1. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –
(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)
2. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]
3. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).
4. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, what is the relationship between \(s\) and \(d\)?
(a) s=6d\(^2\) (b) 3s=7d (c) s\(^3\)=d2\(^2\) (d) d\(^2\)=s/2
5. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).
6. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?
7. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.
(a) undefined (b) positive (c) negative (d) zero
8. If the lengths of the perpendicular sides of a right-angled triangle are \(a\) and \(b\), and the length of the perpendicular drawn from the right-angled vertex to the hypotenuse is \(p\), then –
(a) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} +\cfrac{1}{b^2} \) (b) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} -\cfrac{1}{b^2} \) (c) \(p^2=a^2+b^2\) (d) \(p^2=a^2-b^2\)
9. The areas of the three adjacent faces of a right rectangular prism are \(a\), \(b\), and \(c\) square units. Find the length of the space diagonal (the longest diagonal) of the prism.
10. If a right circular cone has volume \(V\) cubic units, base area \(A\) square units, and height \(H\) units, then determine the value of \(\cfrac{AH}{3V}\).
11. In a cuboid-shaped room, the length, breadth, and height are respectively \(a\), \(b\), and \(c\) units. Given \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), calculate the length of the longest rod that can be placed inside the room.
12. Draw an equilateral triangle with side length 7 cm. Then draw both the circumcircle and the incircle of the triangle. Using a scale, determine the lengths of the circumradius and the inradius. Also, write whether there is any relationship between them.
13. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that PB = AQ, AP = 9 units, and QC = 16 units, what is the length of PB?
(a) 12 cm (b) 6 cm (c) 8 cm (d) 10 cm
14. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).
15. If a right circular cylinder has a radius of 3 cm and a height of 4 cm, then what is the length of the longest rod that can be placed inside it?
16. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).
17. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?
18. A right-angled isosceles triangle in which each of the equal sides has a length of \(a\) units. The perimeter of the triangle is —
(a) \((1+\sqrt{2})a\) units (b) \((2+\sqrt{2})a\) units (c) \(3a\) units (d) \((3+2\sqrt{2})a\) units
19. If the total surface area and the volume of a cube are numerically equal, then the length of its edge will be -- units.
20. A brass plate has a square base with side length \(x\) cm, thickness 1 mm, and a total weight of 4725 grams. If the weight of 1 cubic centimeter of brass is 8.4 grams, then calculate and write the value of \(x\).
21. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.
22. In \(\triangle ABC\), points \(D\) and \(E\) lie on sides \(AB\) and \(AC\) respectively, such that \(DE \parallel BC\) and \(AD : DB = 3 : 1\). If \(EA = 3.3\) cm, determine the length of \(AC\).
(a) 1.1 cm (b) 4 cm (c) 4.4 cm (d) 5.5 cm
23. The radii of two circles are \(r_1\) and \(r_2\) units, respectively. The distance between their centers is \(d\) units. If the circles are externally tangent, which of the following will be correct?
(a) \(r_1-r_2=d\) (b) \(r_1+r_2\gt d\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2\lt d\)
24. If the radius of a circle is \(\sqrt{2}a\) and the perpendicular distance from the center to a chord is \(a\), then determine the length of the chord.
25. If a rectangular prism has \(d\) diagonals, \(v\) vertices, and \(e\) edges, then the value of \(v+d-e\) is ____.
26. A solid rod has a length of \( h \) meters and a diameter of \( r \) meters. It is melted to form 6 spheres, each with a radius of \( r \) meters. Determine the relationship between \( h \) and \( r \).
27. If the volume of a right circular cone is \(V\) cubic units, the area of the base is \(A\) square units, and the height is \(H\), then what is the value of \(\frac{AH}{V}\)?
28. If the volume of a right circular cone is \(V\) cubic units, the area of its base is \(A\) square units, and its height is \(H\) units, then find the value of \(\frac{AH}{V}\).
29. Sathi has drawn a right-angled triangle where the length of the hypotenuse is 6 cm more than twice the length of the smallest side. If the length of the third side is 2 cm less than the hypotenuse, then calculate and write the lengths of all three sides of the triangle.
30. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.
