Athens Academy Islamabad · Unit Model Paper · Mathematics · Grade 11

Athens Academy Islamabad · Unit Model Paper · Mathematics · Grade 11 · Unit 1

Athens Academy Islamabad · Unit Model Paper

MATHEMATICS HSSC–I

Unit 1 — Complex Numbers

Time allowed: 2:35 Hours Total Marks Sections B and C: 80

FBISE · Grade 11 · Session March 2027 · every question set from Unit 1 only

SECTION – A (Objective)

How to prepare Section A

MCQs for this unit are not printed here — they are on the Athens portal, timed and shuffled like the board's objective paper. Work through all three:

Unit MCQs — full chapter bank Online Quiz — timed self-test Flashcards — definitions & traps

Watch the unit's classic traps: $i^{2}=-1$ (never $+1$ ); the conjugate flips only the imaginary sign while the additive inverse flips both; $|3+4i|=5$ , not $7$ ; and the principal argument must lie in $-\pi\lt\theta\le\pi$ .

SECTION – B (Marks 48)

Q. 2 Solve the following Questions. (12 × 4 = 48)

(i)	Simplify $z=\dfrac{(3+i)^{3}}{3-i}$ in the form $a+ib$ where $i=\sqrt{-1}$ , and find the value of $\|z\|$ .	04	OR	Simplify $(5+6i)-(7-4i)$ and $(2-6i)(4+9i)$ , giving each answer in the form $x+iy$ .	04
(ii)	Solve the system of linear equations. $(3-2i)x+(1+2i)y-1=0$ $(3+2i)x-(1-2i)y-1=0$	04	OR	Solve the system $2z-(1-3i)\omega=1+2i$ and $(1+i)z+(2-i)\omega=2+i$ .	04
(iii)	If $z=4-7i$ , find the conjugate $\bar{z}$ and the modulus $\|z\|$ . State also the value of $z\bar{z}$ .	04	OR	For $z=7-6i$ , show that $z\bar{z}=\|z\|^{2}$ and state whether the result is real.	04
(iv)	Using the cycle of powers of $i$ , simplify $i^{31}+i^{58}+i^{100}$ .	04	OR	Simplify $i^{11}+i^{100}$ , showing the remainder method in each case.	04
(v)	Find the multiplicative inverse of $7-5i$ in the form $a+bi$ , and verify your answer.	04	OR	Express $\dfrac{3+7i}{-4+6i}$ in the form $x+iy$ .	04
(vi)	Write $(2-3i)^{2}$ in the form $x+iy$ and state its real part and imaginary part.	04	OR	Find the real and imaginary parts of $\left(\dfrac{12+5i}{3+2i}\right)^{2}$ .	04
(vii)	Factorise over the complex numbers: (a) $z^{2}+16$ (b) $3z^{2}+5$ .	04	OR	Factorise $z^{2}+169$ over $\mathbb{C}$ , and solve $z^{3}+2z^{2}-5z-6=0$ .	04
(viii)	Without expanding the brackets, evaluate $\left\|\dfrac{(3+2i)(1-2i)}{4+3i}\right\|$ .	04	OR	For $z=-3+2i$ , verify that $\|z\|=\|-z\|=\|\bar{z}\|$ . Verify also that $\|z_{1}z_{2}\|=\|z_{1}\|\|z_{2}\|$ for $z_{1}=3+4i$ , $z_{2}=1-2i$ .	04
(ix)	Find the values of the real numbers $x$ and $y$ if $\dfrac{x}{2+3i}-y(1+2i)=1+i$ .	04	OR	Find the real numbers $x$ and $y$ if $(x+iy)(2-i)=3+i$ .	04
(x)	Find the modulus and the principal argument of $-\sqrt{3}+i$ , stating the quadrant used.	04	OR	Find the modulus and the principal argument of $-\sqrt{3}-i$ and of $\sqrt{3}-i$ .	04
(xi)	Express $1+i\sqrt{3}$ in polar form $r(\cos\theta+i\sin\theta)$ , using the principal argument.	04	OR	Convert $4\left(\cos\dfrac{7\pi}{6}+i\sin\dfrac{7\pi}{6}\right)$ to the rectangular form $x+iy$ .	04
(xii)	Solve $z^{2}-6z+25=0$ by completing the square.	04	OR	Solve $z^{2}+4z+13=0$ by completing the square, and state why the roots form a conjugate pair.	04

SECTION – C (Marks 32)

Note: Solve the following Questions. (4 × 8 = 32)

Show full working. A correct method with verified arithmetic earns the marks; verification steps such as $IZ=E$ or substituting back into both equations are part of the answer.

Q.3	Express $\dfrac{2+11i}{3-i}$ in the form $x+iy$ , showing every step of the conjugate method. Hence find the multiplicative inverse of $3-i$ and verify that the product equals $1$ .	08	OR	For $z=x+iy$ , prove that $z\bar{z}=\|z\|^{2}=x^{2}+y^{2}$ and that the result is always real. Verify the result for $z=7-6i$ , and prove that $\overline{z_{1}z_{2}}=\bar{z_{1}}\,\bar{z_{2}}$ for $z_{1}=2+3i$ and $z_{2}=1-4i$ .	08
Q.4	Solve the system $(1+i)z+2\omega=3+i$ and $(1-i)z+(1+i)\omega=5-i$ by elimination, showing the conjugate division in full, and verify both solutions in both equations.	08	OR	Find the real numbers $x$ and $y$ satisfying $\dfrac{x}{1+i}+\dfrac{y}{1-i}=1+3i$ . Hence solve $z^{2}-2z+5=0$ by completing the square, and state why the roots form a conjugate pair.	08
Q.5	Find the modulus and principal argument of $1+i$ , $-1+i$ , $-1-i$ and $1-i$ . State the quadrant rule used in each case, and write each of them in polar form.	08	OR	Find the product $\dfrac{2}{3}\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)\times 6\left(\cos\dfrac{5\pi}{6}+i\sin\dfrac{5\pi}{6}\right)$ , giving the answer in rectangular form. Find also the quotient $\dfrac{6\,\text{cis}\frac{5\pi}{6}}{\frac{2}{3}\,\text{cis}\frac{\pi}{3}}$ , and state its principal argument.	08
Q.6	In an alternating-current circuit the voltage is $(-50+100J)$ volts and the current is $(-6-2J)$ amperes, where $E=IZ$ and $J$ is the imaginary unit used by engineers. Find the impedance $Z$ in the form $a+bJ$ , showing the full conjugate division, and verify your answer by computing $IZ$ . State the unit of $Z$ .	08	OR	Given $z=x+iy$ , express the condition $\|z-2i\|=\|\bar{z}+3\|$ as an equation in $x$ and $y$ , and state what curve it represents. Express also the condition $Re(z-3)\le 2$ in terms of $x$ , and describe the region.	08

Board Marking Pattern — FBISE HSSC–I Mathematics

Section	Attempt	Marks
A — MCQs (prepare on portal)	Compulsory · answered on the bubble sheet	On portal
B — Short questions	All 12, one alternative from each pair	12 × 4 = 48
C — Long questions	All 4, one alternative from each pair	4 × 8 = 32
Total (Sections B and C)	Theory paper — no practical in Mathematics	80

This sheet follows the layout of the FBISE HSSC–I Mathematics paper, but every question is drawn from Unit 1 (Complex Numbers) alone. The board paper draws its twelve short and four long questions from the whole syllabus. Section A marks are earned on the portal quiz until exam day.

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📃 Chapter 1 · Complex Numbers — Chapter Model Paper

FBISE Grade 11 Mathematics — Chapter 1: Complex Numbers Model Paper