Mensuration: SSC CGL Practice Questions

Ans: A

Explanation: First, find the radius of the inner circle. We know the circumference (C) is 264 meters, and C = 2 * pi * r. So, 264 = 2 * (22/7) * r. Solving for r: r = (264 * 7) / (2 * 22) = 42 meters. The path is 3 meters wide, so the radius of the outer circle is 42 + 3 = 45 meters. Now calculate the areas of both circles. Area of the inner circle: A_inner = pi * r^2 = (22/7) * 42^2 = (22/7) * 1764 = 5544 square meters. Area of the outer circle: A_outer = pi * R^2 = (22/7) * 45^2 = (22/7) * 2025 = 6364.29 square meters. The area of the path is the difference between the outer and inner circle areas: 6364.29 – 5544 = 820.29 square meters. Rounding to the nearest whole number gives 820.
Correct Option: A

Ans: C

Explanation: Let r be the radius of the base and h be the height of the cylinder.
The base area is given by πr² = 38.5 cm². Using π = 22/7, we have (22/7)r² = 38.5.
So, r² = 38.5 * (7/22) = (385/10) * (7/22) = (35 * 11 / 10) * (7 / 22) = (7*7) / 4 = 49/4.
Therefore, r = √(49/4) = 7/2 = 3.5 cm.
The curved surface area is given by 2πrh = 616 cm².
Using π = 22/7, we have 2 * (22/7) * (7/2) * h = 616.
So, 22h = 616.
h = 616 / 22 = 28 cm.
The volume of the cylinder is given by πr²h = (22/7) * (7/2)² * 28 = (22/7) * (49/4) * 28 = 22 * (7/4) * 28 = 22 * 7 * 7 = 1078 cm³.

Correct Option: C

Ans: C

Explanation: 1. Find the side of the square field: Since the area of the square field is 1764 square meters, the side is the square root of 1764, which is 42 meters.
2. Calculate the width of the rectangular park: The width is one-sixth the side of the square field, so the width is 42/6 = 7 meters.
3. Calculate the length of the rectangular park: The length is four times the width, so the length is 4 * 7 = 28 meters.
4. Calculate the area of the rectangular park: The area is length times width, so the area is 28 * 7 = 196 square meters.
5. Calculate the cost to level the park: The cost is the area times the rate per square meter, so the cost is 196 * 30 = 5880.

Correct Option: C

Ans: C

Explanation: Since DE is parallel to BC, triangle ADE is similar to triangle ABC. The ratio of AD to AB is AD/(AD+DB) = 2/(2+5) = 2/7. The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, Area(ADE) / Area(ABC) = (AD/AB)^2 = (2/7)^2 = 4/49. We know Area(ADE) = 8. So, 8 / Area(ABC) = 4/49. This implies Area(ABC) = 8 * (49/4) = 98. The area of the quadrilateral BDEC is the difference between the area of triangle ABC and the area of triangle ADE. Thus, Area(BDEC) = Area(ABC) – Area(ADE) = 98 – 8 = 90.

Correct Option: C

Ans: C

Explanation: The area of a triangle is calculated using the formula: Area = (1/2) * base * height. In this case, the base of the triangle is KL (which lies on PQ), and the height is the perpendicular distance from vertex M to side KL. We are given the area of the triangle (120 cm²) and the length of PQ (30 cm). Since KL lies on PQ, the length of KL is not explicitly given, but we only need the base of triangle. Let’s assume KL = x, as KL lies on PQ, PQ=30, x is the base of the triangle. The formula is 120 = (1/2) * x * height. The question does not provide enough information to determine the length of KL, so we have to use the given length of PQ as the base. If we consider PQ to be the base of the triangle, the length is 30 cm. So, 120 = (1/2) * 30 * height. This simplifies to 120 = 15 * height. Solving for height: height = 120 / 15 = 8 cm. Thus, the height of the triangle from vertex M to side KL is 8 cm.

Ans: D

Explanation: Let the initial base be ‘b’ and the initial height be ‘h’. The initial area is (1/2)bh. The base increases by 40%, so the new base is 1.4b. The area is to increase by 60%, making the new area 1.6 * (1/2)bh = 0.8bh. Let the new height be h’. The new area is also (1/2)(1.4b)h’ = 0.7bh’. Therefore, 0.7bh’ = 0.8bh. Dividing by 0.7b, we get h’ = (0.8/0.7)h = (8/7)h = 1.142857h. The increase in height is 1.142857h – h = 0.142857h. The percentage increase is (0.142857h/h) * 100% = 14.2857%. This is approximately 14.29%.

Correct Option: D

Ans: B

Explanation:
The midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, XY = (1/2)PQ, YZ = (1/2)PR, and XZ = (1/2)QR.
The perimeter of triangle XYZ is XY + YZ + XZ = (1/2)PQ + (1/2)PR + (1/2)QR = (1/2)(PQ + PR + QR).
Since the perimeter of triangle PQR is 48 cm, PQ + PR + QR = 48 cm.
Thus, the perimeter of triangle XYZ is (1/2) * 48 cm = 24 cm.

Ans: B

Explanation: First, check if triangle ABC is a right-angled triangle using the Pythagorean theorem. If AB^2 + AC^2 = BC^2, then the triangle is right-angled. 16^2 + 63^2 = 256 + 3969 = 4225, and 65^2 = 4225. Since 16^2 + 63^2 = 65^2, triangle ABC is a right-angled triangle with the right angle at A. In a right-angled triangle, the median to the hypotenuse is half the length of the hypotenuse. Since BC is the hypotenuse, and AM is the median to BC, AM = BC/2 = 65/2 = 32.5 cm.
Correct Option: B

Ans: D

Explanation: Since DE is parallel to QR, triangle PDE is similar to triangle PQR. The ratio of their corresponding sides is given as DE:QR = 3:5. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, Area(PDE) / Area(PQR) = (3/5)^2 = 9/25.

Let Area(PDE) = 9x. Then, Area(PQR) = 25x.
The area of quadrilateral DERQ = Area(PQR) – Area(PDE) = 25x – 9x = 16x.
The ratio of the area of quadrilateral DERQ to the area of triangle PDE is (16x) / (9x) = 16/9.

Ans: A

Explanation: We can use the Angle Bisector Theorem, which states that the angle bisector of a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides. In triangle PQR, QS bisects angle Q, so by the Angle Bisector Theorem, PS/SR = PQ/QR. We are given PQ = 10 cm, QR = 20 cm, and PR = 21 cm. We can rewrite PR as PS + SR = 21. Also, PS/SR = 10/20 = 1/2. This means SR = 2*PS. Substituting this into the equation PS + SR = 21, we get PS + 2*PS = 21, so 3*PS = 21. Therefore, PS = 21/3 = 7.

Ans: D

Explanation: Let the four angles of the quadrilateral be a, a+d, a+2d, and a+3d, where ‘a’ is the smallest angle and ‘d’ is the common difference. We are given that the smallest angle is 60 degrees, so a = 60. The sum of the interior angles of a quadrilateral is 360 degrees. Therefore, a + (a+d) + (a+2d) + (a+3d) = 360. Substituting a = 60, we have 60 + (60+d) + (60+2d) + (60+3d) = 360. Simplifying, we get 240 + 6d = 360. Subtracting 240 from both sides gives 6d = 120. Dividing by 6, we find d = 20. The largest angle is a+3d = 60 + 3(20) = 60 + 60 = 120 degrees.

Ans: C

Explanation: An equilateral triangle has three equal sides. The perimeter is the sum of all sides. Therefore, the length of each side is the perimeter divided by 3. 36 cm / 3 = 12 cm.

Ans: A

Explanation: Since YQ:QZ = 1:1 and YZ = 6√3, then YQ = QZ = (6√3)/2 = 3√3. Triangle XYZ is equilateral, so XY = XZ = YZ = 6√3. Also, each angle in an equilateral triangle is 60 degrees. Therefore, angle XYZ = 60 degrees. Now consider triangle XYQ. We know XY = 6√3, YQ = 3√3, and angle XYQ = 60 degrees. We can use the Law of Cosines to find XQ: XQ^2 = XY^2 + YQ^2 – 2 * XY * YQ * cos(60). XQ^2 = (6√3)^2 + (3√3)^2 – 2 * (6√3) * (3√3) * (1/2). XQ^2 = 108 + 27 – 54. XQ^2 = 81. XQ = √81 = 9.

Ans: B

Explanation: The area of a sector of a circle is calculated using the formula: Area = (θ/360) * πr², where θ is the central angle in degrees and r is the radius. In this case, θ = 60 degrees and r = 21 cm. Plugging the values into the formula: Area = (60/360) * (22/7) * 21 * 21 = (1/6) * (22/7) * 21 * 21 = 11 * 21 = 231.

Ans: D

Explanation: First, calculate the volume of the copper sphere. The radius of the sphere is 6 cm / 2 = 3 cm. Volume of a sphere = (4/3) * pi * r^3 = (4/3) * pi * (3 cm)^3 = 36 * pi cm^3. Next, calculate the volume of the wire. The radius of the wire is 0.01 cm. Let the length of the wire be L. The volume of the wire = pi * r^2 * L = pi * (0.01 cm)^2 * L = 0.0001 * pi * L. Since the volume of the sphere equals the volume of the wire: 36 * pi = 0.0001 * pi * L. Solving for L, L = 36 / 0.0001 = 360000 cm.
Correct Option: D