Geometry and Trigonometry MCQ Quiz in हिन्दी - Objective Question with Answer for Geometry and Trigonometry - मुफ्त [PDF] डाउनलोड करें

Choice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and ∠AOC is a central angle of the circle. From the figure, the two diameters that meet to form ∠AOC are perpendicular, so the measure of ∠AOC is 90°.

Therefore, the length of minor arc \(\rm \overparen{A C}\) is \(\frac{90}{360}\) of the circumference of the circle. Since the circumference of the circle is 64, the length of minor arc \(\rm \overparen{A C}\) is \(\frac{90}{360} \times 64= 16\)

Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc \(\rm \overparen{A C}\) is \(\frac{1}{4}\) of the circumference. However, the lengths in choices B and C are, respectively, \(\frac{1}{3}\) and \(\frac{1}{2}\) the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is \(\frac{1}{4}\) the circumference. 

Choice 4 is correct. Using the volume formula \(V=\frac{7 \pi k^3}{48}\) and the given information that the volume of the glass is 473 cubic centimeters, the value of k can be found as follows:

\(664=\frac{7 \pi k^3}{48}\)

\(k^3=\frac{664(48)}{7 \pi}\)

\(k=\sqrt[3]{\frac{664(48)}{7 \pi}} \approx 11.3151\)

Therefore, the value of k is approximately 11.32 centimeters.

Choice 1 is correct.

Using the volume formula \(V=\frac{7 \pi k^3}{48}\) and the given information that the volume of the glass is 400 cubic centimeters, the value of k can be found as follows:

\(400=\frac{7 \pi k^3}{48}\)

\(k^3=\frac{400(48)}{7 \pi}\)

\(k=\sqrt[3]{\frac{400(48)}{7 \pi}} \approx 9.556\)

Therefore, the value of k is approximately 9.56 centimeters.

Choice 1 is correct.

It's given that the volume of a right rectangular prism is ℓwh. The prism shown has a length of 5 cm, a width of 8 cm, and a height of 4 cm .

Thus, ℓwh = (5)(8)(4), or 160 cubic centimeters.

Choice 4 is Correct.

It's given that the volume of a prism is ℓwh. The prism shown has a length of 5 cm, a width of 5 cm, and a height of 10 cm.

Thus, ℓwh = (5)(5)(10), or 250 cubic centimeters.

Choice 3 is correct.

It's given that the volume of a right rectangular prism is ℓwh. The prism shown has a length of 11 cm, a width of 2 cm, and a height of 3 cm .

Thus, ℓwh = (11)(2)(3), or 66 cubic centimeters.

Choice 2 is correct.

The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths x, y, and z. It's given that the triangle has a perimeter of 25 units. Therefore, x + y + z = 25. If x = 10 units and y = 8 units, the value of z, in units, can be found by substituting 10 for x and 8 for y in the equation x + y + z = 25, which yields 10 + 8 + z = 25, or 18 + z = 25. Subtracting 18 from both sides of this equation yields z = 7.

Therefore, if x = 10 units and y = 8 units, the value of z, in units, is 7.

Choice 2 is correct.

The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths x, y, and z. It's given that the triangle has a perimeter of 20 units. Therefore, x + y + z = 20. If x = 8 units and y = 5 units, the value of z, in units, can be found by substituting 8 for x and 5 for y in the equation x + y + z = 20, which yields 8 + 5 + z = 20, or 13 + z = 20. Subtracting 16 from both sides of this equation yields z = 7.

Therefore, if x = 8 units and y = 5 units, the value of z, in units, is 7 .

Explanation:

The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths x, y, and z. It's given that the triangle has a perimeter of 32 units.

Therefore, x + y + z = 32. If x = 13 units and y = 10 units, the value of z, in units, can be found by substituting 13 for x and 10 for y in the equation x + y + z = 32, which yields 13 + 10 + z = 32, or 23 + z = 32. Subtracting 23 from both sides of this equation yields z = 9.

Therefore, if x = 13 units and y = 10 units, the value of z, in units, is 9.

Hence Option(3) is correct.

The diagonal of the rectangle is equal to the diameter of the circle. The diameter is \(2 \times 10 = 20\) inches.

Using the Pythagorean theorem for the rectangle, where \(d^2 = l^2 + w^2\), we have \(20^2 = 16^2 + w^2\).

Solving for \(w\), \(400 = 256 + w^2\), \(w^2 = 144\), \(w = 12\) inches.

Option 3 is correct. Option 1 is incorrect because it represents an incorrect calculation. Option 2 and Option 4 do not satisfy the Pythagorean theorem with the given length.