Mathematics 11 past paper 2023

The multiplicative inverse of $-i$ is:

MCQ1 marksannual

If $Z_1 = 2 + 3i$ and $Z_2 = 4 + 2i$, then show that $(Z_1 \overline{Z_2} + \overline{Z_1} Z_2)$ is a real number.

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Use Cramer's rule to solve the system of linear equations: $x+y-z=3$; $2x-y-z=1$; $3x+y+2z=0$.

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What is the modulus of a complex number $(8-15i)$?

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Construct a truth table of a logical statement $(p \leftrightarrow q) \wedge (p \rightarrow q)$.

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If three consecutive numbers in an arithmetic progression are increased by 1, 2 and 3 respectively, the resulting numbers are in geometric progression. Find the original numbers if…

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The contrapositive of a conditional $p \rightarrow q$ is:

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Solve for $x$: $\begin{vmatrix} x & -1 \\ 5 & 1-x \end{vmatrix} = \begin{vmatrix} 1 & 0 & -3 \\ 2 & x & -6 \\ 1 & 3 & x-5 \end{vmatrix}$.

Part B4 marksannual

If $y = \frac{1}{4} + \frac{1 \cdot 3}{4 \cdot 8} + \frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12} + \dots$ then prove that $y^2 + 2y - 1 = 0$.

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Which structure in the following is true for the set of natural numbers under multiplication?

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If $\alpha, \beta$ are the roots of $x^2 + px + q = 0$, find the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

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Without using calculator, prove that $\cos 10^\circ \cdot \cos 30^\circ \cos 50^\circ \cdot \cos 70^\circ = \frac{3}{16}$.

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Identify the correct matrix representation from the options provided.

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Using properties of the cube roots of unity, verify that $(1+\omega) + (1+\omega)^2 + (1+\omega)^3 = 2\omega$.

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Solve the following system of equations: $5x^2 - 14xy + 9y^2 = 0$; $4x^2 - 3xy - 16 = 0$.

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Rank of matrix $\begin{bmatrix} -2 \\ 0 \\ -1 \end{bmatrix}$ is:

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Express $\frac{125 + 4x - 9x^2}{(x-1)(x+3)(x+4)}$ in partial fractions.

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Solve triangles $\triangle ABC$ (with usual notations) if: (a) $\alpha = 60^\circ, \beta = 15^\circ$ and $b = 33$ (b) $b = 23, c = 24$ and $\alpha = 75^\circ$

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Second term of a geometric sequence is 9 and its fourth term is 1. Find sum to infinity.

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One of the multiplicative factors of $(x^4 - 5x^2 + 4)$ is:

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Insert six arithmetic means between 15 and -13.

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Which one of the following represents $\frac{x^3 + 2x^2 + 3}{(x^2 + 1)(x + 4)}$?

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Prove that Sine is a periodic function and its period is $2\pi$.

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For what value of $x$, the numbers $\frac{1}{2}, \frac{1}{5}, \frac{1}{x}$ are in harmonic progression?

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A die is thrown twice. Find the probability that the sum of the upper face numbers is a prime number or an odd number.

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If $\binom{n}{8} = \binom{n}{12}$, then value of $n$ is:

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Find the value of $k$, if the constant term in the expansion of $(2x^2 + \frac{k}{x})^6$ is 960.

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If $\cos \theta = \frac{\sqrt{10}}{10}$ with $2\pi < \theta < \frac{5\pi}{2}$, then find values of the remaining five trigonometric ratios.

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The coefficient of the third term in the expansion of $(x - \frac{1}{x})^8$ is:

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Verify that: $\cos 4x \cos x - \sin 6x \sin 3x = \cos 7x \cos 2x$.

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In which quadrant does the terminal side of the angle $-510°$ lie?

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In an oblique triangle $\triangle ABC$ (with usual notations) $a=6$, $c=12$ and $\beta=124^\circ$. Apply law of cosines and law of sines to find the values of $b$, $\alpha$ and $\g…

Part B4 marksannual

Verify that: $2 \tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{7} = \tan^{-1} \frac{31}{17}$.

Part B4 marksannual

The value of $\cos(x + 60^\circ) + \cos(x - 60^\circ)$ is:

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Solve the trigonometric equation $\cos 5\theta + \cos \theta = \cos 3\theta$ where $\theta \in [0, \pi]$.

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The period of $\frac{8}{7} \sec(x - \pi)$ is:

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