Mathematical Reasoning Practice Questions - DPP for JEE

 Page 1


1. Let p, q and r be any three logical statements. Which one of the
following is true?
(a) 
(b)
(c)
(d)
2. ~(p ? q)? [(~p) ? (~ q)] is
(a) a tautology
(b) a contradiction
(c) neither a tautology nor contradicion
(d) cannot come any conclusion.
3. For integers m and n, both greater than 1, consider the following three
statements :
P : m divides n
Q : m divides n
2
R : m is prime,
then
(a)
Page 2


1. Let p, q and r be any three logical statements. Which one of the
following is true?
(a) 
(b)
(c)
(d)
2. ~(p ? q)? [(~p) ? (~ q)] is
(a) a tautology
(b) a contradiction
(c) neither a tautology nor contradicion
(d) cannot come any conclusion.
3. For integers m and n, both greater than 1, consider the following three
statements :
P : m divides n
Q : m divides n
2
R : m is prime,
then
(a)
(b)
(c)
(d)
4. If is false and q and r are both false, then p is
(a) True
(b) False
(c) May be true or false
(d) Data sufficient
5. Consider the following statements
p : x, y ? Z such that x and y are odd.
q : xy is odd. Then,
(a) p ? q is true
(b) is true
(c) Both (a) and (b)
(d) None of these
6. If S*(p, q, r) is the dual of the compound statement S(p,q,r) and S
(p,q,r) = ~ p ? [~ (q ? r)] then S*(~p, ~q, ~r) is equivalent to –
(a) S (p, q, r)
(b) ~ S (~p, ~q, ~r)
(c) ~ S (p, q, r)
(d) S*(p, q, r)
7. The dual of statement (p ? q) ? ~ q p? ~ q is
(a) (p ? q) ? ~ q p ? ~ q
(b) (p ? q) ? ~ q p ? ~ q
(c) (p ? q) ? ~ q p ? ~ q
(d) (p ? q) ? ~ q p ? ~ q.
8. The converse of the statement if x< y then x
2
 < y
2
 is
(a) If x is not less then y then x
2
 is not less than y
2
(b) If x
2 
< y
2 
then x < y
(c) If x
2 
= y
2 
then x = y
Page 3


1. Let p, q and r be any three logical statements. Which one of the
following is true?
(a) 
(b)
(c)
(d)
2. ~(p ? q)? [(~p) ? (~ q)] is
(a) a tautology
(b) a contradiction
(c) neither a tautology nor contradicion
(d) cannot come any conclusion.
3. For integers m and n, both greater than 1, consider the following three
statements :
P : m divides n
Q : m divides n
2
R : m is prime,
then
(a)
(b)
(c)
(d)
4. If is false and q and r are both false, then p is
(a) True
(b) False
(c) May be true or false
(d) Data sufficient
5. Consider the following statements
p : x, y ? Z such that x and y are odd.
q : xy is odd. Then,
(a) p ? q is true
(b) is true
(c) Both (a) and (b)
(d) None of these
6. If S*(p, q, r) is the dual of the compound statement S(p,q,r) and S
(p,q,r) = ~ p ? [~ (q ? r)] then S*(~p, ~q, ~r) is equivalent to –
(a) S (p, q, r)
(b) ~ S (~p, ~q, ~r)
(c) ~ S (p, q, r)
(d) S*(p, q, r)
7. The dual of statement (p ? q) ? ~ q p? ~ q is
(a) (p ? q) ? ~ q p ? ~ q
(b) (p ? q) ? ~ q p ? ~ q
(c) (p ? q) ? ~ q p ? ~ q
(d) (p ? q) ? ~ q p ? ~ q.
8. The converse of the statement if x< y then x
2
 < y
2
 is
(a) If x is not less then y then x
2
 is not less than y
2
(b) If x
2 
< y
2 
then x < y
(c) If x
2 
= y
2 
then x = y
(d) None of these
9. If p and q are true statement and r, s are false statements, then the truth
value of  ~ [(p?~r) ? ( ~q ? s)] is
(a) true
(b) false
(c) false if p is true
(d) None of these
10. Identify the false statement
(a) ~ [p ? (~ q)] = (~ p) ? q
(b) [p ? q] ? (~ p) is a tautology
(c) [p ? q) ?  (~ p) is a contradiction
(d) ~ [p ? q] = (~ p) ? (~ q)
11. The contrapositive of p ? (~q ? ~r) is –
(a) (~ q ? r) ? ~ p
(b) (q ? r) ? ~p
(c) (q ? ~r) ? ~ p
(d) None of these
12. Which of the following is wrong ?
(a) p ? q is logically equivalent to ~ p ? q
(b) If the truth values of p, q, r are T, F, T respectively, then the truth value
of (p ? q) ? (q ? r) is T
(c) ~ (p ? q ? r)  ~ p ? ~ q ? ~ r
(d) The truth value of p ? ~ (p ? q) is always T.
13. The false statement of the following is
(a) is a contradiction
(b) is a contradiction
(c) is a tautology
(d) p is a tautology
14. In the truth table for the statement ( ~ p ? ~ q) ? ( ~ q ? ~ p),  the last
Page 4


1. Let p, q and r be any three logical statements. Which one of the
following is true?
(a) 
(b)
(c)
(d)
2. ~(p ? q)? [(~p) ? (~ q)] is
(a) a tautology
(b) a contradiction
(c) neither a tautology nor contradicion
(d) cannot come any conclusion.
3. For integers m and n, both greater than 1, consider the following three
statements :
P : m divides n
Q : m divides n
2
R : m is prime,
then
(a)
(b)
(c)
(d)
4. If is false and q and r are both false, then p is
(a) True
(b) False
(c) May be true or false
(d) Data sufficient
5. Consider the following statements
p : x, y ? Z such that x and y are odd.
q : xy is odd. Then,
(a) p ? q is true
(b) is true
(c) Both (a) and (b)
(d) None of these
6. If S*(p, q, r) is the dual of the compound statement S(p,q,r) and S
(p,q,r) = ~ p ? [~ (q ? r)] then S*(~p, ~q, ~r) is equivalent to –
(a) S (p, q, r)
(b) ~ S (~p, ~q, ~r)
(c) ~ S (p, q, r)
(d) S*(p, q, r)
7. The dual of statement (p ? q) ? ~ q p? ~ q is
(a) (p ? q) ? ~ q p ? ~ q
(b) (p ? q) ? ~ q p ? ~ q
(c) (p ? q) ? ~ q p ? ~ q
(d) (p ? q) ? ~ q p ? ~ q.
8. The converse of the statement if x< y then x
2
 < y
2
 is
(a) If x is not less then y then x
2
 is not less than y
2
(b) If x
2 
< y
2 
then x < y
(c) If x
2 
= y
2 
then x = y
(d) None of these
9. If p and q are true statement and r, s are false statements, then the truth
value of  ~ [(p?~r) ? ( ~q ? s)] is
(a) true
(b) false
(c) false if p is true
(d) None of these
10. Identify the false statement
(a) ~ [p ? (~ q)] = (~ p) ? q
(b) [p ? q] ? (~ p) is a tautology
(c) [p ? q) ?  (~ p) is a contradiction
(d) ~ [p ? q] = (~ p) ? (~ q)
11. The contrapositive of p ? (~q ? ~r) is –
(a) (~ q ? r) ? ~ p
(b) (q ? r) ? ~p
(c) (q ? ~r) ? ~ p
(d) None of these
12. Which of the following is wrong ?
(a) p ? q is logically equivalent to ~ p ? q
(b) If the truth values of p, q, r are T, F, T respectively, then the truth value
of (p ? q) ? (q ? r) is T
(c) ~ (p ? q ? r)  ~ p ? ~ q ? ~ r
(d) The truth value of p ? ~ (p ? q) is always T.
13. The false statement of the following is
(a) is a contradiction
(b) is a contradiction
(c) is a tautology
(d) p is a tautology
14. In the truth table for the statement ( ~ p ? ~ q) ? ( ~ q ? ~ p),  the last
column has the truth value in the following order is
(a) TTTF
(b) FTTF
(c) TFFT
(d) TTTT
15. If p is any statement, t is tautology and c is a contradiction, then which
of the following is not correct?
(a) p? (~ p) = c
(b) p? t = t
(c) p ? t = p
(d) p ? c = c.
16. The logically equivalent proposition of  is
(a)
(b)
(c)
(d)
17. The inverse of the statement (p ? ~ q) ? r is
(a) ~ (p ? ~q) ? ~ r
(b) (~p ? q) ? ~ r
(c) (~p ? q) ? ~ r
(d) None of these
18. If x = 5 and y = – 2, then x – 2y = 9. Then contrapositive of this
proposition is
(a) If x – 2y ? 9, then x ? 5 or y ? –2.
(b) If x – 2y = 9 then x ? 5 and y ? –2
(c) x – 2y = 9 if and only if x = 5 and y = – 2
(d) None of these
19. The statement p ? (q?p) is equivalent to
(a) p ? (p? q)
(b) p ? (p q)
Page 5


1. Let p, q and r be any three logical statements. Which one of the
following is true?
(a) 
(b)
(c)
(d)
2. ~(p ? q)? [(~p) ? (~ q)] is
(a) a tautology
(b) a contradiction
(c) neither a tautology nor contradicion
(d) cannot come any conclusion.
3. For integers m and n, both greater than 1, consider the following three
statements :
P : m divides n
Q : m divides n
2
R : m is prime,
then
(a)
(b)
(c)
(d)
4. If is false and q and r are both false, then p is
(a) True
(b) False
(c) May be true or false
(d) Data sufficient
5. Consider the following statements
p : x, y ? Z such that x and y are odd.
q : xy is odd. Then,
(a) p ? q is true
(b) is true
(c) Both (a) and (b)
(d) None of these
6. If S*(p, q, r) is the dual of the compound statement S(p,q,r) and S
(p,q,r) = ~ p ? [~ (q ? r)] then S*(~p, ~q, ~r) is equivalent to –
(a) S (p, q, r)
(b) ~ S (~p, ~q, ~r)
(c) ~ S (p, q, r)
(d) S*(p, q, r)
7. The dual of statement (p ? q) ? ~ q p? ~ q is
(a) (p ? q) ? ~ q p ? ~ q
(b) (p ? q) ? ~ q p ? ~ q
(c) (p ? q) ? ~ q p ? ~ q
(d) (p ? q) ? ~ q p ? ~ q.
8. The converse of the statement if x< y then x
2
 < y
2
 is
(a) If x is not less then y then x
2
 is not less than y
2
(b) If x
2 
< y
2 
then x < y
(c) If x
2 
= y
2 
then x = y
(d) None of these
9. If p and q are true statement and r, s are false statements, then the truth
value of  ~ [(p?~r) ? ( ~q ? s)] is
(a) true
(b) false
(c) false if p is true
(d) None of these
10. Identify the false statement
(a) ~ [p ? (~ q)] = (~ p) ? q
(b) [p ? q] ? (~ p) is a tautology
(c) [p ? q) ?  (~ p) is a contradiction
(d) ~ [p ? q] = (~ p) ? (~ q)
11. The contrapositive of p ? (~q ? ~r) is –
(a) (~ q ? r) ? ~ p
(b) (q ? r) ? ~p
(c) (q ? ~r) ? ~ p
(d) None of these
12. Which of the following is wrong ?
(a) p ? q is logically equivalent to ~ p ? q
(b) If the truth values of p, q, r are T, F, T respectively, then the truth value
of (p ? q) ? (q ? r) is T
(c) ~ (p ? q ? r)  ~ p ? ~ q ? ~ r
(d) The truth value of p ? ~ (p ? q) is always T.
13. The false statement of the following is
(a) is a contradiction
(b) is a contradiction
(c) is a tautology
(d) p is a tautology
14. In the truth table for the statement ( ~ p ? ~ q) ? ( ~ q ? ~ p),  the last
column has the truth value in the following order is
(a) TTTF
(b) FTTF
(c) TFFT
(d) TTTT
15. If p is any statement, t is tautology and c is a contradiction, then which
of the following is not correct?
(a) p? (~ p) = c
(b) p? t = t
(c) p ? t = p
(d) p ? c = c.
16. The logically equivalent proposition of  is
(a)
(b)
(c)
(d)
17. The inverse of the statement (p ? ~ q) ? r is
(a) ~ (p ? ~q) ? ~ r
(b) (~p ? q) ? ~ r
(c) (~p ? q) ? ~ r
(d) None of these
18. If x = 5 and y = – 2, then x – 2y = 9. Then contrapositive of this
proposition is
(a) If x – 2y ? 9, then x ? 5 or y ? –2.
(b) If x – 2y = 9 then x ? 5 and y ? –2
(c) x – 2y = 9 if and only if x = 5 and y = – 2
(d) None of these
19. The statement p ? (q?p) is equivalent to
(a) p ? (p? q)
(b) p ? (p q)
(c) p ? (p q)
(d) p ? (p ?q)
20. The negation of (p ? q) ? (q ? ~ r) is
(a) (~ p ? ~ q) ? (q ? ~ r)
(b) (~ p ? ~ q) ? (~ q ? r)
(c) (~ p ? ~ q) ? (~ q ? r)
(d) (p ? q) ? (~ q ? ~ r)
21. Let p: Kiran passed the examination,
q: Kiran is sad
The symbolic form of a statement “It is not true that Kiran passed therefore
she is sad” is
(a) (~ p? q)
(b) (p ? ~q)
(c) ~ (p? ~ q)
(d) ~ ( p?q)
22. The conditional  ? p is
(a) A tautology
(b) A fallacy i.e., contradiction
(c) Neither tautology nor fallacy
(d) None of these
23. If p, q are true and r is false statement, then which of the following is
true statement?
(a) (p  q)  r is F
(b) (p  q) ? r is T
(c) (p  q)  (p  r) is T
(d) (p ? q) ? (p ? r) is T
24. Let p, q, r be  three statements. Then  is equal to
(a)
(b)