12th cbse 2020 - mathematics : 9 days 80/80 plan · 2020. 3. 10. · 12th cbse 2020 - mathematics :...

12th CBSE 2020 - MATHEMATICS : 9 Days 80/80 Plan

Name of Units Weightage

Inverse Trigonometry ( Ch-2 ) 7th Mar (COMPLETED) 08

Calculus ( Ch- 5,6,7,8,9 ) Ch-5 ( COMPLETED) 8, 9, 10th Mar 35

Vectors and Three Dimensional Geometry ( Ch -10, 11) 11th Mar 14

Linear Programming ( Ch - 12) 12th Mar 05

Probability ( Ch- 13) 10th Mar 08

Relations and Functions ( Ch-1 ) 12th Mar 08

Algebra - Matrices and Determinants ( Chs 3 & 4) 13th Mar 10

2015 2016 2017 2018 2019 2020(Sample)

Very Short Answer (1 mark) 1 Q

Short Answer (2 marks) 2 Q 1 Q 1 Q

Long Answer Type I (4 marks)

1 Q 1 Q 1 Q

Long Answer Type II (6 marks)

1 Q 1 Q 1 Q 1 Q 1 Q 1 Q

Application of Derivatives TREND for LAST 5 Years

The ratio ΔyΔx is average rate of change of ‘y’ with respect to ‘x’

Derivative as a Rate Measure

where Δx represents change in ‘x’& Δy represents change in ‘y’

Now, if we apply the limit Δx → 0 represents

inst. rate of change of ‘y’

w.r.t. x, at x = x0

dydx x0

lim ΔyΔx = dy

dxΔx→0

Class 12th - 4 Marks (CBSE 2017)

Q1. The volume of a cube is increasing at the rate of 9 cm3/s.How fast is its surface area increasing when the length of an edge is 10 cm?

12th CBSE: APPLICATION OF DERIVATIVES FULL REVISION

SOLUTION

Q2. The total cost C(x) associated with the product of x units of an item is given by Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of cost at any level of output.



SOLUTION

TANGENTS AND NORMALS

Hence, equation of a tangent at (x0 , y0) is

TangentsWe know that = f ′(x)

dydx

f ′(x0) = Slope of tangent at (x0, y0)

y – y0 = f ′(x0)(x – x0)

P(x0, y0)

y = f(x)

Hence, the equation of a Normal at (x0 , y0) is

Normals

Normal is perpendicular to Tangent

–1 /f ′(x0)=Slope of Normal at (x0, y0)

y – y0 = –1 /f ′(x0) (x – x0)

P(x0, y0)

y = f(x)

Q3. Find the equations of the tangent and the normal, to the curve 16x2+9y2=145 at the point (x1, y1), where x1=2 and y1>0 Class 12th - 4 Marks (CBSE 2016)


SOLUTION

Q4. Find the equation of tangents to the curve y=x3+2x - 4, which are perpendicular to line x+14y+3=0



SOLUTION

Q5. Find the equation of tangent to curves x = sin 3t, y = cos2t at .

(CBSE 2016 - 4 Marks)


SOLUTION

Q6. Find the equation of tangent to the curve at the

point, where it cuts the X-axis.



SOLUTION

Q7. Find the equations of tangents to the curve 3x2 - y2 = 8, which passes

through the point .



SOLUTION

A function f (x) is said to be increasing if

A function f (x) is said to be decreasing if

INCREASING AND DECREASING FUNCTIONS

x2

> x1 ⇒ f (x

2)

> f (x

1)

A function f (x) is said to be non decreasing if

x2

> x1 ⇒

f (x

2) ≥

f (x

1)

x2

> x1 ⇒

f (x

2) <

f (x

1)

A function f (x) is said to be non increasing if

x2

> x1 ⇒

f(x

2) ≤

f(x

1)

x2

x1

f(x1)

f(x2)

f(x1)

f(x2)

x2

x1

DERIVATIVE TEST

If function is differentiable then

f ′(x) > 0 ⇒ f is increasing f ′(x) < 0 ⇒ f is decreasing

Let us study a test to check if the function is increasing or decreasing

We need to take care that set of values for which f ′(x) = 0 do not form an interval.

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Q8. Find the intervals in which the function

is (a) Strictly increasing, (b) Strictly decreasing.

(CBSE 2018 - 4 marks)


SOLUTION

Q9. Find the values(s) of x for which y= [x(x-2)]2 is an increasing function.



SOLUTION

Q10. Prove that is an increasing function of θ on



SOLUTION


MAXIMA AND MINIMA

Q11. Show that the surface area of a closed cuboid with square base and given volume is minimum, when is a cube.



SOLUTION

Q12. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is . Also show that the maximum volume of the cone is of the volume of the sphere.

CBSE 2014

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is . Also show that the maximum volume of the cone is of the volume of the sphere.

CBSE 2014Solution

Solution

Q13. If the length of three sides of a trapezium other than the base are each equal to 10 cm, then find the area of the trapezium, when it is maximum.

(CBSE 2014, 2010 - 6 Marks)


SOLUTION

Q14. A manufacturer can sell x items at a price of Rs each.The cost of x items is Rs .Find the number of items he should sell to reach maximum profit.



SOLUTION

Errors and Approximations

Approximation is, knowing the value of a function at one point and using it to decide the value of the function at another point.

Error: If error in x is Δx, and y is a function of x, what will be the error in y ?

E.g. We know the value of f(2) And we approximate the value of f(2.001)

However, if Δx is small, then

We know, if Δx → 0, =ΔyΔx

dydx

then the quantity

ΔyΔx

≈dydx

Δy ≈dydx

×Δx

f(a + h) = f(a) + h.f ′(a) It is used to find f(a + h);knowing f(a) and f ′(a)

Smaller the magnitude of Δx , better is theapproximation

f ′ (a) =f(a + h) – f(a)

h1)

2)

Results

If Absolute Error in x = Δx

Some definitions

Relative Error in x Δxx=

Percentage Error in xΔxx= × 100

Q15. Using differentials, find the appropriate value of (3.968)3/2 .



SOLUTION

Q16. Using differentials, find approximate value of .

(CBSE Delhi 2012 - 4 marks)


SOLUTION

MEAN VALUE THEOREMS

If a function f(x) satisfies

1) f(x) is continuous in [a ,b]

2) f(x) is differentiable in (a ,b)

then ∃ c ∈ (a , b) s.t.

LMVT-Lagrange mean value Theorem

f ′(c) =b – a

f(b) – f(a)

Becomes equal to instantaneous rate of change at some point inside the interval

Average rate of change over the interval

LMVT-Lagrange mean value Theorem

Consider a chord joining A and B

c ∈(a,b)

B(b, f(b))

(a, f (a))

A

c f ′(c) =b – a

f(b) – f(a)

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Rolle’s Theorem

If y = f (x) is a function which is

1. Continuous in the interval [a, b]2. Differentiable in interval (a, b)

Then, there exists at least one c∈ (a , b) such that f ′(c) = 0.

A(a, f (a))

B(b, f(b))

C

3. f (a) = f (b)

There may be more than one such value of ‘c’

B

Rolle’s theorem says atleastone ‘c’ where f ′(c) = 0

Example

Ac

2

c1

c3

Q1. Verify Rolle's theorem for the function

Q2. Verify Rolle's theorem for the following functions on indicated intervals,

Q3. Verify Lagrange's mean value theorem for the following functions on the indicated intervals. Also, find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem:

Q. Find the area of the greatest rectangle that can be inscribed in an ellipse



SOLUTION

8th Jan 2020-(Shift 1)

8th Jan 2020-(Shift 1)

Thank You

12th cbse 2020 - mathematics : 9 days 80/80 plan · 2020. 3. 10. · 12th cbse 2020 - mathematics :...

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