मराठी

F is a Real Valued Function Given by F ( X ) = 27 X 3 + 1 X 3 and α, β Are Roots of 3 X + 1 X = 12 . Then,(A) F(α) ≠ F(β) (B) F(α) = 10 (C) F(β) = −10 (D) None of These - Mathematics

Advertisements
Advertisements

प्रश्न

f is a real valued function given by \[f\left( x \right) = 27 x^3 + \frac{1}{x^3}\] and α, β are roots of \[3x + \frac{1}{x} = 12\] . Then,

 
 

पर्याय

  • (a) f(α) ≠ f(β)

  • (b) f(α) = 10

  • (c) f(β) = −10

  • (d) None of these

     
MCQ

उत्तर

(d) None of these

Given: \[f\left( x \right) = 27 x^3 + \frac{1}{x^3}\] \[\Rightarrow f\left( x \right) = \left( 3x + \frac{1}{x} \right)\left( 9 x^2 + \frac{1}{x^2} - 3 \right)\]
\[\Rightarrow f\left( x \right) = \left( 3x + \frac{1}{x} \right)\left( \left( 3x + \frac{1}{x} \right)^2 - 9 \right)\]
\[\Rightarrow f\left( \alpha \right) = \left( 3\alpha + \frac{1}{\alpha} \right)\left( \left( 3\alpha + \frac{1}{\alpha} \right)^2 - 9 \right)\]
Since α and β are the roots of
 
\[3x + \frac{1}{x} = 12\]
\[3\alpha + \frac{1}{\alpha} = 12 \text{ and  } 3\beta + \frac{1}{\beta} = 12\]\[\Rightarrow f\left( \alpha \right) = 12\left( \left( 12 \right)^2 - 9 \right)\] and \[f\left( \beta \right) = 12\left( \left( 12 \right)^2 - 9 \right)\] \[\Rightarrow f\left( \alpha \right) = f\left( \beta \right) = 12\left( \left( 12 \right)^2 - 9 \right)\]
shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 3: Functions - Exercise 3.6 [पृष्ठ ४४]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 11
पाठ 3 Functions
Exercise 3.6 | Q 26 | पृष्ठ ४४

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine

(b) {x : f(x) = −2}


If  \[y = f\left( x \right) = \frac{ax - b}{bx - a}\] , show that x = f(y).

 

 


If  \[f\left( x \right) = \frac{x + 1}{x - 1}\] , show that f[f[(x)]] = x.

 

 


If f(x) = (a − xn)1/na > 0 and n ∈ N, then prove that f(f(x)) = x for all x.

 

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(ii) g − 


If f(x) = loge (1 − x) and g(x) = [x], then determine function:

(ii) fg


Write the domain and range of function f(x) given by

\[f\left( x \right) = \frac{1}{\sqrt{x - \left| x \right|}}\] .
 

Which of the following are functions?


If \[f\left( x \right) = \log \left( \frac{1 + x}{1 - x} \right)\] , then \[f\left( \frac{2x}{1 + x^2} \right)\]  is equal to

 

 


If  \[f\left( x \right) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}\] for x ∈ R, then f (2002) = 


If f : R → R be given by for all \[f\left( x \right) = \frac{4^x}{4^x + 2}\]  x ∈ R, then

 

Check if the following relation is function:


If f(x) = `{(x^2 + 3","  x ≤ 2),(5x + 7","  x > 2):},` then find f(3)


Check if the relation given by the equation represents y as function of x:

3x − 6 = 21


Find x, if g(x) = 0 where g(x) = `(5x - 6)/7`


Check the injectivity and surjectivity of the following function.

f : N → N given by f(x) = x2 


Express the following exponential equation in logarithmic form

10−2 = 0.01


Express the following exponential equation in logarithmic form

e–x = 6


Find the domain of f(x) = log10 (x2 − 5x + 6)


If `log(( x - y)/4) = logsqrt(x) + log sqrt(y)`, show that (x + y)2 = 20xy 


If f(x) = 3x + 5, g(x) = 6x − 1, then find (f + g) (x)


Select the correct answer from given alternative.

The domain and range of f(x) = 2 − |x − 5| is


Answer the following:

Let f: R → R be a function defined by f(x) = 5x3 – 8 for all x ∈ R, show that f is one-one and onto. Hence find f –1 


Answer the following:

Let f : R → R be given by f(x) = x3 + 1 for all x ∈ R. Draw its graph


Answer the following:

Find the domain of the following function.

f(x) = `(x^2 + 4x + 4)/(x^2 + x - 6)`


Find the domain of the following function.

f(x) = `sqrtlog(x^2 - 6x + 6)`


Let f(x) = 2x + 5. If x ≠ 0 then find `(f(x + 2) -"f"(2))/x`


A function f is defined by f(x) = 2x – 3 find x such that f(x) = f(1 – x)


The range of 7, 11, 16, 27, 31, 33, 42, 49 is ______.


The domain of the function f(x) = log3+x (x2 - 1) is ______.


If f(x) = `1/sqrt(4 - 3x)`, then dom(f) = ______..


If f(x) = `{{:(x^2",", x ≥ 0),(x^3",", x < 0):}`, then f(x) is ______.


If f(x) = `x^3 - 1/x^3`, then `f(x) + f(1/x)` is equal to ______.


Find the domain of the following functions given by f(x) = x|x|


If f(x) = `(x - 1)/(x + 1)`, then show that `f(1/x)` = – f(x)


Range of f(x) = `1/(1 - 2 cosx)` is ______.


Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x, and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ______.


If f: R `rightarrow` R be a function defined by f(x) = 4x3 – 7. Then ______.


If f : R – {2} `rightarrow` R i s a function defined by f(x) = `(x^2 - 4)/(x - 2)`, then its range is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×