Mathematics 11 past paper 2022

For a complex number $z$, all the following formulas are true **EXCEPT**:

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Solve the following system by reducing their augmented matrix to the echelon form: $$\begin{aligned} x_1 + 4x_2 + 2x_3 &= 2 \\ 2x_1 + x_2 - 2x_3 &= 9 \\ 2x_1 + 2x_2 - 2x_3 &= 12 \e…

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If $a, b, c$ are real numbers such that $a < b$, $c < 0$, $a \neq 0$, $b \neq 0$, then which of the following inequalities holds:

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Separate $\frac{(2-3i)^2}{1-i}$ into real and imaginary parts.

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Find the real and imaginary parts of the complex number $\frac{(\sqrt{3} - i)^5}{(\sqrt{3} + i)^5}$.

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Which of the following sets forms an **abelian group** under the operation of multiplication?

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If $U = \text{the set of the English alphabets}$, $A$ and $B$ are subsets of $U$, where $A = \{x \mid x \text{ is a vowel}\}$ and $B = \{y \mid y \text{ is a consonant}\}$, then ve…

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Solve the system of simultaneous equations: $$3x + 2y = 7$$ $$3x^2 = 25 + 2y^2$$

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What is the converse of $p \rightarrow q$?

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Determine whether $p \rightarrow (q \rightarrow p)$ is a tautology, a contingency or an absurdity.

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Find the value of $\lambda$ for which the system $$\begin{cases} x + y + z = 0 \\ 2x + y - \lambda z = 0 \\ x + 2y - 2z = 0 \end{cases}$$ has a non-trivial solution. Also solve the…

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Suppose the number of players that play cricket and hockey are $15$ and $13$ respectively. If the total number of players is $21$, what is the number of players that play **both**…

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Construct the truth table for the biconditional $p \leftrightarrow q$.

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Resolve $\frac{2x^4}{(x+3)(x-2)^2}$ into partial fractions.

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The set of non-zero rational numbers is a group under the operation of:

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If $A = \{1, 2, 3, 4\}$, state the domain and range of the relation $R = \{(x, y) \mid x + y = 5\}$.

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Resolve $\frac{x^2}{(x^2 + 4)(x + 2)}$ into partial fractions.

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If $A$ is a matrix of order $3 \times 4$, then which of the following equalities is **TRUE**?

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If $A = [1 \quad 1+i \quad i]$, then find $(\overline{A})' A$.

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Find the sum $S_n$ of the Arithmetic Series $a + (a+d) + (a+2d) + \dots + (a+(n-1)d)$.

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Under the operation "*", complete the following table to obtain a semigroup: | * | a | b | c | |---|---|---|---| | a | c | a | b | | b | ... | ... | c | | c | b | c | a |

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Prove that ${}^nC_k + {}^nC_{k-1} = {}^{n+1}C_k$.

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$\left|\begin{array}{ccc} 1 & 0 & 0 \\ 2 & -i & 0 \\ 3 & -2 & i \end{array}\right| =$

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Without expansion, show that $\left|\begin{array}{ccc} 2 & 3 & -1 \\ 1 & 1 & 0 \\ 2 & -3 & 5 \end{array}\right| = 0$.

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Find the sum of the following series to $n$-terms: $1 + (1+2) + (1+2+3) + \dots

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If $A$ is a skew-symmetric matrix then:

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Find the matrix $A$ if $\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} A = \begin{bmatrix} 0 & -3 & 8 \\ 3 & 3 & -7 \end{bmatrix}$.

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Expand $(1 - 2x)^{\frac{1}{3}}$ to four terms and apply it to evaluate $(0.8)^{\frac{1}{3}}$ correct to three places of decimal.

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If $f(x)$ is a polynomial with only two roots $1$ and $2$, then $f(x) =$

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Find the numerical value of $k$ if the polynomial $x^3 + kx^2 - 7x + 6$ has remainder $4$ when divided by $x - 2$.

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If $2y = \frac{1}{2^2} + \frac{1 \cdot 3}{2!} \cdot \frac{1}{2^4} + \frac{1 \cdot 3 \cdot 5}{3!} \cdot \frac{1}{2^6} + \dots$ then prove that $4y^2 + 4y - 1 = 0$.

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If the polynomial $f(x)$ is divided by $x+2$, the quotient is $x-2$ and the remainder is 2, then $f(x)$ will be:

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Find the inverse of matrix $A = \begin{bmatrix} 2i & i \ i & -i \end{bmatrix}$, hence show that $AA^{-1} = I_2$.

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If $\sin \alpha = \frac{4}{5}$ and $\sin \beta = \frac{12}{13}$, where $\frac{\pi}{2} < \alpha < \pi$ and $\frac{\pi}{2} < \beta < \pi$. Find (i) $\cos(\alpha + \beta)$ (ii) $\sin(…

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If one root of the equation $f(x) = 0$ is $-1$, then $5 - f(-1) =$

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Find the two consecutive numbers whose product is 72.

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Without using calculator/table prove that $\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ = \frac{1}{16}$.

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If $w$ is a cube root of unity, then which of the following equations is true?

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Show that $R = \frac{abc}{4\Delta}$.

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The partial fraction of $\frac{1}{1-x^3}$ will be in the form of:

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If $5, 8$ are two arithmetic means between $a$ and $b$, then find $a$ and $b$.

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What is the partial fraction decomposition of $\frac{x^2 + 2x - 1}{x^2 - 1}$?

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Resolve $\frac{3x - 11}{(x + 3)(x^2 + 1)}$ into partial fractions.

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Solve the equation $\sqrt{3} \tan x - \sec x - 1 = 0$ for its general solution.

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If $a_1 = -1$ in a sequence with general term $a_n = n + a_{n-1}$, then the sum of the first two terms $S_2$ is:

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Find the $9^{\text{th}}$ term of the harmonic sequence $-\frac{1}{5}, -\frac{1}{3}, -1, \dots$.

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Find the second term of the sequence whose general term is $a_n = 2n^2 - 3$.

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If $y = 1 - \frac{x}{2} + \frac{x^2}{4} - \dots$, then show that $x = 2\left(\frac{1-y}{y}\right)$.

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If $b$ is a harmonic mean between $-2$ and $4$, then $b = \dots$

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Find values of $n$ and $r$, when ${}^nC_r = 56$ and ${}^nP_r = 336$.

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If $s_{\infty} = \frac{2}{3}$ and $a = \frac{2}{7}$ in an infinite geometric progression, then the common ratio is:

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Find values of $n$ and $r$, when ${}^nC_r = 10$ and ${}^nP_r = 60$.

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$\binom{8}{7} + \binom{8}{6} =$

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If $x$ is so small that its square and higher powers can be neglected, then show that $\frac{\sqrt{4+x}}{(1+x)^3} \cong 2 - \frac{23}{4}x$.

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For what values of $x$, the binomial expansion of $(1 - \frac{x}{2})^{-1}$ is convergent (valid)?

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There are 9 green and 6 red balls in a box. A ball is drawn (taken out). What is the probability that (i) the ball is green (ii) the ball is red.

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If a fair die is rolled, then what is the probability that the top is an even number?

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Show that the area of a sector of a circular region of radius $r$ is $\frac{1}{2}r^2\theta$, where $\theta$ is the circular measure of the central angle of the sector.

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What is the radius of the circle whose arc-length of measure 4 has a central angle of $\frac{\pi}{2}$?

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Expand and simplify $(2 + i)^4 - (2 - i)^4$.

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Which of the following expressions is the sum of the series $1 - x + x^2 - x^3 + \dots$?

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If $\cot \theta = \frac{4}{3}$ and the terminal arm of the angle is not in the quadrant-I, find the values of $\cos \theta$ and $\csc \theta$.

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If $D(-5, 5\sqrt{2})$ lies on the terminal side of $\theta$, then find the value of $\tan \theta$.

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Find the remaining trigonometric functions if $\cos \theta = -\frac{1}{2}$ and the terminal arm of angle $\theta$ is in quad-III.

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What is the length of the arc that subtends an angle of measure $60^\circ$ at the centre of a circle with radius $6$?

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Show that $\frac{\cos(\pi+\theta) \sec(\pi-\theta)}{\sin^2(\pi+\theta) \cdot \tan(\pi-\theta)} = -\cot \theta \cdot \csc^2 \theta$.

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If ${}^nC_4 = {}^nC_{10}$, then $n = \dots$

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Show that $\frac{\sin(\alpha - \beta)}{\sin(\alpha + \beta)} = \frac{\tan \alpha - \tan \beta}{\tan \alpha + \tan \beta}$.

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$\sin\left(\frac{7\pi}{6}\right) =$

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Prove that $\cot 2x = \frac{\sin x - \sin 3x}{\cos 3x - \cos x}$.

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How many distinct three-digit numbers can be formed from the integers $1, 2, 3, 4, 5, 6$ if each digit is used at most once?

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Which of the following trigonometric expressions is identically equal to $1 - \cos 2\theta$?

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Show that $\tan^{-1}\left(\frac{27}{11}\right) - \tan^{-1}\left(\frac{8}{19}\right) = \frac{\pi}{4}$.

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What is the middle term in the expansion of $(x + x^{-1})^{14}$?

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Show that $2 \cos^{-1} \frac{4}{5} = \sin^{-1} \frac{24}{25}$.

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What is the primary period of $\tan\left(\frac{x}{3}\right)$?

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$\sin(\frac{3\pi}{2} - \alpha) =$

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The circumradius $R$ of a triangle with sides $a, b, c$ is equal to:

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What is the primary period of $\frac{\sin 2x}{1 + \cos 2x}$?

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For what value of $x$, $\tan(x - 30^\circ) = \cot x$?

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A ladder makes an angle of $30^\circ$ with the wall of height $8\text{ m}$. What is the length of the ladder?

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What is the solution of $\sec x = 2$ in the interval $[0, \pi]$?

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What is the value of $\sin^{-1}(-\frac{1}{2})$?

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