Question 1

(1) Which term of the AP : $121, 117, 113, \dots$, is its first negative term? [Hint : Find $n$ for $a_n < 0$]

(2) The sum of the third and the seventh terms of an AP is $6$ and their product is $8$. Find the sum of first sixteen terms of the AP.

(3) A ladder has rungs $25$ cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from $45$ cm at the bottom to $25$ cm at the top. If the top and the bottom rungs are $\dfrac{12}{2}$ m apart, what is the length of the wood required for the rungs? [Hint : Number of rungs = $\dfrac{250}{125} + 1$]

(4) The houses of a row are numbered consecutively from $1$ to $49$. Show that there is a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$. [Hint : $S_{x-1} = S_{49} - S_x$]

(5) A small terrace at a football ground comprises of $15$ steps each of which is $50$ m long and built of solid concrete. Each step has a rise of $\dfrac{1}{4}$ m and a tread of $\dfrac{1}{2}$ m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = $\dfrac{1}{3} \times 50 \times \dfrac{1}{4} \times \dfrac{1}{2}$ m$^3$]