10 Questions
The total surface area (TSA) of a cube is given by the formula $6l^2$.
True
The volume of a cuboid is calculated by multiplying its length, breadth, and height.
True
The curved surface area (CSA) of a cylinder is calculated using the formula $2\pi r^2$.
False
The volume of a cone is one-third of the volume of a cylinder with the same base area and height.
True
The total surface area (TSA) of a sphere is given by $4\pi r^2$.
True
The volume of a hemisphere is two-thirds of the volume of a sphere with the same radius.
True
The lateral surface area (LSA) of a cube is $6l^2$.
False
The perimeter of a cuboid is calculated by the formula $4(l + b + h)$.
True
For the assumed mean method in statistics, mean is given by $a + \frac{\Sigma f_i d_i}{\Sigma f_i}$.
True
The base area of a hemisphere is given by $2\pi r^2$.
False
Study Notes
Number Systems
- HCF(a, b) × LCM(a, b) = a × b
Algebra
Polynomials
- For zeroes of quadratic polynomial p(x) = ax² + bx + c, a ≠ 0:
- Sum of zeroes = -b/a
- Product of zeroes = c/a
- For zeroes of quadratic polynomial p(x) = ax³ + bx² + cx + d, a ≠ 0:
- Sum of zeroes = -b/a
- Product of zeroes = -d/a
- Sum of product of zeroes taken two at a time = c/a
- Identities:
- (a + b)² = a² + b² + 2ab
- (a - b)² = a² + b² - 2ab
- a² - b² = (a + b)(a - b)
- (a + b)³ = a³ + b³ + 3a²b + 3ab²
- (a - b)³ = a³ - b³ - 3a²b + 3ab²
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- a³ - b³ = (a - b)(a² + b² + ab)
- a³ + b³ = (a + b)(a² + b² - ab)
- a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
- If a + b + c = 0, then a³ + b³ + c³ = 3abc
Linear Equations in Two Variables
- a1/a2 ≠ b1/b2: intersecting, consistent, 1 solution
- a1/a2 = b1/b2 ≠ c1/c2: parallel, inconsistent, 0 solution
- a1/a2 = b1/b2 = c1/c2: coincide, infinite solutions, consistent
- Methods for solving linear equations:
- Substitution Method
- Elimination Method
- Reduction Method
Quadratic Equations
- ax² + bx + c = 0, where 'a' ≠ 0
- Quadratic Formula: x = (-b ± √b² - 4ac) / 2a
- Discriminant (D) = b² - 4ac
- If D > 0, then two distinct real roots
- If D = 0, then two equal roots
- If D < 0, then no real roots
Arithmetic Progression
- General form of AP: a, a + d, a + 2d, a + 3d,...
- an = a + (n-1)d
- Sn = [2a + (n-1)d]/2
- Sn = n/2[a + l]
- Sn = Sn-1 + an
- If a, b, c are in AP then 2b = a + c
Coordinate Geometry
- Distance Formula: √(x2 - x1)² + (y2 - y1)²
- Distance of point (p, x) from origin: √p² + x²
- Section Formula: {(m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2)}
- Midpoint Formula: {(x1 + x2)/2, (y1 + y2)/2}
Trigonometry
Trigonometric Identities
- Complementary angles:
- sin θ = cos (90° - θ)
- cos θ = sin (90° - θ)
- tan θ = cot (90° - θ)
- Other identities:
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
- sec θ = 1/cos θ
- cosec θ = 1/sin θ
- cot θ = 1/tan θ
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- 1 + cot²θ = cosec²θ
- sin²θ / cos²θ = 1/cos²θ
- sin²θ / sin²θ = 1/sin²θ
Surface Area and Volume
Cube
- Perimeter: 12l
- Base Area: l²
- LSA: 4l²
- TSA: 6l²
- Volume: l³
- d: √3 l
Cuboid
- Perimeter: 4(l + b + h)
- Volume: l × b × h
- Base Area: lb
- LSA: 2h(l + b)
- TSA: 2(lb + bh + lh)
- d: √(l² + b² + h²)
Cylinder
- Base Area: πr²
- CSA: 2πrh
- TSA: 2πr(h + r)
- Volume: πr²h
Cone
- Base Area: πr²
- CSA: πrl
- TSA: πr(l + r)
- Volume: 1/3πr²h
Sphere
- TSA: 4πr²
- Volume: 4/3πr³
Hemisphere
- Base Area: πr²
- CSA: 2πr²
- TSA: 3πr²
- Volume: 2/3πr³
Statistics
Mean
- Direct Method: mean = Σfmi / Σfi
- Assumed Mean Method: mean = a + (Σfidi / Σfi)
Test your knowledge of algebraic concepts, including number systems, polynomials, and identities.
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