Kit Library / Mathematics / Class 9 Ganita Manjari
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
💡 Key Idea
Division breaks closure for rationals.
Rational numbers are closed under addition, subtraction, and multiplication, meaning the result is always a rational number. However, they are not closed under division because division by zero is undefined, which is not a rational number.
↳ Rational numbers are closed under +, -, × but not ÷ (due to zero).
📖 Definition
Zero has no multiplicative inverse.
The multiplicative inverse of a number $x$ is $\frac{1}{x}$, such that $x \times \frac{1}{x} = 1$. Every non-zero rational number has a multiplicative inverse.
↳ Zero is the only rational number without a multiplicative inverse.
💡 Key Idea
Subtraction and division are not associative.
An operation is associative if $(a \circ b) \circ c = a \circ (b \circ c)$. Addition and multiplication of rationals are associative, but subtraction and division are not.
↳ For rationals, + and × are associative; − and ÷ are not.
➗ Formula
GP grows by common ratio.
In a geometric progression (GP), each term is found by multiplying the previous term by a fixed common ratio $r$. The $n$th term is given by $a_n = a_1 \cdot r^{n-1}$.
↳ GP nth term = first term × (common ratio)^(n-1).
➗ Formula
a² − b² = (a+b)(a−b).
The difference of squares identity states $a^2 - b^2 = (a+b)(a-b)$. This is useful for quick mental math with large numbers.
↳ Use (a+b)(a−b) to quickly compute differences of squares.
💡 Key Idea
Diagonals bisect each other.
In a parallelogram, the diagonals bisect each other, meaning they intersect at their midpoints. The intersection point is the midpoint of both diagonals.
↳ Parallelogram diagonals intersect at their common midpoint.
✏️ Example
Find relation between a and b.
Given A(-5, 6) and C(9, 1) as opposite vertices of a parallelogram, the intersection P(a, b) is the midpoint of AC.
↳ Midpoint of diagonal gives intersection point coordinates.
💡 Key Idea
Find term 130 more than 31st.
In an arithmetic progression (AP), the $n$th term is $a_n = a + (n-1)d$. To find which term is $k$ more than the 31st term, set up the equation.
↳ Set up equation aₙ = a₃₁ + 130 to solve for n.
➗ Formula
Convert 1.272727... to fraction.
To convert a recurring decimal to a fraction, let $x$ equal the decimal, multiply by a power of 10 to shift the repeating part, then subtract.
↳ Subtract shifted decimals to eliminate the repeating part.
💡 Key Idea
Zero is the additive identity.
Zero (0) is the identity for addition of rational numbers because adding 0 to any rational number leaves it unchanged: $x + 0 = x$.
↳ 0 is the additive identity; 1 is the multiplicative identity.
✏️ Example
Find three numbers in GP.
For three numbers in GP, let them be $\frac{a}{r}, a, ar$. Their product is $a^3$ and their sum is given.
↳ Represent GP terms as a/r, a, ar to solve sum and product.
💡 Key Idea
P(E) + P(not E) = 1.
The complement rule states that the probability of an event happening plus the probability of it not happening equals 1: $P(E) + P(\text{not } E) = 1$.
↳ Probability of complement = 1 − probability of event.
➗ Formula
Find x given distance = 10.
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula.
↳ Square both sides to solve distance problems.
💡 Key Idea
All coordinates are negative.
In the III quadrant, both x and y coordinates are negative. A rectangle with length $p$ on X-axis and breadth $q$ on Y-axis placed here has all vertices with negative coordinates.
↳ III quadrant points have both coordinates negative.
➗ Formula
Use perpendicular distance from center.
The length of a chord in a circle can be found using the radius and the perpendicular distance from the center to the chord.
↳ Chord length = 2√(r² − d²).
💡 Key Idea
Degree = highest power of variable.
The degree of a polynomial is the highest power of the variable in the polynomial. A constant has degree 0.
↳ Degree is the highest exponent; constants have degree 0.
✏️ Example
Use distributive property.
Multiply polynomials using the distributive property (FOIL method for binomials).
↳ Multiply each term in first bracket by each in second.
💡 Key Idea
Sign rules for integer operations.
Brahmagupta's laws define operations with negative numbers: product of same signs is positive, different signs is negative; subtracting a negative is addition.
↳ Same signs multiply to positive; different signs to negative.
💡 Key Idea
37 terms → middle is 19th.
For an AP with 37 terms, the three middle terms are the 18th, 19th, and 20th terms. Their sum and the sum of last three terms can be used to find the AP.
↳ Middle terms of odd-count AP are centered around the middle term.
This mock test covers key Class 9 Maths topics including rational numbers (closure, identities, inverses), arithmetic and geometric progressions, coordinate geometry, polynomials, and circle properties. It tests both con…
Full explanation, worked examples & memory tricks in the app
All 18 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. In the standard form of a rational number, the common factor of numerator and denominator is always:
BeginnerB · 1
A rational number is in standard form when its denominator is positive and the numerator and denominator have no common factor other than 1.
Q2. Which of the following is not true for rational numbers?
BeginnerC · Rational numbers are closed under division.
Rational numbers are not closed under division because division by zero is not defined.
Q3. Which of the following numbers does not have a multiplicative inverse?
BeginnerB · 0
0 does not have a multiplicative inverse because there is no rational number which when multiplied by 0 gives 1.
Q4. Which of the following operations is not associative for rational numbers?
BeginnerC · Both (B) and (C)
Both subtraction and division are not associative for rational numbers. For example, (a - b) - c ≠ a - (b - c).
Q5. By using suitable formula, the value of 102² - 98² is:
BeginnerC · 400
Q6. The numerical expression 3/8 + (-5)/7 = -19/56 shows that:
BeginnerD · rational numbers are closed under addition.
The sum of two rational numbers is also a rational number, demonstrating closure under addition.
Q7. Zero (0) is:
BeginnerB · the identity for addition of rational numbers.
0 is the additive identity because adding 0 to any rational number leaves it unchanged.
Q8. Rational numbers are closed under division.
BeginnerB · False
Rational numbers are not closed under division because division by zero is undefined.
Q9. The value of 102² - 98² is 800.
BeginnerA · True
102² - 98² = (102-98)(102+98) = 4 × 200 = 800.
Q10. In a square fractal pattern, the number of smaller squares formed at each step follows a GP with first term 1 and common ratio 8. The number of squares in the 5th step is:
IntermediateB · 4096
The nth term of a GP is ar^(n-1). Here a=1, r=8, and n=5, so the 5th term is 1 × 8^4 = 4096.
Q11. A(-5, 6) and C(9, 1) are the two opposite vertices of a parallelogram ABCD. Its diagonals intersect each other at P(a, b). The relation between a and b is:
IntermediateA · b = a - 4.5
The diagonals of a parallelogram bisect each other. So P is the midpoint of AC: a = (-5+9)/2 = 2, b = (6+1)/2 = 3.5. Then b = a + 1.5.
Q12. Which term of the AP 5, 15, 25, ..... will be 130 more than its 31st term?
IntermediateC · 44
The AP has first term a=5 and common difference d=10. The 31st term = 5 + 30×10 = 305. We need the term = 305 + 130 = 435. So 5 + (n-1)×10 = 435, giving n = 44.
Q13. 1.272727..... can be expressed in rational form as:
IntermediateC · 14/11
Let x = 1.272727... Then 100x = 127.272727... Subtracting gives 99x = 126, so x = 126/99 = 14/11.
Q14. If b is the multiplicative inverse of a number x, then the multiplicative inverse of a/b is:
IntermediateD · (xb)/a
The multiplicative inverse of a/b is b/a. Since b = 1/x, we get b/a = (1/x)/a = 1/(ax). But the question asks for the inverse of a/b, which is b/a = (xb)/a when substituting b = 1/x.
Q15. Assertion (A): If x² + 4x + a = (x + 2)² then the value of a is 4. Reason (R): (a + b)² = a² + 2ab + b²
IntermediateB · Both A and R are true and R is the correct explanation of A.
(x+2)² = x² + 4x + 4, so a = 4. The reason correctly explains why.
Q16. Assertion (A): In a cricket match, a batsman hits a boundary 5 times out of 45 balls he plays. The probability that in a given ball, he does not hit the boundary is 8/9. Reason (R): P(E) + P(not E) = 1.
IntermediateC · Both A and R are true and R is the correct explanation of A.
P(hitting boundary) = 5/45 = 1/9. So P(not hitting) = 1 - 1/9 = 8/9. The reason correctly explains the assertion.
Q17. The sum of diameters of two circles is 35 cm and the difference of their circumferences is 22 cm. The area of the smaller circle will be:
AdvancedA · 154 cm²
Let radii be r1 and r2. 2r1 + 2r2 = 35, so r1 + r2 = 17.5. Also 2π(r1 - r2) = 22, so r1 - r2 = 3.5. Solving gives r1 = 10.5 and r2 = 7. Area of smaller circle = π × 7² = 154 cm².
Q18. If the sum of three numbers in GP is 38 and their product is 1728, then the numbers are:
AdvancedA · 8, 12, 18
Let the numbers be a/r, a, ar. Product = a³ = 1728, so a = 12. Sum = 12/r + 12 + 12r = 38. Solving gives r = 2/3 or 3/2. So the numbers are 8, 12, 18.
18 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
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Word Match
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True False
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Memory Match
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Flashcard Battle
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Speed Quiz
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Guess Term
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Sequence Builder
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Revision Battle
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