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QUESTION: 1

The area bounded by [x] +[y] = 8 such that x, y __>__ 0 is .... sq. units

Where [.] is G.I.F.

Solution:

Area of 9 unit squares = 9 sq.units

QUESTION: 2

Let f : [0, ∞)→ R be a continuous and strictly increasing function such that The area enclosed by y = f (x), the x-axis and the ordinate at x = 3, is

Solution:

Given, f^{3}(x) = ∫(0 to x)tf(t)dt

Differentiating throughout w.r.t x,

3f^{2}(x) f′(x) = xf^{2}(x)

⇒ f′(x)= x/3

So, f(x)=∫(0 to x) x/3 dx = (x^{2})/6

Area = ∫f(x)dx=∫(0 to 3) (x^{2})/6 dx

= {(x^{3})/18}(0 to 3)

= 3/2

QUESTION: 3

The area bounded by x = x_{1}, y = y_{1} and y = -(x + 1)^{2} where x_{1}, y_{1} are the values of x, y satisfying the equation sin^{-1}x + sin^{-1}y = -π will be, (nearer to origin)

Solution:

QUESTION: 4

Area bounded between the curves

Solution:

QUESTION: 5

and be two functions and let f_{1}(x) = max {f(t), 0 __<__ t __<__ x, 0 __<__ x __<__} and g_{1}(x) = min {g(t), 0 __<__ t __<__ x, 0 __<__ x __<__ 1}. Then the area bounded by f_{1}(x) __<__ 0, g_{1}(x) __<__ 0 and x-axis is

Solution:

QUESTION: 6

The values of the parameter a(a __>__ 1) for which the area of the figure bounded by the pair of straight lines y^{2} – 3y + 2 = 0 and the curves is greatest is.

(Here [.] denotes the greatest integer function).

Solution:

The curves represent parabolas which are symmetric about yaxis. The equation y^{2} – 3y + 2 = 0 gives a pair of straight lines y = 1, y = 2 which are parallel to x-axis. The shaded region in figure determines the area bounded by the two parabolas and two lines. Let us slice this region into horizontal strips. For the approximating rectangle shown in figure. We have Length = x_{2} – x_{1}, width = Δy, Area = (x_{2} - x_{1})dy

AT it can move vertically y = 1 and y = 2. So, Required area

QUESTION: 7

Area of region bounded by x^{2} + y^{2} __<__ 4 and (|x| + |y|) __<__ 2 is ____ square units.

Solution:

Using circles representation, required area = 8 sq. units

QUESTION: 8

The area of a circle is A_{1} and the area of a regular pentagon inscribed in the circle is A_{2} .Then A_{1} : A_{2} is

Solution:

QUESTION: 9

The area bounded by the curve and x-axis is

Solution:

QUESTION: 10

the area of the region bounded by y = x and y = x+ sin x is

Solution:

QUESTION: 11

The area bounded by the curves y = x^{2}, y = [x+1], x __<__ 1 and the y - axis, where[.] denotes the greatest integer not exceeding x, is

Solution:

QUESTION: 12

The area of the smaller region in which the curve denotes the greatest integer function, divides the circle (x – 2)^{2} + (y + 1)^{2} = 4, is equal to

Solution:

Circle has (2, -1) as its centre and radius of this circle is 2.

Thus if P(x, y) be any point on it, then x ∈ [0, 4].

The g(x) is increasing in [0, 4]

QUESTION: 13

The area bounded by the curves y = 2- |x -1| , y = sin x ; x = 0 and x =2 is

Solution:

QUESTION: 14

A curve passes through the point (0,1)and has the property that the slope of the curve at every point P is twice the y–coordinate of P . If the area bounded by the curve, the axes of coordinates and the line

Solution:

QUESTION: 15

The area bounded by the curve (y – arc sinx)^{2} = x – x^{2}, is

Solution:

QUESTION: 16

The area of the region enclosed by the curve 5x^{2} + 6xy + 2y^{2} + 7x + 6y + 6 = 0 is

Solution:

Comparing the given equation with the general equation of second degree i.e.

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0,

We have, abc + 2fgh – af^{2} – bg^{2} – ch^{2} = 60 – 45

And h^{2} – ab = 9 – 10 < 0.

So, the given equation represents an ellipse.

Rewriting the given equation as a quadratic in y, we obtain

Clearly, the values of y are real for x ∈ [1, 3]

When x = 1, we get y = -3 and x = 3 ⇒ y = -6.

QUESTION: 17

Let f(x) be a continuous function such that the area bounded by the curve y = f(x), the xaxis and the two ordinates x = 0 and

Solution:

Differentiating both sides w.r.t. a, we get

QUESTION: 18

The area enclosed between the curves y = sin^{4}x cos^{3}x, y = sin^{2}xcos^{3}x between x = 0 and

Solution:

Required area is equal to

QUESTION: 19

The area of the region bounded by the curves which contains (1, 0) point in its interior is

Solution:

We observe that all powers of x in the above equation are even, so it is symmetric about y-axis. The curve intersects y-axis at (0, 1). Also,

Therefore, as x → ∞, y → -1 i.e. y = -1 is an asymptote to the curve.

This means that x is imaginary for y > 1 or y < -1 i.e. the curve lies between y = -1 and y = 1. At (0, 1) the tangent to the curve

QUESTION: 20

Area bounded by the curves y = e^{x} , y= log_{e} x and the lines x = 0, y = 0, y = 1 is

Solution:

Area = Area of rectangle

QUESTION: 21

A circular arc of radius ‘1’ subtends an angle of ‘x’ radians, as shown in the figure. The point ‘R’ is the point of intersection of the two tangent lines at P & Q. Let T(x) be the area of triangle PQR and S(x) be area of the shaded region. Then

Solution:

QUESTION: 22

A point P moves inside a triangle formed by such that min {PA, PB, PC} = 1. The area formed by the curve traced by P is ....... sq. units

Solution:

Re quired area = Area of equilateral

QUESTION: 23

Area of the triangle formed by the tangent and normal at (1, 1) on the curve and the y-axis is

Solution:

Find equation of the tangent and normal and then put x=0 to evaluate vertices of triangle.

Then find area of triangle .

QUESTION: 24

Area bounded by the curve and area bounded by latus rectum of

Solution:

Area bounded by C_{1} & C_{2} is λ_{1} = 16/3* *sq.units

λ_{2} = Area bounded by C_{1} & its latus rectum = 8/3 sq.units

QUESTION: 25

The area of the part of circle x^{2} + y^{2} - 2x - 4y -1 = 0 above 2x - y = 0 is ...... sq. units

Solution:

Required area = Area of semi-circle

Since given line is diameter of circle

QUESTION: 26

The area bounded by the curves y = |x| – 1 and y – |x| 1is

Solution:

QUESTION: 27

Area of closed curve 3(x -1)^{2} + 4(y^{2} - 3) = 0 is where [.] is G.I.F

Solution:

QUESTION: 28

Let f(x) = x + sin x. The area bounded by y = f^{-1} (x), y = x, x ∈ [0, π] is

Solution:

The curves given by y = x + sin x and y = f^{-1}(x) are images of each other in the line y = x.

Hence required area

QUESTION: 29

The area of the region bounded by the curves |x + y| __<__ 2, |x – y| __<__ 2 and 2x^{2} + 6y^{2} __>__ 3 is

Solution:

2x^{2} + 6y^{2} __>__ 3. . . . . . (1) area of ellipse

QUESTION: 30

Area enclosed by the closed curve 5x^{2} + 6xy + 2y^{2} + 7x + 6y + 6 = 0 is

Solution:

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