JEE Advanced (One or More Correct Option): Definite Integrals & Application of Integrals | Chapter-wise Tests for JEE Main & Advanced PDF Download

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Q.1. Area enclosed between the curves
| y | = 1 – x² and x² + y² = 1 is
(a) enclosed area is symmetric with respect to coordinate axis
(b) π –8/3 sq. unit
(c) 2π –8/3 sq. unit
(d) enclosed area is symmetric about y = 2 line

Correct Answer is option (a, b)
Graph

Q.2. Area enclosed between the curves y² = x and x² = y is equal to -
(a)
(b) 1/3
(c) area of region {(x, y) : x² ≤y ≤ |x|}
(d) 2/3

Correct Answer is option (a, b, c)
 

Q.3. The area bounded by curves y = x³ ; y = (x)^1/3
(a) lies in first quadrant
(b) lies is third quadrant
(c) is equal to + 1 unit² 
(d) is equal to 4 unit²

Correct Answer is option (a, b, c)
y = x³, y = x^1/3 is mirror image about y = x line

Q.4. Area bounded by y = x – x² and line y = mx is equal to 9/2. Then possible values of ‘m’ are
(a) – 4 
(b) – 2
(c) 2 
(d) 4

Correct Answer is option (b, d)

Q.5. Let   If the ratio in which x-axis divides the area bounded by x-axis and the curve y = (x –1) (x – 2) (x – 3) in two parts is a/b then -
(a) a = |A₁|
(b) b = |A₂|
(c) a = A₁ 
(d) b = A₂

Correct Answer is option (a, b, c)
Since y > 0 for x ∈ (1, 2) & y < 0 for x ∈ (2, 3)

So A₁ > 0 A₂ < 0 ⇒So a = A₁ = |A₁| & b = |A₂|

Q.6. If A_i is the area bounded by |x –a_i| + |y| = b_i, i ∈ N, where a_i+1 = a_i + 3/2b_i and b_i+1 = b_i/2, a₁ = 0, b₁= 32, then:
(a) A₃ = 128
(b)A₃ = 256
(c)

(d)

Correct Answer is option (a, c)
a₁ = 0, b₁ = 32, a₂ = a₁ + 3/2b₁ = 48, b₂ = b₁/2 = 16
a₃ = 48 + 3/2 x 16 = 72, b₃ = 16/2 = 8

So the three loops from i = 1 to i = 3 are alike

Now area of i^th loop (square) =1/2(diagonal)²

So the area form a G.P. series
So, the sum of the G.P. upto infinite terms

Q.7. The curve y = a√x +bx passes through the point (1,2)and the area enclosed by the curve, the x-axis and the line x = 4 is 8 sq. units. Then
(a) a + 2b = 1
(b) 2a + b = 1
(c) a = -1, b = 3

(d) a = 3, b = -1

Correct Answer is option (a, d)
a + b = 2 and  implies 2a + 3b = 3. 

Solving a = 3 and b = -1

Q.8. A complex number z satisfies the equation | z² - 9 | + | z² |= 41 then the true statements, among the following are
(a) |z +3|+| z-3|=10
(b) |z +3|+| z-3|=8
(c) Maximum value |z| is 5
(d) Maximum value |z| is 6

Correct Answer is option (a, c)
|z² - 9|+| z²| =41 ⇒ |z +3|+| z-3|=10

Q.9. If A₁, A₂, A₃ be the area of the regions as shown in figure given below, then which of the following options is/are true?
(a) A₂ = (e^e - 2) SQ. unit
(b) A₂: A₃ = (e^e - 2) SQ. unit 
(c) A₁ - A₂ = (e^e - 2) SQ. unit
(d) A₂ > A₁ > A₃ 

Correct Answer is option (a, b, c)

So,


Clearly
A₁ > A₂ > A₃

Q.10. Area bounded by curve y² = x and x = 4 is divided into 4 equal parts by the lines x = a and y = b then.
(a) Area of each part = 8/3
(b) b = 0
(c) a= √2
(d) a = (16)^1/3

Correct Answer is option (a, b, c)
Total area =  

Area of each part = 8/3