CDS 2014(I) Mathematics Question Answer with fully Solved Part-1

CDS(I) 2014 was already held and Result was also published..Here If you  are Preparing yourself for the UPSC'S Combined Defense Service 15 feb,2015 then you will must follow the previous years question and answer to know the best way to crack Combined Defense Service Conducted By UPSC.I have included some CDS 2014 Mathematics Question and answer with fully solved.You may get small help from those question. Here the question's:-

1.what is the number of integral solutions of the equations HCF(a,b)=5 and a+b=65?

   (a) None  (b) Infinitely many  (c)Less Than 65  (d) Exactly one

   ANSWER:-

    As HCF(a,b)=5, Let  me Consider a=5x and b=5y, So, 5x+5y=65 

    And now x+y=13

   So, numbers of pairs of(x,y)=(1,12),(2,11),(3,10),(4,9),(5,8),(6,7) 

   Hence the number of solutions is (C)less than 65

2.The difference of two consecutive cubes 

(a) is odd or even (b) is never divisible by 2   (c)Less Than 65  (d) Exactly one

Ans:- Let the two consecutive numbers be 4 and 5

 So, 5³-4³ =125-64=61 and now let the two conswcutive numbers be 9 and 10

 So, 10³-9³ =1000-729=271

Hence the answer is (b) is never divisible by 2 

3. If a and b be positive integers,then what is the value of [(a/HCF(a,b),b/HCF(a,b)] ???

(a) a  (b)b  (c) 1  (d) a/HCF(a,b)

Ans:- (c) HCF [(a/HCF(a,b),b/HCF(a,b)] is always equal to 1.

 4.The product of four consecutive natural numbers plus one is:
(a) A none-square (b) Always sum of two square numbers (c) A square (d) None of the above

Ans:(c) A square

Let four consecutive numbers are 3,4,5 and 6.

So,   =(3×4×5×6) + 1 =361=(19×19)

Now let four consecutive numbers be 9,10,11 and 12

  So,  ==(9×10×11×12) +1=11881 =  (109×109)

5. A positive number, when increased by 10, equals 200 times its reciprocal. What is that number?
(a) 100 (b) 10 (c) 20 (d) 200

Ans:- (b) 10

Let  the positive number x

then,according to the question

  As  x+10=200/x

So, X²+ 10x=200 or X²+ 10x - 200=0 or,(x-10)(x-20)=0 or,x=10 and x=-20

 but x# -20 .so b is the right answer

6. 2X³ + X² - 2x -1 is divisible by

(a) x+2 (b)2x+1 (c) x-2 (d) 2x-1

Ans:- (b)2x+1 

Now 2X³ + X² - 2x -1=X²(2x+1)-1(2x+1) or (2x+1)(X²-1)

So the answer is 2x+1

                The Next Post Will be about to rest of the question.

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