Airforce X Y / Indian Navy SSR Exam  >  Airforce X Y / Indian Navy SSR Notes  >  Mathematics for Airmen Group X  >  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR PDF Download

Q. 33. Determine the value of  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR       (1997 - 5 Marks)

Ans. π2

Solution. 

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR                  .....(1)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
Putting cos x = t, – sin x dx = dt

When x → 0,t →1 and when x → p,t → -1

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
From the figure it is clear that

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

The required area A is given by

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


Q. 35. Prove that JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRHence or otherwise, evaluate the integral  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans. log 2

Solution.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR          ...... (1)
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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).

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

∴ R is [t, f (t)]

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR              ...(1)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Differentiating both sides,we get,

t2 - t3 =- f (t)

∴ f (t) = x3 - x2.


Q. 37. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans. π/2

Solution.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Q. 38. Let f(x) be a continuous function given by JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solution.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∵ 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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

The required area is the shaded region in the figure.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ Required area

NOTE THIS STEP :

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Q. 39. For x > 0, let JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR Find the function  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Here, lnt = loget.

Solution.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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, JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRShow th at S0, S1, S2, …, Sn are in geometric progression. Also, find their sum for a = -1 and b = π.

Ans. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Integrating by parts,  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR           .......(2)

a downward parabola with vertex at (0, 2)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR            ..........(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.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ Required area = BCDEB

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR           .......(2)

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

R.H.S. Hence proved.


Q. 43. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans. 

Solution. We have, 

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

[∵ cos x  is independent of θ]

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR (Using Leibnitz thm.)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


Q. 44. Find the value of JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solution.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

{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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


Q. 45. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Ans.  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solution. Let

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

= I1+I2

Now using the property that

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
Integrating by parts, we get

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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

Ans. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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:

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR 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)

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Similarly

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

First let us consider g (0) >  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

a contradiction as  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ 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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

a contradiction as  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ 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) <  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

a contradiction as  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ 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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

a contradiction as as  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

∴ 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. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR 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. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solution. JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

⇒ 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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solving the above we get  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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.

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Equation of chord AB is
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Required area is the area of shaded region given by

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSRJEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR


Q. 49.  JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

Solution. 

JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
JEE Advanced (Subjective Type Questions): Definite Integrals & Applications of Integrals - 3 | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

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