New Millennium Academy

Pokhara-17, Birauta, Kaski, Nepal

First Terminal Examination — 2083 — ANSWER KEY

Subject: Additional Mathematics Class: 9 (Nine) Date: 2083-03-30

F.M.: 75

⚠ Q.N. 17 and 21 reference embedded figures (i.ibb.co images) that could not be opened for verification. Both are solved fully from the numeric/geometric data given in the question text, which is self-sufficient — confirm point labels match your printed figure.

MCQ — Answers1 × 11 = 11

1. (b) \(2\times3\) — order = rows × columns.

2. (a) \(x=5,\ y=4\) — \(x-2=3,\ 5=y+1\).

3. (a) \(1°=\dfrac{1}{360}\)th of a full rotation.

4. (b) \(133.33^{\text{g}}\) — \(G=D\times\dfrac{10}{9}=120\times\dfrac{10}{9}\).

5. (c) \(\dfrac{4\pi}{5}\) — largest \(=\dfrac{4}{10}\times360°=144°=\dfrac{4\pi}{5}^{\text{c}}\).

6. (c) Locus

7. (a) \((2,3)\) — midpoint of \((1,4)\) and \((3,2)\).

8. (c) \(h=-3\) — \(2h-2=-8\Rightarrow h=-3\).

9. (a) \(\binom{4}{-3}\) — \((2,-3)+2(1,0)\).

10. (d) \(45\) — \(40+0.5(50-40)\).

11. (c) \(0\)

WCA — Answers40 marks

Q.N. 12[3]

a)Cartesian product \(A\times B\) is the set of all ordered pairs \((a,b)\) with \(a\in A,\ b\in B\).

b)\(F\times P=\{\)(Apple,320),(Apple,220),(Apple,120),(Banana,320),(Banana,220),(Banana,120),(Mango,320),(Mango,220),(Mango,120)\(\}\) — 9 pairs.

c)Domain \(=\{\)Apple, Banana, Mango\(\}\); Range \(=\{320,220,120\}\)

Q.N. 13[4]

a)\(R_1=\{(1,1),(2,2),(3,3)\}\) (minimal reflexive relation).

b)Arrow diagram: self-loop only at each point — \(1\to1,\ 2\to2,\ 3\to3\); no arrow between distinct points.

c)Yes, R is an equivalence relation. Listing the 9 pairs shows R \(=\{1,2,3\}\times\{1,2,3\}\) (the universal relation) — it contains all \((1,1),(2,2),(3,3)\) (reflexive), every \((a,b)\) has \((b,a)\) present (symmetric), and it is trivially transitive. Hence reflexive + symmetric + transitive ⇒ equivalence relation.

Q.N. 14[3]

a)\(m_{12}+s_{22}=3+2=5\)

b)\(N=\begin{pmatrix}-3&-1\\-8&0\end{pmatrix}\)

\(N=S-2M=\begin{pmatrix}1&5\\0&2\end{pmatrix}-\begin{pmatrix}4&6\\8&2\end{pmatrix}\)

Q.N. 15[4]

a)\(90°=100^{\text{g}}\) (i.e. \(\dfrac{D}{90}=\dfrac{G}{100}\))

b)\(27°\) — \(D=30\times\dfrac{9}{10}=27°\)

c)\(51°\) and \(102°\)

Sum of triangle \(=180°\Rightarrow\) remaining sum \(=180-27=153°\); ratio 1:2 → parts \(=51°,102°\)

Q.N. 16[3]

a)\(l=r\theta\) (\(\theta\) in radians)

b)\(11\) m

\(l=14\times\dfrac{\pi}{4}=14\times\dfrac{22}{7\times4}=11\) m

Q.N. 17[2]

a)Since central angle is proportional to arc length (same radius):

\(\dfrac{\angle AOB}{\angle AOC}=\dfrac{\text{arc }AB}{\text{arc }ABC}\Rightarrow\dfrac{1^{\text{c}}}{180°}=\dfrac{r}{\pi r}=\dfrac{1}{\pi}\Rightarrow 1^{\text{c}}=\dfrac{180°}{\pi}\)

b)Statement: The radian is a constant angle, independent of the circle's radius; i.e. \(\pi^{\text{c}}=180°\).

Q.N. 18[4]

a)Locus: the path traced by a point moving under a given geometrical condition.

b)Midpoint of AC \(=(3.5,\ 4)\)

c)\(D=(3,5)\)

Diagonals of a parallelogram bisect each other: midpoint BD = midpoint AC \(\Rightarrow D=A+C-B=(1+6-4,\ 2+6-3)\)

Q.N. 19[3]

a)Point dividing \(PQ\) internally in ratio \(m:n\): \(\left(\dfrac{mx_2+nx_1}{m+n},\ \dfrac{my_2+ny_1}{m+n}\right)\)

b)\((0,4)\) and \((3,2)\)

Ratio 1:2 → \(\left(\tfrac{6-6}{3},\tfrac{0+12}{3}\right)=(0,4)\); Ratio 2:1 → \(\left(\tfrac{12-3}{3},\tfrac{0+6}{3}\right)=(3,2)\)

Q.N. 20[4]

a)\(A'(-3,-2),\ B'(-6,-4),\ C'(-5,-1)\)

b)Plot \(\triangle ABC\) with \((2,3),(4,6),(1,5)\) and \(\triangle A'B'C'\) with \((-3,-2),(-6,-4),(-5,-1)\) on the same axes — the image lies in the third quadrant, point-symmetric to the original about the line \(y=-x\).

Q.N. 21[3]

a)Triangle Law: if two vectors are represented in magnitude and direction by two sides of a triangle taken in order, their resultant is given by the third side taken in reverse order — \(\vec{AB}+\vec{BC}=\vec{AC}\).

b)\(\binom{2}{6}\) — \((3,4)+(-1,2)\)

c)T is midpoint of BC, so \(\vec{BT}=\tfrac12\vec{BC}\).

\(\vec{AT}=\vec{AB}+\vec{BT}=\vec{AB}+\tfrac12(\vec{AC}-\vec{AB})=\tfrac12\vec{AB}+\tfrac12\vec{AC}=\tfrac12(\vec{AB}+\vec{AC})\) — proved.

Q.N. 22[3]

a)A quartile is one of the three values that divide ordered data into four equal parts.

b)\(Q_1=45\)

Position \(=\dfrac{n+1}{4}=\dfrac{8}{4}=2^{\text{nd}}\) item \(=45\)

c)No effect — \(Q_1\) stays \(45\). The changed value (55→85) moves from the 3rd position to the 7th (largest); the 2nd-position value used for \(Q_1\) is unaffected.

Q.N. 23[4]

\(f(x)=\dfrac{x^2-4}{x-2}=x+2\) for \(x\neq2\)

x	1.9	1.99	2.001	2.01
f(x)	3.9	3.99	4.001	4.01

a)Approaches \(2\) (but never reaches it).

b)See table above.

c)\(\displaystyle\lim_{x\to2}f(x)=4\)

d)At \(x=2\), the denominator \(x-2=0\), giving \(\tfrac{0}{0}\) — division by zero — so \(f(2)\) is undefined even though the limit exists.

CCA — Answers24 marks

Q.N. 24[6]

a)\(x=3350,\ y=500\)

\(7500=15y\Rightarrow y=500\); \(2x+500=7200\Rightarrow x=3350\)

b)\(P_{37}=7410\) l; \(Q_3=8100\) l

Sorted (n=9): 5550,6000,7200,7500,7500,7600,8000,8200,9000.
\(P_{37}\): position \(=\dfrac{37(n+1)}{100}=3.7^{\text{th}}\Rightarrow7200+0.7(7500-7200)=7410\)
\(Q_3\): position \(=\dfrac{3(n+1)}{4}=7.5^{\text{th}}\Rightarrow8000+0.5(8200-8000)=8100\)

c)Not defined — this is a finite, real (non-formulaic) 9-day dataset with no defined general term; the limit-as-\(n\to\infty\) concept applies only to sequences generated by a mathematical rule, not arbitrary finite data.

Q.N. 25[6]

a)\(\angle P=90°,\ \angle A=75°,\ \angle B=15°\)

\(A+B=90°\), \(A-B=\left(\tfrac{\pi}{3}\right)^{\text{c}}=60°\Rightarrow A=75°,\ B=15°\)

b)\(x^2+y^2=49\) (excluding points A, B)

Angle in a semicircle is \(90°\) ⇒ locus is the circle on AB as diameter: centre \((0,0)\), radius \(7\).

c)\(A'(11,0),\ B'(-3,0)\)

\(x'=2(2)-x=4-x\): for A, \(4-(-7)=11\); for B, \(4-7=-3\)

Q.N. 26[6]

a)\(D=(2,2)\)

\(\vec{AB}=(4,-2)\); \(D=C+\vec{AB}=(-2+4,\ 4-2)\)

b)\(3:1\) externally (point is \((10,0)\), beyond B)

\(0=\dfrac{m(1)+n(3)}{m+n}\Rightarrow m:n=-3:1\) (negative ⇒ external division)

c)\(K=-\dfrac{43}{4}\)

\(16+9+4K+12+6=0\Rightarrow4K=-43\)

d)Standard relation: \(\dfrac{D}{180}=\dfrac{R}{\pi}\). Multiplying both sides by 2: \(\dfrac{2D}{180}=\dfrac{2R}{\pi}\Rightarrow\dfrac{D}{90}=\dfrac{2R}{\pi}\) — proved.

Q.N. 27[6]

a)Gita is correct. \(\infty-\infty\) is an indeterminate form — its value depends on the specific expressions involved, so it cannot be fixed as \(0\).

b)\(1°-1^{\text{g}}\) is positive (Sita is correct). Since \(1^{\text{g}}=0.9°\): \(1°-0.9°=0.1°>0\), while \(1^{\text{g}}-1°=-0.1°<0\).

c)Yes, they are parallel. \(\vec b=2\vec a\) (a scalar multiple of \(\vec a\)); vectors are parallel whenever one is a scalar multiple of the other — they need not be equal in magnitude.

d)\(A=\{1,2\},\ B=\{a,b\}\Rightarrow B\times A=\{(a,1),(a,2),(b,1),(b,2)\}\)

e)Because order matters in an ordered pair: e.g. \((1,a)\in A\times B\) but \((1,a)\notin B\times A\) (unless \(A=B\)). So in general \(A\times B\neq B\times A\).

— End of Answer Key —