Negative Questions MCQs for Sub-Topics of Topic 7: Mensuration

Question 1. Which of the following is NOT a concept primarily dealt with in Mensuration?

(A) Calculating the length of a boundary

(B) Determining the amount of surface within a boundary

(C) Finding the space occupied by a solid object

(D) Analyzing the properties of prime numbers

Question 2. Perimeter is defined as the total length of the boundary of a closed plane figure. Which of the following is NOT a valid unit for measuring perimeter?

(A) Metres ($\text{m}$)

(B) Centimetres ($\text{cm}$)

(C) Kilometres ($\text{km}$)

(D) Square metres ($\text{m}^2$)

Question 3. Area is the amount of surface enclosed by a closed plane figure. Which of the following is NOT a standard unit for measuring area?

(A) Square kilometres ($\text{km}^2$)

(B) Hectares ($\text{ha}$)

(C) Cubic centimetres ($\text{cm}^3$)

(D) Acres

Question 4. Which statement about the scope of Mensuration is INCORRECT?

(A) Mensuration deals with measurements of 2D shapes.

(B) Mensuration deals with measurements of 3D shapes.

(C) Mensuration helps calculate the volume of hollow objects only.

(D) Mensuration helps calculate the surface area of solid objects.

Question 5. Consider units of measurement. Which conversion is NOT correct?

(A) $1\ \text{cm} = 10\ \text{mm}$

(B) $1\ \text{m} = 100\ \text{cm}$

(C) $1\ \text{km} = 1000\ \text{m}$

(D) $1\ \text{m}^2 = 100\ \text{cm}^2$

Question 6. Perimeter is a measure of length. Which of the following is NOT a valid unit for measuring length?

(A) Inch

(B) Foot

(C) Acre

(D) Mile

Question 7. Area is a two-dimensional measure. Which statement is NOT true about area?

(A) Area is always positive for a real object.

(B) Area can be measured in square units.

(C) Area is a property of plane figures.

(D) Area measures the outer boundary length.

Question 8. Which of the following figures does NOT necessarily have a perimeter?

(A) A closed triangle

(B) An open line segment

(C) A square field

(D) A circular track

Question 9. Which of the following conversions for area is NOT correct in the Indian context (common units)?

(A) $1\ \text{hectare} = 10000\ \text{m}^2$

(B) $1\ \text{acre} \approx 4046.86\ \text{m}^2$

(C) $1\ \text{km}^2 = 100\ \text{hectares}$

(D) $1\ \text{are} = 10\ \text{m}^2$

Question 10. Volume measures the space occupied by a solid. Which is NOT a valid unit for volume?

(A) Litre ($\text{L}$)

(B) Cubic metre ($\text{m}^3$)

(C) Millilitre ($\text{mL}$)

(D) Square foot ($\text{ft}^2$)

Question 1. Which of the following is NOT a correct formula for the perimeter of a rectangle with length $l$ and width $w$?

(A) $l+w+l+w$

(B) $2l + 2w$

(C) $2(l+w)$

(D) $l \times w$

Question 2. The perimeter of a square is $4s$, where $s$ is the side length. Which statement is NOT true?

(A) If the side is $5\ \text{cm}$, the perimeter is $20\ \text{cm}$.

(B) If the perimeter is $36\ \text{m}$, the side is $9\ \text{m}$.

(C) If the area is $100\ \text{cm}^2$, the perimeter is $40\ \text{cm}$.

(D) The perimeter is $s^2$.

Question 3. Which statement about the perimeter of a parallelogram is NOT correct?

(A) Opposite sides are equal in length.

(B) Adjacent sides are equal in length.

(C) The perimeter is twice the sum of its adjacent sides.

(D) If the adjacent sides are $a$ and $b$, the perimeter is $2(a+b)$.

Question 4. The perimeter of a polygon is the sum of the lengths of its sides. Which polygon's perimeter is NOT calculated simply by adding its side lengths?

(A) Triangle

(B) Quadrilateral

(C) Regular Pentagon

(D) Circle

Question 5. A wire of length $L$ is bent to form different shapes. For which shape is the total length of the wire NOT equal to the perimeter of the shape?

(A) A square

(B) A rectangle

(C) An open semi-circle

(D) A triangle

Question 6. If the perimeter of an isosceles triangle is $25\ \text{cm}$ and its unequal side is $7\ \text{cm}$, which statement is NOT true?

(A) The sum of the two equal sides is $25-7=18\ \text{cm}$.

(B) Each of the equal sides is $9\ \text{cm}$.

(C) The sides of the triangle are 7 cm, 9 cm, 9 cm.

(D) The triangle is equilateral.

Question 7. The perimeter of a rhombus is $40\ \text{m}$. Which statement is NOT necessarily true?

(A) All four sides are equal.

(B) The length of each side is $10\ \text{m}$.

(C) The diagonals are equal in length.

(D) The diagonals are perpendicular bisectors.

Question 8. If the perimeter of a regular hexagon is $48\ \text{cm}$, which statement is NOT correct?

(A) The hexagon has 6 equal sides.

(B) Each side length is $8\ \text{cm}$.

(C) All interior angles are $60^\circ$.

(D) The perimeter is $6 \times 8\ \text{cm}$.

Question 9. Which of the following can NOT be the side lengths of a triangle?

(A) 3 cm, 4 cm, 5 cm

(B) 6 m, 8 m, 10 m

(C) 5 cm, 5 cm, 10 cm

(D) 7 m, 9 m, 11 m

Question 10. Which statement regarding the perimeter of a shape is INCORRECT?

(A) Perimeter is a linear measurement.

(B) Perimeter is always greater than the longest side (for non-degenerate shapes).

(C) Perimeter is always an integer value.

(D) Perimeter is the length of the boundary.

Question 1. Which is NOT a correct formula for the area of a square with side length $s$?

(A) $s^2$

(B) side $\times$ side

(C) $\frac{1}{2} d^2$ where $d$ is the diagonal

(D) $4s$

Question 2. Which statement about the area of a rectangle is NOT true?

(A) Area = length $\times$ width.

(B) If length is $10\ \text{cm}$ and width is $5\ \text{cm}$, area is $50\ \text{cm}^2$.

(C) Area is measured in linear units.

(D) If perimeter is constant, the area is maximum for a square.

Question 3. Which is NOT a correct formula for the area of a triangle?

(A) $\frac{1}{2} \times \text{base} \times \text{height}$

(B) $\sqrt{s(s-a)(s-b)(s-c)}$

(C) $\text{base} \times \text{height}$

(D) $\frac{1}{2}ab \sin C$ (where a, b are sides and C is included angle)

Question 4. Which statement about the area of a parallelogram is NOT correct?

(A) Area = base $\times$ height.

(B) Area is equal to the area of a rectangle with the same base and height.

(C) Area is half the product of its diagonals.

(D) Area is measured in square units.

Question 5. Which of the following units is NOT suitable for measuring the area of a floor?

(A) Square metres ($\text{m}^2$)

(B) Square feet ($\text{ft}^2$)

(C) Acres

(D) Cubic metres ($\text{m}^3$)

Question 6. If the area of a rectangle is $100\ \text{cm}^2$, and the length is $20\ \text{cm}$, which statement is NOT true?

(A) The width is $5\ \text{cm}$.

(B) The perimeter is $50\ \text{cm}$.

(C) The area is length divided by width.

(D) If the length was halved, the area would be $50\ \text{cm}^2$ (width unchanged).

Question 7. The area of a triangle with base $10\ \text{m}$ and height $8\ \text{m}$ is $40\ \text{m}^2$. If the base is doubled and the height is halved, which statement is NOT true?

(A) The new base is $20\ \text{m}$.

(B) The new height is $4\ \text{m}$.

(C) The new area is $\frac{1}{2} \times 20 \times 4 = 40\ \text{m}^2$.

(D) The area is halved.

Question 8. Which statement regarding the relationship between area and perimeter is NOT always true?

(A) Figures with the same perimeter can have different areas.

(B) Figures with the same area can have different perimeters.

(C) For a fixed perimeter, a square has the maximum area among rectangles.

(D) For a fixed area, a square has the minimum perimeter among rectangles.

Question 9. The area of a trapezium with parallel sides $a$ and $b$ and height $h$ is $\frac{1}{2}(a+b)h$. Which statement is NOT correct about the components of this formula?

(A) $a$ and $b$ are the lengths of the non-parallel sides.

(B) $h$ is the perpendicular distance between the parallel sides.

(C) $\frac{a+b}{2}$ represents the average length of the parallel sides.

(D) The formula involves the sum of the parallel sides.

Question 10. Which statement about calculating area is INCORRECT?

(A) Area is a measure of the space inside a 2D boundary.

(B) Area can be found by multiplying two perpendicular linear dimensions for some shapes.

(C) Area is always calculated using only one formula for all shapes.

(D) Units for area are derived from units of length.

Question 1. Which statement about Heron's formula is NOT correct?

(A) It is used to find the area of a triangle when all three sides are known.

(B) The formula is $\sqrt{s(s-a)(s-b)(s-c)}$.

(C) $s$ in the formula represents the perimeter of the triangle.

(D) $a, b, c$ are the lengths of the sides of the triangle.

Question 2. For a triangle with sides $a, b, c$, the semi-perimeter is $s = \frac{a+b+c}{2}$. Which statement is NOT true?

(A) $s$ is always positive for a non-degenerate triangle.

(B) $s$ is a length measurement.

(C) $s$ is always greater than any single side of the triangle.

(D) $s$ can be equal to one of the side lengths for a valid triangle.

Question 3. If the sides of a triangle are $8\ \text{cm}$, $15\ \text{cm}$, and $17\ \text{cm}$, which statement is NOT correct about using Heron's formula?

(A) The semi-perimeter $s = (8+15+17)/2 = 20\ \text{cm}$.

(B) $s-a = 12\ \text{cm}$, $s-b = 5\ \text{cm}$, $s-c = 3\ \text{cm}$.

(C) Area $= \sqrt{20 \times 12 \times 5 \times 3} = \sqrt{3600} = 60\ \text{cm}^2$.

(D) This triangle is isosceles.

Question 4. Which statement is NOT true about finding the area of an equilateral triangle with side $a$ using Heron's formula?

(A) $s = 3a/2$.

(B) $s-a = a/2$.

(C) Area $= \sqrt{\frac{3a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}$.

(D) The simplified formula derived from Heron's is $a^2$.

Question 5. Heron's formula can be applied to different types of triangles. For which type is using Heron's formula usually NOT the simplest method if other information is available?

(A) Scalene triangle

(B) Isosceles triangle

(C) Right-angled triangle

(D) Triangle where only side lengths are given

Question 6. If the sides of a triangle are $a, b, c$, under which condition can Heron's formula NOT be used to find a non-zero area?

(A) $a+b > c$, $b+c > a$, $c+a > b$

(B) $a^2+b^2=c^2$

(C) $a+b = c$ (Triangle inequality is violated)

(D) $a=b=c$

Question 7. An isosceles triangle has equal sides of length $x$ and base $y$. Which statement is NOT correct?

(A) Perimeter $= 2x+y$.

(B) Semi-perimeter $s = x + y/2$.

(C) $s-x = y/2$.

(D) $s-y = x + y/2 - y = x - y/2$.

Question 8. If the area of a triangle is calculated using Heron's formula, and the result is $0$, which statement is NOT true about the triangle?

(A) The three vertices are collinear.

(B) It is a degenerate triangle.

(C) It has a non-zero area.

(D) The triangle inequality condition is violated ($a+b \le c$ or similar).

Question 9. Which quantity is NOT required when calculating the area of a triangle using Heron's formula?

(A) Length of side $a$

(B) Length of side $b$

(C) Height corresponding to a base

(D) Semi-perimeter

Question 10. The sides of a triangular field are $100\ \text{m}$, $120\ \text{m}$, and $140\ \text{m}$. Which is NOT a step in finding the area using Heron's formula?

(A) Calculate the perimeter: $360\ \text{m}$.

(B) Calculate the semi-perimeter: $180\ \text{m}$.

(C) Find the product $(s-a)(s-b)(s-c)$.

(D) Divide the result by $s$ and take the square root.

Question 1. Which is NOT a correct formula for the area of a rhombus with diagonals $d_1$ and $d_2$?

(A) $\frac{1}{2} d_1 d_2$

(B) base $\times$ height

(C) $s^2$ where $s$ is the side length

(D) Area of 4 congruent right triangles formed by diagonals

Question 2. Which property is NOT always true for a kite, relevant to its area calculation?

(A) Diagonals are perpendicular.

(B) One diagonal is the perpendicular bisector of the other.

(C) Area is half the product of diagonals.

(D) All four sides are equal.

Question 3. Which is NOT a valid method to find the area of a general quadrilateral?

(A) Divide it into two triangles using a diagonal and sum their areas.

(B) Use the formula $\frac{1}{2}d(h_1+h_2)$ where $d$ is a diagonal and $h_1, h_2$ are perpendiculars to it from opposite vertices.

(C) Multiply the perimeter by the average height.

(D) Divide it into a rectangle and triangles/trapezia.

Question 4. To find the area of a quadrilateral ABCD using Heron's formula by dividing it with diagonal AC, which quantity is NOT needed?

(A) Length of side AB

(B) Length of side BC

(C) Length of diagonal BD

(D) Length of diagonal AC

Question 5. Which statement is NOT true about finding the area of a regular polygon (with more than 4 sides)?

(A) It can be divided into congruent isosceles triangles meeting at the center.

(B) Its area is $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$.

(C) Its area is the sum of the areas of the triangles formed by triangulation from one vertex.

(D) Its area can be calculated using a formula based on its side length and the number of vertices without needing apothem or triangulation.

Question 6. A rhombus has diagonals $12\ \text{cm}$ and $16\ \text{cm}$. Which statement about this rhombus is NOT true?

(A) Its area is $96\ \text{cm}^2$.

(B) Its area is half the product of its diagonals.

(C) Its diagonals are equal in length.

(D) Its diagonals bisect each other at right angles.

Question 7. Which is NOT a type of quadrilateral?

(A) Parallelogram

(B) Rhombus

(C) Pentagon

(D) Trapezium

Question 8. To find the area of a quadrilateral field ABCD, a surveyor takes measurements. Which set of measurements is NOT sufficient to find the area?

(A) Lengths of all four sides and one diagonal.

(B) Length of a diagonal and the perpendiculars from the other two vertices to this diagonal.

(C) Lengths of all four sides only.

(D) Coordinates of all four vertices on a coordinate plane.

Question 9. The area of a regular hexagon with side $a$ is $\frac{3\sqrt{3}}{2}a^2$. Which statement is NOT correct?

(A) The area is $6 \times$ Area of an equilateral triangle with side $a$.

(B) The hexagon can be divided into 6 equilateral triangles.

(C) The formula is $\frac{\sqrt{3}}{4}(6a)^2$.

(D) The formula is $\frac{1}{2} \times (6a) \times \text{apothem}$.

Question 10. Which statement about the area of a polygon is INCORRECT?

(A) The area of a polygon can always be found by dividing it into triangles.

(B) The area of a concave polygon can be calculated by summing the areas of non-overlapping triangles formed by triangulation.

(C) The area of a polygon is always positive.

(D) The area unit for polygons is the same as the unit for perimeter squared.

Question 1. Which of the following is NOT a correct formula for the circumference $C$ of a circle with radius $r$ and diameter $d$?

(A) $C = 2\pi r$

(B) $C = \pi d$

(C) $C = \pi r^2$

(D) $C = \pi (2r)$

Question 2. Which of the following is NOT a correct formula for the area $A$ of a circle with radius $r$ and diameter $d$?

(A) $A = \pi r^2$

(B) $A = \pi (d/2)^2$

(C) $A = \frac{C \times r}{2}$ where $C$ is circumference

(D) $A = \pi d$

Question 3. Which statement about the value of $\pi$ is NOT true?

(A) It is an irrational number.

(B) It is approximately $3.14$.

(C) It is exactly $\frac{22}{7}$.

(D) It is the ratio of circumference to diameter.

Question 4. If the radius of a circle is $10\ \text{cm}$, which statement is NOT correct?

(A) Its diameter is $20\ \text{cm}$.

(B) Its circumference is $20\pi\ \text{cm}$.

(C) Its area is $100\pi\ \text{cm}^2$.

(D) Its area is $20\pi\ \text{cm}^2$.

Question 5. If the circumference of a circle is $C$, which expression does NOT represent its area $A$?

(A) $A = \frac{C^2}{4\pi}$

(B) $A = \pi (\frac{C}{2\pi})^2$

(C) $A = \frac{C \times r}{2}$ where $r$ is radius

(D) $A = C^2 / (2\pi)$

Question 6. If the diameter of a circle is tripled, which statement is NOT true?

(A) The radius is tripled.

(B) The circumference is tripled.

(C) The area becomes six times the original area.

(D) The ratio of circumference to diameter remains constant.

Question 7. Which unit is NOT suitable for measuring the circumference of a wheel?

(A) Metres

(B) Centimetres

(C) Inches

(D) Square feet

Question 8. If the area of a circle is $A$, which expression does NOT represent its radius $r$?

(A) $r = \sqrt{A/\pi}$

(B) $r = A/\pi$

(C) $r = \frac{C}{2\pi}$ where $C$ is circumference

(D) $r = \sqrt{\frac{A}{\pi}}$

Question 9. Which statement about the relationship between circumference and area is NOT correct?

(A) Both depend on the radius (or diameter).

(B) Circumference is a linear measure, area is a square measure.

(C) If circumference increases, area decreases.

(D) If area increases, circumference increases.

Question 10. A circular park has a diameter of $42\ \text{m}$. Which statement is NOT correct? (Use $\pi = \frac{22}{7}$)

(A) Its radius is $21\ \text{m}$.

(B) Its circumference is $132\ \text{m}$.

(C) Its area is $1386\ \text{m}^2$.

(D) Its area is $2772\ \text{m}^2$.

Question 1. Which statement about a sector of a circle is NOT true?

(A) It is bounded by two radii and an arc.

(B) Its area is proportional to the central angle.

(C) Its perimeter includes the length of the chord connecting the endpoints of the radii.

(D) It is a part of the circle.

Question 2. Which of the following is NOT a correct formula for the area of a sector with radius $r$ and central angle $\theta$ (in degrees)?

(A) $\frac{\theta}{360} \times \pi r^2$

(B) $\frac{1}{2} r^2 \theta_{radians}$

(C) $\frac{1}{2} \times \text{arc length} \times r$

(D) $\frac{\theta}{360} \times 2\pi r$

Question 3. Which statement about a segment of a circle is NOT correct?

(A) It is bounded by a chord and an arc.

(B) A diameter divides the circle into two segments (semicircles).

(C) Its area is the area of the corresponding sector plus the area of the triangle formed by the radii and chord.

(D) There is a minor segment and a major segment for any chord (unless it's a diameter).

Question 4. If a sector has a central angle of $90^\circ$ and radius $7\ \text{cm}$, which statement is NOT true? (Use $\pi = \frac{22}{7}$)

(A) It is also called a quadrant.

(B) Its area is $\frac{1}{4} \times \pi (7)^2 = 38.5\ \text{cm}^2$.

(C) The length of its arc is $\frac{1}{4} \times 2\pi (7) = 11\ \text{cm}$.

(D) Its perimeter is $11\ \text{cm}$.

Question 5. The area of a minor segment is Area of corresponding sector $-$ Area of triangle formed by radii and chord. Which statement is NOT true about this calculation?

(A) The triangle is formed by the two radii bounding the sector and the chord connecting their endpoints.

(B) The area of the triangle can be found using $\frac{1}{2}r^2 \sin \theta$ where $\theta$ is the central angle.

(C) The area of the major segment is found by subtracting the area of the minor segment from the area of the sector.

(D) The central angle must be less than $180^\circ$ for a minor segment.

Question 6. Which is NOT a correct formula for the length of an arc of a sector with radius $r$ and central angle $\theta$?

(A) $l = \frac{\theta_{degrees}}{360} \times 2\pi r$

(B) $l = r \theta_{radians}$

(C) $l = \frac{\theta_{degrees}}{180} \times \pi r$

(D) $l = \frac{\theta_{radians}}{2\pi} \times \pi r^2$

Question 7. If a sector has an area equal to the area of the corresponding triangle formed by the radii and chord, which statement is NOT necessarily true?

(A) The area of the segment is zero.

(B) The central angle is $0^\circ$.

(C) The area of the sector is non-zero.

(D) The radius is non-zero.

Question 8. A chord divides a circle into two segments. Which statement is NOT true?

(A) The sum of the areas of the minor and major segment equals the area of the circle.

(B) The perimeter of a segment is the arc length plus the chord length.

(C) A diameter divides the circle into two equal segments.

(D) The area of the major segment is always greater than the area of the minor segment.

Question 9. The area of a sector with radius $r$ and arc length $l$ is $\frac{1}{2}rl$. Which statement is NOT correct?

(A) This formula is derived from the proportional relationship between sector area and total area/angle.

(B) This formula requires the arc length to be given directly.

(C) This formula works only when the central angle is $180^\circ$.

(D) This formula requires radius and arc length to be in consistent units.

Question 10. Which of the following statements is NOT correct?

(A) The angle swept by the minute hand of a clock in 60 minutes is $360^\circ$.

(B) The area swept by the minute hand in 10 minutes is $\frac{60}{360}$ of the area of the circle formed by the minute hand.

(C) The distance moved by the tip of the minute hand in 15 minutes is $\frac{90}{360}$ of the circumference.

(D) The area swept by the hour hand in 1 hour is $\frac{30}{360}$ of the area of the circle formed by the hour hand.

Question 1. When finding the area of a composite figure formed by combining basic shapes, which statement is NOT always true?

(A) The total area is the sum of the areas of the individual shapes if they do not overlap.

(B) If shapes overlap, the area of the overlapping region must be accounted for.

(C) The area is found by adding the perimeters of the individual shapes.

(D) Subtracting areas might be required if a region is removed from a larger shape.

Question 2. To find the perimeter of a composite figure, which approach is NOT correct?

(A) Trace along the outer boundary of the combined figure and measure its length.

(B) Sum the perimeters of all the individual shapes involved.

(C) Exclude the lengths of any line segments that are internal to the combined figure.

(D) Add the lengths of all segments that form the outer boundary.

Question 3. A rectangular park is $50\ \text{m}$ by $30\ \text{m}$. A path $2\ \text{m}$ wide runs inside it, along the boundary. Which statement about the area of the path is NOT correct?

(A) The outer dimensions of the path are $50\ \text{m}$ by $30\ \text{m}$.

(B) The inner dimensions of the path are $(50-4)\ \text{m}$ by $(30-4)\ \text{m}$, i.e., $46\ \text{m}$ by $26\ \text{m}$.

(C) The area of the outer rectangle is $50 \times 30 = 1500\ \text{m}^2$.

(D) The area of the path is the sum of the areas of the outer and inner rectangles.

Question 4. A circular park of radius $20\ \text{m}$ has a concentric path of width $5\ \text{m}$ inside it. Which statement about the area of the path is NOT correct? (Use $\pi = 3.14$)

(A) The radius of the outer circle is $20\ \text{m}$.

(B) The radius of the inner circle is $15\ \text{m}$.

(C) The area of the path is the area of the outer circle minus the area of the inner circle.

(D) The area of the path is $\pi (20^2 - 15^2) = \pi (400 - 225) = 175\pi \approx 549.5\ \text{m}^2$.

Question 5. A square piece of cloth of side $10\ \text{cm}$ has a circular hole of radius $3\ \text{cm}$ cut out from its center. Which statement about the area of the remaining cloth is NOT correct? (Use $\pi = 3.14$)

(A) Area of the square is $100\ \text{cm}^2$.

(B) Area of the circular hole is $\pi (3)^2 = 9\pi \approx 28.26\ \text{cm}^2$.

(C) The area of the remaining cloth is the sum of the areas of the square and the circle.

(D) The area of the remaining cloth is $100 - 9\pi \approx 100 - 28.26 = 71.74\ \text{cm}^2$.

Question 6. Two crossing paths, each $1\ \text{m}$ wide, run perpendicular to each other through the center of a rectangular park $30\ \text{m}$ by $20\ \text{m}$, parallel to its sides. Which statement about the area of the paths is NOT correct?

(A) Area of path along length is $30 \times 1 = 30\ \text{m}^2$.

(B) Area of path along width is $20 \times 1 = 20\ \text{m}^2$.

(C) The area of the central overlap is $1 \times 1 = 1\ \text{m}^2$.

(D) The total area of the paths is $30 + 20 + 1 = 51\ \text{m}^2$.

Question 7. A design is made by attaching a semicircle to one side of an equilateral triangle. If the side of the triangle is $a$, which statement about the perimeter of the composite figure is NOT correct?

(A) The perimeter includes the lengths of the three sides of the triangle.

(B) The side where the semicircle is attached is internal to the figure's boundary.

(C) The perimeter is the sum of the lengths of the two free sides of the triangle and the arc length of the semicircle.

(D) The perimeter is $2a + \frac{1}{2}(2\pi \times a/2) = 2a + \pi a/2$.

Question 8. Which calculation method is NOT appropriate for finding the area of the shaded region in a figure?

(A) Summing the areas of distinct, non-overlapping shapes forming the region.

(B) Subtracting the area of an unshaded region from the area of a larger enclosing shape.

(C) Multiplying the perimeter of the shaded region by a constant factor.

(D) Using integration (in higher mathematics).

Question 9. A circular garden of radius $10\ \text{m}$ has a square lawn in its center, with vertices on the circumference. Which statement is NOT correct? (Use $\pi = 3.14$)

(A) The diameter of the circle is the diagonal of the square.

(B) The diagonal of the square is $20\ \text{m}$.

(C) If the diagonal is $d$, the side of the square is $d/\sqrt{2}$.

(D) The area of the square is $10\sqrt{2} \times 10\sqrt{2} = 200\ \text{m}^2$. The area of the circle is $100\pi \approx 314\ \text{m}^2$. The area of the remaining garden is $314 - 200 = 114\ \text{m}^2$.

Question 10. Which statement about the perimeter of a shape formed by joining basic shapes is INCORRECT?

(A) The perimeter is the total length of the outer boundary.

(B) Overlapping or internal segments are not part of the perimeter.

(C) If two shapes are joined along an edge, that edge contributes to the perimeter.

(D) The perimeter calculation involves identifying which parts of the original shapes form the new boundary.

Question 1. Which of the following is NOT a three-dimensional shape?

(A) Cuboid

(B) Sphere

(C) Circle

(D) Cylinder

Question 2. Which of the following terms is NOT used to describe a component of a solid shape's boundary?

(A) Face

(B) Edge

(C) Vertex

(D) Perimeter

Question 3. A cuboid has length, width, and height. Which statement about its faces, edges, or vertices is NOT correct?

(A) It has 6 faces.

(B) It has 12 edges.

(C) It has 8 vertices.

(D) All its faces are squares.

Question 4. Which of the following solid shapes does NOT have a flat base?

(A) Cone

(B) Cylinder

(C) Sphere

(D) Pyramid

Question 5. Which statement about a prism is NOT correct?

(A) It has two identical and parallel polygonal bases.

(B) Its lateral faces are rectangles or parallelograms.

(C) Its lateral faces meet at a single apex.

(D) The shape of the base determines the name of the prism (e.g., triangular prism).

Question 6. Which statement about a pyramid is NOT correct?

(A) It has a polygonal base.

(B) Its lateral faces are triangles.

(C) All its faces are congruent triangles.

(D) Its lateral faces meet at a common vertex (apex).

Question 7. Which of the following solid shapes does NOT have any vertices?

(A) Cube

(B) Cylinder

(C) Sphere

(D) Cone

Question 8. Which statement about the faces of a solid is NOT correct?

(A) Faces are the flat or curved surfaces of the solid.

(B) Faces are always polygonal.

(C) The total surface area is the sum of the areas of its faces.

(D) Faces meet at edges.

Question 9. Euler's formula $F + V - E = 2$ applies to polyhedrons. Which statement about this formula is NOT true?

(A) F represents the number of faces.

(B) V represents the number of vertices.

(C) E represents the number of edges.

(D) It applies to shapes with curved surfaces like spheres and cylinders.

Question 10. Which statement is NOT true about the terms edges and vertices?

(A) Edges are line segments where faces meet.

(B) Vertices are points where edges meet.

(C) A cube has more edges than vertices.

(D) A cone has multiple edges and vertices.

Question 1. Which of the following is NOT a correct formula for the Total Surface Area (TSA) of a cuboid with length $l$, width $w$, and height $h$?

(A) $2(lw+wh+hl)$

(B) $2lw + 2wh + 2hl$

(C) Lateral Surface Area + 2 $\times$ Area of base

(D) $lwh$

Question 2. Which statement about the Lateral Surface Area (LSA) or Curved Surface Area (CSA) is NOT correct?

(A) LSA of a cube is the area of its four side faces.

(B) CSA of a cylinder is the area of its curved side.

(C) LSA/CSA includes the area of the bases.

(D) TSA = LSA/CSA + Area of bases.

Question 3. Which is NOT a correct formula for the Total Surface Area (TSA) of a solid cylinder with radius $r$ and height $h$?

(A) $2\pi r h + 2\pi r^2$

(B) $2\pi r (h+r)$

(C) Curved Surface Area + Area of two bases

(D) $\pi r^2 h$

Question 4. Which is NOT a correct formula for the Curved Surface Area (CSA) of a cone with radius $r$ and slant height $l$?

(A) $\pi r l$

(B) $\pi r \sqrt{r^2 + h^2}$ where $h$ is height

(C) $\frac{1}{3}\pi r^2 h$

(D) Area obtained by unrolling the curved surface into a sector

Question 5. Which is NOT a correct formula related to the surface area of a sphere or hemisphere of radius $r$?

(A) Surface Area of sphere $= 4\pi r^2$

(B) CSA of hemisphere $= 2\pi r^2$

(C) TSA of solid hemisphere $= 2\pi r^2$

(D) Area of great circle of sphere $= \pi r^2$

Question 6. If the side length of a cube is tripled, which statement about its surface area is NOT true?

(A) The new side length is $3s$.

(B) The original surface area is $6s^2$.

(C) The new surface area is $6(3s)^2 = 54s^2$.

(D) The new surface area is three times the original surface area.

Question 7. A cylindrical tank is open at the top, with base radius $r$ and height $h$. Which statement is NOT correct about the surface area of the tank?

(A) The surface area includes the area of the base.

(B) The surface area includes the area of the top (incorrect).

(C) The surface area is $\pi r^2 + 2\pi r h$.

(D) The surface area is the sum of the base area and the curved surface area.

Question 8. Which unit is NOT suitable for measuring the surface area of a solid object?

(A) Square metres ($\text{m}^2$)

(B) Square feet ($\text{ft}^2$)

(C) Hectares ($\text{ha}$)

(D) Cubic centimetres ($\text{cm}^3$)

Question 9. Which statement about the surface area of a solid cone is NOT correct?

(A) TSA = CSA + Area of base.

(B) CSA = $\pi r l$, where $l$ is slant height.

(C) Area of base = $\pi r^2$.

(D) TSA = $\pi r (h+r)$, where $h$ is height.

Question 10. Which statement is NOT true about the surface area of a sphere?

(A) It is proportional to the square of the radius.

(B) Doubling the radius doubles the surface area.

(C) It is $4\pi r^2$.

(D) It is a measure of the outer boundary of the sphere.

Question 1. Which of the following is NOT a correct formula for the volume $V$ of a cuboid with length $l$, width $w$, and height $h$?

(A) $l \times w \times h$

(B) Area of base $\times$ height

(C) $2(lw+wh+hl)$

(D) Volume occupied by the solid

Question 2. Which statement about the volume of a cube is NOT correct?

(A) Volume $= s^3$ where $s$ is side length.

(B) If the side is $3\ \text{cm}$, the volume is $27\ \text{cm}^3$.

(C) If the volume is $64\ \text{m}^3$, the side is $8\ \text{m}$.

(D) Volume is measured in cubic units.

Question 3. Which is NOT a correct formula for the volume $V$ of a cylinder with radius $r$ and height $h$?

(A) $V = \pi r^2 h$

(B) $V = (\text{Area of base}) \times \text{height}$

(C) $V = 2\pi r h$

(D) $V = \pi (d/2)^2 h$ where $d$ is diameter

Question 4. Which statement about the volume of a cone is NOT correct?

(A) Volume $= \frac{1}{3}\pi r^2 h$ where $r$ is radius and $h$ is height.

(B) The volume is one-third the volume of a cylinder with the same base and height.

(C) Volume is proportional to the slant height.

(D) Volume is measured in cubic units.

Question 5. Which is NOT a correct formula for the volume $V$ of a sphere of radius $r$?

(A) $V = \frac{4}{3}\pi r^3$

(B) $V = \frac{1}{6}\pi d^3$ where $d$ is diameter

(C) $V = 4\pi r^2$

(D) $V = \frac{4}{3}\pi (\frac{d}{2})^3$

Question 6. If the radius of a sphere is tripled, which statement about its volume is NOT true?

(A) The new radius is $3r$.

(B) The original volume is $\frac{4}{3}\pi r^3$.

(C) The new volume is $\frac{4}{3}\pi (3r)^3 = 36\pi r^3$.

(D) The new volume is 27 times the original volume.

Question 7. Which unit is NOT suitable for measuring the volume of liquid in a tank?

(A) Litres ($\text{L}$)

(B) Millilitres ($\text{mL}$)

(C) Cubic metres ($\text{m}^3$)

(D) Square metres ($\text{m}^2$)

Question 8. The volume of a hemisphere of radius $r$ is $\frac{2}{3}\pi r^3$. Which statement is NOT correct?

(A) It is half the volume of a sphere of radius $r$.

(B) Its volume is equal to its total surface area.

(C) Volume is measured in cubic units.

(D) The volume depends on the cube of the radius.

Question 9. Which statement is NOT true about the relationship between the volume of a cone and a cylinder?

(A) If they have the same base radius and height, the cone's volume is $\frac{1}{3}$ the cylinder's volume.

(B) If they have the same volume and base radius, the cone's height is three times the cylinder's height.

(C) If they have the same volume and height, the cone's radius is three times the cylinder's radius.

(D) $\frac{V_{cone}}{V_{cylinder}} = \frac{1}{3}$ if bases and heights are equal.

Question 10. A cubic tank has a side length of $2\ \text{m}$. Which statement is NOT correct?

(A) Its volume is $8\ \text{m}^3$.

(B) It can hold $8000\ \text{L}$ of water.

(C) Its total surface area is $24\ \text{m}^2$.

(D) Its volume is equal to its surface area.

Question 1. When calculating the volume of a solid formed by combining two basic solids (without overlap), which statement is NOT true?

(A) The total volume is the sum of the individual volumes.

(B) Volume is an additive property.

(C) The area of the joint surface needs to be subtracted from the volume sum.

(D) The units of the total volume are the same as the units of the individual volumes ($\text{m}^3$, $\text{cm}^3$, etc.).

Question 2. To find the total surface area (TSA) of a solid formed by joining a cone on top of a cylinder (same base radius), which statement is NOT correct?

(A) The area of the common base is included in the TSA calculation.

(B) The TSA includes the CSA of the cone and the CSA of the cylinder.

(C) The TSA includes the area of the base of the cylinder on the ground.

(D) TSA = CSA of cone + CSA of cylinder + Area of cylinder base.

Question 3. A cylindrical container has a hemispherical bottom. Which statement about its total volume is NOT correct?

(A) Volume = Volume of cylinder + Volume of hemisphere.

(B) If the radius is $r$ and cylinder height is $h$, Volume = $\pi r^2 h + \frac{2}{3}\pi r^3$.

(C) Volume = $\pi r^2 (h + \frac{2}{3}r)$.

(D) Volume = Area of cylinder base $\times$ total height.

Question 4. A solid is formed by scooping out a cone from a cylinder of the same base and height. Which statement about the volume of the remaining solid is NOT correct?

(A) Volume of remaining solid = Volume of cylinder $-$ Volume of cone.

(B) If cylinder volume is $V_{cyl}$, the volume of the remaining solid is $\frac{2}{3} V_{cyl}$.

(C) The volume of the removed cone is $\frac{1}{3}$ the volume of the cylinder.

(D) Volume of remaining solid = Area of base $\times$ height.

Question 5. When a solid is formed by placing one object on top of another, the area of the common boundary surface is not exposed to the outside. Which statement about calculating the total surface area is NOT correct?

(A) Sum the total surface areas of the individual objects and subtract the area of the common boundary twice.

(B) Sum the exposed surface areas of the individual objects.

(C) Sum the curved surface areas and the areas of any exposed flat bases/tops.

(D) The TSA is simply the sum of the CSAs of the objects.

Question 6. A decorative block is made by placing a hemisphere on the top of a cube. The base of the hemisphere is equal to the top face of the cube. If the side of the cube is $a$, which statement is NOT correct about the total surface area?

(A) Area of base of cube $= a^2$.

(B) Area of 4 side faces of cube $= 4a^2$.

(C) Curved surface area of hemisphere $= 2\pi (a/2)^2 = \pi a^2/2$.

(D) Total Surface Area = Area of 6 faces of cube + CSA of hemisphere.

Question 7. A solid is composed of a cylinder with hemispherical ends. The radius of the hemisphere is equal to the radius of the cylinder. Which statement about the total volume is NOT correct?

(A) Volume = Volume of cylinder + 2 $\times$ Volume of hemisphere.

(B) If radius is $r$ and cylinder height is $h$, Volume = $\pi r^2 h + 2 \times \frac{2}{3}\pi r^3 = \pi r^2 h + \frac{4}{3}\pi r^3$.

(C) Volume = $\pi r^2 (h + \frac{4}{3}r)$.

(D) Volume is proportional to the total height of the solid.

Question 8. When a solid object is submerged in a liquid, the volume of the displaced liquid is equal to the volume of the submerged part of the object. Which situation does NOT demonstrate this principle directly?

(A) Measuring the volume of an irregular stone by water displacement.

(B) A boat floating on water.

(C) A solid cylinder fully submerged in a measuring cylinder.

(D) Calculating the weight of an object from its density and volume.

Question 9. A hollow cylinder is open at both ends. Which statement is NOT correct about calculating the surface area of the hollow cylinder?

(A) It has an inner curved surface area and an outer curved surface area.

(B) It has a top annular (ring) area and a bottom annular area.

(C) The total surface area is the sum of inner CSA, outer CSA, top ring area, and bottom ring area.

(D) The total surface area is $2\pi R h + 2\pi r h + 2\pi (R^2 - r^2)$.

Question 10. A toy is in the shape of a cone on a hemisphere. The radius of the hemisphere is $3\ \text{cm}$ and the height of the cone is $4\ \text{cm}$. Which statement is NOT correct about the total surface area of the toy?

(A) The slant height of the cone is $\sqrt{3^2+4^2} = 5\ \text{cm}$.

(B) CSA of cone $= \pi (3)(5) = 15\pi\ \text{cm}^2$.

(C) CSA of hemisphere $= 2\pi (3)^2 = 18\pi\ \text{cm}^2$.

(D) Total Surface Area = CSA of cone $+$ CSA of hemisphere $+$ Area of cone base.

Question 1. When a solid is melted and recast into another solid, which property is NOT conserved?

(A) Volume

(B) Mass (assuming density is constant and no material is lost)

(C) Surface Area (generally changes)

(D) Amount of material

Question 2. Which statement about the conversion of solids is NOT correct?

(A) A sphere can be melted and recast into a cylinder.

(B) A cube can be melted and recast into a number of smaller cubes.

(C) The sum of the volumes of the smaller shapes equals the volume of the original shape.

(D) The total surface area remains the same before and after conversion.

Question 3. Which statement about a frustum of a cone is NOT correct?

(A) It has two circular bases that are parallel and congruent.

(B) It is a part of a cone.

(C) It has a curved lateral surface.

(D) The radii of its two bases are different.

Question 4. For a frustum of a cone with radii $r_1, r_2$ and height $h$, which is NOT a correct formula related to its properties?

(A) Slant height $l = \sqrt{h^2 + (r_1-r_2)^2}$.

(B) Volume $V = \frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)$.

(C) Curved Surface Area CSA $= \pi (r_1 + r_2) l$.

(D) Total Surface Area TSA $= \pi (r_1 + r_2) l + \pi r_1^2 + \pi r_2^2$.

Question 5. A cone is cut by a plane parallel to the base at a certain height. Which statement about the parts formed is NOT correct?

(A) The upper part is a smaller cone, similar to the original cone.

(B) The lower part is a frustum of a cone.

(C) The plane section is a circle.

(D) The ratio of the volumes of the upper part and the original cone is equal to the ratio of their heights squared.

Question 6. A bucket is in the shape of a frustum of a cone, open at the top. The metal sheet required to make it does NOT include the area of which part?

(A) Curved surface area of the frustum.

(B) Area of the bottom circular base.

(C) Area of the top circular base.

(D) Area of the side walls.

Question 7. If a solid cylinder is melted and recast into a hollow cylinder, which statement is NOT necessarily true?

(A) The volume of the solid cylinder equals the volume of the material in the hollow cylinder.

(B) The height of the solid cylinder equals the height of the hollow cylinder.

(C) The outer volume of the hollow cylinder is equal to the volume of the solid cylinder.

(D) Volume is conserved in the conversion.

Question 8. A solid metallic sphere is melted and recast into a wire of a certain diameter. Which statement is NOT correct?

(A) The volume of the sphere is equal to the volume of the wire.

(B) The wire is cylindrical in shape.

(C) The length of the wire can be calculated if the sphere's radius and the wire's diameter are known.

(D) The surface area of the wire is equal to the surface area of the sphere.

Question 9. Which of the following cannot be in the shape of a frustum of a cone?

(A) A bucket

(B) A drinking glass (usually cylindrical, but some are frustums)

(C) A cone-shaped funnel (usually full cone)

(D) The top part of a pencil after sharpening (cone or frustum?)

Question 10. The volume of a frustum is the difference between the volumes of the original cone and the smaller cone removed. Which statement is NOT true about this relationship?

(A) The original cone and the smaller cone are similar.

(B) The ratio of their heights is the same as the ratio of their radii.

(C) The ratio of their volumes is the cube of the ratio of their heights (or radii).

(D) The volume of the frustum is simply the volume of the larger cone minus the volume of the smaller cone's base.