1983 年

In the figure, the equation of the straight line is $y=mx+c$. Which one of the following is true?

If $a$ and $b$ are positive numbers, which of the following is/are true? I. $\log_{10}(a+b)=\log_{10}a+\log_{10}b$ II. $\log_{10}\dfrac{a}{b}=\log_{10}a-\log_{10}b$ III. $\dfrac{\log_{10}a}{\log_{10}b}=\dfrac{a}{b}$

A function $\mathrm{f}(x)$ is called an even function if $\mathrm{f}(x)=\mathrm{f}(-x)$. Which of the following functions is/are even functions? I. $\mathrm{f}_1(x)=\dfrac1x$ II. $\mathrm{f}_2(x)=x^2$ III. $\mathrm{f}_3(x)=x^3$

In an arithmetic progression, the first term is $3$ and the common difference is $2$. If the sum of the first $n$ terms of the arithmetic progression is $143$ then $n=$

Three positive numbers $a$, $b$ and $c$ are in geometric progression. Which of the following are true? I. $\dfrac1a,\ \dfrac1b,\ \dfrac1c$ are in geometric progression. II. $a^2,\ b^2,\ c^2$ are in geometric progression. III. $\log_{10}a,\ \log_{10}b,\ \log_{10}c$ are in arithmetic progression.

The scale of a map is $1:20\,000$. On the map the area of a farm is $2\text{ cm}^2$. The actual area of the farm is

It took Paul $40$ minutes to walk from Town $A$ to Town $B$. If the return journey took him $30$ minutes, the percentage increase in his speed was

A merchant sold $2$ articles each at $\$1000$. For the first article, he gained $25\%$ on the cost price. For the second article, he lost $20\%$ on the cost price. Altogether

Three numbers are in the ratio $2:3:5$. The ratio of their average to the largest of the three numbers is

A solid sphere is cut into two hemispheres. The percentage increase in the total surface area is