Class XII ISC Chapter Continuity and Differentiability
Explore our extensive collection of Continuity and Differentiability questions, tailored for class 12 students following the ISC curriculum. This topic is vital for understanding how to optimize a particular outcome given certain constraints. Our practice questions cover various aspects of Linear Programming, including formulating linear inequalities, graphical methods, and finding optimal solutions. Whether you're looking to strengthen your problem-solving skills or prepare for your exams, these questions provide the perfect resource to master Continuity and Differentiability concepts with ease.
class 12 C&D exercise5-1
Continuity & Differentiability
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Q1 Examine the following functions for continuity at the indicated points :
(i) x3+3 , x≠0 1 , x=0
at x=0
(ii) f(x) = x3 + 2x2-1 at x =1.
| x3+3 , x≠0 |
| 1 , x=0 |
Q2 Prove that the function f(x) = 5x - 3 is continuous at x = 0, at x = -3 and at x = 5.
Q3 (i) If f(x) = x2- 1 / x - 1 for x ≠ 1 and f(x) = 2 when x = 1, show that the function is continous at x = 1
(ii) A function f is defined as f(x) = x2 - x - 6 / x - 3 , x ≠ 3 5 x = 3
Show that f is continuous at x = 3
| x2 - x - 6 / x - 3 , x ≠ 3 |
| 5 x = 3 |
Q4 Is the function f defined by f(x) = x , if x≤1 5, if x > 1
continous at
(i) x = 0 (ii) x = 1 (ii) x = 2 ?
| x , if x≤1 |
| 5, if x > 1 |
Q5 Is the function f defined by f(x) = 3x + 5, if x≥1 x2, if x<2< /td>
continuous at x = 2?
| 3x + 5, if x≥1 |
| x2, if x<2< /td> |
Q6 Is the function f defined by f(x) = 2x2 - 3x - 2 / x - 2 , x ≠ 2 5, x = 2
continuous at x = 2?
| 2x2 - 3x - 2 / x - 2 , x ≠ 2 |
| 5, x = 2 |
Q7 Is the function f defined by f(x)= x / sin2x ,when x ≠ 0 2, x = 0
continuous at x = 0 ?
| x / sin2x ,when x ≠ 0 |
| 2, x = 0 |
Q10 If f(x) = x2 - x -6 / x2 - 2x -3 , x ≠ 3 k, x = 3
find k so that the function f may be continuous at x = 3.
| x2 - x -6 / x2 - 2x -3 , x ≠ 3 |
| k, x = 3 |
Q11 If f(x) = tan 3x / kx , x ≠ 0 1, x = 0
find k so that the function f may be continuous at x = 0.
| tan 3x / kx , x ≠ 0 |
| 1, x = 0 |
Q16 Examine the following functions for continuity at the indicated points :
(i)f(x) = x2, x≥0 -x, x<0< /td>
at x=0
(ii) f(x) = 5x-4, if x<1< /td> 4x2-3x, if x≥1
at x=0
| x2, x≥0 |
| -x, x<0< /td> |
| 5x-4, if x<1< /td> |
| 4x2-3x, if x≥1 |
Q17 Is the function f defined by
f(x) = 2x2- 3, x≤1 5, x>1
continuous at x = 0? What about its continuity at x =1 and at x =2?
| 2x2- 3, x≤1 |
| 5, x>1 |
Q18 Examine the following functions for continuity :
(i) f(x) = x / |x| , x ≠ 0 0, x = 0
at x = 0
(ii) f(x)= x - 4 / 2|x - 4| , x ≠ 4 0, x = 4
at x = 4
| x / |x| , x ≠ 0 |
| 0, x = 0 |
| x - 4 / 2|x - 4| , x ≠ 4 |
| 0, x = 4 |
Q19 Examine the following functions for continuity :
(i) f(x)= x / sin3x , x ≠ 0 3, x = 0
at x = 0
(ii) f(x)= sin2x / sin3x , x ≠ 0 2, x = 0
at x = 0
| x / sin3x , x ≠ 0 |
| 3, x = 0 |
| sin2x / sin3x , x ≠ 0 |
| 2, x = 0 |
Q20
(i) If f(x) = x2 - 25 / x - 5 , x ≠ 5 k, x = 5
is continuous at x = 5, find the value of k.
(ii) Find the value of k so that the function : kx + 1, if x≤5 3x-5, if x>5
is continuous at x = 5.
(iii) For what value of k is the following function continuous at x = 2?
2x + 1, x < 2 k , x = 2 3x - 1, x > 2
(iv) Find the value of k so that the function f defined by
kx + 1, if x≤π cosx, if x>π
is continuous at x = π.
(v) For what value of k is the function
f(x) = tan 5x / sin 2x , x ≠ 0 k, x = 0
is continuous at x = 0.
(vi) Find the value of the constant k so that the function
f(x) = 1 - cos4x / 8x2 , x ≠ 0 k, x = 0
is continuous at x = 0.
| x2 - 25 / x - 5 , x ≠ 5 |
| k, x = 5 |
| kx + 1, if x≤5 |
| 3x-5, if x>5 |
| 2x + 1, x < 2 |
| k , x = 2 |
| 3x - 1, x > 2 |
| kx + 1, if x≤π |
| cosx, if x>π |
| tan 5x / sin 2x , x ≠ 0 |
| k, x = 0 |
| 1 - cos4x / 8x2 , x ≠ 0 |
| k, x = 0 |
Q21 Find the relationship between a and b so that the function f defined by
ax + 1, if x≤3 bx + 3, if x>3
is continuous at x = 3.
| ax + 1, if x≤3 |
| bx + 3, if x>3 |
Q22 If the function f defined by f(x) = 3ax + b, x > 1 11 , x = 1 5ax - 2b, x < 1
is continuous at x = 1, find the values of a and b.
| 3ax + b, x > 1 |
| 11 , x = 1 |
| 5ax - 2b, x < 1 |
