Toppers Study Toppers Study Toppers StudyDetailed NCERT Solutions for 11 Mathematics 5. Complex Numbers and Quadratic Equations to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Detailed NCERT Solutions for 11 Mathematics 5. Complex Numbers and Quadratic Equations to simplify learning. Understand chapters clearly and practice with free solutions for better results.
Preparing for exams becomes easier with Miscellaneous Exercise on Chapter - 5. Whether you are studying for board exams or mid-term exams, 11 Mathematics Chapter 5. Complex Numbers and Quadratic Equations solutions provide quick revising points, well-structured answers, and additional practice material to help you score better.
ncert_solutionsMiscellaneous Exercise on Chapter 5
Q2. For any two complex numbers z1 and z2, prove that
Re (z1z2) = Re z1 Re z2 – Im z1 Im z2
Solution:
Let z1 = a + ib, z2 = c + id
Re z1 = a, Re z2 = c, Im z1 = b, Im z2 = d ….. (1)
z1z2 = (a + ib) (c + id)
= ac + iad + ibc + bd i2
= ac + iad + ibc + bd (-1)
= ac + iad + ibc - bd
= ac - bd + i(ad + bc)
Comparing real and imaginary part we obtain,
Re(z1z2) = ac - bd, Im(z1z2) = ad + bc
Now we take real part
⇒ Re(z1z2) = ac - bd
⇒ Re(z1z2) = Re z1 Re z2 – Im z1 Im z2 [using (1) ]
Hence, proved
Multiplying numerator and denominator by 28 + 10i
On multiplying numerator and denominator by (2 – i), we get
On comparing real and imaginary part we obtain
Q14. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.
Solution:
Let z = (x – iy) (3 + 5i)
= 3x + 5xi - 3yi - 5yi2
= 3x + 5xi - 3yi + 5y
= 3x + 5y + 5xi - 3yi
= (3x + 5y) + (5x - 3y)i
Comparing both sides and equating real and imaginary parts, we get
3x + 5y = – 6 …… (1)
5x - 3y = 24 …… (2)
Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them,
(x + iy)3 = u + iv
⇒ x3 + 3 . x2 . iy + 3 . x . (iy)2 + (iy)3 = u + iv
⇒ x3 + 3x2 y i + 3 x y2 i2 + y3i3 = u + iv
⇒ x3 + 3x2 y i - 3 x y2 - i y3 = u + iv
⇒ x3 - 3 x y2 + 3x2 y i - iy3 = u + iv
⇒( x3 - 3 x y2) + i(3x2 y - y3) = u + iv
On equating both sides real and imaginary parts, we get;
u = x3 - 3 x y2, …… (1)
v = 3x2 y - y3, ……(2)
Q18. Find the number of non-zero integral solutions of the equation |1 - i|x = 2x
Thus, 0 is the only integral solution of the given equation. Therefore, the number of nonzero integral solutions of the given equation is 0.
Q19. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
(a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B2
Solution :
(a + ib) (c + id) (e + if) (g + ih) = A + iB ……….. (1) given
Replacing i by (-i) we have
(a - ib) (c - id) (e - if) (g - ih) = A - iB ……….. (2)
Multiplying (1) and (2)
(a + ib) (a - ib) (c + id) (c - id) (e + if) (e - if) (g + ih) (g - ih) = (A + iB) (A - iB)
(a2 + b2) (c2 + d2) (e2 + f 2) (g2 + h2) = A2 + B2 [ ∵ (x + iy) (x - iy) = x2 + y2]
Hence, proved
Or m = 4k
Hence, least positive integer is 1.
Therefore, the least positive integral value of m is 4 × 1 = 4
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In short, Miscellaneous Exercise on Chapter - 5 for 11 Mathematics Chapter 5. Complex Numbers and Quadratic Equations are a complete study package. With quick revising points, NCERT exercises, and additional important questions, you can prepare effectively for exams. Make these solutions your study companion and excel in your academic journey.
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