HSC Science (General) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

For what values of a and b is the function

f(x) `{:(= (x^2 - 4)/(x - 2)",", "for"  x < 2),(= "a"x^2 - "b"x + 3",", "for"  2 ≤ x < 3),(= 2x - "a" + "b"",", "for"  x ≥ 3):}}` continuous for every x on R?

Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for"  -3 ≤ x ≤ 2),(= |x - 5|",", "for"  2 < x ≤ 7):}`.

Discuss the continuity of f(x) at x = `pi/4` where, 

f(x) `{:(= ((sinx + cosx)^3 - 2sqrt(2))/(sin 2x - 1)",", "for"  x ≠ pi/4),(= 3/sqrt(2)",", "for"  x = pi/4):}`

Determine the values of p and q such that the following function is continuous on the entire real number line.

f(x) `{:(= x + 1",", "for"   1 < x < 3),(= x^2 + "p"x + "q"",", "for"  |x - 2| ≥ 1):}`

Show that there is a root for the equation 2x³ − x − 16 = 0 between 2 and 3.

Show that there is a root for the equation x³ − 3x = 0 between 1 and 2.

Let f(x) = ax + b (where a and b are unknown)

= x² + 5 for x ∈ R

Find the values of a and b, so that f(x) is continuous at x = 1

Suppose f(x) `{:(= "p"x + 3",", "for"  "a" ≤ x ≤ "b"),(= 5x^2 − "q"",", "for"  "b" < x ≤ "c"):}`

Find the condition on p, q, so that f(x) is continuous on [a, c], by filling in the blanks.

f(b) = ______

`lim_(x -> "b"^+) "f"(x)` = _______

∴ pb + 3 = _______ − q

∴ p = `"_____"/"b"` is the required condition

Select the correct answer from the given alternatives:

f(x) = `{:(= (2^(cotx) - 1)/(pi - 2x)",", "for"  x ≠ pi/2),(= log sqrt(2)",", "for"  x = pi/2):}`

Select the correct answer from the given alternatives:

If f(x) = `(1 - sqrt(2) sinx)/(pi - 4x), "for"  x ≠ pi/4` is continuous at x = `pi/4`, then `"f"(pi/4)` =

Select the correct answer from the given alternatives:

If f(x) = `((sin2x)tan5x)/("e"^(2x) - 1)^2`, for x ≠ 0 is continuous at x = 0, then f(0) is

Select the correct answer from the given alternatives:

f(x) = `(x^2 - 7x + 10)/(x^2 + 2x - 8)`, for x ∈ [– 6, – 3]

Select the correct answer from the given alternatives:

If f(x) `{:(= "a"x^2 + "b"x + 1",", "for"  |x −1| ≥ 3), (= 4x + 5",", "for"  -2 < x < 4):}` is continuous everywhere then,

Select the correct answer from the given alternatives:

f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for"  x ≠ 0),(= "k"",", "for"  x = 0):}` is continuous at x = 0, then ‘k’ =

Select the correct answer from the given alternatives:

f(x) `{:(= (32^x - 8^x - 4^x + 1)/(4^x - 2^(x + 1) + 1)",", "for"  x ≠ 0),(= "k""," , "for"  x = 0):}` is continuous at x = 0, then value of ‘k’ is

Select the correct answer from the given alternatives:

If f(x) = `(12^x - 4^x - 3^x + 1)/(1 - cos 2x)`, for x ≠ 0 is continuous at x = 0 then the value of f(0) is ______.

Select the correct answer from the given alternatives:

If f(x) = `((4 + 5x)/(4 - 7x))^(4/x)`, for x ≠ 0 and f(0) = k, is continuous at x = 0, then k is

Select the correct answer from the given alternatives:

If f(x) = [x] for x ∈ (–1, 2) then f is discontinuous at

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= (x^2 - 3x - 10)/(x - 5)",", "for"  3 ≤ x ≤ 6","  x ≠ 5),(= 10",", "for"  x = 5),(=(x^2 - 3x - 10)/(x - 5)",", "for"  6 < x ≤ 9):}`

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= 2x^2 - 2x + 5",", "for"  0 ≤ x ≤ 2),(= (1 - 3x - x^2)/(1 - x) "," , "for"  2 < x < 4),(= (x^2 - 25)/(x - 5)",", "for"  4 ≤ x ≤ 7 and x ≠ 5),(= 7",", "for"  x = 5):}`