**1. If x and y are positive integers then the following is always true?**

2x-3Y<0

(1) x = (y-1)

(2) x> y

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**2.** **Given: l/(m+n)= m/(n+l)=n/(l+m) = K , where k is a real number. If p > k where p is a positive integer then the following is always true?**

(p+k)/2= an odd integer.

(1) p = 3/2

(2) p = 5/2

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**3.** ** Find if { u _{n} } is an arithmetic sequence. **

(1) u

_{n}= 5n-1, n Ñ” N.

(2) u

_{n}= (n-1)/2, n Ñ” N.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**4. A restaurants sells fish rolls and salads.**

Let F be the event a customer chooses a fish roll.

Let S be the event a customer chooses a salad.

Let N be the event a customer chooses neither a fish roll nor a salad.

Find P (FUS).

(1) P (N) = 0.14

(2) If P(F) = 0.31 and P(S) = 0.62

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**5. In a right angled triangle,the ratio of sides is 5:13:12. If the sides are x, y and h, where h is the hypotenuse, then we can find the area if **

(1) h is known

(2) x + y is known

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**6. If a and b are two distinct real numbers then the following is always true?**

(a + b) is always anirrational number.

(1) Both a and b areirrational numbers

(2) a is rational but b is irrational

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**7. f(x) =ax ^{2} + 4x -c has a maximum value.**

(1) a<0

(2) c>0

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**8. If C _{1}, C_{2}and r_{1 }and r_{2} are respectively the centres and radii of two given circles and circles touch each other if **

(1) C

_{1}C

_{2}= r

_{1}+r

_{2}

(2) C

_{1}C

_{2}= r

_{1}-r

_{2}

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**9. p ^{2} + r(q-r)^{2}Is an odd number? **

(1) p, q and r are even numbers

(2) p, q and r are odd numbers

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**10. Is (7 ^{a-1} – 5^{a})>0?**

(1) 5 (2) a is a positive integer

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**11. If a and b are realnumbers then a ^{2}/(bc) +b^{2}/(ca) +c^{2}/(ab)= 3.**

(1) a + b + c = 0

(2) a, b and c are in A.P.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**12. If x is an integer, find its value?**

(1) x>2

(2) x<4

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**13. If n is a two digit prime number, find the value of n.**

(1) The sum of digits is 16.

(2) When the digits of n are reversed the number obtained is a prime number.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**14. What is the value of x, if x is a digit?**

(1) 1x2x6x3 is divisible by 11.

(2) 12226x3 is divisible by 9

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**15. What is the value of (x + y) in base 2?**

(1) x = (100)_{2}

(2) y = (20)_{8}

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient.

E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**Solutions**

**1. A**

Since x = (y-1), : 2x -3y = 2(y-1)-3y = - y-2

Now, since y is a positive integer, : - y – 2 < 0

Now if x > y, 2x -2y > 0, : 2x – 3y > -y. Therefore, It does not prove 2x - 3y > 0.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

**2. A**

l/(m+n) =m/(n+l) =n/(l+m) = k( Each ratio = sum of antecedents/sum of consequents )

k=(l+m+n)/2(l+m+n)=1/2

When K =1/2 and p=3/2, (p+k)/2 = 1 , which is an odd integer.

When K =1/2 and p=5/2, (p+k)/2 = 1 , which is not an integer.

Therefore, statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

**3. D**

The nth (general term) of any arithmetic progression is always an expression of first degree in n.

Here, both expressions are of first degree in n.

Therefore, Each statement ALONE is sufficient.

**4. A**

Therefore, statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

**5. C**

EACH statement ALONE is sufficient.

**6. B**

Since a is rational and b is irrational, therefore (a + b) is always an irrational.

On the other hand sum of 2 irrational numbers may or may not be irrational.

Therefore, statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

**7. C**

Since a < 0, therefore graph of the given parabola is concave down and also as

c> 0, Discriminant of the quadratic polynomial is always positive. Hence the quadratic polynomial has a maximum value.

**8. A**

Since the distance between the centres of the circles is equal to the sum of their radii, therefore circles must touch each other externally. The same is not always true with statement (2).

Therefore, statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

**9. E**

Difference of two odd numbers is always odd.

Square of an odd number is always odd.

Product of two numbers is always odd.

But the sum of two odd number is always even.

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

**10. D**

EACH statement ALONE is sufficient.

**11. A**

The same is not always true when a, b and c are in A.P.

Therefore, statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

**12. C**

Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

**13. D**

EACH statement ALONE is sufficient.

**14. D**

EACH statement ALONE is sufficient.

**15. (c)**. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient as we need the value of both x and y.

From (1), x = (100)_{2}. Therefore it is given in base 2.

From (2), y = (16)_{10} = (10000)_{2}

Therefore, y + x = (10000)_{2} + (100)_{2}= (10100)_{2}