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JEE Application of Derivatives PYQs with Solutions

795 previous year questions from Application of Derivatives (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

795 questionsMathematics2006–2026Answers + solutions

About Application of Derivatives PYQs

Application of Derivatives has 795 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (153) and JEE Advanced (67), spanning 2006 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 12
  • 2025: 15
  • 2024: 22
  • 2023: 24
  • 2022: 38
  • 2021: 47
  • 2020: 11
  • 2019: 3
  • 2018: 2
  • 2017: 6
  • 2016: 2
  • 2015: 3

Application of Derivatives questions with solutions

1

Consider the function f: (0, ∞) arrow (-∞, ∞) given by f(x) = x \, _e(x) - x + 1 . Then which one of the following statements is TRUE?

JEE Advanced 2026 Solution View → 2

Let P be the point on the parabola y = x^2 such that the slope of the tangent to the parabola at the point P is 4 . Let Q be the point in the first quadrant lying on the circle

JEE Advanced 2026 Solution View → 3

Let f: ℝ arrow ℝ be a differentiable function such that f((x+y)/(3)) = (f(x) + f(y))/(3) for all x, y ∈ ℝ , and f'(0) = 3 . Then the minimum value of the function g(x) = 3 + e^x

JEE Main 2026 Solution View → 4

_ 0 ≤ x ≤ π (16 ((x)/(2)) ^3((x)/(2))) is equal to:

JEE Main 2026 Solution View → 5

Let f(x) be a polynomial of degree 5 , and have extrema at x = 1 and x = -1 . If _ x → 0 ((f(x))/(x^3)) = -5 , then f(2) - f(-2) is equal to:

JEE Main 2026 Solution View → 6

The number of critical points of the function f(x) = cases |( x)/(x)|, & x ≠ 0 1, & x = 0 cases in the interval (-2π, 2π) is equal to :

JEE Main 2026 Solution View → 7

Let f be a differentiable function satisfying f(x)=1-2 x+∫_ 0 ^ x e^ (x-t) f(t) dt, x ∈ R and let g(x)=∫_ 0 ^ x (f(t)+2)^ 15 (t-4)^ 6 (t+12)^ 17 dt, x ∈ R . If p and q are

JEE Main 2026 Solution View → 8

Consider the following three statements for the function f:(0, ∞) arrow ℝ defined by f(x)=| _ e x|-|x-1| : (I) f is differentiable at all x>0 . (II) f is increasing in (0,1) .

JEE Main 2026 Solution View → 9

Let (2 α, α) be the largest interval in which the function f(t)= |t+1| t^ 2 , t 2 , is \_\_\_\_

JEE Main 2026 Solution View → 10

Let f(x)=x^ 2025 -x^ 2000 , x ∈[0,1] and the minimum value of the function f(x) in the interval [0,1] be (80)^ 80 (n)^ -81 . Then n is equal to

JEE Main 2026 Solution View → 11

Let f: ℝ → ℝ be a twice differentiable function such that f''(x) > 0 for all x ∈ ℝ and f'(a-1) = 0 , where a is a real number. Let g(x) = f( ^ 2 x - 2 x + a ), \; 0 Consider the

JEE Main 2026 Solution View → 12

Let f: R arrow R be a twice differentiable function such that the quadratic equation f(x) m^ 2 -2 f^ (x) m+f^ (x)=0 in m, has two equal roots for every x ∈ R . If f(0)=1, f^ (0)=2

JEE Main 2026 Solution View → 13

Let ℝ denote the set of all real numbers. Define the function f: R arrow R by f(x)=\ array cc 2-2 x^2-x^2 (1)/(x) & if x ≠ 0 2 & if x=0 array . Then which one of the following

JEE Advanced 2025 Solution View → 14

Let ℝ denote the set of all real numbers. For a real number x , let [x] denote the greatest integer less than or equal to x . Let n denote a natural number. Match each entry in

JEE Advanced 2025 Solution View → 15

Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ be defined by f(x)= cases (6 x+ x)/(2 x+ x) & if x ≠ 0 (7)/(3) & if x=0 cases Then which of the following statements is

JEE Advanced 2025 Solution View → 16

Let the function f(x)=(x)/(3)+(3)/(x)+3, x ≠ 0 be strictly increasing in (-∞, α_1) U(α_2, ∞) and strictly decreasing in (α_3, α_4) U(α_4, α_5) . Then ∑_ i = 1 ^5 α_ i ^2 is equal

JEE Main 2025 Solution View → 17

Let f: R arrow R be a polynomial function of degree four having extreme values at x=4 and x=5 . If _ x arrow 0 (f(x))/(x^2)=5 , then f(2) is equal to :

JEE Main 2025 Solution View → 18

Let x=-1 and x=2 be the critical points of the function f(x)=x^3+ax^2+b _ c |x|+1, x ≠ 0 . Let m and M respectively be the absolute minimum and the absolute maximum values of f in

JEE Main 2025 Solution View → 19

Let a 0 . If the function f(x)=6 x^3-45 a x^2+108 a^2 x+1 attains its local maximum and minimum values at the points x_1 and x_2 respectively such that x_1 x_2=54 , then a+x_1+x_2

JEE Main 2025 Solution View → 20

Let f be a differentiable function on R such that f(2) = 1 , f^ (2)=4 . Let _ x arrow 0 (f(2+x))^ 3 / x =e^α . Then the number of times the curve y=4 x^3-4 x^2-4(α-7) x-α meets x

JEE Main 2025 Solution View → 21

The shortest distance between the curves y^2=8 x and x^2+y^2+12 y+35=0 is :

JEE Main 2025 Solution View → 22

Let f: R arrow R be a function defined by f(x)=\|x+2|-2| x\| . If m is the number of points of local minima and n is the number of points of local maxima of f , then m+n is

JEE Main 2025 Solution View → 23

If the function f(x)=2 x^3-9 ax^2+12 a^2 x+1 , where a 0 , attains its local maximum and local minimum values at p and q , respectively, such that p^2=q , then f(3) is equal to:

JEE Main 2025 Solution View → 24

The sum of all local minimum values of the function f(x)=\ array lr 1-2 x, & x -1 (1)/(3)(7+2|x|), & -1 ≤ x ≤ 2 (11)/(18)(x-4)(x-5), & x 2 array . is

JEE Main 2025 Solution View → 25

Let (2,3) be the largest open interval in which the function f(x)=2 _ e (x-2)-x^2+a x+1 is strictly increasing and (b, c) be the largest open interval, in which the function

JEE Main 2025 Solution View → 26

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 ~cm^3

JEE Main 2025 Solution View → 27

If the set of all values of a, for which the equation 5 x^3-15 x-a=0 has three distinct real roots, is the interval (α, β) , then β-2 α is equal to ______

JEE Main 2025 Solution View → 28

Let the function f: ℝ arrow ℝ be defined by f(x)= x e^ π x (x^ 2023 +2024 x+2025) (x^2-x+3) + 2 e^ π x (x^ 2023 +2024 x+2025) (x^2-x+3) . Then the number of solutions of f(x)=0 in

JEE Advanced 2024 Solution View → 29

Let the set of all values of p , for which f(x)=(p^2-6 p+8)( ^2 2 x- ^2 2 x)+2(2-p) x+7 does not have any critical point, be the interval (a, b) . Then 16 a b is equal to _______

JEE Main 2024 Solution View → 30

A variable line L passes through the point (3,5) and intersects the positive coordinate axes at the points A and B . The minimum area of the triangle OAB , where O is the origin

JEE Main 2024 Solution View → 31

Let the set of all positive values of λ , for which the point of local minimum of the function (1+x (λ^2-x^2) ) satisfies (x^2+x+2)/(x^2+5 x+6) < 0 , be (α, β) . Then α^2+β^2 is

JEE Main 2024 Solution View → 32

If the function f(x)=2 x^3-9 x^2+12 a^2 x+1, a>0 has a local maximum at x=α and a local minimum at x=α^2 , then α and α^2 are the roots of the equation :

JEE Main 2024 Solution View → 33

Let A be the region enclosed by the parabola y^2=2 x and the line x=24 . Then the maximum area of the rectangle inscribed in the region A is________

JEE Main 2024 Solution View → 34

For the function f(x)=( x)-x+1, x ∈ ℝ , between the following two statements (S1) f(x)=0 for only one value of x in [0, π] . (S2) f(x) is decreasing in [0, (π)/(2)] and increasing

JEE Main 2024 Solution View → 35

Let f(x)=4 ^3 x+3 3 ^2 x-10 . The number of points of local maxima of f in interval (0,2 π) is

JEE Main 2024 Solution View → 36

The number of critical points of the function f(x)=(x-2)^ 2 / 3 (2 x+1) is

JEE Main 2024 Solution View → 37

If the function f(x)=((1)/(x))^ 2 x ; x>0 attains the maximum value at x= 1 e then :

JEE Main 2024 Solution View → 38

The interval in which the function f(x)=x^x, x>0 , is strictly increasing is

JEE Main 2024 Solution View → 39

Let the maximum and minimum values of ( 8 x-x^2-12 -4)^2+(x-7)^2, x ∈ R be M and m , respectively. Then M^2-m^2 is equal to _________

JEE Main 2024 Solution View → 40

Let a rectangle A B C D of sides 2 and 4 be inscribed in another rectangle P Q R S such that the vertices of the rectangle A B C D lie on the sides of the rectangle P Q R S . Let

JEE Main 2024 Solution View → 41

For the function f(x)= x+3 x-(2)/(π)(x^2+x), where x ∈[0, (π)/(2)], consider the following two statements : (I) f is increasing in (0, (π)/(2)) . (II) f^ is decreasing in (0

JEE Main 2024 Solution View → 42

Let f(x)=3 x-2 + 4-x be a real valued function. If α and β are respectively the minimum and the maximum values of f , then α^2+2 β^2 is equal to

JEE Main 2024 Solution View → 43

Let f: ℝ arrow ℝ be a thrice differentiable function such that f(0)=0, f(1)=1, f(2)=-1, f(3)=2 and f(4)=-2 . Then, the minimum number of zeros of (3 f^ f^ +f f^ )(x) is _______

JEE Main 2024 Solution View → 44

If 5f(x) + 4f((1)/(x)) = x ^ 2 - 2, ∀ x ≠ 0 and y = 9 x ^ 2 f(x), then y is strictly increasing in:

JEE Main 2024 Solution View → 45

Let f(x) = (x + 3) ^ 2 (x - 2) ^ 3 , x ∈ [ - 4, 4 ] . If M and m are the maximum and minimum values of f , respectively in [ - 4, 4 ] , then the value of M - m is :

JEE Main 2024 Solution View → 46

Let g: R → R be a non constant twice differentiable such that g ^ ' ((1)/(2)) = g ^ ' ((3)/(2)) . If a real valued function f is defined as f ( x ) = (1)/(2) [ g ( x ) + g ( 2 - x

JEE Main 2024 Solution View → 47

The function f(x) = 2x + 3x(2)/(3), x ∈ R , has

JEE Main 2024 Solution View → 48

The function f(x) = x x ^ 2 - 6x - 16 , x ∈ ℝ - \ - 2, 8\

JEE Main 2024 Solution View → 49

Let g ( x ) = 3f((x)/(3)) + f ( 3 - x ) and f ^ " ( x ) > 0 for all x ∈ ( 0, 3 ) . If g is decreasing in ( 0, α ) and increasing in ( α, 3 ) , then 8α is

JEE Main 2024 Solution View → 50

Let S be the set of all twice differentiable functions f from ℝ to ℝ such that d ^ 2 f d x ^ 2 (x) > 0 for all x ∈ (- 1, 1) . For f ∈ S , let X _ f be the number of points x ∈ (

JEE Advanced 2023 Solution View → 51

max _ 0 ≤ x ≤ π \ x - 2sinxcosx + (1)/(3)sin3x\ =

JEE Main 2023 Solution View → 52

The set of all a ∈ ℝ for which the equation x | x - 1 | + | x + 2 | + a = 0 has exactly one real root, is

JEE Main 2023 Solution View → 53

If the total maximum value of the function f(x) = ( 3e 2sinx ) ^ ^ 2 x , x ∈ (0, (π)/(2)), is (k)/(e), then ((k)/(e)) ^ 8 + k ^ 8 e ^ 5 + k ^ 8 is equal to

JEE Main 2023 Solution View → 54

Let f: [2, 4] → ℝ be a differentiable function such that (x _ e x) f ^ ' (x) + ( _ e x)f(x) + f(x) ≥ 1, x ∈ [2, 4] with f(2) = (1)/(2) and f(4) = (1)/(2) . Consider the following

JEE Main 2023 Solution View → 55

Let g(x) = f(x) + f(1 - x) and f ^ " (x) > 0, x ∈ (0, 1) . If g is decreasing in the interval (0, α) and increasing in the interval (α, 1) , then ^ - 1 2α + ^ - 1 ((1)/(α)) + ^

JEE Main 2023 Solution View → 56

In the figure, θ _ 1 + θ _ 2 = (π)/(2) and 3 (BE) = 4(AB) . If the area of ∆ CAB is 2 3 - 3 unit ^ 2 , when θ _ 2 θ _ 1 is the largest, then the perimeter (in unit) of ∆ CED is

JEE Main 2023 Solution View → 57

Let the quadratic curve passing through the point (- 1, 0) and touching the line y = x at (1, 1) be y = f (x). Then the x-intercept of the normal to the curve at the point (α, α +

JEE Main 2023 Solution View → 58

A square piece of tin of side 30cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is

JEE Main 2023 Solution View → 59

If a _ α is the greatest term in the sequence a _ n = n ^ 3 n ^ 4 + 147 , n = 1, 2, 3 . . . . , then α is equal to ______

JEE Main 2023 Solution View → 60

Let a curve y = f(x), x ∈ (0, ∞) pass through the points P(1, (3)/(2)) and Q(a, (1)/(2)) . If the tangent at any point R ( b, f ( b ) ) to the given curve cuts the y -axis at the

JEE Main 2023 Solution View → 61

The number of points, where the curve y = x ^ 5 - 20 x ^ 3 + 50x + 2 crosses the x -axis, is _____.

JEE Main 2023 Solution View → 62

Let the tangent to the curve x ^ 2 + 2x - 4y + 9 = 0 at the point P(1, 3) on it meet the y - axis at A . Let the line passing through P and parallel to the line x - 3y = 6 meet

JEE Main 2023 Solution View → 63

The sum of the abosolute maximum and minimum values of the function f(x) = | x ^ 2 - 5x + 6| - 3x + 2 in the interval [- 1, 3] is equal to :

JEE Main 2023 Solution View → 64

Let f(x) = 2x + ^ - 1 x and g(x) = _ e ( 1 + x ^ 2 + x), x ∈ [0, 3] . Then

JEE Main 2023 Solution View → 65

Let f(x) = | cases 1 + ^ 2 x & ^ 2 x & sin2x ^ 2 x & 1 + ^ 2 x & sin2x ^ 2 x & ^ 2 x & 1 + sin2x cases |, x ∈ [(π)/(6), (π)/(3)] . If α and β respectively are the maximum and the

JEE Main 2023 Solution View → 66

If f(x) = x ^ 2 + g ^ ' (1)x + g ^ " (2) and g(x) = f(1) x ^ 2 + x f ^ ' (x) + f ^ " (x) , then the value of f(4) - g(4) is equal to _____ .

JEE Main 2023 Solution View → 67

The absolute minimum value, of the function f(x) = | x ^ 2 - x + 1| + [ x ^ 2 - x + 1], where [t] denotes the greatest integer function, in the interval [- 1, 2], is

JEE Main 2023 Solution View → 68

A wire of length 20m is to be cut into two pieces. A piece of length ℓ _ 1 is bent to make a square of area A _ 1 and the other piece of length ℓ _ 2 is made into a circle of area

JEE Main 2023 Solution View → 69

If the functions f(x) = x ^ 3 3 + 2bx + a x ^ 2 2 and g(x) = x ^ 3 3 + ax + b x ^ 2 , a ≠ 2b have a common extreme point, then a + 2b + 7 is equal to

JEE Main 2023 Solution View → 70

The number of points on the curve y = 54 x ^ 5 - 135 x ^ 4 - 70 x ^ 3 + 180 x ^ 2 + 210x at which the normal lines are parallel to x + 90y + 2 = 0 is:

JEE Main 2023 Solution View → 71

If the equation of the normal to the curve y = (x - a)/((x + b)(x - 2)) at the point (1, - 3) is x - 4y = 13 then the value of a + b is equal to ______

JEE Main 2023 Solution View → 72

Let the function f ( x ) = 2 x ^ 3 + ( 2p - 7 ) x ^ 2 + 3 ( 2p - 9 ) x - 6 have a maxima for some value of x 0 . Then, the set of all values of p is

JEE Main 2023 Solution View → 73

Let x = 2 be a local minima of the function f(x) = 2 x ^ 4 - 18 x ^ 2 + 8x + 12, x ∈ (- 4, 4) . If M is local maximum value of the function f in (- 4, 4), then M =

JEE Main 2023 Solution View → 74

If the tangent to the curve y = x ^ 3 - x ^ 2 + x at the point (a, b) is also tangent to the curve y = 5 x ^ 2 + 2x - 25 at the point (2, - 1) , then |2a + 9b| is equal to ______.

JEE Main 2022 Solution View → 75

Let f(x) = 3 ^ ( x ^ 2 - 2) ^ 3 + 4 , x ∈ R . Then which of the following statements are true? P: x = 0 is a point of local minima of f Q: x = 2 is a point of inflection of f R: f

JEE Main 2022 Solution View → 76

The function f(x) = x e ^ x(1 - x) , x ∈ R , is

JEE Main 2022 Solution View → 77

The sum of the absolute maximum and absolute minimum values of the function f(x) = ^ - 1 (sinx - cosx) in the interval [0, π] is

JEE Main 2022 Solution View → 78

Let f: [0, 1] → R be a twice differentiable function in (0, 1) such that f(0) = 3 and f(1) = 5 . If the line y = 2x + 3 intersects the graph of f at only two distinct points in

JEE Main 2022 Solution View → 79

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is ^ - 1 (3)/(4) . Water is poured in it at a constant rate of

JEE Main 2022 Solution View → 80

Let M and N be the number of points on the curve y ^ 5 - 9xy + 2x = 0 , where the tangents to the curve are parallel to x -axis and y -axis, respectively. Then the value of M + N

JEE Main 2022 Solution View → 81

Let P and Q be any points on the curves (x - 1) ^ 2 + (y + 1) ^ 2 = 1 and y = x ^ 2 , respectively. The distance between P and Q is minimum for some value of the abscissa of P in

JEE Main 2022 Solution View → 82

If the maximum value of a , for which the function f _ a (x) = ^ - 1 2x - 3ax + 7 is non-decreasing in (- (π)/(6), (π)/(6)) , is ¯ a , then f ¯ a ((π)/(8)) is equal to

JEE Main 2022 Solution View → 83

Let f(x) = \ cases x ^ 3 - x ^ 2 + 10x - 7, & x ≤ 1 - 2x + _ 2 ( b ^ 2 - 4), & x > 1 cases ) Then the set of all values of b , for which f(x) has maximum value at x = 1 , is:

JEE Main 2022 Solution View → 84

The number of distinct real roots of the equation x ^ 5 ( x ^ 3 - x ^ 2 - x + 1) + x(3 x ^ 3 - 4 x ^ 2 - 2x + 4) - 1 = 0 is

JEE Main 2022 Solution View → 85

Let the function f(x) = 2 x ^ 2 - _ e x, x > 0 , be decreasing in (0, a) and increasing in (a, 4) . A tangent to the parabola y ^ 2 = 4ax at a point P on it passes through the

JEE Main 2022 Solution View → 86

Let the area enclosed by the x -axis, and the tangent and normal drawn to the curve 4 x ^ 3 - 3x y ^ 2 + 6 x ^ 2 - 5xy - 8 y ^ 2 + 9x + 14 = 0 at the point (- 2, 3) be A . Then 8A

JEE Main 2022 Solution View → 87

If the absolute maximum value of the function f(x) = ( x ^ 2 - 2x + 7) e ^ (4 x ^ 3 - 12 x ^ 2 - 180x + 31) in the interval [- 3, 0] is f(α), then

JEE Main 2022 Solution View → 88

The curve y(x) = a x ^ 3 + b x ^ 2 + 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

JEE Main 2022 Solution View → 89

Let f: R → R be a function defined by f(x) = (x - 3) ^ n _ 1 (x - 5) ^ n _ 2 , n _ 1 , n _ 2 ∈ N . The, which of the following is NOT true?

JEE Main 2022 Solution View → 90

Let f and g be twice differentiable even functions on (- 2, 2) such that f((1)/(4)) = 0, f((1)/(2)) = 0, f(1) = 1 and g((3)/(4)) = 0, g(1) = 2 Then, the minimum number of

JEE Main 2022 Solution View → 91

A wire of length 22m 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

JEE Main 2022 Solution View → 92

The number of real solutions of x ^ 7 + 5 x ^ 3 + 3x + 1 = 0 is equal to _____.

JEE Main 2022 Solution View → 93

Let l be a line which is normal to the curve y = 2 x ^ 2 + 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

JEE Main 2022 Solution View → 94

If m and n respectively are the number of local maximum and local minimum points of the function f(x) = ∫ _ 0 ^ x ^ 2 t ^ 2 - 5t + 4 2 + e ^ t dt , then the ordered pair (m, n) is

JEE Main 2022 Solution View → 95

The number of distinct real roots of x ^ 4 - 4x + 1 = 0 is

JEE Main 2022 Solution View → 96

The lengths of the sides of a triangle are 10 + x ^ 2 , 10 + x ^ 2 and 20 - 2 x ^ 2 . If for x = k , the area of the triangle is maximum, then 3 k ^ 2 is equal to

JEE Main 2022 Solution View → 97

If the sum of all the roots of the equation e ^ 2x - 11 e ^ x - 45 e ^ - x + (81)/(2) = 0 is _ e P , then P is equal to _____.

JEE Main 2022 Solution View → 98

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r . If the sum of their surface areas is constant k , then the ratio x: r , for which the sum of their

JEE Main 2022 Solution View → 99

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x - x ^ 2 + 2| - x in the interval [- 1, 2] is

JEE Main 2022 Solution View → 100

Let S be the set of all the natural numbers, for which the line (x)/(a) + (y)/(b) = 2 is a tangent to the curve ((x)/(a)) ^ n + ((y)/(b)) ^ n = 2 at the point (a, b), ab ≠ 0 . Then

JEE Main 2022 Solution View → 101

Let f(x) = 2 ^ - 1 x + 4 cot ^ - 1 x - 3 x ^ 2 - 2x + 10, x ∈ [- 1, 1]. If [a, b] is the range of the function, then4a - b is equal to

JEE Main 2022 Solution View → 102

Water is being filled at the rate of 1 cm ^ 3 sec ^ - 1 in a right circular conical vessel (vertex downwards) of height 35cm and diameter 14cm . When the height of the water level

JEE Main 2022 Solution View → 103

If the line y = 4 + kx, k > 0 , is the tangent to the parabola y = x - x ^ 2 at the point P and V is the vertex of the parabola, then the slope of the line through P and V is

JEE Main 2022 Solution View → 104

Let f(x) = |(x - 1)( x ^ 2 - 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

JEE Main 2022 Solution View → 105

If the angle made by the tangent at the point ( x _ 0 , y _ 0 ) on the curve x = 12 (t + sintcost), y = 12 (1 + sint) ^ 2 , 0 < t < (π)/(2), with the positivex -axis is (π)/(3)

JEE Main 2022 Solution View → 106

Let f: R → R and g: R → R be two functions defined by f(x) = _ e ( x ^ 2 + 1) - e ^ - x + 1 and g(x) = 1 - 2 e ^ 2x e ^ x · Then, for which of the following range of α , the

JEE Main 2022 Solution View → 107

The number of distinct real roots of the equation x ^ 7 - 7x - 2 = 0 is

JEE Main 2022 Solution View → 108

Let λ ^ * be the largest value of λ for which the function f _ λ (x) = 4λ x ^ 3 - 36λ x ^ 2 + 36x + 48 is increasing for all x ∈ ℝ . Then f λ ^ * (1) + f _ λ, * (- 1) is equal to:

JEE Main 2022 Solution View → 109

For the function f(x) = 4 _ e (x - 1) - 2 x ^ 2 + 4x + 5, x > 1 , which one of the following is NOT correct?

JEE Main 2022 Solution View → 110

If the tangent at the point ( x _ 1 , y _ 1 ) on the curve y = x ^ 3 + 3 x ^ 2 + 5 passes through the origin, then ( x _ 1 , y _ 1 ) does NOT lie on the curve

JEE Main 2022 Solution View → 111

The sum of absolute maximum and absolute minimum values of the function f(x) = |2 x ^ 2 + 3x - 2| + sinxcosx in the interval [0, 1] is

JEE Main 2022 Solution View → 112

Let f_ 1 :(0, ∞) arrow ℝ and f_ 2 :(0, ∞) arrow ℝ be defined by f_ 1 (x)=∫_ 0 ^ x ∏_ j=1 ^ 21 (t-j)^ j d t, x>0 and f_ 2 (x)=98(x-1)^ 50 -600(x-1)^ 49 +2450, x 0 where, for any

JEE Advanced 2021 Solution View → 113

Let f_ 1 :(0, ∞) arrow ℝ and f_ 2 :(0, ∞) arrow ℝ be defined by f_ 1 (x)=∫_ 0 ^ x ∏_ j=1 ^ 21 (t-j)^ j d t, x > 0 and f_ 2 (x)=98(x-1)^ 50 -600(x-1)^ 49 +2450, x > 0 where, for

JEE Advanced 2021 Solution View → 114

Paragraph: Let ψ_ 1 :[0, ∞) arrow ℝ, ψ_ 2 :[0, ∞) arrow ℝ, f:[0, ∞) arrow ℝ and g:[0, ∞) arrow ℝ be functions such that f(0)=g(0)=0 , ψ_ 1 (x)=e^ -x +x, x ≥ 0 , ψ_ 2 (x)=x^ 2 -2

JEE Advanced 2021 Solution View → 115

Paragraph: Let ψ_ 1 :[0, ∞) arrow ℝ, ψ_ 2 :[0, ∞) arrow ℝ, f:[0, ∞) arrow ℝ and g:[0, ∞) arrow ℝ be functions such that f(0)=g(0)=0 , ψ_ 1 (x)=e^ -x +x, x ≥ 0 , ψ_ 2 (x)=x^ 2 -2

JEE Advanced 2021 Solution View → 116

Let f: R → R be defined by f(x) = x ^ 2 - 3x - 6 x ^ 2 + 2x + 4 Then which of the following statements is(are) TRUE?

JEE Advanced 2021 Solution View → 117

The function f(x) = x ^ 3 - 6 x ^ 2 + ax + b is such that f(2) = f(4) = 0 . Consider two statements: ( S _ 1 ) there exists x _ 1 , x _ 2 ∈ (2, 4), x _ 1 ( S _ 2 ) there exists x

JEE Main 2021 Solution View → 118

An angle of intersection of the curves, x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 and x ^ 2 + y ^ 2 = ab, a > b, is :

JEE Main 2021 Solution View → 119

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2) . If f(0) = 0, f(1) = 1 and f(2) = 2, then :

JEE Main 2021 Solution View → 120

Let f(x) be a cubic polynomial with f(1) = - 10, f(- 1) = 6, and has a local minima at x = 1, and f ^ ' (x) has a local minima at x = - 1 . Then f(3) is equal to .

JEE Main 2021 Solution View → 121

The number of real roots of the equation e ^ 4x + 2 e ^ 3x - e ^ x - 6 = 0 is :

JEE Main 2021 Solution View → 122

If R is the least value of a such that the function f(x) = x ^ 2 + ax + 1 is increasing on [ 1, 2 ] and S is the greatest value of a such that the function f ( x ) = x ^ 2 + ax +

JEE Main 2021 Solution View → 123

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

JEE Main 2021 Solution View → 124

Let M and m respectively be the maximum and minimum values of the function f ( x ) = ^ - 1 ( sinx + cosx ) in [0, (π)/(2)] . Then the value of tan ( M - m ) is equal to:

JEE Main 2021 Solution View → 125

A wire of length 20m 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

JEE Main 2021 Solution View → 126

The number of distinct real roots of the equation 3 x ^ 4 + 4 x ^ 3 - 12 x ^ 2 + 4 = 0 is _________.

JEE Main 2021 Solution View → 127

The local maximum value of the function, f(x) = ((2)/(x)) ^ x ^ 2 , x > 0, is

JEE Main 2021 Solution View → 128

A wire of length 36m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is

JEE Main 2021 Solution View → 129

Let f: (a, b) → R be twice differentiable function such that f(x) = ∫ _ a ^ x g(t)dt for a differentiable function g(x) . If f (x) = 0has exactly five distinct roots in (a, b)

JEE Main 2021 Solution View → 130

If a rectangle is inscribed in an equilateral triangle of side length 2 2 as shown in the figure, then the square of the largest area of such a rectangle is _____.

JEE Main 2021 Solution View → 131

Let a curve y = f(x) pass through the point (2, ( _ e 2) ^ 2 ) and have slope 2y x _ e x for all positive real values of x . Then the value of f(e) is equal to _____.

JEE Main 2021 Solution View → 132

Let f(x) = 3 ^ 4 x + 10 ^ 3 x + 6 ^ 2 x - 3 , x ∈ [- (π)/(6), (π)/(2)] . Then, f is :

JEE Main 2021 Solution View → 133

The number of real roots of the equation e ^ 6x - e ^ 4x - 2 e ^ 3x - 12 e ^ 2x + e ^ x + 1 = 0 is:

JEE Main 2021 Solution View → 134

Let f: R → R be defined as f(x) = \ cases - (4)/(3) x ^ 3 + 2 x ^ 2 + 3x, & x > 0 3x e ^ x , & x ≤ 0 cases .) Then f is increasing function in the interval

JEE Main 2021 Solution View → 135

The sum of all the local minimum values of the twice differentiable function f: R → R defined by f(x) = x ^ 3 - 3 x ^ 2 - 3 f ^ " ( 2 ) 2 x + f ^ " (1) is:

JEE Main 2021 Solution View → 136

Let A = [ a _ ij ] be a 3 × 3 matrix, where a _ ij = \ cases 1 & , & ifi = j - x & , & if|i - j| = 1 2x + 1 & , & otherwise cases ) Let a function f: R → R be defined as f(x) =

JEE Main 2021 Solution View → 137

Let a be a real number such that the function f ( x ) = a x ^ 2 + 6x - 15, x ∈ R is increasing in (- ∞ , (3)/(4)) and decreasing in ((3)/(4), ∞) . Then the function g ( x ) = a x

JEE Main 2021 Solution View → 138

Let P ( x ) be a real polynomial of degree 3 which vanishes at x = - 3 . Let P ( x ) have local minima at x = 1 , local maxima at x = - 1 and ∫ _ - 1 ^ 1 P(x)dx = 18 , then the

JEE Main 2021 Solution View → 139

Consider the function f: R → R defined by f(x) = \ cases (2 - sin((1)/(x))) | x | , & x ≠ 0 0, & x = 0 cases .) Then f is:

JEE Main 2021 Solution View → 140

Let f: [ - 1, 1 ] → R be defined as f ( x ) = a x ^ 2 + bx + c for all x ∈ [ - 1, 1 ] , where a, b, c ∈ R such that f ( - 1 ) = 2, f ^ ' ( - 1 ) = 1 and for x ∈ ( - 1, 1 ) the

JEE Main 2021 Solution View → 141

The maximum value of z in the following equation z = 6xy + y ^ 2 , where 3x + 4y ≤ 100 and 4x + 3y ≤ 75 for x ≥ 0 and y ≥ 0 is

JEE Main 2021 Solution View → 142

Let f be a real valued function, defined on R - \ - 1, 1\ and given by f(x) = 3 _ e |(x - 1)/(x + 1)| - (2)/(x - 1) . Then in which of the following intervals, function f(x) is

JEE Main 2021 Solution View → 143

The range of a ∈ R for which the function f(x) = (4a - 3)(x + _ e 5) + 2(a - 7)cot((x)/(2)) ^ 2 ((x)/(2)), x ≠ 2nπ, n ∈ N , has critical points, is :

JEE Main 2021 Solution View → 144

The triangle of maximum area that can be inscribed in a given circle of radius &#39; r &#39; is :

JEE Main 2021 Solution View → 145

Let the normals at all the points on a given curve pass through a fixed point (a, b) . If the curve passes through (3, - 3) and (4, - 2 2 ), given that a - 2 2 b = 3 , then ( a ^

JEE Main 2021 Solution View → 146

Let a be an integer such that all the real roots of the polynomial 2 x ^ 5 + 5 x ^ 4 + 10 x ^ 3 + 10 x ^ 2 + 10x + 10 lie in the interval (a, a + 1) . Then, | a | is equal to

JEE Main 2021 Solution View → 147

The maximum slope of the curve y = (1)/(2) x ^ 4 - 5 x ^ 3 + 18 x ^ 2 - 19x occurs at the point

JEE Main 2021 Solution View → 148

The shortest distance between the line x - y = 1 and the curve x ^ 2 = 2y is:

JEE Main 2021 Solution View → 149

If the curves x = y ^ 4 and xy = k cut at right angles, then (4k) ^ 6 is equal to ___ .

JEE Main 2021 Solution View → 150

If Rolle&#39;s theorem holds for the function f(x) = x ^ 3 - a x ^ 2 + bx - 4, x ∈ [1, 2] with f ^ ' ((4)/(3)) = 0 , then ordered pair (a, b) is equal to :

JEE Main 2021 Solution View → 151

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x ^ 6 is unity and it has extrema at x = - 1 and x = 1 . If _ x → 0 f(x) x ^ 3 = 1, then 5 · f(2) is equal to

JEE Main 2021 Solution View → 152

For which of the following curves, the line x + 3 y = 2 3 is the tangent at the point ( 3 3 2 , (1)/(2)) ?

JEE Main 2021 Solution View → 153

Let f: R → R be defined as f(x) = \ cases - 55x, & ifx 4 cases ) Let A = x ∈ R: f is increasing . Then A is equal to:

JEE Main 2021 Solution View → 154

If the curve y = a x ^ 2 + 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 Solution View → 155

If the area of the triangle formed by the x -axis, the normal and the tangent to the circle (x - 2) ^ 2 + (y - 3) ^ 2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

JEE Main 2021 Solution View → 156

The function f(x) = 4 x ^ 3 - 3 x ^ 2 6 - 2sinx + (2x - 1)cosx :

JEE Main 2021 Solution View → 157

If the tangent to the curve y = x ^ 3 at the point P(t, t ^ 3 ) 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 Solution View → 158

The minimum value of α for which the equation (4)/(sinx) + (1)/(1 - sinx) = α has at least one solution in (0, (π)/(2)) is______.

JEE Main 2021 Solution View → 159

Let the function f: (0, π) → ℝ be defined by f(θ) = (sinθ + cosθ) ^ 2 + (sinθ - cosθ) ^ 4 . Suppose the function f has a local minimum at θ precisely when θ ∈ \ λ _ 1 π, … , λ _ r

JEE Advanced 2020 Solution View → 160

Let a, b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y ^ 2 = 4 , and suppose the ellipse x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 passes

JEE Advanced 2020 Solution View → 161

Consider all rectangles lying in the region \ (x, y) ∈ R × R: 0 ≤ x ≤ (π)/(2)and0 ≤ y ≤ 2sin(2x)\ and having one side on the x-axis. The area of the rectangle which has the

JEE Advanced 2020 Solution View → 162

For a polynomial g(x) with real coefficients, let m _ g denote the number of distinct real roots of g(x) . Suppose S is the set of polynomials with real coefficients defined by S

JEE Advanced 2020 Solution View → 163

The set of all real values λ for which the function f(x) = (1 - ^ 2 x) . (λ + sinx), xε(- (π)/(2), (π)/(2)) , has exactly one maxima and exactly one minima, is :

JEE Main 2020 Solution View → 164

For all twice differentiable functions f: R → R , with f(0) = f(1) = f ' (0) = 0 ,

JEE Main 2020 Solution View → 165

If the tangent to the curve, y = f(x) = x _ e x, (x > 0) at a point (c, f (c)) is parallel to the line-segment joining the points (1, 0) and (e, e) ,then c is equal to :

JEE Main 2020 Solution View → 166

The position of a moving car at time t is given by f ( t ) = a t ^ 2 + bt + c, t > 0 , where a, bandc are real numbers greater than 1 . Then the average speed of the car over the

JEE Main 2020 Solution View → 167

Let AD and BC be two vertical poles at AandB respectively on a horizontal ground. If AD = 8m , BC = 11m , AB = 10m; then the distance (in meters) of a point M lying in between AB

JEE Main 2020 Solution View → 168

If x = 1 is a critical point of the function f ( x ) = (3 x ^ 2 + ax - 2 - a) e ^ x , then

JEE Main 2020 Solution View → 169

Which of the following points lies on the tangent to the curve x ^ 4 e ^ y + 2 y + 1 = 3 at the point ( 1, 0 ) ?

JEE Main 2020 Solution View → 170

Let f(x) = sin ⁡ x ^ 2 , x > 0 . Let x _ 1 Then which of the following options is/are correct?

JEE Advanced 2019 Solution View → 171

Let f: R → R be given by f(x) = (x - 1)(x - 2)(x - 5) . Define F(x) = ∫ _ 0 ^ x f(t)dt, x > 0 . Then which of the following options is/are correct?

JEE Advanced 2019 Solution View → 172

Let f: R → R be given by f(x) = \ cases x ^ 5 + 5 x ^ 4 + 10 x ^ 3 + 10 x ^ 2 + 3x + 1, & x Then which of the following options is/are correct?

JEE Advanced 2019 Solution View → 173

For every twice differentiable function f: ℝ → [- 2, 2]with (f(0)) ^ 2 + ( f ^ ′ (0)) ^ 2 = 85 , which of the following statement(s) is (are) TRUE?

JEE Advanced 2018 Solution View → 174

The number of real solutions of the equation ^ - 1 ⁡ ( ∑ _ i = 1 ^ ∞ x ^ i + 1 - x ∑ _ i = 1 ^ ∞ ((x)/(2)) ^ i ) = (π)/(2) - ^ - 1 ⁡ ( ∑ _ i = 1 ^ ∞ (- (x)/(2)) ^ i - ∑ _ i = 1 ^

JEE Advanced 2018 Solution View → 175

If f: R → R is a twice differentiable function such that f ^ ′′ (x) > 0 for all x ∈ R and f((1)/(2)) = (1)/(2), f(1) = 1 then

JEE Advanced 2017 Solution View → 176

If f(x) = | cases cos ⁡ 2x & cos ⁡ 2x & sin ⁡ 2x - cos ⁡ x & cos ⁡ x & - sin ⁡ x sin ⁡ x & sin ⁡ x & cos ⁡ x cases |, then

JEE Advanced 2017 Solution View → 177

Let f: R → (0, 1) , be a continuous function then, which of the following function(s), has(have) the values zero at some point in the interval, (0,1)?

JEE Advanced 2017 Solution View → 178

Let f(x) = x + _ e ⁡ x - x _ e ⁡ x, x ∈ (0, ∞) • Column 1 contains information about zeros of f( x ) , f ' ( x ) and f ' ' ( x ) • Column 2 contains information about the limiting

JEE Advanced 2017 Solution View → 179

Let f(x) = x + _ e ⁡ x - x _ e ⁡ x,x ∈ (0, ∞) . • Column 1 contains information about zeros of f (x) , f ^ ′ (x) and f ^ ′′ (x) . • Column 2 contains information about the

JEE Advanced 2017 Solution View → 180

Let f(x) = x + _ e ⁡ x - x _ e ⁡ x,x ∈ (0, ∞) . • Column 1 contains information about zeros of f (x) , f ^ ′ (x) and f ^ ′′ (x) . • Column 2 contains information about the

JEE Advanced 2017 Solution View → 181

The least value of α ∈ R for which, 4α x ^ 2 + (1)/(x) ≥ 1, for all x > 0, is

JEE Advanced 2016 Solution View → 182

The total number of distinct x ∈ [0, 1] for which, ∫ _ 0 ^ x t ^ 2 1 + t ^ 4 dt = 2x - 1 is

JEE Advanced 2016 Solution View → 183

Let f, g: [- 1, 2] → ℝ be continuous functions which are twice differentiable on the interval (- 1, 2) . Let the values of f and g at the points - 1, 0 and 2 be as given in the

JEE Advanced 2015 Solution View → 184

Paragraph: Let F: ℝ arrow ℝ be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F^ (x) Question: The correct statement(s) is(are)

JEE Advanced 2015 Solution View → 185

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of Vm m ^ 3 , has a 2 mm thick solid wall and is open

JEE Advanced 2015 Solution View → 186

Let f: (0, ∞) → R be given by f(x) = ∫ _ (1)/(x) ^ x e ^ - (t + (1)/(t)) (dt)/(t), then

JEE Advanced 2014 Solution View → 187

Let a ∈ R and let f: R → R be given by f(x) = x ^ 5 - 5x + a, then

JEE Advanced 2014 Solution View → 188

For every pair of continuous functions f, g: [0, 1] → R such that max ⁡ \ f(x): x ∈ [0, 1]\ = max ⁡ g(x): x ∈ [0, 1] , then the correct statement (s) is (are)

JEE Advanced 2014 Solution View → 189

The slope of the tangent to the curve (y - x ^ 5 ) ^ 2 = x (1 + x ^ 2 ) ^ 2 at the point (1, 3) is ________.

JEE Advanced 2014 Solution View → 190

The function f(x) = 2|x| + |x + 2| - ||x + 2| - 2|x|| has a local minimum or a local maximum at x =

JEE Advanced 2013 Solution View → 191

Paragraph: Let f:[0,1] arrow ℝ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f^ (x)-2 f^ (x)+f(x) ≥ e^ x

JEE Advanced 2013 Solution View → 192

Paragraph: Let f:[0,1] arrow ℝ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f^ (x)-2 f^ (x)+f(x) ≥ e^ x

JEE Advanced 2013 Solution View → 193

A line L : y = m x + 3 meets y - a x is at E(0,3) and the arc of the parabola y ^ 2 = 1 6 x , 0 ≤ y ≤ 6 at the point F ⁡ ( x _ 0 y _ 0 ) .The tangent to the parabola at F ⁡ ( x _

JEE Advanced 2013 Solution View → 194

The number of points in (- ∞ , ∞) , for which x ^ 2 - xsinx - cosx = 0 , is:

JEE Advanced 2013 Solution View → 195

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8: 15 is converted into an open rectangular box by folding after removing squares of equal area

JEE Advanced 2013 Solution View → 196

Let f(x)=(1-x)^ 2 ^ 2 x+x^ 2 for all x ∈ I R and let g(x)=∫_ 1 ^ x ((2(t-1))/(t+1)- t) f(t) d t for all x ∈(1, ∞) . Question: Which of the following is true?

JEE Advanced 2012 Solution View → 197

If f(x)=∫_ 0 ^ x e^ t^ 2 (t-2)(t-3) d t for all x ∈(0, ∞) , then

JEE Advanced 2012 Solution View → 198

Let f: I R arrow I R be defined as f(x)=|x|+|x^ 2 -1| . The total number of points at which f attains either a local maximum or a local minimum is

JEE Advanced 2012 Solution View → 199

Let p(x) be a real polynomial of least degree which has a local maximum at x=1 and a local minimum at x=3 . If p(1)=6 and p(3)=2 , then p^ (0) is

JEE Advanced 2012 Solution View → 200

The number of distinct real roots of x^4-4 x^3+12 x^2+x-1=0 is

JEE Advanced 2011 Solution View →
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Related JEE Mathematics chapters

Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Binomial Theorem (550)Quadratic Equation (517)Determinants (508)

Application of Derivatives in other exams

NDA Application of Derivatives (66)BITSAT Application of Derivatives (67)