2022 MEB Local Question

2022 (Local)
MATHEMATICS Time Allowed : (3) Hours

(Answer ALL questions)

1.(a) Let I be the identity function on $R$ and $f: R \rightarrow R$ be defined by $f(x)=3+x-2 x^{2}$; Show that $f\circ I=I\circ f=f$. (3 marks)

(b) When the expression $(x+k)^{4}+(2 x+1)^{2}$ is divided by $x+2$, the remainder is 10 . Find the possible values of $k$. (3 marks)

2.(a) Find the term independent of ${x}$ in the expansion of $\left(\frac{1}{2 {x}^{2}}-{x}\right)^{9}$. (3 marks)

(b) A semicircle is divided into $n$ sectors such that the angles of the sectors form an arithmetic progression. If the smallest angle is $20^{\circ}$ and the largest angle is $52^{\circ}$, calculate n. (3 marks)

3.(a) Given that ${A}=\left(\begin{array}{cc}2 & -2 \\ 2 & 3\end{array}\right), {B}=\left(\begin{array}{ll}5 & {a} \\ {c} & 4\end{array}\right)$ and ${C}=\left(\begin{array}{ll}{b} & 6 \\ 4 & {~d}\end{array}\right)$, find the values of ${a}, {b}, {c}$ and ${d}$ when $3 {~A}-2 {~B}=4 {C}$. (3 marks)

(b) Make a table for the toss of two coins, putting first coin on the left, second coin at the top. Find ${P}({H}, {H})$ and ${P}({a}$ head and a tail in any order $)$. (3 marks)

4.(a) Given: ${OM}, {AB}={CD}$, Prove: $\triangle {DBE}$ is isosceles with base BD. (3 marks)

(b) Find the matrix which will translate a distance of $-2$ units horizontally and 2 units vertically. What is the map of $(1,-4)$ ? (3 marks)

5.(a) Using basic acute angle, find $\cos 120^{\circ}$ and $\sec120^{\circ}$. (3 marks)

(b) Differentiate the function $3^{2 x} \cdot \log _{50}\left(x^{2}+2\right)$ with respect to $x$. (3 marks)

(Answer any FOUR questions)

6.(a) A function $f$ is defined by $f(x)=2-3 x$. Find $(f\circ f)(x)$ and $f^{-1}(x)$. Determine whether $(f \circ f)^{-1}(x)$ is the same as $\left(f^{-1}\circ f^{-1}\right)(x)$. (5 marks)

(b) Show that $2 x+1$ is a factor of $6 x^{4}-5 x^{3}-10 x^{2}+5 x+4$ and find the other factors. (5 marks)

7.(a) Show that the mapping $O$ defined by $x \odot y=x^{2}+y^{2}$ is a birary operation on the set $R$ of real numbers and also $(x \cap y) \odot x=x \odot(y \odot x)$. Is the binary operation associative? Why? (5 marks)

(b) Evaluate the coefficients of $x^{5}$ and $x^{4}$ in the expansion of $\left(\frac{x}{3}-3\right)^{7}$. Hence evaluate the coefficient of $x^{5}$ in the expansion of $\left(\frac{x}{3}-3\right)^{7}(x+6)$. (5 marks)

8.(a) Find the solution set in $R$ of the incquation $3-x+x^{2} \geq 3 x^{2}$ by algebraic method and illustrate it on the number line. (5 marks)

(b) If the fourth term of an A.P. $81 \frac{1}{2}, 91,100 \frac{1}{2}, \ldots$ is equal to the sum of the first $n$ terms of an $A.P. \quad 5,7 \frac{1}{2}, 10, \ldots$, find $n$. (5 marks)

9.(a) Find the smallest value of ${n}$ for which the sum to $n$ terms and the sum to infinity of a G.P. 1, $\frac{1}{5}, \frac{1}{25}, \ldots$ differ by less than $\frac{1}{1000}$ (5 marks)

(b) Given that $A=\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right)$ and $B=\left(\begin{array}{rr}3 & 4 \\ -2 & 1\end{array}\right)$, write down the inverse matrix of $A$ and use it to solve the equation $Y A=3 B+2 A$. (5 marks)

10.(a) Find the solution set of the system of equations $4 x+2 y=3$ $3 x-4 y=5$ by matric method, the variables are on the set of real numbers. (5 marks)

(b) A spinner is equally likely to point to any one of the numbers $1,2,3,4,5,6,7,8$. Find the probabilities of scoring a number divisible by 4 and scoring a prime number. If the arrow is spun 200 times, how many would you expect scoring a number not divisible by 4 ? (5 marks)

(Answer any THREE questions)

11.(a) ${M}$ is the midpoint of a chord ${AB}$ of a given circle, ${C}$ is any point on the major arc ${AB}$ and ${CM}$ meets the circle at ${D}$. The circle tangent to ${AB}$ at ${A}$ and passes through ${C}$ cuts ${CD}$ at B. Prove that DM = ME. (5 marks)

(b) ${ABC}$ is a triangle and ${BE}, {CF}$ are the perpendiculars drawn to the sides ${AC}, {AB}$. Prove that $\alpha(\triangle {ABE}): \alpha(\triangle {ACF})={AB}^{2}: {AC}^{2}$. (5 marks)

12.(a) In $\triangle {ABC}, {AB}={AC}, {P}$ is any point on ${BC}$, and $Y$ is any point on ${AP}$. The circles ${BPY}$ and $C P Y$ cut $A B$ and $A C$ respectively at $X$ and Z. Prove that $X Z$ // $B C$. (5 marks)

(b) If $\sin \theta=a$, where $\theta$ is an acute angle, express $\tan ^{2} \theta$ and $\sin 4 \theta$ in terms of a. (5 marks)

13.(a) Solve $\triangle {ABC}$ with $\angle {A}=154^{\circ}, \angle {B}=15^{\circ} 30^{\circ}$ and ${AB}=20$. (5 marks)

(b) Find the equations of the tangent and the normal lines to the curve $y=x^{2}-3 x+2$ at the point where ${x}=3$. (5 marks)

14.(a) Given that $\overrightarrow{{OP}}=2 \vec{a}+\vec{b}, \overrightarrow{O Q}=3 \vec{a}-2 \vec{b}$ and $\overrightarrow{O R}=h \vec{a}+5 \vec{b}$, find, in terms of $\vec{a}$ and $\vec{b}$, the vectors $\overrightarrow{{PQ}}$ and $\overrightarrow{{PR}}$. If ${P}, {Q}$ and ${R}$ are collinear, find the value of ${h}$. (5 marks)

(b) If the radins of a circle incteases from $4 {~cm}$ to $4.01 {~cm}$, find the approximate increase in the area. (5 marks)

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