Class 9 Maths: Number Systems, Polynomials, Coordinate Geometry, Linear Equations, Euclid's Geometry & Lines and Angles: 30 flashcards
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1. What is the set of real numbers made of?
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Real numbers = rational numbers + irrational numbers.
2. What is the condition for a number to be rational?
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It can be written as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
3. Give three examples of irrational numbers.
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$\sqrt{2}$, $\sqrt{3}$ and $\pi$ are irrational numbers.
4. How many rational numbers lie between any two rational numbers?
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Infinitely many rational numbers lie between any two rational numbers.
5. State the law $a^m \cdot a^n$ for real numbers.
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$a^m \cdot a^n = a^{m+n}$
6. What is the degree of a polynomial?
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The degree is the highest power of the variable in the polynomial.
7. What are the degrees of linear, quadratic and cubic polynomials?
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Linear = 1, quadratic = 2, cubic = 3.
8. State the Remainder Theorem.
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When $p(x)$ is divided by $x-a$, the remainder is $p(a)$.
9. State the Factor Theorem.
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$x-a$ is a factor of $p(x)$ if and only if $p(a) = 0$.
10. Expand $(x+y)^2$.
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$(x+y)^2 = x^2 + 2xy + y^2$
11. Factorise $x^2 - y^2$.
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$x^2 - y^2 = (x+y)(x-y)$
12. Factorise $x^3 + y^3$.
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$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
13. What are the four quadrants of the Cartesian plane?
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Quadrant I $(+,+)$, Quadrant II $(-,+)$, Quadrant III $(-,-)$, Quadrant IV $(+,-)$.
14. In the ordered pair $(x, y)$, what are $x$ and $y$ called?
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$x$ is the abscissa (x-coordinate) and $y$ is the ordinate (y-coordinate).
15. What are the coordinates of the origin?
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The origin is $O(0, 0)$.
16. How many solutions does a linear equation in two variables have?
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It has infinitely many solutions.
17. What is the graph of a linear equation in two variables?
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It is always a straight line.
18. What is Euclid's book on geometry called?
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It is called *Elements*.
19. What is the difference between an axiom and a postulate?
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Axioms are common notions accepted without proof; postulates are geometry-specific assumptions.
20. State Euclid's fifth postulate.
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If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side.
21. What are complementary angles?
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Two angles whose sum is $90^\circ$ are complementary.
22. What are supplementary angles?
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Two angles whose sum is $180^\circ$ are supplementary.
23. What is true about vertically opposite angles?
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Vertically opposite angles are always equal.
24. When a transversal cuts two parallel lines, what is true about corresponding angles?
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Corresponding angles are equal.
25. When a transversal cuts two parallel lines, what is true about alternate interior angles?
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Alternate interior angles are equal.
26. When a transversal cuts two parallel lines, what is true about co-interior angles?
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Co-interior angles are supplementary (they add to $180^\circ$).
27. Classify angles by size: acute, obtuse, reflex.
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Acute is less than $90^\circ$; obtuse is between $90^\circ$ and $180^\circ$; reflex is greater than $180^\circ$ but less than $360^\circ$.
28. What is the sum of angles around a point?
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The sum of angles around a point is $360^\circ$.
29. What is the sum of angles on a straight line?
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The sum of angles on a straight line is $180^\circ$.
30. How do you prove that $\sqrt{2}$ is irrational?
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By contradiction: assume $\sqrt{2} = \frac{p}{q}$ in lowest terms, derive that both $p$ and $q$ must be even, which contradicts lowest terms.
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