(a) The 6th term of an A.P is 35 and the 13th term is 77. Find the 20th term. (b) The Venn diagram represents three subsets P, Q and R of the universal set ...
Question 1 Report
(a) The 6th term of an A.P is 35 and the 13th term is 77. Find the 20th term.
(b) 
The Venn diagram represents three subsets P, Q and R of the universal set U. Copy the Venn diagram. Shade and indicate the regions represented by (i) \(P \cap Q' \cap R\) ; (ii) \(P' \cap Q \cap R'\).
Answer Details
(a) The 20th term. For an A.P., \(T_n=a+(n-1)d\).
\[T_6=a+5d=35,\qquad T_{13}=a+12d=77.\]
Subtracting the first from the second: \(7d=42\Rightarrow d=6\). Then \(a=35-5(6)=5\).
\[T_{20}=a+19d=5+19(6)=5+114=119.\]
(b) Shading the Venn diagram. The three circles \(P\) (top left), \(Q\) (top right) and \(R\) (bottom) divide \(U\) into eight regions.
(i) \(P\cap Q'\cap R\): shade the region lying inside both \(P\) and \(R\) but outside \(Q\). This is the overlap (lens) of the \(P\) and \(R\) circles, with the central piece that also lies in \(Q\) left unshaded.
(ii) \(P'\cap Q\cap R'\): shade the region lying inside \(Q\) only, that is, the part of the \(Q\) circle that is outside both \(P\) and \(R\).
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