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JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

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Q.1. Let JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 
let JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE be the function defined by JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Then, which of the following statements is/are TRUE ?         (JEE Advance 2022)
(a) The minimum value of g(x) isJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) The maximum value of g(x) is JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) The function g(x) attains its maximum at more than one point
(d) The function g(x) attains its minimum at more than one point

Ans. (a,b,c)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
And, derivative changes sign from negative to positive at x = 1/2, hence x = 1/2 is point of local minimum as well as absolute minimum of g(x) for JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, minimum value of JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Maximum value of g(x)  is either equal to g(0) or g(1).
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence (B) and (C) are also correct


Q. 2. If the tangent to the curve y=x3−x2+x at the point (a,b) is also tangent to the curve y=5x2+2x−25 at the point (2, −1), then |2a+9b| is equal to __________.     (JEE Main 2022)

Ans. 195
Slope of tangent to curve y=5x2+2x−25
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

∴ Equation of tangent :y+1 = 22(x−2)

∴ y=22x−45.

Slope of tangent to y=x3−x2+x at point (a,b)

=3a2−2a+1

3a2−2a+1=22

3a2−2a−21=0

∴ a=3 or −73

Also b=a3−a2+a

Then (a,b)=(3,21) or (−7/3,−151/9).

(−7/3,−151/9) does not satisfy the equation of tangent

∴a=3,b=21

∴|2a+9b|=195


Q.3. For the curve C:(x2+y2−3)+(x2−y2−1)5=0, the value of 3y′−y3y′′, at the point (α,α), α>0, on C, is equal to ____________.    (JEE Main 2022)

Ans. 16

∵ C:(x2+y2−3)+(x2−y2−1)5=0 for point (α, α)

α22−3+(α2−α2−1)5=0

∴ α=2

On differentiating (x2+y2−3)+(x2−y2−1)5=0 we get

x+yy′+5(x2−y2−1)4(x−yy′)=0 ...... (i)

When x=y=√2 then y′=3/2

Again on differentiating eq. (i) we get :

1+(y′)2+yy″+20(x2−y2−1)(2x−2yy′)(x−y′y)+5(x2−y2−1)4(1−y′2−yy″)=0

For x=y=√2 and y′=3/2 we get JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.4. A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is tan-1 3/4. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ______________.    (JEE Main 2022)

Ans. 5
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.5. Let M and N be the number of points on the curve y5 - 9xy + 2x = 0, where the tangents to the curve are parallel to -axis and -axis, respectively. Then the value of  M + N equals ___________.    (JEE Main 2022)

Ans. 2
Here equation of curve is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
On differentiating :
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
When tangents are parallel to x-axis then 9y - 2 = 0
∴ M=1.

For tangent perpendicular to x-axis

5y4−9x=0 ...... (ii)

From equation (i) and (ii) we get only one point.

∴ N=1.

∴ M+N=2.


Q.6. Let the function f(x)=2x2−loge⁡x, x>0, be decreasing in (0,a) and increasing in (a,4). A tangent to the parabola y2=4ax at a point P on it passes through the point (8a, 8a−1) but does not pass through the point (−1a,0). If the equation of the normal at P is :JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE, then α+β is equal to ________________.    (JEE Main 2022)

Ans. 45
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE so f(x) is decreasing in (0,1/2) and increasing in
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
It is passing through (4,3)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
ButJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEpasses through (-2,0) so rejected.
Equation of normal
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

 

Q.7. A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4th day is 30, then number of infected students on 8th day will be __________.   (JEE Main 2022)

Ans. 90

Total students = 100

At t = 0 (zero day), infected student = 2

Let at t = t day infected student = x

 At t = t day non infected student = (100  x)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Putting value of c in equation (1), we get
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now, when t = 8, then r = ?
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.8. Let f(x)=|(x−1)(x2−2x−3)|+x−3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ____________.    (JEE Main 2022)

Ans. 3
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
So, 3 points.


Q.9. Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to ___________.    (JEE Main 2022)

Ans. 13

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.10. Let  JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE Then which of the following statements are true?    (JEE Main 2022)

P : x=0 is a point of local minima of f

Q : x=√2 is a point of inflection of f

R : f′ is increasing for x>√2
(a) Only P and Q  
(b) Only P and R  
(c) Only Q and R  
(d) All P, Q and R

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.11. If the minimum value of JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is 14 , then the value of α is equal to:    (JEE Main 2022)
(a) 32
(b) 64
(c) 128
(d) 256

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.12. Let JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Then the set of all values of b, for which f(x) has maximum value at x = 1, is:    (JEE Main 2022)
(a) (-6,-2)
(b) (2,6)
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. c

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
If  f(x) has maximum value at x =1 then JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Intersection of above two sets
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.13. The curve y(x)=ax3+bx2+cx+5 touches the x-axis at the point P(−2,0) and cuts the y-axis at the point Q, where y′ is equal to 3 . Then the local maximum value of y(x) is:    (JEE Main 2022)
(a) 
27/4
(b) 
29/4
(c) 
37/4
(d) 
9/2

Ans. a


Q.14. Let f(x)=4x3−11x2+8x−5, x ∈ R. Then f:    (JEE Main 2022)
(a) has a local minina at x=1/2
(b) has a local minima at x=3/4
(c) is increasing in (1/2,3/4)
(d) is decreasing in (1/2,4/3)

Ans. 4
Given,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEf′(x) is positive before 1/2 means slope of f(x) is positive before 1/2 and f'(x) is negative after 1/2 means slope of f(x) is negative after 1/2.

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

∴ At 4/3,f(x) is local minimum

From wavy curve method,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.15. If xy4 attains maximum value at the point (x, y) on the line passing through the points (50+, 0) and (0, 50 + α), α > 0, then (x, y) also lies on the line:    (JEE Main 2022)
(a) y= 4x
(b) x = 4y
(c) y = 4x+ α
(d) x = 4y - α

Ans. a
Equation of line passing through the point (50 + α, 0) and (0, 50 +α ) is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
For maximum or minimum value of p,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
or
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.16. Let λx−2y=μ be a tangent to the hyperbola a2x2−y2=b2. Then JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is equal to:    (JEE Main 2022)
(a) -2
(b) -4
(c) 2
(d) 4

Ans. d


Q. 17. If the tangent at the point (x1, y1) on the curve y=x3+3x2+5 passes through the origin, then (x1, y1) does NOT lie on the curve :    (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Given curve,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Slope of tangent,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Slope of line joined by (x1, y1) and (0, 0) is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Point (x1, y1) is on the line,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

x = 1 satisfy the equation

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ (x1, y1) = (1, 9)

Option D does not satisfy point (1, 9)


Q.18. The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is:     (JEE Main 2022)
(a) 9
(b) 10
(c) 11
(d) 12

Ans. a

We know,

Surface area of balloon (s) = 4πr2

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Given that, surface area of balloon is increasing in constant rate.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Given at t = 0, radius r = 3
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ Equation (1) becomes
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Also given, at t = 5, radius r = 7
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ Equation (1) is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.19. Let λ be the largest value of λ for which the function fλ(x)=4λx3−36λx2+36x+48 is increasing for all x ∈ R. Then fλ∗(1)+fλ∗(−1) is equal to:    (JEE Main 2022)
(a) 36
(b) 48
(c) 64
(d) 72

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.20. The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE If the curve passes through the point (1, 1), then e . y(e) is equal to    (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) tan(1)
(c) 1
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

∵ y(1)=1⇒tan−1(xy)=ln ⁡x + tan−1(1)

Put x=e and y=y(e) we get

tan−1(e.y(e)) = 1+tan−11

tan−1(e.y(e))−tan−11=1

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.21. If the angle made by the tangent at the point (x0, y0) on the curve x=12(t+sin⁡t cos⁡t), y=12(1+sin⁡t)2, 0<t < π/2, with the positive x-axis is π/3, then y0 is equal to:    (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) 27
(d) 48

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.22. Water is being filled at the rate of 1 cm3/ sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm2/ sec) at which the wet conical surface area of the vessel increases is    (JEE Main 2022)
(a) 5
(b) √21/5
(c) √26/5
(d) √26/10

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.23. Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is:    (JEE Main 2022)
(a) 2:5
(b) 19:45
(c) 3:8
(d) 19:15

Ans. b
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.24. Let S be the set of all the natural numbers, for which the line JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is a tangent to the curve JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE at the point (a, b), ab ≠ 0. Then:     (JEE Main 2022)
(a) S = ϕ
(b) n(S) = 1
(c) S = {2k : k ∈ N}
(d) S = N

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Differentiating both sides with respect to x, we get
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Equation of tangent at (a, b) is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ It is tangent for all value of n.


Q.25. The sum of the absolute minimum and the absolute maximum values of the

function f(x) = |3x - x2 + 2|  x in the interval [1, 2] is:     (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) 5
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. a


Q.26. Let JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is:     (JEE Main 2022)
(a) 10
(b) 8
(c) 7

(d) 5

Ans. b
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Above equation is circle
⇒  a =  -c and b = 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Passes through (2, 5)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.27. A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is:     (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. b
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.28. The number of real roots of the equation
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE    (JEE Main 2021)
(a) 2
(b) 4
(c) 1
(d) 0

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Number of real roots = 1


Q.29. A box open from top is made from a rectangular sheet of dimension a  b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to:        (JEE Main 2021)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
V = l . b . h = (a − 2x)(b − 2x) x
⇒ V(x) = (2x − a)(2x − b) x
⇒ V(x) = 4x3 − 2(a + b)x2 + abx
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.30. A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:        (JEE Main 2021)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. d
Let the wire is cut into two pieces of length x and 20 - x.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Length of side of regular HexagonJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.31. Let 'a' be a real number such that the function f(x) = ax2 + 6x − 15, x ∈ R is increasing in (−∞,3/4) and decreasing in (3/4,∞). Then the function g(x) = ax2 − 6x + 15,  x ∈ R has a:           (JEE Main 2021)
(a) local maximum at x = − 3/4
(b) local minimum at x = −3/4
(c) local maximum at x = 3/4
(d) local minimum at x = 3/4

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.32. The maximum value of        (JEE Main 2021)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(a) √5
(b) 3/4
(c) 5
(d) √7

Ans.  a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
We know, maximum value of acos x ±  bsin x
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.33. Let slope of the tangent line to a curve at any point P(x, y) be given by JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE If the curve intersects the line x + 2y = 4 at x = 2, then the value of y, for which the point (3, y) lies on the curve, is :        (JEE Main 2021)
(a) -18/19
(b) -4/3
(c) 18/35
(d) -18/11

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Curve intersect the line x + 2y = 4 at x =  2
So,  2 + 2y = 4  y = 3
So the curve passes through (2, 3) 

JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
It also passes through (3, y)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.34. Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = π/2  in the first quadrant. Then,          (JEE Main 2021)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.35. The maximum slope of the curve JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE occurs at the point:          (JEE Main 2021)
(a) (3,21/3)
(b) (0,0)
(c) (2,9)
(d) (2,2)

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
At x = 2,

y = 8 - 40 + 72 - 38 = 2

∴ Required point = (2, 2)


Q.36. The minimum value of JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEwhere a, x ∈ R and a > 0, is equal to:          (JEE Main 2021)
(a) a+ 1/a
(b) 2a
(c) a+1
(d) 2√a

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.37. If the curves, JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE intersect each other at an angle of 90°, then which of the following relations is TRUE?          (JEE Main 2021)
(a) a -c = b + d
(b) a + b = c + d
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) a - b = c - d

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Differentiating both sides:
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Differentiating both sides :
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.38. If the curve y = ax2 + bx + c, x ∈ R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are:          (JEE Main 2021)
(a) a = − 1, b = 1, c = 1  
(b) a = 1, b = 1, c = 0  
(c) a = 1/2, b = 1/2, c = 1  
(d) a = 1, b = 0, c = 1

Ans. b

Given curve y = ax2 + bx + c, x∈R
This curve passes through the point (1, 2)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

 Given, slope of tangent at origin is 1
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

(0, 0) lie on curve
∴ c = 0, a = 1


Q.39. Let JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE be defined as
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Let A = {x  R : f is increasing}. Then A is equal to:          (JEE Main 2021)
(a) (−5,∞)
(b) (−∞,−5)∪(4,∞)
(c) (−5,−4)∪(4,∞)
(d) (−∞,−5)∪(−4,∞)

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, f(x) is monotonically increasing in interval (−5,−4)∪(4,∞)


Q.40 For which of the following curves, the line JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is the tangent at the point JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE          (JEE Main 2021)
(a) 2x2 - 18y2 = 9
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) x2 + 9y2 = 9
(d) x2 + y2 = 7

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.41. If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is:          (JEE Main 2021)
(a) 0
(b) 2t3
(c) -2t3
(d) -t
3

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Slope of tangent at point p,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
This slope is same as slope of line PQ.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.42. The function
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE          (JEE Main 2021)
(a) increases in (−∞,12]  
(b) decreases in (−∞,12]  
(c) increases in [12,∞)  
(d) decreases in [12,∞)

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.43. Let f (x) be a polynomial of degree 5 such that x =±1 are its critical points. IfJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEthen which one of the following is not true?    (2020)
(a)  f is an odd function.
(b) f(1) - 4f(-1) = 4.
(c) x = 1 is a point of maxima and x = −1 is a point of minimum of f.
(d) x = 1 is a point of minima and x = - 1 is a point of maxima of f.

Ans. d
Given,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE    (1)
Let,
f(x) = 2x3 + ax4 + bx5 ⇒ f'(x) = 6x2 + 4ax3 + 5bx4
Since x = ±1 are the critical points of f (x), then
f'(1) = 6 + 4a + 5b = 0
⇒ 4a + 5b = -6...   (2)
Now, f'(-1) = 0 ⇒ 6 - 4a + 5b = 0
⇒ -4a + 5b = - 6....    (3)
On solving Eqs. (2) and (3), we get JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Therefore, JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, x = 1 is a point of maxima and x = -1 is a point of minima of  f(x).


Q.44. Let f(x) = x cos-1 JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEthen which of the following is true?    (2020)
(a) f' is increasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEand decreasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

(b) f' (0) =JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) f is not differentiable at x = 0.
(d) f' is decreasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEand increasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. d
We have,
f(x) = x[π - cos-1 JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, f'(x) is decreasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEand increasing in JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.45. The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy - 3yat the point (2, 2) is    (2020)
(a) √2
(b) 4√2
(c) 2
(d) 2√2

Ans. d
The equation of the given curve is
x+ 2xy - 3y2 = 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now, the slope of normal at (2,2) is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
The equation of the normal is
y −2 = −1(x−2)⇒ x+ y - 4 = 0
Hence, the length of perpendicular from origin to the normal is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.46. Let f(x) be a polynomial of degree 3 such that f(−1) = 10, f(1) = −6, f(x) has a critical point at x = −1 and f'(x) has a critical point x = 1. Then f(x) has a local minima at x = _________.    (2020)

Ans. 3.00
Given, f'(x) has a critical point x = 1, let
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now, f '(-1) = 0 JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Therefore, JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now, f(1) = - 6
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ −11k + 6d = −36...    (1)
Now, f(-1) = 10 ⇒ 5k + 6d = 60...    (2)
From Eqs. (1) and (2), we get
k=5,d=6
f(x) = x- 3x- 9x + 5 ⇒ f'(x) = 3(x + 1)(x-3)
⇒ f"(x) = 6x - 6 ⇒ f"(3) = 6 x 3 - 6 = 12
Hence, f(x) has a local minima at x =3.


Q.47. A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which the thickness of ice decreases, is    (2020)
(a) 5/6π
(b) 1/54π
(c) 1/36π
(d) 1/18π

Ans. d
Let the initial thickness of the ice is x cm, then
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
At x = 5 cm and JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE= 50 cm3/ min
We have,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.48. Let a function f :[0,5] →JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE be continuous f (1) = 3 and F be defined as JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEwhere JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEThen, for the function F, the point x = 1 is    (2020)
(a) a point of local minima.
(b) not a critical point.
(c) a point of local maxima.
(d) a point of inflection.

Ans. a
We have,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE ....   (since f (1) = 0)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE    (since g (1) = 0)
⇒ F'(1) = 1.g (1) = 0....    (1)
⇒ F" (x) = 2x g (x) + x2 g' x
⇒ F" (1) = 2g (1) + 1 . g ' (1) = f(1) = 3....    (2)
From Eqs. (1) and (2), it is clear that x = 1 is the point of minima for function F.


Q.49. The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:    (2019)
(a) 6π
(b) 3√3π
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) 2√3π

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE    ...(1)
Volume of cone
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE   ...(2)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
For maxima/minima,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE      JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 
Then, h = √3 is point of maxima
Hence, the required maximum volume is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.50. Let d ∈ R, and JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE θ ∈ [0, 2π]. If the minimum value of det (A) is 8, then a value of d is:     (2019)
(a) -5
(b) -7
(c) 2(√2 + 1)
(d) 2(√2 + 2)

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Minimum value of det (A) is attained when sin2θ = 1
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ d = -5 or 1


Q.51.JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE If I is minimum then the ordered pair (a, b) is:     (2019)
(a) (0,√2)
(b) (-√2,0)
(c) (√2,-√2)
(d) (-√2,√2) 

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Also,  JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ I is minimum when (a, b) = (-√2, √2)


Q.52. The tangent to the curve,JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEpassing through the point (1, e) also passes through the point:     (2019)
(a) (2, 3e)
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) (3, 6e)

Ans. b
The equation of curve JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Since (1, e) lies on the curveJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEthen equation of tangent at (1, e) is
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
So, equation of tangent to the curve passes through the pointJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.53. A helicopter is flying along the curve given by y - x3/2 = 7, (x ≥ 0). A soldier positioned at the point JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:     (2019)
(a)  JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) 1/2

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ f(x) is increasing function ∀ x > 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE   JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.54. The maximum value of the function f(x) = 3x3 -18x+ 27x - 40 on the set S = {x ∈ R: x2 + 30 ≤ 11x } is     (2019)
(a) -122
(b) -222
(c) 122 
(d) 222

Ans. c
Consider the function,
f(x) = 3x(x - 3)2 - 40
Now S = {x ∈ R : x2 + 30 ≤ 11x}
So x2 - 11x + 30 < 0
⇒ x ∈ [5, 6]
∴ f(x) will have maximum value for x = 6
The maximum value of function is,
f(6) = 3 x 6 x 3 x 3 - 40 = 122.


Q.55. Let x,y be positive real numbers and m, n positive integers. The maximum value of the expression JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is:      (2019)
(a) 1
(b) 1/2
(c) 1/4
(d) None of these

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.56.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE where a, b and d are non-zero real constants. Then:     (2019)
(a) f is an increasing function of x
(b) f is a decreasing function of x
(c) f is not a continuous function of x
(d) f is neither increasing nor decreasing function of x

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ f(x) is increasing function.
Hence, f(x) is increasing function.


Q.57. The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 - x2 such that the rectangle lies inside the parabola, is:     (2019)
(a) 36 
(b) 20√2
(c) 32
(d) 18√3

Ans. c
Given, the equation of parabola is,
x2 = 12 - y
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Area of the rectangle = (2t) (12 - t2)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
At t = 2, area is maximum = 24(2) - 2(2)3
= 48 - 16 = 32 sq. units


Q.58. Let P(4, -4) and Q(9, 6) be two points on the parabola, y2 = 4x and let this X be any point arc POQ of this parabola, where O is vertex of the parabola, such that the area of ΔPXQ is maximum. Then this minimum area (in sq. units) is:     (2019)
(a) 75/2
(b) 125/4
(c) 625/4
(d) 125/2

Ans. b
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Parametric equations of the parabola y2 = 4x are, x = t2 and y = 2t.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
For maximum area t = 1/2
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.59. The maximum value of JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE for any real value of θ is:     (2019)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. a
 Let, the functions is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.60. If a curve passes through the point (1, -2) and has slope of the tangent at any point (x, y) on it as JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE then the curve also passes through the point:     (2019)
(a) (3, 0)
(b) (√3, 0)
(c) (-1, 2)
(d)  (-√2, 1)

Ans. b
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Solution of equation
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ curve passes through point (1, -2)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ C = -9/4
Then, equation of curve
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Since, above curve satisfies the point.
Hence, the curve passes through (√3, o).


Q.61. The tangent to the curves y = x2 - 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point:     (2019)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. b

∵ Tangent to the given curve is parallel to line 2y = 4x + 1
∴ Slope of tangent (m) = 2
Then, the equation of tangent will be of the form
y = 2x + c    ...(1)
∵ Line (1) and curve y = x2 - 5x + 5 has only one point of intersection.
∴ 2x + c = x2 - 5x + 5
x2 - 7x + (5 - c) = 0
∴ D = 49 - 4(5 - c) = 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 
Hence, the equation of tangent: JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.62. If the function/given by f(x) = x3 - 3(a - 2)x2 + 3ax + 7, for some a ∈ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation,  JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE     (2019)
(a) -7
(b) 5
(c) 7
(d) 6

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ a = 5
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Now,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.63. The shortest distance between the line y = x and the curve y2 = x - 2 is:     (2019)
(a) 2
(b) 7/8
(c) 7/4√2
(d) 11/4√2

Ans. c
The shortest distance between line y = x and parabola = the distance LM between line y = x and tangent of parabola having slope 1.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Let equation of tangent of parabola having slope 1 is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Distance between the line y - x = 0 and JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.64. If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x4 + 12x3 - 36x2 + 25, x∈R, then :     (2019)
(a) S1 = {-2}; S2 = {0, 1}
(b) S1 = {-2, 0}; S2 = {1}
(c) S1 = {-2, 1}; S2 = {0}
(d) S1 = {-1}; S2 = {0,2}

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Here at -2 & 1, f'(x) changes from negative value to positive value.
⇒ -2 & 1 are local minimum points. At 0, f'(x) changes from positive value to negative value.
⇒ 0 is the local maximum point.
Hence, S1 = {-2, 1} and S2 = {0}


Q.65. Let f : [0 : 2] → R be a twice differentiable function such that f"(x) > 0, for all x∈(0, 2). If φ(x) = (x) + f(2 - x), then φ is:     (2019)
(a) increasing on (0, 1) and decreasing on (1, 2).
(b) decreasing on (0, 2)
(c) decreasing on (0, 1) and increasing on (1,2).
(d) increasing on (0, 2)

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 
But f" (x) > 0 ⇒ f'(x) is an increasing function
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, φ (x) is increasing on (1, 2) and decreasing on (0, 1).


Q.66. Given that the slope of the tangent to a curve y = y(x) at any point (x, y) isJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE. If the curve passes through the centre of the circle x2 + y2 - 2x - 2y = 0, then its equation is :     (2019)
(a) x loge|y| = 2(x - 1)
(b) x loge |y| = - 2(x - 1)
(c) x|loge|y| = -2(x - 1)
(d) x loge | y | = x - 1

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Equation (i) passes through the centre of the circle x2 + y2 - 2x - 2y = 0, i.e., (1,1)
∴ C = 2
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.67. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:     (2019)
(a) √6
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) 2√3
(d) √3

Ans. c
Let radius of base and height of cylinder be r and h respectively.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE     ...(1)
Now, volume of cylinder, V = πr2h
Substitute the value of r2 from equation (i),
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Differentiating w.r.t. h,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
For maxima/minima,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
⇒ Volume is maximum when h = 2√3


Q.68. If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1; then the set S = {x ∈ R : f(x) = f(0)} contains exactly:     (2019)
(a) four irrational numbers.
(b) four rational numbers.
(c) two irrational and two rational numbers.
(d) two irrational and one rational number.

Ans. d
Since, function f(x) have local extreme points at x = -1,0, 1. Then
f(x) = K(x+ l)x(x - 1)
= K (x3 - x)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.69. If the tangent to the curve, y = x3 + ax - b at the point (1, -5) is perpendicular to the line, - x + y + 4 = 0, then which one of the following points lies on the curve?     (2019)
(a) (-2,1) 
(b) (-2,2)
(c) (2,-1)
(d) (2,-2)

Ans. d
y = x3 + ax - b
Since, the point (1, -5) lies on the curve.
⇒ 1 + a - b = - 5
⇒ a - b = -6    ...(1)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Since, required line is perpendicular to y = x - 4, then slope of tangent at the point P (1, -5) = -1
∴ 3 + a = - 1
⇒ a = -4
⇒ b = 2
the equation of the curve is y = x3 - 4x - 2
⇒ (2, -2) lies on the curve


Q.70. Let S be the set of all values of x for which the tangent to the curve y = f(x) = x3 - x2 - 2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (- 1, f(- 1)), then S is equal to:     (2019)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. d
y = f(x) = x3 - x2 - 2x
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
f(1) = 1 - 1 -2 = -2, f(-1) = -1 -1+2 = 0
Since the tangent to the curve is parallel to the line segment joining the points (1, -2) and (-1, 0)
And their slopes are equal.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, the required set JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.71. A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is:     (2019)
(a) 1/15 π
(b) 1/10 π
(c) 2/π
(d) 1/5 π

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Given that water is poured into the tank at a constant rate of 5 m3/minute.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Volume of the tank is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE   ...(i)
where r is radius and h is height at any time. By the diagram,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE   ...(ii)
Differentiate eq. (i) w.r.t. ‘t’, we get
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Putting h = 10, r = 5 and dV/dt = 5 in the above equation.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.72. Let f(x) = ex - x and g(x) = x2 - x,  x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing, is:     (2019)
(a)  JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. b
Given functions are, f(x) = ex - x and g(x) = x2 - x
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Given f(g (x)) is increasing function.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE are either both positive or negative
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.73. If the tangent to the curve JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE at a point (α, β) is parallel to the line 2x + 6y - 11 = 0, then :     (2019)

(a) |6α+2β|= 19
(b) |6α+2β|= 9
(c) |2α+6β|= 19
(d) |2α+6β|= 11 

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
These values of α and β satisfies |6α + 2β| = 19


Q.74. A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is:     (2019)
(a) 1/18π
(b) 1/36π
(c) 5/6π
(d) 1/9π

Ans. a
Given that ice melts at a rate of 50 cm3/min
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Substitute r = 5,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.75. Let a1, a2, a3,.... be an A. P. with a6 = 2. Then the common difference of this A.P., which maximises the product a4 a4 a5 is:     (2019)
(a) 3/2
(b) 8/5
(c) 6/5
(d) 2/3

Ans. b
a6 = a + 5d = 2
Here, a is first term of A.P and d is common difference
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.76. If m is the minimum value of k for which the functionJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE is increasing in the interval [0, 3] and M is the maximum value off in [0,3] when k = m, then the ordered pair (m, M) is equal to:     (2019)
(a) (4, 3√2)
(b) (4, 3√3)
(c) (3, 3√3)
(d) (5, 3√6)

Ans. b
Given function JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Differentiating w. r. t. x,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
[∴ f(x) is increasing in [0, 3]]
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.77. The equation of a common tangent to the curves, y2 = 16x and xy = - 4, is:     (2019)
(a) x - y + 4 = 0
(b) x + y + 4 = 0
(c) x - 2y + 16 = 0
(d) 2x - y + 2 = 0

Ans. a
Given curves, y2 = 16x and xy = - 4
Equation of tangent to the given parabola;
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∵ This is common tangent.
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
∴ equation of common tangent is y = x + 4


Q.78. The tangents to the curve y = (x - 2)2 -1 at its points of intersection with the line x - y = 3, intersect at the point:     (2019)

(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. c
Tangent to the curve y = (x - 2)2 - 1 at any point (h, k) is,
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.79. JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE then the local minimum value of value of h(x) is:     (2018)
(a) 3
(b) -3
(c) -2√2
(d) 2√2

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE    JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence, local minimum value is 2√2


Q.80. If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is:    (2018)
(a) 8√2 π
(b) 6√2 π
(c) 6√3 π
(d) 8√3 π

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
where r is radius and h is height of coin
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.81. Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f (x) = 2x3 - 9x2 + 12x + 5 in the interval [0, 3]. Then M - m is equal to:     (2018)
(a) 1
(b) 9
(c) 5
(d) 4

Ans. a
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
For maxima or minima put f'(x) = 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.82. Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is     (2017)
(a) 30
(b) 12.5
(c) 10
(d) 25

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
2r + θr =20 ... (i)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEEJEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
A to be maximum
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Hence for r = 5, A is maximum
Now, 10 + θ·5 = 20 ⇒ θ = 2 (radian)
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.83. The function f defined by f(x) = x3 – 3x2 + 5x + 7, is:     (2017)
(a) decreasing in R
(b) increasing in R
(c) increasing in (0, ∞) and decreasing in (–∞, 0)
(d) decreasing in (0, ∞) and increasing in (–∞, 0)

Ans. b
f(x) = x3 – 3x2 + 5x + 7
f'(x) = 3x2 – 6x + 5 > 0 ( ∵D< 0, a>0)
⇒x ∈ ϕ


Q.84. A tangent to the curve, y = f(x) at P(x,y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1)= 1, then the curve also passes through the point:    (2017)
(a) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
Let y = f(x) be a curve
slope of tangent = f'(x)
Equation of tangent (Y-y) = f'(x) (X- x)
Put Y = 0
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
y = 1/x3 is required curve (2, 1/8) passing through
y = 1/x3


Q.85. Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β then tan β is    (2016)
(a) 4/9
(b) 6/7
(c) 1/4
(d) 2/9

Ans. d
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE


Q.86. If m and M are the minimum and the maximum values of JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE  then M - m is equal to    (2016)
(a) 7/4
(b) 15/4
(c) 9/4
(d) 1/4

Ans. c
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
M = maximum value = 17/4
m = minimum value = 2
M-m = 17/4 - 2 = 9/4.

Q.87. Let f(x) = sin4x + cos4x. Then f is an increasing function in the interval    (2016)
(a)JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

(b)JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(c)JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE

Ans. c
f(x) = sin4x + cos4x
f'(x) = 4sin3x cosx - 4cos3xsinx
= 4sinx cosx(sin2x - cos2x)
= - 2sin2x. cos2x
= - sin4x > 0
⇒ sin4x < 0
⇒ π < 4x < 2π
JEE Main Previous year questions (2016-22): Applications of Derivatives Notes | Study Mathematics For JEE - JEE 

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