TN Maths, 12th Chapter 7 Important Questions

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Class : XII

Subject : Maths

Chapter : 7

Chapter : Applications Of Differential Calculus

IMPORTANT QUESTIONS

      

2 & 3 Marks Questions

1. The temperature T in Celsius in a long rod of length 10m , insulated art both ends , is a function of length x given by T = x ( 10 – x) . Prove that the rate of change of temperature at the mid point of the rod is zero.

2). A person learnt 100 words for an English test. The number of words the person remembers in t in days after learning is given by w( i ) = 100 X (1-0.1 t)², 0 ≤ t ≤ 10 what is the rate at which the person forgets the words 2 days after learning?

3). A particle moves so that the distance moved is according to the law s(t) = (t³/3)-t²+3. At what time the velocity and acceleration are zero respectively?

4). A camera is accidently knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t²  in t seconds.

i) How long does the camera full before at bits the ground?

ii) What is the average velocity with which the camera full’s during the last 2

seconds.

iii) What is the instantaneous velocity of the camera when it hits the ground?

5). If the volume of a cube of a side length x is v=x³. Find the rate of change of the volume with respect to x when x = 5 units.

6). If the mass m( x) (in kilograms) of a thin rod of length x ( in meters ) is given by , m( x) = √3x then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 meters.

7). Find the equations of tangent and normal to the curve y=x²+3x+2 at the point (1, 2).

8). Find the points on the curve y = x³–3x²+x-2 at which the forget is parallel to the line y = x.

9). Find the angle of intersection of the curve y = sin x with the positive x- axis.

10. Find the points on the curve y = x³-6x²+x+3 where the normal is parallel to the line

 x + y = 1729 .

11). Find the slope of the tangent to the following curves at the respective given points .

i) y = x⁴ + 2x² - x at x = 1,

ii) x = a cos³ t , y = b sin³ t at t = (π/2)

12). Find the points on the curve y²– 4xy = x²+5 for which the forgent is horizontal .

13. Find the tangent and normal to the following curves at the given points on the curve.

i) y = x sinx at ( π/2 , π/2)

ii) x = cos t ,y = 2 sin³t at t (π/3)

14. Find the absolute extrema of the following functions on the given closed interval.

 i) f (x) = 3x⁴ – 4x³; [ -1 ,2]

 ii) f(x) = x² – 12x +10 ; [1,2]

 iii)f(x) = 6x⁴/³ - 3x⅓ ; [-1 ,1]

 iv)f(x) = 2 cos x + sin 2x ; [o , π/2]

15. Find the local extremum of the function f(x)=x⁴+32x.

16. Find two positive numbers whose sum is 12 and their product is maximum.

17. Find two p[ositive numbers whose product is 20 and their sum is minimum.

5 Mark Questions

1. A ladder 17 metre long is leaning against the wall . The base of the ladder is pulled away from the wall at a rate of 5m/s . when the base of the ladder is 8 meters from the wall.

i). how fast is the top of the ladder moving down the wall?

ii) at what rate the area of the triangle formed by the ladder , wall and the floor is changing?

2. Find the equation of the tangent and normal at any point to the lissajous curve given by x = 3 sin 26 , t € R.

3. Find the angle between y=x² and y=(x-3)².

4. If the curves ax²+by²=1 and cx²+dx²=1 intersect each other orthogonally then show that

 1/a - 1/b = 1/c - 1/d

5. A particle moves along a horizontal line such that its position at time t ≥ 0 is given by s(t) = t³-6t²+ 9t +1 , where s is measured in meters and t in seconds?

i). At what time the particle is at rest?

ii). At what time the particle changes its direction ?

iii). Find the total distance travelled by the particle in the first 2 seconds?

6. For the function f(x) = 4x³+3x²-6x+1 find the intervals of monotonicity, local extrema intervals of concavity and points of inflection.

7. A steel plant is capable of producing x tones per day of a low- grade steel and y tones per day of a high grade steel , where y=[(40-5x)/(10–x)] . If the fixed market price of low –grade steel is half That of high – grade steel, then what should be optimal productions in low- grade steel and High –grade steel in order to have maximum receipts.

8. A hollow cone with base radius a cm and height b cm is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is 4/9 times volume of the cone.

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