Olympiad Test: Achieves Section-4 - Class 8 MCQ

• (P) The Curved Surface Area (C.S.A.) of a cylinder is 2πrh.
• (Q) The Curved Surface Area (C.S.A.) of a cube is 4 × Side².
• (R) The volume of a cylinder is πr²h.
• (S) The volume of a cube is Side³.

• (P) The perimeter of a square is 4 × Side.
• (Q) The area of a rectangle is l × b.
• (R) The circumference of a circle is 2πr.
• (S) The perimeter of a rectangle is 2(l + b).

• (P) The area of a triangle is 1/2 × Base × Height.
• (Q) The area of a parallelogram is Base × Height.
• (R) The area of a trapezium is 1/2 × (Base₁ + Base₂) × Height.
• (S) The area of a rhombus is 1/2 × Diagonal₁ × Diagonal₂​.

• (P) The total surface area of a cube is 6a².
• (Q) The total surface area of a cylinder is 2πrh + 2πr² = 2πr(h + r).
• (R) The volume of a sphere is 4/3​πr³.
• (S) The surface area of a sphere is 4πr².

• (P) The sum of the interior angles of a triangle is 180^∘.
• (Q) The sum of the exterior angles of any polygon is 360^∘.
• (R) The sum of the interior angles of a pentagon is 540^∘.
• (S) The sum of the interior angles of a quadrilateral is 360^∘.

Let the speed of the man in still water be x km/hr and the speed of the stream be y km/hr.
Upstream speed =  x − y = 5 km/hr
Downstream speed = x + y = 9 km/hr
Adding the two equations:
(x − y) + (x + y) = 5 + 9
2x = 14
x =7km/hr
Subtracting the two equations:
(x + y) − (x − y) = 9 − 5
2y = 4
y = 2km/hr

(i) The volume of a sphere is indeed given by the formula (4/3)πr³, which is correct.
(ii) If the radius of a sphere is doubled, the volume becomes (2r)³ = 8r³, making it eight times the original volume, which is also correct.
(iii) The surface area of a sphere is given by 4πr², which is correct.

(i) The determinant of a 2x2 matrix [a b; c d] is indeed ad - bc, which is correct.
(ii) The inverse of a non-singular matrix (a matrix with a non-zero determinant) always exists, which is also correct.
(iii) If a matrix is singular, its determinant is zero, which is correct.

(i) The roots of the quadratic equation ax² + bx + c = 0 are indeed given by (-b ± √(b²-4ac)) / 2a, which is correct.
(ii) The discriminant of the quadratic equation ax² + bx + c = 0 is b² - 4ac, which is also correct.
(iii) If the discriminant is negative, the quadratic equation has two complex conjugate roots, not two real and distinct roots, which makes this statement false.

(i) The sum of the first n natural numbers is indeed given by n(n+1)/2, which is correct.
(ii) The sum of the squares of the first n natural numbers is given by n(n+1)(2n+1)/6, which is also correct.
(iii) The sum of the cubes of the first n natural numbers is given by (n(n+1)/2)², which is correct.