Sets and Relations MCQ Questions & Answers in Calculus | Maths

$$\eqalign{ & {\text{Given that, }}f\left( x \right) = \left[ x \right]{\text{ and }}g\left( x \right) = \left| x \right| \cr & {\text{Now, }}f\left( {g\left( {\frac{8}{5}} \right)} \right) = g\left( {\frac{8}{5}} \right) = \left[ {\frac{8}{5}} \right] = 1 \cr & {\text{and }}g\left( {f\left( { - \frac{8}{5}} \right)} \right) = g\left( {\left[ { - \frac{8}{5}} \right]} \right) = g\left( { - 2} \right) = \left| { - 2} \right| = 2 \cr & \therefore \,f\left( {g\left( {\frac{8}{5}} \right)} \right) - g\left( {f\left( { - \frac{8}{5}} \right)} \right) = 1 - 2 = - 1 \cr} $$

$$C$$ stands for set of students talking Economics.

$$\eqalign{ & a + b + c + d + e + f + g = 40;\,\,\,a + b + d + g = 16 \cr & b + c + e + g = 22;\,\,d + e + f + g = 26 \cr & b + g = 5;\,\,e + g = 14;\,\,g = 2 \cr & {\text{Go by backward substitution}} \cr & e = 12,\,b = 3,\,d + f = 12,\,c + e = 17\,\, \Rightarrow c = 5;\,\,a + d = 11 \cr & a + d + f = 18\,\, \Rightarrow \,f = 7 \cr & \therefore \,\,d = 12 - 7 = 5 \cr} $$

$$f\left( x \right) = \left| x \right|$$   is many-one and into.
$$f\left( x \right) = x\left| x \right|$$   is one-one but into as $$f\left( x \right)$$  will have only rational values.
$$f\left( x \right) = \sin \,\pi x$$   is onto but many-one $$\left[ {\because f\left( {\frac{3}{4}} \right) = f\left( {\frac{1}{4}} \right)} \right]$$

$$\eqalign{ & \left( {f - ig} \right)\left( z \right) \cr & = f\left( z \right) - ig\left( z \right) \cr & = \sin \,z - i\,\cos \,z \cr & = - i\left( {\cos \,z + i\,\sin \,z} \right) \cr & = - i{e^{iz}} = \theta \,\,\left( {{\text{say}}} \right) \cr & {\text{Now, }}\left( {f + ig} \right)*\left( {f - ig} \right)\left( z \right) \cr & = \left( {f + ig} \right)\left( {f - ig} \right)\left( z \right) \cr & = \left( {f + ig} \right)\left( \theta \right) \cr & = f\left( \theta \right) + ig\left( \theta \right) \cr & = \sin \,\theta + i\,\cos \,\theta \cr & = i\left( {\cos \,\theta - i\,\sin \,\theta } \right) \cr & = i{e^{ - i\theta }} \cr & = i{e^{ - i\left( { - i{e^{iz}}} \right)}} \cr & = i{e^{ - {e^{iz}}}} \cr} $$

$$\eqalign{ & n\left( A \right) = p,\,n\left( B \right) = q;\,n\left( {A \times B} \right) = pq = 7 \cr & {\text{So, possible values of }}p,\,q\,{\text{are}}\,7,\,1 \cr & \Rightarrow {p^2} + {q^2} = {\left( 7 \right)^2} + {\left( 1 \right)^2} = 50 \cr} $$