CDS Exam (I)- 2015 - Elementary Mathematics Defence Question Paper

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CDS Exam (I)- 2015 - Elementary Mathematics

CDS Exam (I)- 2015 - Elementary Mathematics Defence Questions

31.
A light was seen regularly at an interval of 13 seconds. It was seen for the first time at 1 hour 54 minutes 50 seconds (a.m.) and the last time at 3 hours 17 minutes 49 seconds (a.m.). How many times was the light seen?
A.
375
B.
378
C.
383
D.
384
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ANSWER :
D. 384
32.
An automobile financer claims to be lending money at simple interest, but he includes the interest every six months for calculating the principal. If he is charging an interest at the rate of 10%, the effective rate of interest becomes
A.
10.25%
B.
10.5%
C.
10.75%
D.
11%
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ANSWER :
A. 10.25%
33.
What are the possible solutions for x of the equation \( x^{\sqrt[n]{x}} = \sqrt[n]{x^{x}} \), where x and n are positive integers?
A.
0, n
B.
1, n
C.
\( n, n^{\frac{n}{n-1}} \)
D.
\( 1, n^{\frac{n}{n-1}} \)
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ANSWER :
D. \( 1, n^{\frac{n}{n-1}} \)
34.
Let A and B be finite non-empty sets with the number of elements in A = m and number of elements in B = n. Let \( m > n \). If for some integer \( k \geq 1 \), the number of non-empty subsets of A = \( 2^{k} \) + the number of non-empty subsets of B, then which one of the following is correct?
A.
\( m = n + 2 \)
B.
\( m = n + 1 \)
C.
\( m = n + p \) for some odd prime number p
D.
\( m = n + t \) for some composite number t
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ANSWER :
B. \( m = n + 1 \)
35.
If n is a natural number and \( n = p_{1}^{x_{1}} p_{2}^{x_{2}} p_{3}^{x_{3}} \), where \( p_{1}, p_{2}, p_{3} \) are distinct prime factors, then the total number of prime factors (counted with repetition) for n is
A.
\( x_{1} + x_{2} + x_{3} \)
B.
\( x_{1}x_{2}x_{3} \)
C.
\( (x_{1} + 1)(x_{2} + 1)(x_{3} + 1) \)
D.
None of the above
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ANSWER :
A. \( x_{1} + x_{2} + x_{3} \)
36.
Consider the following statements for the sequence of numbers given below:
11, 111, 1111, 11111, ...
1. Each number can be expressed in the form \( (4m + 3) \), where m is a natural number.
2. Some numbers are squares.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
A. 1 only
37.
The HCF and LCM of two polynomials are \( (x + y) \) and \( (3x^{5} + 5x^{4}y + 2x^{3}y^{2} - 3x^{2}y^{3} - 5xy^{4} - 2y^{5}) \) respectively. If one of the polynomials is \( (x^{2} - y^{2}) \), then the other polynomial is
A.
\( 3x^{4} - 8x^{3}y + 10x^{2}y^{2} + 7xy^{3} - 2y^{4} \)
B.
\( 3x^{4} - 8x^{3}y - 10x^{2}y^{2} + 7xy^{3} + 2y^{4} \)
C.
\( 3x^{4} + 8x^{3}y + 10x^{2}y^{2} + 7xy^{3} + 2y^{4} \)
D.
\( 3x^{4} + 8x^{3}y - 10x^{2}y^{2} + 7xy^{3} + 2y^{4} \)
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ANSWER :
C. \( 3x^{4} + 8x^{3}y + 10x^{2}y^{2} + 7xy^{3} + 2y^{4} \)
38.
Let p be a prime number other than 2 or 5. One would like to express the vulgar fraction \( \frac{1}{p} \) in the form of a recurring decimal. Then the decimal will be
A.
a pure recurring decimal and its period will be necessarily \( (p - 1) \)
B.
a mixed recurring decimal and its period will be necessarily \( (p - 1) \)
C.
a pure recurring decimal and its period will be some factor of \( (p - 1) \)
D.
a mixed recurring decimal and its period will be some factor of \( (p - 1) \)
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ANSWER :
C. a pure recurring decimal and its period will be some factor of \( (p - 1) \)
39.
Consider the following statements:
1. There exists only one prime number p such that \( (17p + 1) \) is a square.
2. If x is the product of 10 consecutive prime numbers starting from 2, then \( (x + 1) \) is also a prime number.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
A. 1 only
40.
The sign of the quadratic polynomial \( ax^{2} + bx + c \) is always positive if
A.
a is positive and \( b^{2} - 4ac < 0 \)
B.
a is positive and \( b^{2} - 4ac > 0 \)
C.
a can be any real number and \( b^{2} - 4ac < 0 \)
D.
a can be any real number and \( b^{2} - 4ac \geq 0 \)
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ANSWER :
A. a is positive and \( b^{2} - 4ac < 0 \)
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