Untitled

┬а2 mark

|┬а-0.25 mark |

┬а60 minutes

00:00:00

Question 1:

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 2:

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

Question 3:

If \(x=(1111)_2\), \(y=(1001)_2\) and \(z=(110)_2\), then what is \(x^3 - y^3 - z^3 - 3xyz\) equal to?

Question 4:

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 5:

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 6:

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

Question 7:

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

Question 8:

Consider the following statements in respect of the determinant
\[
\Delta = \begin{vmatrix}
k(k+2) & 2k+1 & 1 \\
2k+1 & k+2 & 1 \\
3 & 3 & 1
\end{vmatrix}
\]
I. ╬Ф is positive if k>0.
II. ╬Ф is negative if k<0.
III. ╬Ф is zero if k=0.

How many of the statements given above are correct?

Question 9:

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 10:

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

Question 11:

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

Question 12:

If
\[
\left|\begin{array}{ccc}
2 & 3+i & -1 \\
3-i & 0 & i-1 \\
-1 & -1-i & 1
\end{array}\right| = A + i\,B
\]
where \(i=\sqrt{-1}\), then what is \(A+B\) equal to?

Question 13:

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 14:

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

Question 15:

Convert the binary number (1111)_2 to its decimal equivalent.

Question 16:

Which of the following is the binary representation of the decimal number 9?

Question 17:

If a binary number is represented as (110)_2, what is its decimal equivalent?
serach-cds
x