CDS Exam (I)- 2017 - Elementary Mathematics Defence Question Paper

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CDS Exam (I)- 2017 - Elementary Mathematics

CDS Exam (I)- 2017 - Elementary Mathematics Defence Questions

11.
If \( x = t^{\frac{1}{t-1}} \) and \( y = t^{\frac{t}{t-1}} \), \( t > 0 \), \( t \neq 1 \), then what is the relation between x and y?
A.
\( y^{x} = x^{\frac{1}{y}} \)
B.
\( x^{\frac{1}{y}} = y^{x} \)
C.
\( x^{y} = y^{x} \)
D.
\( x^{y} = y^{\frac{1}{x}} \)
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ANSWER :
C. \( x^{y} = y^{x} \)
12.
If \( A : B = 3 : 4 \), then what is the value of the expression \( \left(\frac{3A^{2} + 4B^{2}}{3A^{2} - 4B^{2}}\right) \)?
A.
\( -\frac{91}{37} \)
B.
\( -\frac{43}{55} \)
C.
\( \frac{47}{55} \)
D.
Cannot be determined
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ANSWER :
A. \( -\frac{91}{37} \)
13.
If \( A = \{x : x \) is a multiple of \( 7\} \), \( B = \{x : x \) is a multiple of \( 5\} \) and \( C = \{x : x \) is a multiple of \( 35\} \), then which one of the following is a null set?
A.
\( (A - B) \cup C \)
B.
\( (A - B) - C \)
C.
\( (A \cap B) \cap C \)
D.
\( (A \cap B) - C \)
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ANSWER :
D. \( (A \cap B) - C \)
14.
If \( x = 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}} \), then what is the value of \( x^{3} - 6x^{2} + 6x \)?
A.
3
B.
2
C.
1
D.
0
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ANSWER :
B. 2
15.
If \( \sqrt{\frac{x}{y}} = \frac{26}{5} - \sqrt{\frac{y}{x}} \) and \( x + y = 26 \), then what is the value of xy?
A.
5
B.
15
C.
25
D.
30
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ANSWER :
C. 25
16.
What is the solution of the equation \( x \log_{10}\left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10}(2 + 3^{x}) + x \)?
A.
10
B.
3
C.
1
D.
0
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ANSWER :
D. 0
17.
If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2} + px + q = 0 \), then what is \( \alpha^{2} + \beta^{2} \) equal to?
A.
\( p^{2} - 2q \)
B.
\( q^{2} - 2p \)
C.
\( p^{2} + 2q \)
D.
\( q^{2} - q \)
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ANSWER :
A. \( p^{2} - 2q \)
18.
If \( a^{3} = 335 + b^{3} \) and \( a = 5 + b \), then what is the value of \( a + b \) (given that \( a > 0 \) and \( b > 0 \))?
A.
7
B.
9
C.
16
D.
49
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ANSWER :
B. 9
19.
If \( 9^{x}3^{y} = 2187 \) and \( 2^{3x}2^{2y} - 4^{xy} = 0 \), then what can be the value of \( (x + y) \)?
A.
1
B.
3
C.
5
D.
7
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ANSWER :
C. 5
20.
The pair of linear equations \( kx + 3y + 1 = 0 \) and \( 2x + y + 3 = 0 \) intersect each other, if
A.
\( k = 6 \)
B.
\( k \neq 6 \)
C.
\( k = 0 \)
D.
\( k \neq 0 \)
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ANSWER :
B. \( k \neq 6 \)
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