entrance exam
Question : If $x+y+z=1, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1,$ and $xyz=-1$, then $x^3+y^3+z^3 $ is equal to:
Option 1: –1
Option 2: 1
Option 3: –2
Option 4: 2
Correct Answer: 1
Solution : Given: $x+y+z=1,\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$, $xyz=-1$ Consider, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ ⇒ $xy+yz+zx=xyz=-1$ Now, $x+y+z=1$ ⇒ $(x+y+z)^2=1^2$ ⇒ $x^2+y^2+z^2+2(xy+yz+zx)=1$ ⇒ $x^2+y^2+z^2+2(-1)=1$ ⇒ $x^2+y^2+z^2=3$ We know, $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-(xy+yz+zx)$ So, putting all the values, we get, ⇒ $x^3+y^3+z^3-3×(-1)=1×[3-(-1)]$ ⇒ $x^3+y^3+z^3=4-3$ $\therefore x^3+y^3+z^3=1$ Hence, the correct answer is 1.
Question : Direction: Study the bar graph given below, which shows the percent distribution of total expenditures of a company under various expenses, and answer the question. The expenditure on interest on loans is more than the expenditure on transport by:
Option 1: 5%
Option 2: 10%
Option 3: 40%
Option 4: 30%
Correct Answer: 40%
Solution : From the graph, The percentage of money spent on transit = 12.5 The percentage of money spent on loan interest = 17.5 Percentage increase = $\frac{\text{( Interest on loans expense – Transport expenditure)}}{\text{(Transport expenditure)}} \times 100$ So, the required percentage $=\frac{(17.5 – 12.5)}{12.5}\times100=\frac{5}{12.5} \times 100=40$%
Question : Which Article of the Indian Constitution prohibits the State from conferring "Titles"?
Option 1: Article 14
Option 2: Article 16
Option 3: Article 17
Option 4: Article 18
Correct Answer: Article 18
Solution : The correct answer is Article 18.
All titles are eliminated by Article 18. It forbids the State from bestowing titles onto any individual, citizen or non-citizen. However, the ban does not apply to academic or military distinctions. Additionally, it forbids an Indian national
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