Read the following mathematical statements carefully: I. There ca… - Vidyadip

MCQ

Read the following mathematical statements carefully:

$I.$ There can exist two triangles such that the sides of one triangle are all less than $1$ cm while the sides of the other triangle are all bigger than $10$ metres, but the area of the first triangle is larger than the area of second triangle.

$II$ .If $x, y, z$ are all different real numbers, then $\frac{1}{{{{(x - y)}^2}}} + \frac{1}{{{{(y - z)}^2}}} + \frac{1}{{{{(z - x)}^2}}}$ $=$ ${\left( {\frac{1}{{x - y}} + \frac{1}{{y - z}} + \frac{1}{{z - x}}} \right)^2}$

$III$. $log_3x · log_4x · log_5x = (log_3x · log_4x) $$+ (log_4x · log_5x) + (log_5x · log_3x)$ is true for exactly for one real value of $x.$

$IV$. $A$ matrix has $12$ elements. Number of possible orders it can have is six. Now indicate the correct alternatively.

✓
exactly one statement is $INCORRECT.$
B
exactly two statements are $INCORRECT.$
C
exactly three statements are $INCORRECT.$
D
All the four statements are $INCORRECT$ .

✓

Answer

Correct option: A.

exactly one statement is $INCORRECT.$

a
$I$ .Consider $2\Delta _1 = b_1h_1 $ $2\Delta_ 2 = b_2h_2$ $b_1h_2 > b_2h_2$

first $D$ is each with $1$ cm, $2\Delta 1$ $\frac{{\sqrt 3 }}{2}$ $=$ $2^{nd}$ triangle

$2\Delta_ 2$ $= 10^{-6} ==>\Delta _1 > \Delta_2$

$II$. $RHS =$ $\frac{1}{{{{(x\, - \,y)}^2}}}\,\, + \,\,\frac{1}{{{{(y\, - \,z)}^2}}}\,\, + \,\,\frac{1}{{{{(z\, - \,x)}^2}}}$ $+$ $\frac{2}{{(x - y)(y - z)}}$ $+$ $\frac{2}{{(y - z)(z - x)}}$ $+$ $\frac{2}{{(x - y)(z - x)}}$

$=$ $\frac{1}{{{{(x\, - \,y)}^2}}}\,\, + \,\,\frac{1}{{{{(y\, - \,z)}^2}}}\,\, + \,\,\frac{1}{{{{(z\, - \,x)}^2}}}$ $+$ $\frac{{2[\not z - \not x + \not x - \not y + \not y - \not z]}}{{(x - t)(z - x)}}$ $=$ $LHS$ ==> True

$III$.$x = 1$ and $60$ are two solutions.

$IV$.Possible orders $(1 × 12) ; (12 × 1) ; (2 × 6) ; (6 × 2) ; (3 × 4) ; (4 × 3)$

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