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This section covers key concepts in applied mathematics including equations, series, geometry, and vector analysis. Students should focus on solving quadratic equations, understanding arithmetic progressions, and applyin…
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Prove that |a²/bc a/c a/b| = 0 |b²/ac b/a b/c| |c²/ab c/b c/a|.
Use properties of determinants to prove.
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a) Prove that the roots of the equation are rational: x² - 2(m + 1/m) x + 3 = 0.
Show discriminant is a perfect square.
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b) Evaluate 'K' if one roots of 9x² - 15x + k = 0, exceeds the other by 3 also find the roots.
K = 12; Roots are 3 and 1.
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a) Find 'x' and y if: [x+3 1; -3 3y-4] = [2 1; -3 2].
x = -1, y = 2.
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b) Show that: a = î - ĵ + k, b = 2î - 3ĵ - 5k are perpendicular to each other.
Dot product = 0, proving perpendicularity.
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The diagonal of rhombus are 56ft and 30ft respectively. Find the length of a side and the area of a rhombus.
Side = 40ft; Area = 840ft².
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The volume of the cone is 800cu.m. Find its surface area and the length of the diagonal.
Surface area = 400π; Diagonal = 20m.
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If a = 3î - 2ĵ + 5k and b = -2î - ĵ + k. Find the value of 2a - 3b and it is unit vector.
Resultant vector = 9î - 3ĵ + 12k.
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Solve 2x(4x-1)=15 (by completing square method).
Roots are x = 2 and x = -1.5.
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Solve the equation by the method of factorization: abx² + (b² - ac) x-bc=0.
Factor the equation to find the values of x.
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For what value of 'K' the roots of the given equation are equal x²-10x-k=0.
K must equal 25 for equal roots.
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For what value of 'p' and 'q' will both roots of the equation x² + (2p-4)x=3q+5, vanish.
Set 2p-4 = 0 and 3q+5 = 0.
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Which term in the A.P 4, 1, -2, ......... is -77.
The term is the 40th term.
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Sum if infinite series 1 + 1/3 + 1/9.
The sum is 1.5.
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Expand by the binomial formula (x²/2 - 2/x)⁴.
Use the binomial theorem to expand.
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Find the arc length of a unit circle corresponding to the central angle measuring 60°.
Arc length = π/3.
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Prove that (Cosecθ - Cotθ)² = (1-Cosθ)/(1+Cosθ).
Use trigonometric identities to prove.
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Prove that: (Sin 75°-Sin 15°)/(Cos 75°+Cos 15°) = 1/√3.
Apply sine and cosine addition formulas.
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If a=8, b=10 and c=5. Find m<B.
Use the cosine rule to find angle B.
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