{\pi }{2}}$
10.Find the intervals in which the function f given $f\left(x\right)=\mathrm{sin}x+\mathrm{cos}x,0\le x\le 2\pi$ 1. is strictly increasing or strictly decreasing.
11.Water is running into a conical tank of height 10 m and diameter 10 m at the top, at a constant rate of 18 m3/min. How fast is the water rising in the tank at any instant.
12.Show that the surface area of a closed cuboid with a square base and given volume is minimum, when it is a cube.

Long answer type questions – II( 6 marks)

13.Determine, the points on the curve $y=\frac{1}{4}{x}^{2}$ which are nearest to the point (0, 5).
14.A point on the hypotenuse of a right – angle triangle is at distances a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is ${\left({a}^{\frac{2}{3}}+{b}^{\frac{2}{3}}\right)}^{{}^{\frac{3}{2}}}$

Also Read : CBSE Class 12 Maths Sample Paper 2018

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