NDA Practice SET - 1

The sum of the first \( k \) terms of a sequence \( S \) is \( 3 k^{2 } + 5 k \). Which of the following is correct ?

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms . If \( r \ neq 1 \) is the common ratio, how many possible real values can \( r \ ) have?

If one root of the equation \( x^{2} - k x + k = 0 \) is \( 2 \sqrt{3 } \) more than the other , then which of the following is the value of \( k \ )?

If \( x+\frac {5}{y}=4 \ ) and \( y +\frac{5}{ x}=-4 \), then what is the value of \( (x+y) \)?

If the 5th , 7th, and 13th terms of an AP are in GP, what is the ratio of its first term to its common difference ?

If \( p, 1 , q \) are in AP and \( p, 2, q \) are in GP, which of the following statements is/are true ?
I. \( p, 4, q \ ) are in HP .
II. \( (1 / p), 1 / 4, (1 / q) \) are in AP.
Use the code given below to choose the correct answer.

If \( x=(111 1)_{2}, y=( 1001)_{2} \ ), and \( z =(110)_{2} \), then what is the value of \( x^{ 3}-y^{3}-z ^{3}-3 x y z \)?

If \( \Delta = \begin{ vmatrix}a & b & c \\ d & e & f \\ g & h & i \end{vmatrix } \) and \( A, B, C, D , G\) are the cofactors of the elements \(a, b, c , d, g\) respectively , then what is the value of \(b B + c C - d D - g G\)?

Summary \ [\Delta = \ left|\begin {array}{ccc }k(k+2) & 2k+1 & 1 \ \2k+1 & k+ 2 & 1 \\3 & 3 & 1\end {array}\right |\]Consider the following statements regarding \(\Delta\ ): I. \(\ Delta\) is positive if \(k > 0\). II. \(\Delta \) is negative if \(k < 0\). III. \(\Delta\ ) is zero if \(k = 0\). How many of the above statements are correct ?

If \(\left |\begin{array }{ccc}2 & 3+i & -1\\ 3-i & 0 & i -1\\-1 & - 1-i & 1\end {array}\right | = A + iB \) where \( i=\sqrt{-1 }\), then what is the value of \(A+B\ )?

If \( A^{2 }+B^{2}+C^{ 2}=0 \), then
\[
\ left|\begin {array}{ccc }
1 & \ cos C & \cos B \\\
\cos C & 1 & \cos A \\\
\ cos B & \cos A & 1
\ end{array}\ right|
\ ]

what is the value of?

If \( \omega \) is a non -real cube root of unity , then what is a root of the equation
\[
\ left|\begin {array}{ccc }
x+1 & \omega & \ omega^{2} \\
\omega & x+\omega ^{2} & 1 \\
\omega ^{2} & 1 & x+\omega
\end{array }\right|=0
\]
?

What is \( \left(\frac {\sqrt{3}+ i}{\sqrt{3 }-i}\right )^{3} \) equal to?

If \( x^{2 }-x+1=0 \), then what is the value of \(\left (x-\frac{1 }{x}\right )^{2}+\left (x-\frac{1 }{x}\right )^{4}+\left (x-\frac{1 }{x}\right )^{8}\)?