Download App

Functions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSFunctions1 Mark
MCQ
If $\text{f(x)}=\frac{2^{\text{x}}+2^{-\text{x}}}{2},$ then f(x + y)f(x - y) is equal to:
  • ✓
    $\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
  • B
    $\frac{1}{2}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$
  • C
    $\frac{1}{4}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
  • D
    $\frac{1}{4}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$
✓

Answer

Correct option: A.
$\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
Given,
$\text{f(x)}=\frac{2^{\text{x}}+2^{-\text{x}}}{2}$
Now,
$\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\Big(\frac{2^{\text{x}+\text{y}}+2^{-\text{x}-\text{y}}}{2}\Big)\Big(\frac{2^{\text{x}-\text{y}}+2^{-\text{x}+\text{y}}}{2}\Big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{4}\big(2^{2\text{x}}+2^{-2\text{y}}+2^{2\text{y}}+2^{-2\text{x}}\big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\Big(\frac{2^{2\text{x}}+2^{-2\text{x}}}{2}+\frac{2^{2\text{y}}+2^{-2\text{y}}}{2}\Big)$
$\Rightarrow\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\big[\text{f}(2\text{x})+\text{f}(2\text{y})\big]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Functions→Functions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $x ^2-5| x +2|-4=0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^2+ mn + n ^2$ is equal to $..............$.
→
What is the coefficient of $x^{100}$ in $(1 + x + x^2 + x^3 +.... + x^{100})^3$ ?
→
Let $P$ be an arbitrary point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $a > b > 0$. Suppose $F_1$ and $F_2$ are the foci of the ellipse. The locus of the centroid of the $\Delta P F_1 F_2$ as $P$ moves on the ellipse is
→
If for real values of $x,\cos \theta = x + \frac{1}{x},$ then
→
A problem of mathematics is given to three students whose chances of solving the problem are $\frac{{1}}{{3}} , \frac{{1}}{{4}}$ and $\frac{{1}}{{5}}$ respectively. The probability that the question will be solved is
→
Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is $................$.
→
The roots of the quadratic equation $2{x^2} + 3x + 1 = 0$, are
→
Circles ${(x + a)^2} + {(y + b)^2} = {a^2}$ and ${(x + \alpha )^2}$ $ + {(y + \beta )^2} = $ ${\beta ^2}$ cut orthogonally, if
→
A person wants to climb a $n-$ step staircase using one step or two steps. Let $C_n$ denotes  the number of ways of climbing the $n-$ step staircase. Then $C_{18} + C_{19}$ equals
→
Let $n(A) = 3, \,n(B) = 3$ (where $n(S)$ denotes number of elements in set $S$), then number of subsets of $(A \times B)$ having odd number of elements, is-
→