NDA & NA Exam (II)- 2023 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2023 - Mathematics

NDA & NA Exam (II)- 2023 - Mathematics Defence Questions

31.
Directions (Qn. 31-32): Consider the following for the next two (02) items that follow:
Consider the equation \( (1 - x)^{4} + (5 - x)^{4} = 82 \).

What is the number of real roots of the equation?
A.
0
B.
2
C.
4
D.
8
😑 View Answer Error
ANSWER :
B. 2
32.
What is the sum of all the roots of the equation?
A.
24
B.
12
C.
10
D.
8
😑 View Answer Error
ANSWER :
B. 12
33.
Directions (Qn. 33-35): Consider the following for the next three (03) items that follow:
Consider equation-I: \( z^{3} + 2z^{2} + 2z + 1 = 0 \) and equation-II: \( z^{1985} + z^{100} + 1 = 0 \).

What are the roots of equation-I?
A.
\( 1, \omega, \omega^{2} \)
B.
\( -1, \omega, \omega^{2} \)
C.
\( 1, -\omega, \omega^{2} \)
D.
\( -1, -\omega, -\omega^{2} \)
😑 View Answer Error
ANSWER :
B. \( -1, \omega, \omega^{2} \)
34.
Which one of the following is a root of equation-II?
A.
−1
B.
\( -\omega \)
C.
\( -\omega^{2} \)
D.
\( \omega \)
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ANSWER :
D. \( \omega \)
35.
What is the number of common roots of equation-I and equation-II?
A.
0
B.
1
C.
2
D.
3
😑 View Answer Error
ANSWER :
C. 2
36.
Directions (Qn. 36-37): Consider the following for the next two (02) items that follow:
A quadratic equation is given by \( (a + b)x^{2} - (a + b + c)x + k = 0 \), where \( a, b, c \) are real.

If \( k = \frac{c}{2} \), \( (c \neq 0) \), then the roots of the equation are:
A.
Real and equal
B.
Real and unequal
C.
Real iff \( a > c \)
D.
Complex but not real
😑 View Answer Error
ANSWER :
B. Real and unequal
37.
If \( k = c \), then the roots of the equation are:
A.
\( \frac{a + c}{a + b} \) and \( \frac{b}{a + b} \)
B.
\( -\frac{a + c}{a + b} \) and \( -\frac{b}{a + b} \)
C.
1 and \( \frac{c}{a + b} \)
D.
−1 and \( -\frac{c}{a + b} \)
😑 View Answer Error
ANSWER :
C. 1 and \( \frac{c}{a + b} \)
38.
Directions (Qn. 38-40): Consider the following for the next three (03) items that follow:
Let \( (1 + x)^{n} = 1 + T_{1}x + T_{2}x^{2} + T_{3}x^{3} + \ldots + T_{n}x^{n} \).

What is \( T_{1} + 2T_{2} + 3T_{3} + \ldots + nT_{n} \) equal to?
A.
0
B.
1
C.
\( 2^{n} \)
D.
\( n2^{n-1} \)
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ANSWER :
D. \( n2^{n-1} \)
39.
What is \( 1 - T_{1} + 2T_{2} - 3T_{3} + \ldots + (-1)^{n}nT_{n} \) equal to (\( n \geq 2 \))?
A.
0
B.
\( -2^{n-1} \)
C.
\( n2^{n-1} \)
D.
1
😑 View Answer Error
ANSWER :
D. 1
40.
What is \( T_{1} + T_{2} + T_{3} + \ldots + T_{n} \) equal to?
A.
\( 2^{n} \)
B.
\( 2^{n} - 1 \)
C.
\( 2^{n-1} \)
D.
\( 2^{n} + 1 \)
😑 View Answer Error
ANSWER :
B. \( 2^{n} - 1 \)
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