JEE Main Previous Year Questions (2016- 2024): Differentiation | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR PDF Download

Q.1. The number of real roots of the equation is    (2020)
(1) 1
(2) 3
(3) 2
(4) 4
Ans. (1)
The given equation is

Let e^x = t, so
Dividing both sides of the equation by t²,we get

Again, 
Let 
Then, y² +y −3 =0 ⇒ (y − 2)(y +3) = 0
⇒ y = -2, 3
Now, and (not possible)
Hence, e^x = 1 ⇒ x = 0 

Q.2. If x = 2 sinθ - sin2θ and y = 2cosθ - cos2θ, θ∈[0,2π], then at θ = π is    (2020)
(1) 3/4
(2)
(3) 3/2
(4)
Ans. (Bonus)
We have,
    (1)
    (2)
From Eqs. (1) and (2), we have

Now,


Q.3. If x = 3 tan t and y = 3 sec t, then the value of at t = π/4, is:     (2019)



(4) 1/6
Ans. (2)
Solution.




Q.4. Let f : R → R be a function such that f(x) = x³ + x²f'(1) + xf"(2) + f"'(3), x∈R Then f(2) equals:     (2019)
(1) -4
(2) 30
(3) -2
(4) 8
Ans. (3)
Solution.







Q.5. If x log_e (log_e x) - x² + y² = 4(y > 0), then dy/dx at x = e is equal to:     (2019)




Ans. (2)
Solution. Consider the equation,
x log_e (log_e x) - x² + y² = 4
Differentiate both sides w.r.t. x,




Q.6. For x > 1, if (2x)^2y = 4e^2x-2y, then (1 + log_e 2x)² dy/dx equal to:     (2019)

(2) log_e 2x

(4) x log_e 2x
Ans. (1)
Solution.
(2x)^2y = 4e^2x-2y
⇒




Q.7. Let f be a differentiable function such that f(1) = 2 and f' (x) = f(x) for all x ∈ R. If h (x) = f(f(x)), then h' (1) is equal to:     (2019)
(1) 2e²
(2) 4e
(3) 2e
(4) 4e²
Ans. (2)
Solution. 
Since, f'(x) = f(x)



Since, the given condition
f(1) = 2
From eqⁿ (1)
f(x) = e^x+c = e^ce^x
Then, f(1) = e^c.e¹
⇒ 2 = e^c.e
⇒ 2/e = e^c
Then, from eqⁿ (1)

⇒
Now
⇒


Q.8. is equal to:     (2019)




Ans. (Bouns)
Solution.




Q.9. If f(1) = 1, f'(1) = 3, then the derivative of f(f(f(x))) + (f(x))² at x = 1 is:     (2019)
(1) 33
(2) 12
(3) 15
(4) 9
Ans. (1)
Solution.


Q.10. If e^y + xy = e, the ordered pair  at x = 0 is equal to:    (2019)




Ans. (2)
Solution.
Given, e^y +xy = e    ...(i)
Putting x = 0 in (i), ⇒ e^y = e ⇒ y = 1
On differentiating (i) w. r. to x
    ...(ii)
Putting y = 1 and x= 0 in (ii),

On differentiating (ii) w. r. to x,



Q.11. The derivative of  , with respect to x/2, where     (2019)
(1) 1
(2) 2/3
(3) 1/2
(4) 2
Ans. (4)
Solution.



Q.12. If  then dy/dx is equal to:     (2018)
(1) y/x
(1) - y/x
(1) - x/y
(1) x/y
Ans: (2)
Solution:


Q.13. If x² + y² + sin y = 4, then the value of d²y/dx² at the point (-2, 0) is:    (2017)
(1) -34
(2) -2
(3) 4
(4) -32
Ans: (1)
Solution:

 

Q.14. Let f be a polynomial function such that f(3x) = f'(x). f"(x), for all x ∈ R. Then:    (2017)
(1) f(2) + f'(2) = 28
(2) f"(2) - f'(2) = 0
(3) f(2) - f'(2) + f"(2) = 10
(4) f"(2) - f(2) = 4
Ans. (2)
Solution.










Q.15. For x ∈ R, x ≠ 0, x ≠ 1, let f₀(x)= 1/1-x and f_{n + 1}(x) = f₀ (f(_n(X)), n = 0, 1, 2, ....... Then the value of  is equal to:    (2016)
(1) 4/3
(2) 1/3
(3) 5/3
(4) 8/3
Ans. (3)
Solution.
We have