10th Mathematics Notes
Very Important 10th Mathematics Notes Urdu Medium written by Honorable Mr. Umair Ch Suib. These computerized notes are very helpful in the preparation of Mathematics for 10th Mathematics Notes of the 10th class Mathematics and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Important 10th class mathematics notes for all students of Punjab Textbook Board.
- Important Full Book Multiple Choice Questions ( MCQs ) with correct Answers of Mathematics class 10th Urdu Medium.
- Important Short Notes on Full Book Definitions in Mathematics class 10th Urdu Medium.
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- Multiple Choice Questions. Four possible answers are given for the following questions. Tick (✔) the correct answer.
- (i) If a, Bare the roots of 3x² + 5x - 2 = 0 , then α + β is: (a) 5 / 3 (b) 3 / 5 (c) -5 / 3 (d) -2 / 3
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- (ii) If α , β Bare the roots of 7x² - x + 4 = 0, then α β is: (a) -1 / 7 (b) 4 / 7 (c) 7 / 4 (d) -4 / 7
- (iii) Roots of the equation 4x² - 5x + 2 = 0 are: (a) irrational (b) imaginary (c) rational (d) none of these
- (iv) Cube roots of -1 are: (a) -1, -ω, ω² (b) -1, ω, -ω² (c) -1, -ω, ω² (d) 1, -ω, -ω²
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- (v) Sum of the cube roots of unity is: (a) 0 (b) 1 (c) -1 (d) 3
- (vi) Product of cube roots of unity is: (a) 0 (b) 1 (c) -1 (d) 3
- (vii) If b² - 4ac < 0, then the roots of ax² + bx + c = 0 are: (a) irrational (b) rational (c) imaginary (d) none of these
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- (viii) If b² - 4ac > 0, but not a perfect square then roots of ax² + bx + c = 0 are: (a) imaginary (b) rational (c) irrational (d) none of these
- (ix) 1 / α + 1 / β is equal to: (a) 1 / α (b) 1 / α - 1 / β (c) α - β / α β (d) α + β / α β
- (x) α² + β² is equal to: (a) α² - β² (b) 1 / α² + 1 / β² (c) (α + β)² - 2 α β (d) α + β
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- (xi) Two square roots of unity are: (a) 1, - 1 (b) 1, ω (c) 1, - ω (d) ω, ω²
- (xii) Roots of the equation 4x² - 4x + 1 = 0 are: (a) real, equal (b) real. unequal (c) imaginary (d) irrational
- (xiii) If α + β pare the roots of px² + qx + r = 0, then sum of the roots 2α and 2ẞ is: (a) -q / P (b) r / P (c) -2q / P (d) -q / 2p
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- Write short answers of the following questions.
- (i). Discuss the nature of the roots of the following equations. (a) x² +3x+5=0 (b) 2x²-7x+3=0 (c) x²+6x -1=0 (d) 16x²-8x+1=0
- (ii). Find ω², if ω = - 1 + 1 √- 3 / 2
- (iii). Prove that the sum of the all cube roots of unity is zero.
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- (iv). Find the product of complex cube roots of unity.
- (v). Show that x³ + y³ = (x + y) (x + ωy) (x + ω²y)
- (vi). Evaluate ω^37 + ω^38 + 1
- (vii). Evaluate (1 - ω + ω²)^6
- (viii). If ω is cube root of unity, form an equation whose roots are 3ω and 3ω².
- (ix). Using synthetic division, find the remainder and quotient when (x³ + 3x² + 2) ÷ (x - 2)
- (x). Using synthetic division, show that x - 2 is the factor of x³ + 7x + 2.
- (xi). Find the sum and product of the roots of the equation 2px² + 3qx - 4r = 0.
- (xii). Find 1 / α² + 1 / β² of the roots of the equation x² - 4x + 3 = 0