10 Question MCQ (including Assertion)

Question 1 of 10

Assertion: Two parallelograms having the same pair of adjacent side lengths must have equal areas.
Reason: The area of a parallelogram is the product of a base and its corresponding perpendicular height.

A Both are true, and the reason correctly explains the assertion. B Both are true, but the reason does not correctly explain the assertion. C The assertion is true, but the reason is false. D The assertion is false, but the reason is true.

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Question 2 of 10

How does Brahmagupta's formula generalise Heron's formula for the area of a triangle with sides \( a, b, c \)?

A By assuming the triangle is inscribed in a unit circle. B By setting the fourth side \( d = 0 \) in Brahmagupta's formula. C By substituting the diagonals of the quadrilateral. D By averaging the lengths of two adjacent sides.

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Question 3 of 10

Semicircles are drawn on all three sides of a right-angled triangle as diameters. Let Area(A) and Area(B) be the areas of the semicircles on the legs, and Area(C) be the area of the semicircle on the hypotenuse. Which relationship holds true?

A Area(A) + Area(B) > Area(C) B Area(A) + Area(B) = Area(C) C Area(A)^2 + Area(B)^2 = Area(C)^2 D Area(C) = Area(A) \times Area(B)

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Question 4 of 10

If you use Heron's formula to find the area of an equilateral triangle with side \( a \), what is the resulting simplified expression?

A \( \frac{\sqrt{3}}{2}a^2 \) B \( \frac{\sqrt{3}}{4}a^2 \) C \( \frac{3}{4}a^2 \) D \( a^2\sqrt{3} \)

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Question 5 of 10

Two paths connect points \( P \) and \( Q \). Path A is a single large semicircle of diameter \( D \). Path B consists of three adjacent semicircles with diameters \( d_1, d_2, d_3 \) such that \( d_1+d_2+d_3=D \). Which path is longer?

A Path A is longer. B Path B is longer. C Both paths have equal length. D Depends on the ratio of the smaller diameters.

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Question 6 of 10

A triangle is enlarged so that each side becomes three times the corresponding original side. The new area is:

A Three times the original area B Six times the original area C Nine times the original area D Twenty-seven times the original area

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Question 7 of 10

A triangle has sides of 8 cm and 11 cm, and its total perimeter is 32 cm. What is its exact area?

A \( 8\sqrt{30} \text{ cm}^2 \) B \( 4\sqrt{30} \text{ cm}^2 \) C \( 16\sqrt{15} \text{ cm}^2 \) D \( 30\sqrt{8} \text{ cm}^2 \)

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Question 8 of 10

Assertion (A):
The perimeter of a quarter circle (quadrant) of radius \( r \) is exactly \( \frac{\pi r}{2} \).
Reason (R):
The boundary of a sector of a circle includes both the length of the circular arc and the two straight bounding radii.

A Both A and R are true, and R is the correct explanation of A. B Both A and R are true, but R is not the correct explanation of A. C A is true, but R is false. D A is false, but R is true.

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Question 9 of 10

The area of a right-angled triangle is 54 sq cm. One of its legs has a length of 12 cm. Find its perimeter.

A 36 cm B 30 cm C 42 cm D 45 cm

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Question 10 of 10

Assertion: In the same circle, an arc subtending \(120^\circ\) at the centre is twice as long as an arc subtending \(60^\circ\).
Reason: The length of an arc of radius \(r\) and central angle \(\theta\) is \[l=2\pi r\times\frac{\theta}{360^\circ}.\]

A Both are true, and the reason correctly explains the assertion. B Both are true, but the reason does not correctly explain the assertion. C The assertion is true, but the reason is false. D The assertion is false, but the reason is true.

Select one option. Answers are shown after the test.

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10 Question MCQ (including Assertion)