Test o5

 2 mark

| -0.25 mark |

 60 minutes

 Watch Video

Question 1:

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

Question 2:

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

Question 3:

If \( x=(1111)_{2}, y=(1001)_{2} \) and \( z=(110)_{2} \), then what is \( x^{3}-y^{3}-z^{3}-3 x y z \) equal to?

Question 4:

If \( p, 1, q \) are in AP and \( p, 2, q \) are in GP, then which of the following statements is/are correct?
I. \( p, 4, q \) are in HP.
II. \( (1 / p), 1 / 4,(1 / q) \) are in AP.

Question 5:

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

Question 6:

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, then what is the number of possible real values of \( r \) ?

Question 7:

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

Question 8:

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

Question 9:

Consider the following statements in respect of the determinant \(\Delta=\left|\begin{array}{ccc}k(k+2) & 2k+1 & 1\\2k+1 & k+2 & 1\\3 & 3 & 1\end{array}\right|\)
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 statements given above are correct?

Question 10:

If \(\Delta=\left|\begin{array}{lll}a & b & c \\\ d & e & f \\\ g & h & i\end{array}\right|\) and \(A, B, C, D, G\) are the cofactors of the elements \(a, b, c, d, g\) respectively, then what is \(bB + cC - dD - gG\) equal to?

Question 11:

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

Question 12:

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

Question 13:

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 \(A+B\) equal to?

Question 14:

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