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Mathematical Reasoning
The remainder left out when 82n –(62)2n+1 is divided by 9 is
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0
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2
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7
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8
B.
2
82n – (62)2n + 1
⇒ (9 – 1)2n – (63 – 1)2n + 1
⇒ (2nC0 92n–2nC1 92n – 1 + ….. + 2nC2n)
– (2n + 1C0 632n + 1–2n + 1C1 632n + ….
–2n +1C2n + 1
Clearly remainder is ‘2’.
Statement 1: ~ (p ↔ ~ q) is equivalent to p ↔ q
Statement 2 : ~ (p ↔ ~ q) is a tautology
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Statement–1 is true, Statement–2 is true, Statement–2 is a correct explanation for statement–1
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Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.
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Statement–1 is true, statement–2 is false.
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Statement–1 is false, Statement–2 is true
C.
Statement–1 is true, statement–2 is false.
| p | q | ~q | (p ↔ ~q) | ~ (p ↔ ~q) | p ↔ q |
| T | T | F | F | T | T |
| T | F | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | F | T | T |
Clearly, ~ (p ↔ ~q) is not a tautology because it does not contain T in the column of its truth table. Also, ~(p ↔ ~ q) & p ↔ q have the same truth value
Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p
Statement –2: r is equivalent to ∼ (p ↔ ∼ q).
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Statement −1 is false, Statement −2 is true
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Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
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Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
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Statement − 1 is true, Statement − 2 is false.
D.
Statement − 1 is true, Statement − 2 is false.
Given statement r = ∼ p ↔ q
Statement −1 : r1 = (p ∧ ∼ q) ∨ (∼ p ∧ q)
Statement −2 : r2 = ∼ (p ↔ ∼ q) = (p ∧ q) ∨ (∼ q ∧ ∼ p)
From the truth table of r, r1 and r2,
r = r1.
Hence Statement − 1 is true and Statement −2 is false.
The statement p → (q → p) is equivalent to
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p → (p → q)
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p → (p ∨ q)
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p → (p ∧ q)
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p → (p ↔ q)
B.
p → (p ∨ q)
p → (q → p) = ~ p ∨ (q → p)
= ~ p ∨ (~ q ∨ p) since p ∨ ~ p is always true
= ~ p ∨ p ∨ q = p → (p ∨ q).
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Mock Test Series
Mock Test Series