Number System - Study Mode

[#331] How many times the keys of a typewriter have to be pressed in order to write number from 121 to 1346?
Correct Answer

(D) 4025

Explanation

Solution: 121 to 999 - 3 digit Total number of terms = 999 - 121 + 1 = 878 + 1 = 879 Total keys from 121 to 999 = 879 × 3 = 2637 1000 to 1346 - 4 digit Total number of terms = 1346 - 1000 + 1 = 347 Total keys from 1000 to 1346 = 347 × 4 = 1388 Total keys = 2637 + 1388 = 4025

[#332] Among the following statements, the statement which is not correct is:
Correct Answer

(B) Every real number is a rational number

Explanation

Solution: Every real number is a rational number.

[#333] If 1 3 + 2 3 . . . . . . + 10 3 = 3025, then the value of 2 3 + 4 3 . . . . . . + 20 3 is:
Correct Answer

(C) 24200

Explanation

Solution: $$eqalign{
& x08ecause {1^3} + {2^3} + {3^3},......, + {10^3} = 3025 cr
& { ext{To find}} = {2^3} + {4^3} + {6^3},......, + {20^3} = ? cr
& Rightarrow {2^3}left( {{1^3} + {2^3} + {3^3},......, + {{10}^3}}
ight) cr
& left[ {{{left( {frac{{nleft( {n + 1}
ight)}}{2}}
ight)}^2} = {{left( {frac{{10left( {11}
ight)}}{2}}
ight)}^2} = {{left( {55}
ight)}^2} = 3025}
ight] cr
& Rightarrow 8 imes 3025 cr
& Rightarrow 24200 cr} $$

[#334] (46) 2 - (?) 2 = 4398 - 3066
Correct Answer

(B) 28

Explanation

Solution: Let (46) 2 - x 2 = 4398 - 3066 Then, (46) 2 - x 2 = 1332 ⇒ x 2 = (46) 2 - 1332 ∴ x 2 = (50 - 4) 2 - 1332 x 2 = (50) 2 + 4 2 - 2 × 50 × 4 - 1332 x 2 = 2500 + 16 - 400 - 1332 x 2 = 2516 - 1732 x 2 = 784 x = 28

[#335] Given n = 1 + x and x is the product of four consecutive integers. Then which of the following us true ? I. n is an odd integer. II. n is prime. III. n is a perfect square
Correct Answer

(D) Both I and III are correct

Explanation

Solution: Out of four consecutive integers two are even and therefore, their product is even and on adding 1 to it, we get an odd integer. So, n is odd. Some possible values of n are as under : n = 1 + (1 × 2 × 3 × 4) = (1 + 24) = 25 = 5 2 n = 1 + (2 × 3 × 4 × 5) = (1 + 120) = 121 = 11 2 n = 1 + (3 × 4 × 5 × 6) = (1 + 360) = 361 = 19 2 n = 1 + (4 × 5 × 6 × 7) = (1 + 840) = 841 = 29 2 Ans so on..... Hence, n is odd and a perfect square.