CDS Exam (I)- 2016 - Elementary Mathematics Defence Question Paper

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CDS Exam (I)- 2016 - Elementary Mathematics

CDS Exam (I)- 2016 - Elementary Mathematics Defence Questions

11.
Let m be a non-zero integer and n be a positive integer. Let R be the remainder obtained on dividing the polynomial \( x^{n} + m^{n} \) by \( (x - m) \). Then
A.
R is a non-zero even integer
B.
R is odd, if m is odd
C.
\( R = s^{2} \) for some integer s, if n is even
D.
\( R = t^{3} \) for some integer t, if 3 divides n
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ANSWER :
A. R is a non-zero even integer
12.
If \( 4^{x}2^{y} = 128 \) and \( 3^{3x + 2y} - 9^{xy} = 0 \), then the value of \( x + y \) can be equal to
A.
7
B.
5
C.
3
D.
1
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ANSWER :
B. 5
13.
If the linear factors of \( ax^{2} - (a^{2} + 1)x + a \) are p and q, then \( p + q \) is equal to
A.
\( (x - 1)(a + 1) \)
B.
\( (x + 1)(a + 1) \)
C.
\( (x - 1)(a - 1) \)
D.
\( (x + 1)(a - 1) \)
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ANSWER :
A. \( (x - 1)(a + 1) \)
14.
If \( x = \frac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}} \), then \( bx^{2} - ax + b \) is equal to (given that \( b \neq 0 \))
A.
0
B.
1
C.
ab
D.
2ab
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ANSWER :
A. 0
15.
If \( a^{3} = 117 + b^{3} \) and \( a = 3 + b \), then the value of \( a + b \) is (given that \( a > 0 \) and \( b > 0 \))
A.
7
B.
9
C.
11
D.
13
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ANSWER :
A. 7
16.
If the sum of the roots of \( ax^{2} + bx + c = 0 \) is equal to the sum of the squares of their reciprocals, then which one of the following relations is correct?
A.
\( ab^{2} + bc^{2} = 2a^{2}c \)
B.
\( ac^{2} + bc^{2} = 2b^{2}a \)
C.
\( ab^{2} + bc^{2} = ac \)
D.
\( a^{2} + b^{2} + c^{2} = 1 \)
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ANSWER :
A. \( ab^{2} + bc^{2} = 2a^{2}c \)
17.
Consider the following statements in respect of the expression \( S_{n} = \frac{n(n + 1)}{2} \), where n is an integer:
1. There are exactly two values of n for which \( S_{n} = 861 \).
2. \( S_{n} = S_{-(n + 1)} \) and hence for any integer m, we have two values of n for which \( S_{n} = m \).
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
18.
Consider the following statements in respect of two different non-zero integers p and q:
1. For \( (p + q) \) to be less than \( (p - q) \), q must be negative.
2. For \( (p + q) \) to be greater than \( (p - q) \), both p and q must be positive.
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
19.
If \( \frac{a}{b} = \frac{b}{c} = \frac{c}{d} \), then which of the following is/are correct?
1. \( \frac{b^{3} + c^{3} + d^{3}}{a^{3} + b^{3} + c^{3}} = \frac{d}{a} \)
2. \( \frac{a^{2} + b^{2} + c^{2}}{b^{2} + c^{2} + d^{2}} = \frac{a}{d} \)
Select the correct answer using the code given below.
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
A. 1 only
20.
\( 7^{10} - 5^{10} \) is divisible by
A.
5
B.
7
C.
10
D.
11
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ANSWER :
D. 11
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