We present exact results for the transmission coefficient of a linear lattice at one or more sites of which we attach a Fibonacci quasiperiodic chain. Two cases have been discussed, viz, when a single quasiperiodic chain is coupled to a site of the host lattice and, when more than one dangling chains are grafted periodically along the backbone. Our interest is to observe the effect of increasing the size of the attached quasiperiodic chain on the transmission profile of the model wire. We find clear signature that, with a side coupled semi-infinite Fibonacci chain, the Cantor set structure of its energy spectrum should generate interesting multifractal character in the transmission spectrum of the host lattice. This gives us an opportunity to control the conductance of such systems and to devise novel switching mechanism that can act over arbitrarily small scales of energy. The Fano profiles in resonance are observed at various intervals of energy as well. Moreover, an increase in the number of such dangling chains may lead to the design of a kind of spin filters. This aspect is discussed.