Ncert Solutions Maths class 12th

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New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The given eqn of the curve is y=x33x29x+7.

slope of tangent to the given curve, dydx=3x26x9

when the tangent is parallel to x-axis dydx=0

3x26x9=0

x22x3=0

x2+x3x3=0

x(x+1)3(x+1)=0

(x+1)(x3)=0

 x = 3 or x = -1

When x = 3, y=333(3)29(3)+7=272727+7=20

And when x = -1 y=(1)33(1)29(1)+7=13+9+7=12

Hence, the required points are (3,20)(1,12)

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The given eqn of the curves are

x=1asinθy=bcos2θ

so,  dxdθ=acosθdydθ=2bcosθsinθ

dydx=dy/dθdx/dθ=2bcosθsinθacosθ=2basinθ

Slope of tangent to curve at θ=π2 is dydx|θ=π2

=2basinπ2

=2ba

Hence, slope of normal to curve =12b/a=a2b

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The Equation of the given curve are

x=acos3θy=asin3θ

So,  dxdθ=3acos2θ (sinθ)=3acos2θsinθ.

and dydθ=3asin2θcosθ

dydx=dy/dθdx/dθ=3asin2θcosθ3acos2θsinθ=tanθ

So,  dydx|x=π/4=tanπ4=1 which is the slope of the tanget to the curve.

Now, required slope of normal to the curve =1
Slopeoftangent
 
=11=1

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

Slope of tangent to the given curve y=x33x+2 is dydx=3x23.

so,  dydx|x=3=3 (3)23=273=24.

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

Slope of tangent to the given curve y=x3x+1 is

dydx=3x21.

So,  dydx|x=2=3 (2)21=121=11.

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The given eqn of the curve is y=x1x2

Slope of tangent at x = 10 is given by,

dydx|x=10=(x2)ddx(x1)(x1)ddx(x2)(x2)2|x=10

=(x2)(x1)(x2)2|x=10=x2x+1(x2)2|x=10

=1(x2)2|x=10

=1(102)2=182=164

New answer posted

4 months ago

0 Follower 2 Views

V
Vishal Baghel

Contributor-Level 10

The given eqn of the curve is y=3x44x.

Slope of the tangent at x = 4 is given by

dydx]x=4=12x34]x=4=12 (4)34=12*644

=7684

= 764

New answer posted

4 months ago

0 Follower 3 Views

V
Vishal Baghel

Contributor-Level 10

We have, f (x) = x2 e–x

So, f (x) = x2ddxex+exdx2dx

=x2exddx (x)+ex2xdxdx

= -x2 e-x + e-x 2x.

= x e-x ( x + 2).

=x (2x)eex

If f (x) = 0.

x (2x)ex=0.

 x = 0, x = 2.

Hence, we get there disjoint interval

[, 0), (0, 2) (2, )

When, x  (, 0), we have, f (x) = ( -ve) ( + ve) = ( -ve) < 0.

So, f is strictly decreasing.

When x ∈ (0,2), f (x) = ( + ve) ( + ve) = ( + ve) > 0.

So, f is strictly increasing.

And when x ∈ (2, ), f (x) = ( +ve) ( -ve) = ( -ve) < 0.

So, f is strictly decreasing.

Hence, option (D) is correct.

New answer posted

4 months ago

0 Follower 3 Views

V
Vishal Baghel

Contributor-Level 10

We have, f (x) = x3- 3x2 3x- 100.

So, f (x) = 3x2- 6x + 3 = 3 (x2- 2x + 1) = 3 (x- 1)2

For xR.

(x- 10)20 , 0 for x = 1.

3 (x- 1)2 0

 f (x) 0

∴f (x) is increasing on ?

New answer posted

4 months ago

0 Follower 4 Views

V
Vishal Baghel

Contributor-Level 10

We have, f (x) = log |cosx|.

f (x) = 1cosxddxx=sinxcosx=tanx

whenx  (0, π2),  we get.

tanx> 0 (Ist quadrant).

 tanx< 0

 f (x) < 0.

∴f (x) is decreasing on  (0, π2).

When x ∈ (3π2, 2π) we get,

tanx|< |0 (ivth quadrant).

-tanx|>| 0

 f (x) > 0

∴f (x) is increasing on  (3π2, 2π).

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