Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Mathematics (Maths) Class 12

Commerce : Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

The document Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev is a part of the Commerce Course Mathematics (Maths) Class 12.
All you need of Commerce at this link: Commerce

Q.1. Let the function, f :[ 7,0] → Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev be continuous on [−7, 0] and differentiable on  (−7, 0). If f (−7) = −3 and f'(x) < for all x ∈(-7, 0), then for all such functions f, f(-1) + f(0)  lies in the interval    (2020)
(1) (-∞ , 20]
(2) [ -3,11]
(3) ( -∞ ,11]
(4) [ -6, 20]

Ans. (1)
Using LMVT in [ -7, -1], we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev.....(1)
Using LMVT in [ -7, -0], we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev....(2)
From Eqs. (1) and (2), we get f(0) + f(-1) < 20

Q.2. Let S be the set of points where the functionPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis not differentiable. ThenPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to ________.    (2020)

Ans. (3.00)
We have,
f (x) = Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
It is clear from the graph that the function is not differentiable at x =1,3 and 5.
Now, Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
= f(0) + f(2) + f (0) = 1 + 1 + 1 = 3

Q.3. The value of c in the Lagrange’s mean value theorem for the function f(x) = x3 - 4x2 + 8x + 11, when x ∈[0,1] is    (2020)
(1) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3) 2/3
(4) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (2)
f(x) is a polynomial function, so it is continuous and differentiable in [0,1].
Now, f(x) = x3 - 4x2 + 8x + 11 ⇒ f'(x) = 3x2 - 8x + 8
f(0) = 11 and f(1) = 1 − 4 + 8 + 11 = 16
Using Lagrange’s mean value theorem, we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.4. If the function f defined on Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis continuous, then k is equal to ________.    (2020)
Ans. (5.00)
We have,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
The function is continuous, then
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
⇒ k =3+2 =5

Q.5. If c is a point at which Rolle’s theorem holds for the function Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev  in the interval [3, 4], where Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev then f"(c) is equal to    (2020)
(1)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2)1/12
(3) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (2)
Given, Rolle’s theorem holds for the function Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevin the interval [3, 4], Then,Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now,Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.6. Let S be the set of all functions f : [0,1] →Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevwhich are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists a c ∈(0,1), depending on f, such that    (2020)
(1) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Ans. (Bonus)
S is a set of all functions
If we consider a constant function, then option 1, 3 and 4 are incorrect.
For option 2:
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
This may not be true for all the function
If we apply LMVT in (c, 1) then
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev such that
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
None of the option are incorrect

Q.7. Let f be any function continuous on [a,b] and twice differentiable on (a,b). If for all Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev and f"(x)< 0, then for any Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis greater than    (2020)
(1)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2) 1
(3) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (4)
Using LMVT forPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevwe get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Also, using LMVT for x ∈ [c,b], we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Since f"(x)< 0, then f'(x) is decreasing. Hence,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.8. IfPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis continuous at x = 0, then a + 2b is equal to    (2020)
(1) 1
(2) -1
(3) 0
(4) -2
Ans. (3)
Given, the function f(x) is continuous at x = 0, then
f(0-) = f(0) = f(0+)
Therefore,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now, Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Therefore, a + 3 = b = 1 ⇒ a = -2 and b = 1
⇒ a+2b = −2+2= 0

Q.9. Let [t] denote the greatest integer ≤ t and Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevThen, the function f(x) = [x2]sin(πx) is discontinuous, when x is equal to   (2020)
(1)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (1)
We have,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
On solving options (2),(3) and (4), we obtain the integer values and integral multiple of π in sin is always zero.
For option (1), we getPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevHence, f (x) is discontinuous at x = Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.10. Let Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev(a, k >0) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev=0, then k is    (2020)
(1) 3/2
(2) 4/3
(3) 2/3
(4) 1/3

Ans. (3)
Given,
xk + yk = ak , (a, k > 0)... (1)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now, using Eq. (1), we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
On comparing with Eq. (2), we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.11. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal t0 _______.    (2020)
Ans. 
(36.00)
Given
 Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.12. Let y = y (x) be a function of x satisfying Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevwhere k is a constant and Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Then Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to    (2020)
(1) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (2)
Given
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Let x = sin θ and y = sin ϕ, then the equation becomes
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev.....(1)
Differentiating Eq. (1) w.r.t. x, we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.13.Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to    (2020)
(1) 1/e
(2) 1/e2
(3) e2
(4) e

Ans. (2)
We have,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.14. Let f(x) = (sin(tan-1 x) + sin (cot-1 x))2 -1, Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevand Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevthenPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to    (2020)
(1) 2π/3
(2) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3) 5π/6
(4) π/3
Ans.
(3)
Let tan-1 x =θ  ⇒ x = tanθ ⇒ Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now, Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Given,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
 Therefore,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.15. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to    (2020)
(1) 0
(2) 1/10
(3)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Ans. (1)
We have,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev....(using L Hospital rule)

Q.16. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev      (2019)
(1) exists and equals Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

(2) exists and equals Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3) exists and equals Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4) does not exist
Ans. 
(1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.17. For each x∈R, let [x] be greatest integer less than or equal to x. Then

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) - sin 1
(2) 1
(3) sin 1 
(4) 0

Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.18. For each t∈R, let [t] be the greatest integer less than or equal to t. Then,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) equals 1
(2) equals 0
(3) equals - 1 
(4) does not exist

Ans. (2)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.19. Let f : R → R be a function defined as
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Then, f is:       (2019)
(1) continuous if a = 5 and b = 5
(2) continuous if a = - 5 and b = 10
(3) continuous if a = 0 and b = 5
(4) not continuous for any values of a and b

Ans. (4)
Solution.
Let f(x) is continuous at x = 1, then
f(1-) = f(1) = f(1+)
  5 = a + b    ...(1)
Let f(x) is continuous at x = 3, then
f(3-) = f(3) = f(3+)
a + 3b = b + 15    ...(2)
Let f(x) is continuous at x = 5, then
f(5-) = f(5) = f(5+)
  b + 25 = 30
b = 30 - 25 = 5
From (1), a = 0
But a = 0, b = 5 do not satisfy equation (2)
Hence, f(x) is not continuous for any values of a and b

Q.20.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Let S be the set of points in the interval (- 4, 4) at which f is not differentiable. Then S:       (2019)

(1) is an empty set
(2) equals {- 2, - 1, 0, 1, 2}
(3) equals {-2, - 1, 1, 2}
(4) equals {-2,2}
Ans. (2)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
∵ f(x) is not differentiable at -2, -1,0, 1 and 2.
∴ S = {-2,-1,0, 1,2}

Q.21. Let f be a differentiable function such that f'(x) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev (x > 0) and f'(1) ≠ 4. Then Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) exists and equals 4/7.
(2) exists and equals 4.
(3) does not exist.
(4) exists and equals 0.

Ans. (2)
Solution. Let y = f(x)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Solution of differential equation
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.22. Let [x] denote the greatest integer less than or equal to x. Then:
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) does not exist
(2) equals π
(3) equals π + 1
(e) equals 0

Ans. (1)
Solution.

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Since, LHL ≠ RHL
Hence, limit does not exist.

Q.23. 
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 0
(2) 2
(3) 4
(4) 1
Ans. (4)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.24. Let f = R → R be differentiable at c ∈ R and f (c) = 0. If g (x) = |f(x)|, then at x = c, g is:       (2019)
(1) not differentiable if f' (c) = 0

(2) differentiable if f’(c) ≠ 0
(3) differentiable if f’(c) = 0
(4) not differentiable
Ans. (3)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.25.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
is continuous at x = 0, then the ordered pair (p, q) is equal to:       (2019)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (3)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.26. Let f : R → R be a continuously differentiable function such that f(2) = 6 and f'(2) = 1/48.       (2019)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(1) 18
(2) 24
(3) 12
(4) 36
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.27. 
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 4
(2) 4√2
(3) 8√2
(4) 8

Ans. (4)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.28. 
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (2)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.29. If the function f defined on Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
is continuous, then k is equal to:       (2019)
(1) 2
(2) 1/2
(3) 1
(4) 1/√2
Ans. (2)
Solution.
Since, f(x) is continuous, then
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.30. Let f(x) = 15 - |x - 10|; x ∈ R. Then the set of all values of x, at which the function, g(x) = f (f(x)) is not differentiable, is:        (2019)
(1) {5,10,15}
(2) {10,15}
(3) {5, 10,15,20}
(4) {10}

Ans. (1)
Solution.
Since, f(x)= 15 - |(10-x)|
∴ g(x) = f(f(x)) = 15 - |10 - [15 - |10-x|]|
= 15 — || 10 — x| — 5|
∴ Then, the points where function g(x) is Non-differentiable are
10 - x = 0 and |10 — x| = 5
⇒ x = 10 and x - 10 = ± 5
⇒ x = 10 and x = 15, 5

Q.31. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev where [x] denotes the greatest
integer function, then:        (2019)
(1) f is continuous at x = 4.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.32. If the function Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev is continuous at x = 5, then the value of a - b is:       (2019)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (4)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
∵ Function is continuous at x = 5
LHL = RHL
(5 - π) b + 3 = (5 - π) a + 1
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.33. 
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 4√2
(2) 2 
(3) 22 
(4) 4
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.34. Let f : R → R be a differentiable function satisfying f'(3)+f’(2) = 0. Then Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev is equal to:       (2019)
(1) 1
(2) e-1
(3) e 
(4) e2
Ans. 
(1)
Solution.

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.35. If f: R → R is a differentiable function and f(2) = 6, then Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 24f'(2)
(2) 2f'(2)

(3) 0
(4) 12f'(2)

Ans. (4)
Solution. 
Using L' Hospital rule and Leibnitz theorem, we get
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Putting x = 2,
2f(2)f'(2) = 12f'(2) [∵ f(2) = 6]

Q.36.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 8/3
(2) 3/8
(3) 3/2
(4) 4/3
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.37.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
g(x)= |f(x)|+f(|x|). Then, in the interval (-2,2), g is:       (2019)
(1) differentiable at all points
(2) not continuous
(3) not differentiable at two points
(4) not differentiable at one point

Ans. (4)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
⇒ g(x) is non-differentiable at x = 1
⇒ g(x) is not differentiable at one point.

Q.38. Let K be the set of all real values of x where the function f(x) = sin |x| - |x| + 2 (x - π) cos |x| is not differentiable. Then the set K is equal to:       (2019)
(1) φ (an empty set)
(2) {π}
(3) {0}
(4) {0,π}

Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Then, function f(x) is differentiable for all x < 0
Now check for x = 0
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Then, function  f(x) is differentiable for x = 0. So it is differentiable everywhere
Hence, k = φ

Q.39. Let S be the set of all points in (- π , π) at which the function f(x) = min {sinx, cosx} is not differentiable. Then S is a subset of which of the following?       (2019)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (2)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
∵ f(x) is not differentiable at Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.40. Let f: [- 1, 3] → R be defined as
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at:       (2019)
(1) only one point
(2) only two points
(3) only three points
(4) four or more points

Ans. (3)
Solution. Given fumction is,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
f(x) is discontinuous at x = {0, 1, 3}
Hence, f(x) is discontinuous at only three points.

Q.41. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev then a + b is equal to:       (2019)
(1) -4 
(2) 5
(3) -7
(4) 1
Ans. (3)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
⇒ 2 - a = 5 ⇒ a = - 3 and b = - 4
Then a + b = -3-4 = -7

Q.42.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 6
(2) 2 
(3) 3 
(4) 1
Ans. (2)
Solution.
Given limit is,
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.43. Let f(x) = 5-|x - 2| and g(x) = |x+ 1|, x ∈ R. If f(x) attains maximum value at α and g(x) attains minimum value at β, then Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev       (2019)
(1) 1/2 
(2) -3/2 
(3) -1/2 
(4) 3/2
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
By the graph f(x) is maximum at x = 2
∴ α = 2
g (x) = |x + 1|
Graph of y = g (x)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
By the graph g (x) is minimum at x = - 1
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.44. Let f: (- 1, 1) → R be a function defined by f(x) = maxPrevious year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev If K be the set of all points at which f is not differentiable, then K has exactly:       (2019)
(1) five elements
(2) one element
(3) three elements
(4) two elements

Ans. (3)
Solution. Consider the function
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Now, the graph of the function
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
From the graph, it is clear that f(x) is not differentiable at Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Hence, K has exactly three elements.

Q.45. For each t ∈ R, let [t] be the greatest integer less than or equal to t. Than Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev     (2018)
(1) is equal to 0
(2) is equal to 50
(3) is equal to 120
(4) does not exist (in R)
Ans. 
(3)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.46. Let S = {t ∈ R:f(x)= |x - π|.(e|x| - 1) sin |x| is not differentiable at t}. Then the set S is equal to:    (2018)
(1) ϕ (an empty set)
(2) {0}
(3) {π}
(4) {0,π}
Ans. 
(1)
Solution.

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.47. Let S = {λ, µ) ∈ R × R : f(t) = (|λ|e|t| − µ).sin (2|t|), t ∈R, is a differentiable function}. Then S is a subset of∶    (2018)
(1) R × [0, ∞)
(2) R × (-∞, 0)
(3) (-∞, 0) × R
(4) [-∞, 0) × R
Ans: 
(1)
Solution:
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.48. If f(x) = Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev     (2018)
(1) Exists and is equal to -2
(2) Exists and is equal to 0
(3) Does not exist
(4) Exists and is equal to 2
Ans: 
(1)
Solution:

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
= cos x − tan x [x2 − 2x2]
= x2 tan x − x2 cos x
f′(x) = 2x tan x + x2 sec2 x − 2x cos x + x2 sin x
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev = 2 tan x + x sec2 x − 2 cos x + x sin x
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev = 0 + 0 − 2 + 0 = −2

Q.49.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev equals:    (2018)
(1) -1/6
(2) 1/6
(3) 1/3
(4) -1/3
Ans: 
(1)
Solution:
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Here ‘L’ is in the indeterminate i.e., 0/0
∴ using the  L’ Hospital rule we get:
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.50. If the function f defined as f(x) = Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev is continuous at x = 0, then the ordered pair (k, f(0)) is equal  to:    (2018)
(1) (2, 1)
(2) (3, 1)
(3) (3, 2)
(4)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans:
(2)
Solution:

If the function is continuous at x = 0, then
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
For the limit to exist, power of x in the numerator should be greater than or equal to the power of x in the denominator. Therefore, coefficient of x in numerator is equal to zero
⇒ 3 - k = 0
⇒ k = 3
So the limit reduces to
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.51. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevequals     (2017)
(1) 1/4
(2) 1/24
(3) 1/16
(4) 1/8
Ans. 
(3)
Solution.

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
= 1/16.

Q.52. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev     (2017)
(1) 1/√2
(2) 1/2√2
(3) √3/2
(4) √3

Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.53. The value of K for which the function Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev is continuous at x = π/2, is    (2017)
(1) 2/5
(2) Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3) 17/20
(4) 3/5
Ans. (4)
Solution.
k + 2/5 = 1
k = 1 - 2/5 = 3/5

Q.54.Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev for some positive real number a, then a is equal to     (2017)
(1) 17/2
(2) 15/2
(3) 7
(4) 8
Ans. (3)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.55. Let p = Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevthen loge p is equal to:     (2016)
(1) 2
(2) 1
(3) 1/2
(4) 1/4
Ans. 
(3)
Solution:

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.56. For x ∈ R, f(x) = |log 2 - sin x| and g(x) = f(f(x)), then:    (2016)
(1) g is not differentiable at x = 0
(2) g'(0) = cos (log 2)
(3) g'(0) = - cos(log 2)
(4) g is differentiable at x = 0 and g'(0) = -sin(log 2)
Ans. 
(2)
Solution:

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.57.Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRevis equal to:    (2016)
(1)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(2)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3)Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(4) 3 log 3 - 2
Ans. 
(2)
Solution.

Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
P = 27/e2

Q.58. If f(x) is a differentiable function in the interval (0, ∞) such that f(1) = 1 and Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev for each x > 0, then f(3/2) is equal to:    (2016)
(1) 13/6
(2) 23/18
(3) 25/9
(4) 31/18
Ans. (4)
Solution.
Differentiate w.r.t. t
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.59. Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev     (2016)
(1) 2/3
(2) 3/2 
(3) 2
(4) 1/2
Ans. (2)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev must be of the from 1
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
= e2a = e3
a = 3/2

Q.60. Let a,b ∈ R, (a ≠ 0). If the function f defined as     (2016)
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Ans. (1)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Q.61.Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev    (2016)
(1) 2
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
(3) 1/2
(4) -2
Ans. (4)
Solution.
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev
Previous year Questions (2016-20) - Limits, Continuity and Differentiability Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Related Searches

Free

,

study material

,

Previous Year Questions with Solutions

,

Extra Questions

,

practice quizzes

,

Sample Paper

,

Continuity and Differentiability Notes | EduRev

,

past year papers

,

Objective type Questions

,

Summary

,

Exam

,

Semester Notes

,

Important questions

,

Continuity and Differentiability Notes | EduRev

,

Previous year Questions (2016-20) - Limits

,

mock tests for examination

,

MCQs

,

Previous year Questions (2016-20) - Limits

,

Continuity and Differentiability Notes | EduRev

,

pdf

,

video lectures

,

Viva Questions

,

Previous year Questions (2016-20) - Limits

,

ppt

,

shortcuts and tricks

;