test

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

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

If \(p, 1, q\) are in AP and \(p, 2, q\) are in GP, then 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.

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

If one root of the equation \( x^{2} - kx + k = 0 \) 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 \(\Delta = \left|\begin{array}{ccc}a & b & c\\ d & e & f\\ g & h & i\end{array}\right|\) and \(A, B, C, D, G\) are the cofactors of the elements \(a, b, c, d, g\) respectively, then what is the value of \(b B + c C - d D - g G\)?

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

Summary \[\Delta=\left|\begin{array}{ccc}k(k+2) & 2 k+1 & 1 \\2 k+1 & k+2 & 1 \\3 & 3 & 1\end{array}\right|\]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 \( A+B \)?

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

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

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

If a complex number is \( z = \frac{a + ib}{c - id} \), how can \( z^3 \) be expressed?

If \( z = \frac{\text{a} + i\text{b}}{\text{c} - i\text{d}} \), what is the value of \( z \times \bar{z} \)?

If \( z = \frac{\text{a} + i\text{b}}{\text{c} - i\text{d}} \) and \( |z| = 1 \), what is the value of \( z^3 \)?

What is \( (p q + q r + r p) \) equal to?

What is \( \left(p^{2}+q^{2}+r^{2}\right) \) equal to?

Consider the following for the next three (03) questions:

The sides of a triangle \(ABC\) are \(AB=3\mathrm{~cm},\,BC=5\mathrm{~cm}\), and \(CA=7\mathrm{~cm}\).
What is the maximum value of \(p\)?

What is \( (p+q+r) \) equal to?

What is ∠B equal to?

What is the area of the triangle?

Consider the following statements :
I. The triangle is obtuse-angled triangle.
II. The sum of acute angles of the triangle is also acute.

Which of the statements given above is/are correct?

What is the area of the triangle?