Pre Boards

Practice to excel and get familiar with the paper pattern and the type of questions. Check you answers with answer keys provided.

Sample Papers

Download the PDF Sample Papers Free for off line practice and view the Solutions online.

NDA Mathematics Solved Question Paper 2016

Multiple Choice Questions

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 ?

$a^{2} < \frac{1}{2}$
$a^{2} > \frac{1}{2}$
$a^{2} \leq \frac{1}{2}$
$a^{2} \geq \frac{1}{2}$

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

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 infinitely many solutions ?

$λ = 5 and μ \neq 9$

$λ = 5 and μ = 9$

$λ = 0 and μ = 5$

$λ = 9 and μ \neq 5$

$λ = 5 and μ = 9$

$AX = B [\begin{matrix} 2 & 3 & 5 \\ 7 & 3 & - 5 \\ 2 & 3 & λ \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 9 \\ 8 \\ μ \end{matrix}] ρ [A / B] = \frac{[\begin{matrix} 2 & 3 & 5 \\ 7 & 3 & - 5 \\ 2 & 3 & λ \end{matrix}]}{[\begin{matrix} 9 \\ 8 \\ μ \end{matrix}]} Applying R_{1} \to R_{2}, we get \frac{[\begin{matrix} 7 & 3 & - 5 \\ 2 & 3 & 5 \\ 2 & 3 & λ \end{matrix}]}{[\begin{matrix} 8 \\ 9 \\ μ \end{matrix}]} Applying R_{2} \to R_{2} - \frac{2}{7} R_{1} & R_{3} \to R_{3} - R_{2} \frac{[\begin{matrix} 7 & 3 & - 5 \\ 0 & \frac{15}{7} & \frac{45}{7} \\ 0 & 0 & λ - 5 \end{matrix}]}{[\begin{matrix} 8 \\ \frac{47}{7} \\ μ - 9 \end{matrix}]} Applying R_{2} \to R_{2} \times 7 \frac{[\begin{matrix} 7 & 3 & - 5 \\ 0 & 15 & 45 \\ 0 & 0 & λ - 5 \end{matrix}]}{[\begin{matrix} 8 \\ 47 \\ μ - 9 \end{matrix}]} λ - 5 = 0 \Rightarrow λ = 5 and μ - 9 = 0 \Rightarrow μ = 9$

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. $If β, 2, 2 m are in GP, then what is the value of β \sqrt{m} ?$