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800 most-asked Class 11 & 12 (+1 / +2) questions across Physics, Chemistry, Maths and Biology — each with a model answer and the exact marking-scheme points examiners reward. Revise smart, walk in calm.

800 board questions•Model answers•Marking-scheme points•Every chapter•CBSE · ISC · State boards

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MathsClass 113 marksmedium

Complex Numbers and Quadratic Equations

Express (3 + 4i)/(1 - 2i) in the form a + i b.

Reveal model answer + marking points▾

Multiply numerator and denominator by the conjugate of the denominator (1 + 2i): [(3 + 4i)(1 + 2i)]/[(1 - 2i)(1 + 2i)]. Numerator = 3 + 6i + 4i + 8i^2 = 3 + 10i - 8 = -5 + 10i. Denominator = 1^2 + 2^2 = 5. So the result = (-5 + 10i)/5 = -1 + 2i.

multiply by conjugate of denominator

Marking-scheme points

  • ✓Multiply by conjugate (1 + 2i)
  • ✓Numerator = -5 + 10i; denominator = 5
  • ✓Result = -1 + 2i
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MathsClass 113 marksmedium

Complex Numbers and Quadratic Equations

Solve the equation x^2 - 2x + 5 = 0.

Reveal model answer + marking points▾

Here a = 1, b = -2, c = 5. Discriminant = b^2 - 4ac = 4 - 20 = -16. So x = [2 +/- sqrt(-16)]/2 = [2 +/- 4i]/2 = 1 +/- 2i. The roots are 1 + 2i and 1 - 2i.

x = [-b +/- sqrt(b^2 - 4ac)]/(2a)

Marking-scheme points

  • ✓Discriminant = 4 - 20 = -16
  • ✓sqrt(-16) = 4i
  • ✓x = (2 +/- 4i)/2 = 1 +/- 2i
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MathsClass 113 markshard

Complex Numbers and Quadratic Equations

Find the square root of the complex number -5 + 12i.

Reveal model answer + marking points▾

Let sqrt(-5 + 12i) = a + i b. Squaring: a^2 - b^2 + 2ab i = -5 + 12i. So a^2 - b^2 = -5 and 2ab = 12 (ab = 6). Also |z| = sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = 13, so a^2 + b^2 = 13. Adding a^2 - b^2 = -5 and a^2 + b^2 = 13 gives 2a^2 = 8, a^2 = 4, a = +/-2; then b^2 = 9, b = +/-3. Since ab = 6 is positive, a and b have the same sign. Hence sqrt(-5 + 12i) = +/- (2 + 3i).

(a + ib)^2 = a^2 - b^2 + 2ab i

Marking-scheme points

  • ✓Let sqrt = a + ib; a^2 - b^2 = -5, 2ab = 12
  • ✓a^2 + b^2 = |z| = 13
  • ✓Solve: a = +/-2, b = +/-3 -> +/- (2 + 3i)
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MathsClass 113 marksmedium

Linear Inequalities

The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

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