| Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below. |
| Assertion The distance between the points \[\left( 10\,\cos \,30{}^\circ ,\,0 \right)\] and \[\left( 0,\,10\,\cos \,60{}^\circ \right)\]is 10 units |
| Reason Mid-point of line segment joining (a, b) and (c, d) is given by |
| \[\left( \frac{a-c}{2},\,\frac{b-d}{2} \right)\]. |
A) A is true, R is true; R is a correct explanation for A.
B) A is true, R is true; R is not a correct explanation for A.
C) A is true; R is False.
D) A is false; R is true.
Correct Answer: C
Solution :
| Here, \[{{x}_{1}}=10\,\cos 30{}^\circ\]\[{{y}_{1}}=0\] and \[{{y}_{1}}=0\], |
| \[x{{y}_{2}}=10\,\cos \,60{}^\circ\] |
| \[\therefore\] Distance between the points |
| \[=\sqrt{{{\left( 0-10\,\cos \,30{}^\circ \right)}^{2}}+{{\left( 10\,\cos \,60\,{}^\circ -0 \right)}^{2}}}\] |
| \[=\sqrt{{{\left( -10\times \frac{\sqrt{3}}{2} \right)}^{2}}+{{\left( 10\times \frac{1}{2} \right)}^{2}}}\] |
| \[\left[ \because \,\,\,\cos \,30{}^\circ =\frac{\sqrt{3}}{2}\,and\,\cos \,60{}^\circ =\frac{1}{2} \right]\] |
| \[=\sqrt{\frac{300}{4}+\frac{100}{4}}\] |
| \[=\sqrt{\frac{400}{4}}=\sqrt{100}=10\] units |
| Hence, Assertion is true but Reason is false as the mid-point is \[\left( \frac{a+c}{2},\frac{b+d}{2} \right).\] |
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