MP Board Class 12th Maths Important Questions Chapter 5 Continuity and Differentiability

MP Board Class 12th Maths Important Questions Chapter 5 Continuity and Differentiability

MP Board Class 12th Maths Important Questions Chapter 5 Continuity and Differentiability

Important Questions

Short Answer Type Questions

Question 1.
Find all the points of discontinuity of f, when f is defined as:

Solution:
For x< 2, f(x) = 2x + 3 is polynomial function.
Hence, for x < 2, f(x) is continuous function. For x > 2, f(x) = 2x – 3 is polynomial function.
Hence, x > 2, f(x) is continous.
Now, we shall examine the continuty of f(x) at x = 2 only.
Put x = 2 + h,
When x → 2, then h → 0

Question 2.
Find all the points of discontunity of f, when f is defined as follow:

Question 3.
Examine the continuty of function f(x) at point x = 0?

Question 4.
Function f is defined as:

Then prove that function f is continous function for all points except x = 4?
Solution:

Hence, for x = 4 the function f(x) is dicontinous

When x < 4 then f(x) = -1 which is constant function Hence, it is continous function when x > 4 then f(x) = 1 which is constant function.
Hence, it is continous function
The given functions f is continous at all points except x = 4. Proved.

Question 5.
Find the value of k for which the function

Question 6.
Find the value of k, if function

Question 7.
Function f is continous at x = 0:

Question 8.
Find the relationship between a and b so the following function f defined by:

Question 9.
Prove that the funcion f(x) = |x – 1|, x ∈ R is not differentiable at x = 1? (NCERT)
Solution:
Given:

Question 10.
Show that the function:

Solution:
We know that:
RHD =

From the above, it is clear that,
LHD at x = 2 ≠ RHD at x = 2
∴ f(x) is not differentiable at x = 2. Proved.

Question 11.
Determine the function of defined by

Objective Type Questions:

Question 1.
Choose the correct answer:

Question 3.
The value of  (x³ + sin x²) is to be:
(a) 3x² + cos x²
(b) 3x² + x sin x²
(c) 3x² + 2x cos x²
(d) 3x² + x cos x²
Answer:
(c) 3x² + 2x cos x²

Question 4.
The value of  a^x is to be:
(a) a^x
(b) a^x log_a e
(c) a^x log_e a
(d)
Answer:
(c) a^x log_e a

Question 5.
If y = 500e^7x + 600e^-7x, then the value of  will be:
(a) 45 y
(b) 47 y
(c) 49 y
(d) 50 y
Answer:
(c) 49 y

Question 2.
Fill in the blanks:

1. Differential coefficient of cos x⁰ w.r.t x is ……………………………..
2. Differential coefficient of e^log_ea w.r.t x is ……………………………….
3. Differential coefficient of log_e a w.r.t x is ………………………………
4. Differential coefficient of a^x w.r.t x is ……………………………….
5. Differential coefficient of sin 3x w.r.t x is ……………………………….
6. If y = sin^-1  , then  = ………………………………
7. Differential coefficient of sin x w.r.t. cos x is …………………………………..
8. The value of  (log tan x) is ………………………………………
9. Differential coefficient of log (log sin x) ………………………………

Answer:

Question 3.
Write True/False:

1. Any function f(x) is said to be differentiatiable at any point x = a when Lf'(a) ≠ Rf'(a)?
2. Differential coefficient of sec^-1a w.r.t x is 0?
3. Every differentiatiable function is continous?
4. Differential coefficient of a^2x is a^2x log_ea?

Answer:

1. Flase
2. False
3. True
4. Flase

Question 4.
Match the Column:

Answer:

1. (d)
2. (e)
3. (a)
4. (f)
5. (h)
6. (g)
7. (c)
8. (b)

Question 5.
Write the answer in one word/sentence:

SabDekho

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