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If \( \Large f:R \rightarrow R \) satisfies \( \Large f \left(x+y\right) = f \left(x\right) = f \left(x\right)+f \left(y\right) \) for all \( \Large x,\ y,\ \epsilon R\ and\ f \left(1\right)=7 \), then \( \Large \sum^{n}_{r=1}f \left(r\right) \) is:

A) \( \Large \frac{7n}{2} \)
B) \( \Large \frac{7 \left(n+1\right) }{2} \)
C) \( \Large 7n \left(n+1\right) \)
D) \( \Large \frac{7n \left(n+1\right) }{2} \)

Correct Answer:




D) \( \Large \frac{7n \left(n+1\right) }{2} \)

Description for Correct answer:
\( \Large \sum_{r=1}^{n}f \left(r\right) = f \left(1\right)+f \left(2\right)+f \left(3\right)+....+f \left(n\right) \)

= \( \Large f\left(1\right)+2f \left(1\right)+3f \left(1\right)+...nf \left(1\right) \)

\( \Large \left[ since f \left(x+y\right)=f \left(x\right)+f \left(y\right) \right] \)

= \( \Large \left(1+2+3+...+n\right)f \left(1\right) \)

= \( \Large \frac{n \left(n+1\right) }{2}.7 = \frac{7n \left(n+1\right) }{2} \)

Part of solved Set theory questions and answers : >> Elementary Mathematics >> Set theory

Comments

Similar Questions

1). A function of from the set of natural numbers to integers defined by \( \Large f \left(n\right) = \frac{\frac{n-1}{2},\ when\ n\ is\ odd}{-\frac{n}{2},\ when\ n\ is\ even} \) is:

A). one-one but not onto

B). onto but not one-one

C). one-one and onto both

D). neither one-one nor onto

2). The relation "is subset of on the power set P(A) of set A is:"

A). symmetric

B). antisymmetric

C). equivalence relation

D). none of these

3). In a college of 300 students, every student read 5 newspaper and every newspaper is read by 60 students. The number of newspaper is:

A). at least 30

B). at most 20

C). exactly 25

D). none of these

4). Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basket ball, of the total 64 played both basket ball and hockey, 80 played cricket and basketball and 40 played cricket and hockey, 24 played all these three games. The numbers of boys who did not play any game is:

A). 128

B). 216

C). 240

D). 160

5). Let N denotes the set of all natural numbers and R be the relation on \( \Large N \times N \) defined by \( \Large \left(a,\ b\right) R \left(c,\ d\right)\ if\ ad \left(b+c\right)=bc \left(a+d\right) \), then R is:

A). symmetric only

B). reflexive only

C). transitive only

D). an equivalence relation

6). Let \( \Large f : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) defined by \( \Large f \left(x\right)=\frac{1-x}{1+x},\ 0 \le x \le 1 \) and let \( \Large g : \left[ 0,\ 1 \right] \rightarrow \left[ 0,\ 1 \right] \) be defined by \( \Large g \left(x\right) = 4x \left(1-x\right),\ 0 \le x \le 1 \), then fog and gof is:

A). \( \Large \frac{ \left(2x-1\right)^{2} }{1+4x+4x^{2}},\ \frac{8x \left(1-x\right) }{ \left(1+x\right)^{2} } \)

B). \( \Large \frac{ \left(2x-1\right) }{1+4x+4x^{2}},\ \frac{8 \left(1-x\right)x }{ \left(1+x\right)^{2} } \)

C). \( \Large \frac{ \left(2x+1\right)^{2} }{1+4x+4x^{2}},\ \frac{8}{ \left(1+x\right)^{2} } \)

D). \( \Large \frac{ \left(2x+1\right)^{2} }{ \left(1+x\right)^{2} },\ \frac{8 \left(1-x\right) }{ \left(1+x\right)^{2} } \)

7). An integer m is said to be related to another integer n, if m is a multiple of n, Then the relation is:

A). reflexive and symmetric

B). reflexive and transitive

C). symmetric and transitive

D). equivalence relation

8). Which of the following is a null set?

A). \( \Large \{ x : |x| < 1,\ x\ \epsilon\ N \} \)

B). \( \Large \{ x : |x| = 5,\ x\ \epsilon\ N \} \)

C). \( \Large \{ x : x^{2} = 1,\ \epsilon\ Z \} \)

D). \( \Large \{ x : x^{2}+2x+1=0,\ x\ \epsilon\ R \} \)

9). A set contains n elements. Then the power set contains:

A). \( \Large n^{2} elements \)

B). n elements

C). \( \Large \left(2^{n}-1\right) elements \)

D). \( \Large 2^{n} elements \)

10). In what ratio must a mixture of 20% alcohol be mixed with another mixture of 50% alcohol so that the resultant mixture contains 40% alcohol?

A). 3:5

B). 2:6

C). 1:2

D). 3:7