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JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

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Q.1. The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x, is    (2020)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Ans. (4)
The given equation of circle is
x2 + y2 = 2....    (1)
Now, total area enclosed by the circle is πr2 = π(√2)2 = 2π sq. units
The area bounded by the parabola y= x and y = x is
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence, the required area of the region = JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEsq. units

Q.2. The area (in sq. units) of the region JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis    (2020)
(1) 125/3
(2) 128/3
(3) 124/3
(4) 127/3

Ans. (2)
The given curves are JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE 
Now, on solving y = 4x2 and y = 8x + 12, we get
4x2 - 8x - 12 = 0 ⇒ x2 - 2x - 3 = 0
x = -1, 3 and y = 4, 36
Therefore
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence, the area of shaded region is
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.3. For a > 0, let the curves C1 : y2 = ax and C: x2 = ay intersect at origin O and at Point P. Let the line x = b (0 < b < a) intersect the chord OP and the x axis at points Q and R respectively. If the line x = b bisects the area bounded by the curves, C1 and C2, and the area of JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEthen ‘a’ satisfies the equation:    (2020)
(1) x6 - 6x3 + 4 = 0
(2) x6 - 12x3 + 4 = 0
(3) x6 + 6x3 - 4 = 0
(4) x6 - 12x3 - 4 = 0

Ans.(2)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE    (1)
Now, the area of triangle PQR = 1/2
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Substituting the value of b in Eq. (1), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE


Q.4. The equations of given curves are JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE    (2020)
(1) 32/3
(2) 34/3
(3) 29/3
(4) 31/3
Ans. (1)
The equations of given curves are
x2 < y...    (1)
2x + y < 3...    (2)
On solving y = x2 = and 2x + y = 3, we get the intersection point x = − 1, 3.
Now,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence, the area of shaded region is
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE


Q.5. Given: JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE Then, the area (in sq. units) of the region bounded by the curves, y = f(x) and y = g(x) between the lines 2x = 1 and 2x =√3 is    (2020)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (2)
We have,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Required area, A = Area of trapezium ABCD JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE sq. units

Q.6. The value of α for which JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis    (2020)
(1) loge2
(2)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Ans. (1)
We have,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Thus, JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.7. If JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEthen    (2020)
(1)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Ans. (2)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Therefore, f (x) is decreasing in (1,2).
Now,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence, JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.8. If for all real triplets (a, b, c), f(x) = a + bx + cx2 ; then JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis equal to    (2020)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Ans. (4)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE....    (1)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Now, JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.9. Let z be a complex number such that JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE and JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEThen, the value ofJEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis    (2020)
(1) √10
(2) 7/2
(3) 15/4
(4) 2√3
Ans. (2)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
and JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Now, JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.10. If f(a + b + 1 - x) = f(x),  for all x, where a and b are fixed positive real numbers, then JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis equal to    (2020)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. 
Given,
f(x) = f(a + b + 1 - x)...(1)
f(x + 1) = f (a + b - x)...(2)
Now,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE....(3)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE ....(4)
From Eqs. (2) and (3), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Let x + 1 = t ⇒ dx = dt, so JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.11. The value of JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis equal to    (2020)
(1) 2π

(2) 2π2
(3) π2
(4) 4π

Ans. (3)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE....    (1)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE    (2)
From Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE....    (3)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE    (4)
From Eqs. (3) and (4), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.12. If JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEwhere c is a constant of integration, then JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEis equal to    (2020)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) 2
(3) 9/8
(4) -2
Ans. (4)
Given,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE... (1)
Now,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE...(2)
From Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.13. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE     (2019)
(1) 0
(2) 4/3
(3) 2/3
(4) -4/3

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.14. Let f be a differentiable function from R to R such that |f(x) - f(y)|≤ 2|x - y|3/2, for all x, y, ∈ R. If f(0) = 1 then JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE is equal to:     (2019)
(1) 1 
(2) 2
(3) 1/2 
(4) 0
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
  f(x) is a constant function.
Given f(0) = 1     f(x) = 1
Hence, the integral
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.15. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE then the value of k is:     (2019)
(1) 4
(2) 1/2
(3) 1
(4) 2
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Hence, integral becomes.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
∴ k = 2

Q.16.JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE then f'(1/2) is:      (2019)
(1) 24/25
(2) 18/25
(3) 4/5
(4) 6/25
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Then
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.17. Given: JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE and g (x) = JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE Then, the area (in sq. units) of the region bounded by the curves, y = f (x) and y = g (x) between the lines 2x = 1 and 2x = √3 is
(1)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (2)
We have
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Required area, A = Area of trapezium ABCDJEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEsq. units

Q.18. The value of JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE where [t] denotes the greatest integer less than or equal to t, is:     (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.19. The value of the integral JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(where [x] denotes the greatest integer less than or equal to x) is:       (2019)
(1) 0
(2) sin 4
(3) 4
(4) 4 -sin 4

Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE     JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
⇒ f(x) is odd function
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.20.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE        (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Let tan5x = t
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.21. Let f and g be continuous functions on [0, a] such that f(x) =f(a - x) and g(x) + g(a - x) = 4, JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE equal to:      (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Let, the integral,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.22. The integral JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE is equal to:      (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Let JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
 x (ln x - 1) = ln t
On differentiating both sides w.r.t x we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
When x = e then t = 1 and when x = 1 then t = 1/e.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.23.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE     (2019)
(1) π/4
(2) tan-1(3)
(3) π/2
(4) tan-1(2)
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.24.JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE then the value of the 

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE       (2019)
(1) loge3
(2) logee
(3) loge2
(4) loge1

Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Then, equation (1) becomes,
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.24. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE , where g is a non-zero even function. If 

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE        (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (1)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.25. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE        (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (2)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Adding equation (1) and (2), we get
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.26. The value of the integral JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE x ∈ [0,1]   (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.27. The value of JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE where [t] denotes the greatest integer function, is:        (2019)
(1) π  
(2) -π 
(3) -2π
(4) 2π

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
From (1) + (2), we get;
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.28. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE        (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.29.JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE is equal to:       (2019)
(1) 35/6 - 32/3
(2) 34/3 - 31/3
(3) 37/6 - 35/6
(4) 35/3 - 31/3
Ans. 
(3)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.30. JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE then m.n is equal to:      (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) 2
(3) 1/2
(4) -1
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.31. A value of α such thatJEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE       (2019)
(1) -2
(2) 1/2
(3) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4) 2
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.32. The area (in sq. units) bounded by the parabola y = x2 - 1, the tangent at the point (2, 3) to it and the y-axis is:       (2019)
(1) 8/3
(2) 32/3
(3) 56/3
(4) 14/3
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
equation of tangent at (2, 3)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Here the curve cuts Y-axis
∴ required area
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.33. The area of the region A = {(x, y): 0 ≤ y ≤ x |x| + 1 and - 1 ≤ x ≤ 1} in sq. units is:        (2019)
(1) 2/3
(2) 2
(3) 4/3
(4) 1/3

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
∴ Area of shaded region
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.34. If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is:       (2019)
(1) √3/2
(2) 1/√3
(3) √3
(4) 2/√3
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Two curves will intersect in the 1st quadrant at JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
∵ area of shaded region = 1.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.35. The area (in sq. units) of the region bounded by the curve x2 = 4 y and the straight line x = 4y - 2 is:        (2019)
(1) 5/4
(2) 9/8
(3) 7/8
(4) 3/4
Ans. 
(2)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Let points of intersection of the curve and the line be P and Q
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
x2 - x - 2 = 0
x = 2, - 1
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.36. The area (in sq. units) in the first quadrant bounded by the parabola,y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is:        (2019)
(1) 8/3
(2) 37/24
(3) 187/24
(4) 14/3
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
The equation of parabola x2 = y - 1
The equation of tangent at (2, 5) to parabola is
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
y - 5 = 4(x - 2)
4x - y = 3
Then, the required area
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.37. The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is:      (2019)
(1) 15/4
(2) 21/2
(3) 17/4
(4) 15/2
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.38. The area (in sq. units) of the region
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE     (2019)
(1) 53/6
(2) 8
(3) 59/6
(4) 26/3

Ans. (3)
Solution.
Since, the relation y ≤ x2 + 3x represents the region below the parabola in the 1st quadrant
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
∵ y = 4
⇒ x2 + 3x = 4 ⇒ x = 1, -4
the required area = area of shaded region
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.39. Let S(α) = {{x, y): y2 ≤ x, 0 ≤ x ≤ α} and A(α) is area of the region S(α). If for a λ, 0 < λ < 4, A(λ) : A(4) = 2 : 5, then λ equals:     (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.40. The area (in sq. units) of the region A = {(x, y): x2 ≤ y ≤ x + 2} is:     (2019)
(1) 10/3
(2) 9/2
(3) 31/6
(4) 13/6
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Required area is equal to the area under the curves y ≥ x2 and y ≤ x + 2
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.41. The area (in sq. units) of the region JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE      (2019)
(1) 53/3
(2) 30
(3) 16
(4) 18

Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.42. The region represented by |x - y| ≤ 2 and |x + y|≤ 2 is bounded by a:     (2019)
(1) square of side length 2√2 units
(2) rhombus of side length 2 units
(3) square of area 16 sq. units
(4) rhombus of area 8√2 sq. units
Ans.
(1)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
By the diagram, region is square
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.43. The area (in sq. units) of the region bounded by the curves y = 2x and y = |x +1|, in the first quadrant is:     (2019)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2) 3/2
(3) 1/2

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.44. If the area (in sq. units) of the region {(x, y) : y2 ≤ 4x, x + y ≤ 1, x ≥ 0, y ≥ 0} is a √2 + b, then a - b is equal to:     (2019)
(1) 10/3
(2) 6
(3) 8/3
(4) -2/3
Ans. (2)
Solution. 
Consider y2 = 4x and x + y = 1
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Substituting x = 1 - y in the equation of parabola,
y2 = 4(1 - y) ⇒ y2 + 4y - 4 = 0
⇒ (y + 2)2 = 8 ⇒ y + 2 = ±2√2
Hence, required area
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.45. If the area (in sq. units) bounded by the parabola y2 = 4λx and the line y = λx, λ > 0, is 1/9, then λ is equal to:       (2019)
(1) 2√6
(2) 48
(3) 24
(4) 4√3
Ans. (3)
Solution.
Given parabola y2 = 4λx and the line y = λx
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.46. The value of JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEEdx is:    (2018)
(1) π/8
(2) π/2
(3) 4π
(4) π/4
Ans. 
(4)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.47. Let g(x) = cos x2 ,f(x) = √x, and α ,β (α < β) be the roots of the quadratic equation 18x2 - 9πx +π2 = 0. Then the area (in sq. units) bounded by the curve  y = (gof)(x) and the lines x =α , x = β and y = 0 , is:    (2018)
(1)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (1)
Solution.

g(x) = cos x2
f(x) = √x
g(f (x)) = cos x
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.48. The value of the integral    
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE      (2018)
(1) 3/4
(2) 0
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
I = 3π/16

Q.49. The area (in sq. units) of the region {x ϵ R ∶ x ≥ 0, y ≥ x − 2 and y ≥ √x} is∶    (2018)
(1) 8/3
(2) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. 
(2)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.50. If the area of the region bounded by the curves, y = x2, y = 1/x and the lines y = 0 and x = t (t > 1) is 1 sq. unit, then t is equal to:    (2018)
(1) e2/3
(2) e3/2
(3) 3/2
(4) 4/3
Ans. 
(1)
Solution.

The intersection point of y = x2 and y = 1/x is (1, 1)
Area bounded by the curves is the region ABCDA is given as:
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.51. The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x2 ≤ 4y and y ≤ 1 + √x} is     (2017)
(1) 5/2
(2) 59/12
(3) 3/2
(4) 7/3
Ans. 
(1)
Solution:

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Area of shaded region
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
= 5/2 sq. unit

Q.52. The integral JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE is equal to    (2017)
(1) –1
(2) –2
(3) 2
(4) 4
Ans. 
(3)
Solution:

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.53. 
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE      (2017)
(1) 13/256
(2) 15/64
(3) 13/32
(4) 15/128
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.54. The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y= 4 and y2 = 3x, is:    (2017)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE


Q.55.JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE    (2017)
(1) 4
(2) 2
(3) 3
(4) 1
Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.56. The area (in sq. units) of the region {(x, y) : y2 ≥ 2x and x2 + y2 ≤ 4x, x ≥ 0, y ≥ 0} is:     (2016)
(1) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. 
(2)
Solution.

x2 + y2 ≤ 4x  &  y2 ≥ 2x
To find point of intersection,
x2 + y2 = 4x ⇒ x2 + 2x = 4x
⇒ x2 - 2x = 0 ⇒ x (x - 2) = 0
⇒ x = 0  or  x = 2
∴ y = 0 or y = 2
Solve (x, y) = (0, 0) & (x, y) = (2, 2)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Area = JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.57.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE     (2016)
(1) log 2
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
(3) log 4
(4) JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.58. The area (in sq. units) of the region described by A = {(x, y) | y ≥ x2 - 5x + 4, x + y ≥ 1, y ≤ 0} is:    (2016)
(1) 7/2
(2) 13/6
(3) 17/6
(4) 19/6

Ans. (4)
Solution.
A = {(x,y)|y x2 - 5x + 4, x + y 1, y 0}
Here y x2 - 5x + 4, x + y 1 , y 0
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.59. For x ∈ R, x ≠ 0, if y(x) is a differentiable function such that JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE equals     (2016)
(where C is a constant)
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
Ans. 
(4)
Solution.

JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

Q.60. The value of the integral JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE  where [x] denotes the greatest integer less than or equal to x, is     (2016)
(1) 3
(2) 7
(3) 6
(4) 1/3
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Definite Integrals and Applications of Integrals Notes | Study Mathematics For JEE - JEE

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