1. /
2. CBSE
3. /
4. Class 12
5. /
6. Mathematics
7. /
8. NCERT Solutions class 12...

# NCERT Solutions class 12 Maths Miscellaneous

### myCBSEguide App

Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.

## NCERT Solutions for Class 12 Maths Application of Derivatives

1. Using differentials, find the approximate value of each of the following:

(a)

(b)

Ans. (a)

Let  ……….(i)

=  ……….(ii)

Changing  to  and  to  in eq. (i), we have

……….(iii)

Here  and

From eq. (iii),

0.677

(b)

Let  ……….(i)

=  ……….(ii)

Changing  to  and  to  in eq. (i), we have

…….(iii)

Here  and

From eq. (iii),

0.497

### 2. Show that the function given by  has maximum value at

Ans. Here  ……….(i)

……..(ii)

And

=

……….(iii)

Now

From eq. (iii),

=  =  < 0

is a point of local maxima and maximum value of  is at .

### 3. The two equal sides of an isosceles triangle with fixed base  are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Ans. Let BC =  be the fixed base and AB = AC =  be

the two equal sides of given isosceles triangle.

Since  cm/s  ……….(i)

Area of  x BC x AM

=

=

=  [By chain rule]

=  cm2/s

Now when

cm2/s

Therefore, the area is decreasing at the rate of  cm2/s.

### 4. Find the equation of the normal to the curve  at the point (1, 2).

Ans. Equation of the curve is ……….(i)

Slope of the tangent to the curve at the point (1, 2) to curve (i) is  =1

Slope of the normal to the curve at (1, 2) is

Equation of the normal to the curve (i) at (1, 2) is

### 5. Show that the normal at any point  to the curve   is at a constant distance from the origin.

Ans. The parametric equations of the curve are

And

Slope of tangent at point

=

Slope of normal at any point

=

And Equation of normal at any point

i.e., at  =  is

Distance of normal from origin (0, 0)

=  which is a constant.

### 6. Find the intervals in which the function  given by  is (i) increasing (ii) decreasing.

Ans. Given:

=

=

=

=

=

=

= ……….(i)

Now  for all real  as . Also  > 0

(i)  is increasing if , i.e., from eq. (i),

lies in I and IV quadrants, i.e.,  is increasing for

and

and (ii)  is decreasing if , i.e., from eq. (i),

lies in II and III quadrants, i.e.,  is decreasing for

### 7. Find the intervals in which the function  given by  is (i) increasing (ii) decreasing.

Ans. (i) Given:

=  =

= ……….(i)

Now

= 0

= 0

Here,  is positive for all real

or  [Turning points]

Therefore,  or  divide the real line into three sub intervals  and

For ,  from eq. (i) at  (say),

Therefore,  is increasing at

For from eq. (i) at  (say)

Therefore,  is decreasing at

For from eq. (i) at  (say),

Therefore,  is increasing at

Therefore,  is (i) an increasing function for  and for  and (ii) decreasing function for

### 8. Find the maximum area of an isosceles triangle inscribed in the ellipse  with its vertex at one end of the major axis.

Ans. Equation of the ellipse is  ……….(i)

Comparing eq. (i) with  we have  and

and

Any point on ellipse is P

Draw PM perpendicular to axis and produce it to meet the ellipse in the point Q.

OM =  and PM =

We know that the ellipse (i) is symmetrical about axis, therefore, PM = QM and hence triangle APQ is isosceles.

Area of APQ  x Base x Height

=  PQ.AM =  . 2PM.AM = PM (OA – OM)

=

=

=

Now

= 0

or

i.e.,  or

is impossible

At ,

=  [Negative]

is maximum at

From eq. (i), Maximum area

=

=

=  =

### 9. 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  70 per sq. meter for the base and  45 per square meter for sides. What is the cost of least expensive tank?

Ans. Given: Depth of tank = 2 m

Let  m be the length and  m be the breadth of the base of the tank.

Volume of tank = 8 cubic meters

Cost of building the base of the tank at the rate of  70 per sq. meter is

And cost of building the four walls of the tank at the rate of  45 per sq. meter is

= 

Let  be the total cost of building the tank.

and

Now

[length cannot be negative]

At  [Positive]

is minimum at .

Minimum cost =

= 280 + 360 + 360 =  1000

### 10. The sum of the perimeter of a circle and square is  where  is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Ans. Let  be the radius of the circle and  be the side of square.

According to question,  Perimeter of circle + Perimeter of square =

……….(i)

Let  be the sum of areas of circle and square.

[From eq. (i)]

=

and

Now

At  [Positive]

is minimum when

From eq. (i),

=

=

=

=  =

Therefore, sum of areas is minimum when side of the square is double the radius of the circle.

### 11. A window is in the form of a rectangle surmounted 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.

Ans. Let  m be the radius of the semi-circular opening of the window. Then one side of rectangle part of window is  and  m be the other side of rectangle.

Perimeter of window

= Semi-circular arc AB + Length (AD + DC + BC)

……….(i)

Area of window

= Area of semi-circle + Area of rectangle

=

=

=

and  =

Now

At  [Negative]

is maximum at

From eq. (i),

=

=  =  m

Therefore, Length of rectangle =  m and Width of rectangle =  m

And Radius of semi-circle =  m

### 12. A point on the hypotenuse of a triangle is at distances  and  from the sides of the triangle. Show that the maximum length of the hypotenuse is

Ans. Let P be a point on the hypotenuse AC of a right triangle ABC such that PL AB =  and PM BC =  and let BAC = MPC = , then in right angled

AP = PL =

And in right angled PMC,

PM = PM

Let AC =  then

= AP + PC = ……….(i)

Now

……….(ii)

And

[ and  is +ve as  )

is minimum when

=

Also

=

Putting these values in eq. (i),

Minimum length of hypotenuse =

=  =

### 13. Find the points at which the function  given by  has:

(i) local maxima

(ii) local minima

(iii) point of inflexion.

Ans. Given: ……….(i)

=

=

=

Now

= 0

or  or

or  or

Now, for values of  close to  and to the left of . Also for values of  close to  and to the right of .

Therefore,  is the point of local maxima.

Now, for values of  close to 2 and to the left of . Also for values of  close to 2 and to the right of .

Therefore,  is the point of local minima.

Now as the values of  varies through  does not change its sign. Therefore,  is the point of inflexion.

### 14. Find the absolute maximum and minimum values of the function  given by

Ans. Given:  ……….(i)

=  =

Now

= 0

or

or

[Turning points]

Now

= 0 + 1 = 1

1 + 0 = 1

= 1

Therefore, absolute maximum is  and absolute minimum is 1.

### 15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius  is

Ans. Let  be the radius of base of cone and  be the height of the cone inscribed in a sphere of radius

OD = AD – AO =

In right angled triangle OBD,

OD2 + BD2 = OB2

……….(i)

Volume of cone (V) =  = [From eq. (i)]

V =

and

Now

At

=  [Negative]

Volume is maximum at

### 16. Let  be a function defined on  such that  for all  Then prove that  is an increasing function on

Ans. Let I be the interval

Given:  for all  in an interval I. Let  I with

By Lagrange’s Mean Value Theorem, we have,

where

where

Now

……….(i)

Also,  for all  in an interval I

From eq. (i),

Thus, for every pair of points  I with

Therefore,  is strictly increasing in I.

### 17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is  Also find the maximum volume.

Ans. Let  be the radius and  be the height of the cylinder inscribed in a sphere having centre O and radius R.

In right triangle OAM,  AM2 + OM2 = OA2

……….(i)

Volume of cylinder (V) =  ……….(ii)

V =

=  ……….(iii)

and

Now

At

= [Negative]

V is maximum at

From eq. (iii),

Maximum value of cylinder =

=

=

### 18. Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height  and having semi-vertical angle  is one-third that of the cone and the greatest volume of the cylinder is

Ans. Let  be the radius of the right circular cone of height  Let the radius of the inscribed cylinder be  and height

In similar triangles APQ and ARC,

Volume of cylinder (V) = ……….(ii)

V =

= ……….(iii)

and

Now

At

= [Negative]

V is maximum at

From eq. (iii),

Maximum value of cylinder =

=

=

=

=

Choose the correct answer in the Exercises 19 to 24:

### 19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic meter per hour. Then the depth of wheat is increasing at the rate of:

(A) 1 m/h

(B) 0.1 m/h

(C) 1.1 m/h

(D) 0.5 m/h

Ans. Let  be the depth of the wheat in the cylindrical tank of radius 10 m at time

V = Volume of wheat in cylindrical tank at time  cu. m

It is given that  = 314 cu. m/hr

1 m/h

Therefore, option (A) is correct.

### 20. The slope of the tangent to the curve   at the point  is:

(A)

(B)

(C)

(D)

Ans. Equation of the curves are …..(i) and …..(ii)

and

Slope of the tangent to the given curve at point  =  …..(iii)

At the given point   and

At ,  from eq. (i),

At , from eq. (ii),

Here, common value of  in the two sets of values is

From eq. (iii),

Slope of the tangent to the given curve at point  =

Therefore, option (B) is correct.

### 21. The line  is a tangent to the curve  if the value of  is:

(A) 1

(B) 2

(C) 3

(D)

Ans. Equation of the curve is ……….(i)

Slope of the tangent to the given curve at point  =

……….(ii)

Now

…..(iii)

Putting the values of  and  in eq. (i),

Therefore, option (A) is correct.

### 22. The normal at the point (1, 1) on the curve  is:

(A)

(B)

(C)

(D)

Ans. Equation of the given curve is  ……….(i)

Slope of the tangent to the given curve at point (1, 1) =  (say)

Slope of the normal =

Equation of the normal at (1, 1) is

Therefore, option (B) is correct.

### 23. The normal to the curve  passing through (1, 2) is:

(A)

(B)

(C)

(D)

Ans. Equation of the curve is ………..(i)

Slope of the normal at  = ……….(ii)

Again slope of normal at given point (1, 2) = ……….(iii)

From eq. (ii) and (iii), we have

From eq. (i),

Now, at point (2, 1), slope of the normal from eq. (ii) =

Equation of the normal is

Therefore, option (A) is correct.

### NCERT Solutions class 12 Maths Miscellaneous

24. The points on the curve  where the normal to the curve make equal intercepts with axes are:

(A)

(B)

(C)

(D)

Ans. Equation of the curve is ……….(i)

Slope of the tangent to curve (i) at any point  =

Slope of the normal = negative reciprocal =

[ Slopes of lines making equal intercepts on the axes are ]

Taking positive sign,

……….(ii)

From eq. (i) and (ii), we have  and

Taking positive sign,

……….(ii)

From eq. (i) and (ii), we have  and

Required points are

Therefore, option (A) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

## CBSE App for Class 12

To download NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through the best app for CBSE students and myCBSEguide website.

### Test Generator

Create question paper PDF and online tests with your own name & logo in minutes.

### myCBSEguide

Question Bank, Mock Tests, Exam Papers, NCERT Solutions, Sample Papers, Notes