Geometry and Trigonometry MCQ Quiz in मल्याळम - Objective Question with Answer for Geometry and Trigonometry - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Choice A is correct. For the right triangle shown, the lengths of the legs are 20 units and 15 units, and the length of the hypotenuse is x units.

The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

Therefore, 15² + 20² = x².

225 + 400 = x²

625 = x²

Taking the square root of both sides of this equation yields \(x= \pm \sqrt{625} =\pm 25\). Since a is a length, a must be positive. Therefore, \(x= 25\). 

Choice B 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 40, the length of minor arc \(\rm \overparen{A C}\) is \(\frac{90}{360} \times 40=10\)  

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 .