NDA & NA Exam (II)- 2016 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2016 - Mathematics

NDA & NA Exam (II)- 2016 - Mathematics Defence Questions

31.
For the next two (02) items that follow:
Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2} - (1 - 2a^{2})x + (1 - 2a^{2}) = 0 \).

Under what condition does the above equation have real roots?
A.
\( a^{2} < \frac{1}{2} \)
B.
\( a^{2} > \frac{1}{2} \)
C.
\( a^{2} \leq \frac{1}{2} \)
D.
\( a^{2} \geq \frac{1}{2} \)
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ANSWER :
D. \( a^{2} \geq \frac{1}{2} \)
32.
Under what condition does \( \frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} < 1 \)?
A.
\( a^{2} < \frac{1}{2} \)
B.
\( a^{2} > \frac{1}{2} \)
C.
\( a^{2} > 1 \)
D.
\( a^{2} \in \left(\frac{1}{3}, \frac{1}{2}\right) \) only
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ANSWER :
A. \( a^{2} < \frac{1}{2} \)
33.
What is \( \sqrt{\frac{1+\omega^{2}}{1+\omega}} \) equal to, where \( \omega \) is the cube root of unity?
A.
1
B.
\( \omega \)
C.
\( \omega^{2} \)
D.
\( i\omega \), where \( i = \sqrt{-1} \)
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ANSWER :
B. \( \omega \)
34.
In an examination, 70% of students passed in Physics, 80% students passed in Chemistry, 75% students passed in Mathematics and 85% students passed in Biology and x% students failed in all the four subjects. What is the minimum value of x?
A.
10
B.
12
C.
15
D.
None of the above
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ANSWER :
D. None of the above
35.
For the next two (02) items that follow:
For the system of linear equations \( 2x + 3y + 5z = 9 \), \( 7x + 3y - 2z = 8 \) and \( 2x + 3y + \lambda z = \mu \).

Under what condition does the above system of equations have infinitely many solutions?
A.
\( \lambda = 5 \) and \( \mu \neq 9 \)
B.
\( \lambda = 5 \) and \( \mu = 9 \)
C.
\( \lambda = 9 \) and \( \mu = 5 \)
D.
\( \lambda = 9 \) and \( \mu \neq 5 \)
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ANSWER :
B. \( \lambda = 5 \) and \( \mu = 9 \)
36.
Under what condition does the above system of equations have unique solution?
A.
\( \lambda = 5 \) and \( \mu = 9 \)
B.
\( \lambda \neq 5 \) and \( \mu = 7 \) only
C.
\( \lambda \neq 5 \) and \( \mu \) has any real value
D.
\( \lambda \) has any real value and \( \mu \neq 9 \)
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ANSWER :
C. \( \lambda \neq 5 \) and \( \mu \) has any real value
37.
What is the number of odd integers between 1000 and 9999 with no digit repeated?
A.
2100
B.
2120
C.
2240
D.
3331
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ANSWER :
C. 2240
38.
What is the greatest value of the positive integer n satisfying the condition \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{2^{n-1}} < 2 - \frac{1}{1000} \)?
A.
8
B.
9
C.
10
D.
11
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ANSWER :
C. 10
39.
For the next two (02) items that follow:
\( 2x^{2} + 3x - \alpha = 0 \) has roots −2 and \( \beta \) while the equation \( x^{2} - 3mx + 2m^{2} = 0 \) has both roots positive, where \( \alpha > 0 \) and \( \beta > 0 \).

What is the value of \( \alpha \)?
A.
\( \frac{1}{2} \)
B.
1
C.
2
D.
4
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ANSWER :
C. 2
40.
If \( \beta \), 2, 2m are in GP, then what is the value of \( \beta\sqrt{m} \)?
A.
1
B.
2
C.
4
D.
6
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ANSWER :
A. 1
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