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Higher Mathematics (SSC Classes 9-10)NCTB Higher Mathematics (Classes 9-10)Apollonius's Theorem & MediansQuadratic Equations & DiscriminantsInfinite Geometric Series & Recurring DecimalsBinomial Theorem & Pascal TriangleShoelace Area & Vector Analysis

SSC Higher Mathematics Solver & Proof Engine

Comprehensive procedural equation solver and geometric proof engine for NCTB SSC Higher Mathematics (Classes 9-10). Covers all 14 chapters including Apollonius theorem, quadratic equations, infinite geometric series, binomial expansion, coordinate geometry, and probability.

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Board CQ Presets:

Ch 3: Apollonius Theorem & Medians

Calculate medians and verify 3Σa² = 4Σd²

Step-by-Step Proofs & Analysis
Medians (da, db, dc):dₐ = 5.292, dᵦ = 6.021, d_c = 4.272
3Σa² = 4Σd²330
এ্যাপোলোনিয়াসের উপপাদ্য:b2+c2=2(da2+(a2)2)da2=2b2+2c2−a24=2(5)2+2(7)2−(6)24=50+98−364=28.000∴da=5.292 এককবাহু ও মধ্যমার সম্পর্ক যাচাই:3(a2+b2+c2)=4(da2+db2+dc2)=330\begin{aligned} \text{এ্যাপোলোনিয়াসের উপপাদ্য:} &\quad b^2 + c^2 = 2\left(d_a^2 + \left(\frac{a}{2}\right)^2\right) \\[4pt] d_a^2 &= \frac{2b^2 + 2c^2 - a^2}{4} = \frac{2(5)^2 + 2(7)^2 - (6)^2}{4} \\[4pt] &= \frac{50 + 98 - 36}{4} = 28.000 \\[4pt] \therefore d_a &= \mathbf{5.292\text{ একক}} \\[6pt] \text{বাহু ও মধ্যমার সম্পর্ক যাচাই:} &\quad 3(a^2 + b^2 + c^2) = 4(d_a^2 + d_b^2 + d_c^2) = \mathbf{330} \end{aligned}
কাসিও সাইন্টিফিক ক্যালকুলেটর শর্টকাটের জন্য নিচে দেখুনfx-991EX / ClassWiz
Active Result:
dₐ = 5.292, dᵦ = 6.021

Formula & Derivation

Theoretical foundation, dimensional analysis, and governing boundary conditions.

NCTB Higher Mathematics Curricular Framework

Class 9-10 Higher Mathematics requires rigorous mathematical proofs, exact analytical derivations, and bidirectional geometric and algebraic operations. This engine renders complete KaTeX steps for every NCTB textbook problem type across all 14 chapters.

Chapter 3 (Apollonius Theorem):

• b² + c² = 2[da² + (a/2)²]

• da² = (2b² + 2c² - a²) / 4

• 3(a² + b² + c²) = 4(da² + db² + dc²)

Chapters 7 & 10 (Series & Binomial):

• S∞ = a / (1 - r) for |r| < 1

• (ax + by)ⁿ = ∑ ⁿCr (ax)ⁿ⁻ʳ (by)ʳ

• General term Tr+1 = ⁿCr aⁿ⁻ʳ bʳ

Solved Textbook Examples

Three fully worked pedagogical exemplars: standard textbook, advanced edge-case, and authentic past board examination.

Board Exam Standard (CQ Part C — Chapter 3: Geometry)bg-indigo-500/10 text-indigo-400 border border-indigo-500/20
In ΔABC, the sides are a = 6 cm, b = 5 cm, and c = 7 cm. Find the length of median AD on side BC and verify that 3(a² + b² + c²) = 4(da² + db² + dc²).
Given Parameters:
  • Triangle sides: a = 6 cm, b = 5 cm, c = 7 cm
  • Median on side a: da = AD
Procedural Solution:
According to Apollonius's Theorem: b² + c² = 2[da² + (a/2)²]
Substituting given values: 5² + 7² = 2[da² + (6/2)²] ⇒ 25 + 49 = 2[da² + 9]
74 = 2(da² + 9) ⇒ da² + 9 = 37 ⇒ da² = 28 ⇒ da = √28 ≈ 5.292 cm.
Similarly, db² = (2a² + 2c² - b²)/4 = (72 + 98 - 25)/4 = 36.25 and dc² = (2a² + 2b² - c²)/4 = (72 + 50 - 49)/4 = 18.25.
LHS = 3(a² + b² + c²) = 3(36 + 25 + 49) = 3(110) = 330.
RHS = 4(da² + db² + dc²) = 4(28 + 36.25 + 18.25) = 4(82.5) = 330. (Verified)
Final Answer:Median da = 5.292 cm and 3Σa² = 4Σd² = 330 (Verified).
Board Exam Standard (CQ Part B — Chapter 7: Infinite Series)bg-cyan-500/10 text-cyan-400 border border-cyan-500/20
Convert the recurring decimal 0.12̇ into an irreducible common fraction using the infinite geometric series formula.
Given Parameters:
  • Repeating decimal: 0.12̇ = 0.1222...
Procedural Solution:
Decompose the decimal: 0.12̇ = 0.1 + 0.02 + 0.002 + 0.0002 + ...
Non-repeating portion: 0.1 = 1/10.
Repeating portion is an infinite geometric series with first term a = 0.02 = 2/100 and common ratio r = 0.1 = 1/10 (|r| < 1).
Sum to infinity: S∞ = a / (1 - r) = (2/100) / (1 - 1/10) = (2/100) / (9/10) = 2/90 = 1/45.
Total value: 1/10 + 1/45 = (9 + 2)/90 = 11/90.
Final Answer:Common fraction = 11/90.
Board Exam Standard (CQ Part C — Chapter 11: Coordinate Geometry)bg-emerald-500/10 text-emerald-400 border border-emerald-500/20
Find the area of ΔABC with vertices A(2, 3), B(5, 7), and C(-2, 4) using the Shoelace formula.
Given Parameters:
  • Vertices: A(2, 3), B(5, 7), C(-2, 4) in counter-clockwise order
Procedural Solution:
Shoelace area formula: Δ = 1/2 |(x1·y2 + x2·y3 + x3·y1) - (y1·x2 + y2·x3 + y3·x1)|
Step 1 (downward products): (2×7 + 5×4 + (-2)×3) = (14 + 20 - 6) = 28.
Step 2 (upward products): (3×5 + 7×(-2) + 4×2) = (15 - 14 + 8) = 9.
Area calculation: Δ = 1/2 |28 - 9| = 1/2 |19| = 9.5 square units.
Final Answer:Area of ΔABC = 9.5 square units.
Board Exam Standard (CQ Part B — Chapter 10: Binomial Expansion)bg-purple-500/10 text-purple-400 border border-purple-500/20
Expand (1 + 2x)⁵ using the Binomial Theorem and determine the coefficient of x³.
Given Parameters:
  • Binomial expression: (1 + 2x)⁵
  • Index n = 5
Procedural Solution:
General term: T(r+1) = ⁵Cr (1)⁵⁻ʳ (2x)ʳ = ⁵Cr · 2ʳ · xʳ
r = 0: ⁵C0 · 1 = 1
r = 1: ⁵C1 · 2x = 10x
r = 2: ⁵C2 · 4x² = 40x²
r = 3: ⁵C3 · 8x³ = 10 × 8x³ = 80x³ ⇒ Coefficient of x³ is 80
r = 4: ⁵C4 · 16x⁴ = 80x⁴
r = 5: ⁵C5 · 32x⁵ = 32x⁵
Full expansion: 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵.
Final Answer:Expansion = 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵; Coefficient of x³ = 80.

Casio Scientific Calculator Keystroke GuideExam Ready

Exact button sequences permitted in board and university entrance examination halls:

Casio fx-991ES PLUS / 2nd Edition
Vector and coordinate conversion on 991ES.
1. Convert Cartesian to Polar: Pol(x, y)
SHIFT+ (Pol)3SHIFT)4)=
Displays radius r = 5 and angle θ = 53.13°.

Practical Applications

How this scientific principle drives chemical engineering, aerospace, medicine, and research.

Higher mathematics forms the foundational engine of modern computer vision, 3D rendering (coordinate geometry & vectors), structural civil engineering (Apollonius medians & solid geometry), quantitative finance (probability & infinite series), and digital signal processing.

Frequently Asked Questions

Answers to common conceptual misconceptions and board examination guidelines.

When and how is Apollonius's Theorem applied?▼
Apollonius's Theorem relates the lengths of the sides of any triangle to the length of a median: AB² + AC² = 2(AD² + BD²). It also yields the fundamental relation that three times the sum of the squares of the sides equals four times the sum of the squares of the medians: 3(a² + b² + c²) = 4(da² + db² + dc²).
How do recurring decimals convert to fractions using geometric series?▼
A repeating decimal like 0.12̇ splits into a finite decimal 0.1 (1/10) and an infinite geometric series 0.02 + 0.002 + ... with first term a = 0.02 and ratio r = 0.1. Because |r| < 1, the sum S∞ = a / (1 - r) yields 1/45. Adding both portions produces 11/90.
Why must polygon vertices be ordered counter-clockwise in the Shoelace formula?▼
Arranging vertices in counter-clockwise (CCW) order ensures the determinant summation evaluates to a positive area. Clockwise ordering results in an identical magnitude but negative sign, requiring an absolute value operation.
How do Casio fx-991EX and fx-991CW calculators solve quadratic equations?▼
On the fx-991EX ClassWiz, navigate to Menu > A (Equation/Func) > 2 (Polynomial) > 2 (Degree 2), input coefficients a, b, and c, and press = to inspect both real/complex roots and the vertex of the corresponding parabola.