Real & Complex Numbers

Practise Real & Complex Numbers for FBISE Grade 9–10 Mathematics (Grade 9–10): 7 multiple-choice questions with instant checking and worked explanations. Free, no sign-up.

Tap an answer to check it · 7 questions

1. Question 1

   The sum of the squares of two real numbers is $25$ and their product is $12$. What is the value of $(a+b{)}^2$?
   • A25
   • B13
   • C61
   • D49
   Show answer and explanation

   Answer: D) 49

   The identity $(a+b{)}^2=a^2+b^2+2ab$ directly applies. Substituting the given values $a^2+b^2=25$ and $ab=12$ yields $(a+b{)}^2=25+2\times 12=49$. The other options result from common algebraic mistakes.

2. Question 2

   Consider the quadratic equation $x^2-8x+c=0$ whose roots are non‑real complex numbers. If the sum of the squares of its two roots equals $20$, what is the value of $c$?
   • A10
   • B16
   • C5
   • D22
   Show answer and explanation

   Answer: D) 22

   Using Vieta’s formulas, the sum of the roots is $8$ and their product is $c$. The sum of the squares of the roots equals $(r_1+r_2{)}^2-2r_1{r}_2=64-2c$, which is given as $20$. Solving $64-2c=20$ gives $c=22$. This value also makes the discriminant negative, ensuring the roots are non‑real complex numbers, so the correct answer is $22$. The option $5$ results from an arithmetic mistake in solving $64-2c=20$. The option $10$ satisfies the algebraic equation but yields a positive discriminant, giving real roots. The option $16$ makes the discriminant zero, giving a repeated real root, not a non‑real pair.

3. Question 3

   Which of the following represents the set of all complex numbers $z$ satisfying $|z|=0$?
   • AOnly the number $0$
   • BAll real numbers
   • CAll complex numbers
   • DAll purely imaginary numbers
   Show answer and explanation

   Answer: A) Only the number $0$

   The modulus $|z|$ measures how far $z$ is from the origin. A distance of zero occurs only at the origin itself, so the only complex number satisfying $|z|=0$ is $0$. Therefore the correct answer is “Only the number $0$”. The option “All real numbers” is wrong because any non‑zero real number has a positive distance from the origin. “All purely imaginary numbers” is wrong for the same reason—only $0$ (which is both real and imaginary) has modulus zero. “All complex numbers” is wrong because most complex numbers lie away from the origin.

4. Question 4

   The product of a non‑zero real number and its reciprocal is always
   • A0
   • B1
   • Cthe number itself
   • Dundefined
   Show answer and explanation

   Answer: B) 1

   For any non‑zero real number $x$, its reciprocal is $\frac{1}{x}$, and $x\times \frac{1}{x}=1$. Therefore the product is always $1$. Option "0": Misconception: misunderstanding the definition of reciprocal. Why wrong: the reciprocal of a non‑zero number is also non‑zero, so the product cannot be zero. Option "the number itself": Misconception: confusing $x\times \frac{1}{x}$ with $x\times x$. Why wrong: multiplying a number by its reciprocal yields $1$, not the original number. Option "undefined": Misconception: believing the operation is undefined because division by zero is forbidden. Why wrong: the statement explicitly says the number is non‑zero, so the reciprocal exists and the product is defined.

5. Question 5

   A quadratic equation $x^2-17x+q=0$ has roots that are the lengths of the two perpendicular sides of a right‑angled triangle with integer hypotenuse. Which of the following could be the value of $q$?
   • A72
   • B60
   • C84
   • D90
   Show answer and explanation

   Answer: B) 60

   Using the discriminant $D=289-4q$ we find that $q=60$ makes $D$ a perfect square ($49$), giving roots $5$ and $12$. These lengths satisfy $5^2+12^2=13^2$, so the hypotenuse is an integer. The other options either give legs whose squares do not sum to a perfect square ($q=72$ gives $8$ and $9$, and $8^2+9^2=145$) or a negative discriminant, so they cannot work.

6. Question 6

   If a complex number $z$ satisfies $z+\overline{z}=6$ and $z\cdot \overline{z}=13$, which of the following equals $z^2+{\overline{z}}^2$?
   • A12
   • B14
   • C10
   • D16
   Show answer and explanation

   Answer: C) 10

   The sum $z+\overline{z}=6$ gives the real part $3$, and $z\overline{z}=13$ gives the magnitude $\sqrt{13}$, so the imaginary part is $\pm 2$. Substituting $z=3\pm 2i$ and expanding yields $z^2+{\overline{z}}^2=10$. The other options arise from common algebraic mis‑applications.

7. Question 7

   If a quadratic equation with real coefficients has non‑real roots, which of the following must be true about its discriminant?
   • AIt is negative
   • BIt is a perfect square
   • CIt is positive
   • DIt is zero
   Show answer and explanation

   Answer: A) It is negative

   A quadratic with real coefficients has non‑real (complex) roots precisely when its discriminant $\Delta =b^2-4ac$ is negative. Hence the statement "It is negative" must be true. Zero discriminant gives a repeated real root, not non‑real roots. A positive discriminant yields two distinct real roots. Being a perfect square concerns rationality of real roots, not whether they are real or complex.

More FBISE Grade 9–10 Mathematics topics

• Coordinate Geometry (3)
• Linear Equations & Inequalities (6)
• Probability & Statistics (8)
• Quadratic Equations (5)
• Sets, Functions & Groups (8)
• Trigonometry (6)