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NDA Mathematics Solved Question Paper 2016

Multiple Choice Questions

1.
Let $α and β$ be the roots of the equation :
$x^{2} - (1 - 2 a^{2}) x + (1 - 2 a^{2}) = 0$
Under what condition does the above equation have real roots ?

$a^{2} < \frac{1}{2}$

$a^{2} > \frac{1}{2}$

$a^{2} \leq \frac{1}{2}$

$a^{2} \geq \frac{1}{2}$

$a^{2} \geq \frac{1}{2}$

$∵ α, β are the roots of the equation x^{2} - (1 - 2 a^{2}) x + (1 - 2 a^{2}) = 0 ∵ Roots are real ∴ D \geq 0 \Rightarrow b^{2} - 4 ac \geq 0 {[- {(1 - 2 a)}^{2}]}^{2} - 4 (1) (1 - 2 a^{2}) \geq 0 \Rightarrow 1 - 4 a^{2} + 4 a^{4} - 4 + 8 a^{2} \geq 0 4 a^{4} + 4 a^{2} - 3 \geq 0 Put a^{2} = x 4 x^{2} + 4 x - 3 \geq 0 \Rightarrow 4 x^{2} + 6 x - 2 x - 3 \geq 0 \Rightarrow 2 x (2 x + 3) - 1 (2 x + 3) \geq 0 \Rightarrow (2 x - 1) (2 x + 3) \geq 0 x \geq \frac{1}{2} \Rightarrow a^{2} \geq \frac{1}{2}$

Let $α and β$ be the roots of the equation :

$x^{2} - (1 - 2 a^{2}) x + (1 - 2 a^{2}) = 0$

Under what condition $\frac{1}{α^{2}} + \frac{1}{β^{2}} < 1 ?$

$a^{2} < \frac{1}{2}$
$a^{2} > \frac{1}{2}$
$a^{2} > 1$
$a^{2} \in (\frac{1}{3}, \frac{1}{2}) only$

What is $\sqrt{\frac{1 + ω^{2}}{1 + ω}}$ = ?, where $ω$ is cube root of unity

1
$ω$
$ω$ ²
None of the above

In an examination, 70% 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 ?

10
12
15
None of the above

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $λz = μ$

Under what condition does the above system of equations have infinitely many solutions ?

$λ = 5 and μ \neq 9$
$λ = 5 and μ = 9$
$λ = 0 and μ = 5$
$λ = 9 and μ \neq 5$

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $λz = μ$

Under what condition does the above system of equations have unique solution ?

$λ = 5, μ = 9$
$λ = 5 and μ = 7 only$
$λ \neq 5 and μ has any real value$
$λ$ has any real value and $μ \neq 9$

What is the number of odd integers between 1000 and 9999 with no digit repeated ?

2100
2120
2240
None of the above

What is the greatest value of the positive integer n satisfying the condition :

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . . + \frac{1}{2^{n - 1}} < 2 - \frac{1}{1000} ?$

2x²+ 3x - $α$ = 0 has roots - 2 and $β$ while the equation x² - 3mx + 2m²= 0 has both roots positive, where a > 0 and $β$ > 0. What is the value of $α$ ?

$\frac{1}{2}$
1
2
4

10.

2x²+ 3x - $α$ = 0 has roots - 2 and $β$ while the equation x² - 3mx + 2m²= 0 has both roots positive, where a > 0 and $β$ > 0. $If β, 2, 2 m are in GP, then what is the value of β \sqrt{m} ?$