JEE  >  Maths 35 Years JEE Main & Advanced Past year Papers  >  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Document Description: Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The notes and questions for Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced have been prepared according to the JEE exam syllabus. Information about Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced covers topics like and Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Example, for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced.

Introduction of Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced in English is available as part of our Maths 35 Years JEE Main & Advanced Past year Papers for JEE & Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced in Hindi for Maths 35 Years JEE Main & Advanced Past year Papers course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
1 Crore+ students have signed up on EduRev. Have you?

Q. 33. Determine the value of  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       (1997 - 5 Marks)

Ans. π2

Solution. 

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                  .....(1)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Putting cos x = t, – sin x dx = dt

When x → 0,t →1 and when x → p,t → -1

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 34. Let f(x) = Maximum {x2, (1 – x)2, 2x(1 – x)}, where 0 < x < 1. Determine the area of the region bounded by the curves y = f(x), x-axis, x = 0 and x = 1.

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. We draw the graph of y = x2, y = (1– x)2 and y = 2x (1– x) in figure.

Let us find the point of intersection of y = x2 and y = 2x (1– x)

The x – coordinate of the point of intersection satisfies the equation x2 = 2x (1– x), ⇒ 3x2 = 2x ⇒  0 or x = 2 /3

∴ At  B, x = 2/3

Similarly, we find the x coordinate of the points of intersection of y = (1 – x)2 and y = 2x (1– x) are x = 1/3 and x = 1

∴ At A, x = 1/3 and at C x = 1

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
From the figure it is clear that

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The required area A is given by

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 35. Prove that Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEHence or otherwise, evaluate the integral  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. log 2

Solution.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE          ...... (1)
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 36. Let C1 and C2 be the graphs of the functions y = x2 and y = 2x, 0 < x < 1 respectively. Let C3 be the graph of a function y = f(x), 0 < x < 1, f(0) = 0. For a point P on C1, let the lines through P, parallel to the axes, meet C2 and C3 at Q and R respectively (see figure.) If for every position of P (on C1), the areas of the shaded regions OPQ and ORP are equal, determine the function f(x).

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. f (x) = x3-x2

Solution. f (x) = x3-x2

Let P be on C1, y = x2 be (t, t2)

∴ ordinate of Q is also t2.
Now Q lies on y = 2x, and y = t2

∴ x = t2/2

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

For point R, x = t and it is on y = f (x)

∴ R is [t, f (t)]

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE              ...(1)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equating (1) and (2), we get,

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Differentiating both sides,we get,

t2 - t3 =- f (t)

∴ f (t) = x3 - x2.


Q. 37. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. π/2

Solution.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 38. Let f(x) be a continuous function given by Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∵ f (x) is continuous at x = – 1 and x = 1

∴ (–1)2 + a (–1) + b = – 2 and 2 = (1)2 + a . 1 + b i.e. a – b = 3  and  a + b = 1
On solving we get a = 2, b = –1

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Given curves are y = f (x), x = – 2y2 and 8x + 1 = 0

Solving x = – 2 y2 , y = x2 + 2x –1 (x < –1) we get x = – 2

Also y = 2x,  x = – 2 y2 meet at (0, 0)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The required area is the shaded region in the figure.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Required area

NOTE THIS STEP :

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 39. For x > 0, let Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Find the function  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Here, lnt = loget.

Solution.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence Proved.


Q. 40. Let b ≠ 0 and for j = 0, 1, 2, …, n, let Sj be the area of the region bounded by the y-axis and the curve xeay = sin by, Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEShow th at S0, S1, S2, …, Sn are in geometric progression. Also, find their sum for a = -1 and b = π.

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that x = sin by. e-ay ⇒ – e–ay < x < e–ay

The figure is drawn taking a and b both +ve. The given curve oscillates between x = e–ay and x = – e–ay

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integrating by parts,  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 41. Find the area of the region bounded by the curves y = x2, y = |2 – x2| and y = 2, which lies to the right of the line x = 1.

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. The given curves are y = x2 which is an upward parabola with vertex at (0, 0)

  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE           .......(2)

a downward parabola with vertex at (0, 2)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE            ..........(3)

An upward parabola with vertex at (0, – 2) 

y = 2               .........(4)

A straight line parallel to x – axis

x = 1        ..........(5)

A straight line parallel to y – axis

The graph of these curves is as follows.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Required area = BCDEB

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 42. If f is an even function then prove that 

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that f (x)  is an even function, then to prove

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE           .......(2)

[As f is an even function] Adding two values of I in (1) and (2) we get

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let   x - π /4 = t so  that dx = dt

as x → 0, t → -π /4 and as x → π/4, t → π/2-π/4 = π/4

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

R.H.S. Hence proved.


Q. 43. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 

Solution. We have, 

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[∵ cos x  is independent of θ]

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (Using Leibnitz thm.)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 44. Find the value of Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The second integral becomes zero integrand being an odd function of x.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

{using the prop. of even function and also |x| = x  for 0 < x < π /3}

Let x + π /3 = y ⇒ dx = dy

also   as x → 0,y → π /3 as x → π /3 , y → 2π /3

∴ The given integral becomes

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 45. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans.  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= I1+I2

Now using the property that

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integrating by parts, we get

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 46. Find the area bounded by the curves x2 = y, x2 = –y and y2 = 4x – 3.

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. The given curves are, x2 = y .........(i)

x2 = – y .........(ii)

y2 = 4 x –3 .........(iii)

Clearly point of intersection of (i) and (ii) is (0, 0). For point of intersection of (i) and (iii), solving them as follows

x4 -4x+3 = 0 (x-1)(x3 + x2 +x-3) = 0

or ( x - 1)2 ( x2 + 2x + 3)= 0 ;   ⇒  x = 1 and then y = 1

∴ Req. point is ( 1, 1). Similarly point of intersection of (ii) and (iii) is (1, – 1). The graph of three curves is as follows:

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

We also observe that at x = 1 and y = 1

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE for (i) and (iii) is same and hence the two curves touch each other at (1, 1).

Same is the case with (ii) and (iii) at (1, –1).

Required area = Shaded region in figure  = 2 (Ar OPA)

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 47. f(x) is a differ en tiable function an d g(x) is a dou ble differentiable function such that |f(x)| < 1 and f '(x) = g(x). If f2(0) + g2(0) = 9. Prove that there exists some c∈ (-3, 3) such that g (c).g ''(c) < 0 .

Solution. Given that f (x) is a differentiable function such that f’(x) = g (x), then

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Similarly

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

First let us consider g (0) >  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let us suppose that g'' (x) be positive for all x ∈ (–3, 3).
Then g” (x) > 0 ⇒ the curve y = g (x) is open upwards.
Now one of the two situations are possible. (i) g(x) is increasing

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

a contradiction as  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ at least at one of the point c ∈ (–3,3), g'' (x) < 0.

But g (x) > 0 on (– 3, 3)

Hence g(x) g''(x) < 0 at some x ∈ (– 3, 3).

(ii) g (x) is decreasing

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

a contradiction as  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ at least at one of point c ∈ (-3, 3) g "(x) should be – ve. But g(x) > 0 on (–3 , 3).
Hence g (x) g'' (x) < 0 at some x ∈ (–3 , 3).
Secondly let us consider g (0) <  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let us suppose that g'' (x) be – ve on (– 3 , 3). then g'' (x) < 0 ⇒ the curve y = g(x) is open downward.
Again one of the two situations are possible (i) g (x) is decreasing then

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

a contradiction as  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ At least at one of the point c ∈ (– 3, 3), g'' (x) is + ve. But g (x) < 0 on (– 3, 3).

Hence g(x) g'' (x) < 0 for some x ∈ (– 3, 3).

(ii) g (x) is increasing then

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

a contradiction as as  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ At least at one of the point c ∈ (– 3, 3) g'' (x) is + ve.

But g (x) < 0 on ( –3, 3).

Hence g (x) g'' (x) < 0 for some x ∈ (– 3,3).

Combining all the cases, discussed aboe, we can conclude that at least at one point in (– 3, 3), g (x) g”(x) < 0.


Q. 48. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a quadratic function and its maximum value occurs at a point V. A is a point of intersection of y = f(x) with x-axis and point B is such that chord AB subtends a right angle at V. Find the area enclosed by f(x) and chord AB.

Ans. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 4a2 f (-1) + 4af(1) + f (2) =3a2+ 3a
4b2 f (-1) + 4bf (1) + f (2) = 3b2+3a
4c2 f (-1) + 4cf (1) + f (2) = 3c2+3c

Consider the equation

4 x2 f (-1) + 4 xf (1) + f (2) = 3x2+3x or 

[4 f (-1) - 3]x2 + [4 f (1) - 3]x +f (2)= 0

Then clearly this eqn. is satisfied by x  =  a,b,c

A quadratic eqn. satisfied by more than two values of x means it is an identity and hence

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let f (x) = px2 +qx +r [f (x) being a quadratic eqn.]

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solving the above we get  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

It’s maximum value occur at f’ (x) = 0 i.e., x = 0 then f (x) = 1, ∴ V ( 0, 1)

Let A (–2, 0) be the point where curve meet  x –axis.

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equation of chord AB is
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Required area is the area of shaded region given by

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 49.  Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
All you need of JEE at this link: JEE

Related Searches

pdf

,

video lectures

,

MCQs

,

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

Exam

,

Summary

,

Semester Notes

,

ppt

,

shortcuts and tricks

,

study material

,

Extra Questions

,

Important questions

,

mock tests for examination

,

Viva Questions

,

Free

,

Sample Paper

,

past year papers

,

Subjective Type Questions: Definite Integrals and Applications of Integrals - 3 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

,

Previous Year Questions with Solutions

,

Objective type Questions

,

practice quizzes

;