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Federal (FBISE) · Chapter test · Free

1st Year Mathematics Chapter 1 Test1. Complex Numbers

30 marks · 40 minutes · 21 questions. Covers: 1. Complex Numbers.

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1st Year Mathematics Chapter 1 Test

1st Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 30Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 10 × 1 = 10)

  1. What is the position of the particle from mean position when amplitude is 0.0004mm and angle is π3\frac{\pi}{3}
    1. 1500(1+3i)\frac{1}{500}(1 +\sqrt{3i})
    2. 15000(1+3i)\frac{1}{5000}(1 +\sqrt{3i})
    3. 15000(3+i)\frac{1}{5000}(\sqrt{3} + i)
    4. 1500(3+i)\frac{1}{500}(\sqrt{3} + i)
  2. Part of complex number which is along y-axis is called
    1. Real part
    2. Imaginary part
    3. Complex part
    4. Both a and c
  3. Algebraic form of z=2(cos⁡315∘+isin⁡315∘)z = \sqrt{2} (\cos 315^\circ + i \sin 315^\circ) is
    1. 1+i1 + i
    2. 1−i1 - i
    3. i i
    4. 2+i2 + i
  4. If z=(3+4i)(2−3i)z = ( 3 + 4i ) ( 2 - 3i) then ∣z∣|{z}|
    1. 727\sqrt{2}
    2. 5135\sqrt{13}
    3. 13513\sqrt{5}
    4. 272\sqrt{7}
  5. Write z=−2−2iz = - 2 - 2i in polar form.
    1. 22(cos⁡3π4+isin⁡3π4)2\sqrt{2} \left( \cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)
    2. 22(cos⁡3π4−isin⁡3π4)2\sqrt{2} \left( \cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)
    3. 22(−cos⁡3π4+isin⁡3π4)2\sqrt{2} \left( -\cos\frac{3\pi}{4} + i \sin\frac{3\pi}{4} \right)
    4. 22(−cos⁡3π4−isin⁡3π4)2\sqrt{2} \left( -\cos\frac{3\pi}{4} - i \sin\frac{3\pi}{4} \right)
  6. If z1=2−3i z₁= 2 - 3i and [ z₁z₂] = 13 then value of [−z2ˉ][-\bar{z₂} ] is
    1. 13
    2. 13\sqrt{13}
    3. 1313\frac{\sqrt{13}}{13}
    4. 113 \frac{1}{13}
  7. The argument θ of a complex number that lies in the interval −π<θ≤π-\pi < \theta \leq \pi is called
    1. Principal argument
    2. Limited argument
    3. Minimum argument
    4. Maximum argument
  8. For any two complex numbers z₁, z₂ which of the following can be true?
    1. z₁> z₂
    2. z₁ < z₂
    3. z1ˉ>z2ˉ\bar{z₁}>\bar{z₂}
    4. [ z₁] > [ z₂]
  9. Im(x1+y1ix2+y2i)−1=\text{Im} \left( \frac{x_1 + y_1 i}{x_2 + y_2 i} \right)^{-1} =
    1. x1y2−x2y1x12+y12 \frac{x_1 y_2 - x_2 y_1}{x_1^2 + y_1^2} \,
    2. x1y2−x2y1x12+y12 i\frac{x_1 y_2 - x_2 y_1}{x_1^2 + y_1^2} \, i
    3. x1y2−x2y1x22+y22 i\frac{x_1 y_2 - x_2 y_1}{x_2^2 + y_2^2} \, i
    4. x1y2−x2y1x22+y22 \frac{x_1 y_2 - x_2 y_1}{x_2^2 + y_2^2} \,
  10. Factors of 2 z³+ 9 z² - 11 z - 30 are
    1. ( z - 2 ) ( z + 5 ) ( z - 5 )
    2. ( z - 2 ) ( z + 2 ) ( z + 5 )
    3. ( z - 2 ) ( z + 5 ) ( 2 z + 3 )
    4. ( z - 3 ) ( z + 5 ) ( 2 z + 3 )

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Write the following equation and inequation in term of x and y by taking z = x + 𝑖y. ∣2z−𝑖∣=4| 2z - 𝑖 | = 4
  2. Find the magnitude of the following complex number.(3−−8)(3+−8)\left( \sqrt{3} - \sqrt{-8} \right) \left( \sqrt{3} + \sqrt{-8} \right)
  3. Factorize the following:4x2+404x^2 + 40
  4. Write the following equation and inequation in term of x and y by taking z = x + 𝑖y. 12Re(𝑖zˉ)=4\frac{1}{2}Re (𝑖\bar{z}) = 4
  5. When particle is at a position of x = 2 + 3𝑖 from its mean position and xₘₐₓ = 1 + 4𝑖 is the position at maximum distance from mean position as it can be seen under microscope at this point.Calculate the angle at time t = 0 and find the position of the particle.
  6. Convert the following equation and inequation in Cartesian form.0≤arg⁡(z−41+i)≤π60 \leq \arg\left(\frac{z - 4}{1 + i}\right) \leq \frac{\pi}{6}
  7. Find the real and imaginary part of the following.(3+4𝑖)−2( 3 + 4𝑖 )^{-2}
  8. Find the conjugate of the following complex number.5i2−78\frac{5i}{2} - \frac{7}{8}

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. Verify that 2∣z∣≥∣Re (z)∣+∣Im (z)∣Hint: (Start with (∣x∣−∣y∣)2≥0) \sqrt{2} |z| \geq |\text{Re } (z)| + |\text{Im } (z)| \quad \text{Hint: (Start with } (|x| - |y|)^2 \geq 0)
  2. Solve the following equation by completing square method.4z2+5z=144z^2 + 5z = 14
  3. If x₁ + 𝑖y₁ = 12 + 5𝑖 and x₂ + 𝑖y₂ = 3 + 2𝑖 then find the real and imaginary part of the following.(x1+iy1x2+iy2)2 \left( \frac{x_1 + iy_1}{x_2 + iy_2} \right)^{2}