# CBSE Class 10 Maths Question Paper | 2018

###### Class 10 Board Paper Solution | 3 Mark Questions

Section C contains 10 questions of 3 marks each. Scroll down for explanatory answer and video solution to all ten 3-mark questions that appeared in Class 10 Maths CBSE board exam in 2018.

1. Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Hint to solve this Real Numbers problem

Step 1: Prime factorize 404 and 96.
Step 2: Compute LCM of 404 and 96. LCM is the product of the highest power of all prime factors.
Step 3: Compute HCF of 404 and 96. HCF is the product of the lowest power of common prime factors of 404 and 96.
Step 4: Find product of 404 and 96. Verify that the value is the same as the product of the LCM and HCF of 404 and 96.

2. Find all zeroes of the polynomial (2x4 – 9x3 + 5x2 + 3x – 1) if two of its zeroes are (2 + $$sqrt3$) and$2 - $$sqrt3$). Hint to solve this Polynomials problem Step 1: Two of its zeroes are given. If the zeroes are p and q, then$x - p)(x - q) will divide the polynomial.
Step 2: Compute the quotient of dividing the given polynomial by (x - p)(x - q)
Step 3: The quotient of the division will be a quadratic expression. Factorize the quadratic expression to find the remaining two zeroes.

3. A(-2, 1), B(a, 0), C(4, b) D(1, 2) are the vertices of a parallelogram ABCD. Find the values of a and b. Hence, find the lengths of its sides.

Hint to solve this Coordinate Geometry Parallelogram problem

Concept: In a parallelogram, the diagonals bisect each other.

Step 1: The diagonals are AC and BD. Using coordinates of A and C, compute coordinates of midpoint of AC.
Step 2: Using coordinates of B and C, compute the coordinates of midpoint of BD.
Step 3: The midpoint of AC and midpoint of Bd are the same point. So equate the cooridantes obtained in step 1 and step 2 to find the values of 'a' and 'b'
Step 4: Now that we have the coordinates of all 4 vertices, compute length of AB and BC to find the lengths of the sides of the parallelogram.

OR

If A(- 5, 7), B(- 4, - 5), C(- 1, - 6) and D(4, 5) are the vertices of a quadrilateral. Find the area of the quadrilateral ABCD.

Hint to solve this Coordinate Geometry problem

Step 1: Join AC. The quadrilateral gets divided into two triangles ABC and ADC.
Step 2: Using coordinates of vertices A, B, and C compute area of triangle ABC.
Step 3: Using coordinates of vertices A, D, and C compute area of triangle ADC.
Step 4: Add areas computed in steps 2 and 3 to compute area of quadrilateral.

4. A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed, Find its usual speed.

Hint to solve this Quadratic Equations problem

Step 1: Assign variables for usual speed and usual time taken.
Step 2: Frame equation for distance travelled at usual speed.
Step 3: Frame equation for distance travelled at higher speed when the plane reached its destination in 30 minutes lesser time.
Step 4: Equate expressions in steps 2 and 3 and solve for the usual speed of the plane.

5. Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonals.

Hint to solve this Similar Triangles problem

Step 1: Compute ratio of the lengths of the side and the diagonal of a square.
Step 2: Ratio of the sides of the two equilateral triangles (both will be similar by AAA rule of similarity) will be in the ratio computed in step 1.
Step 3: Use theorem "Ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides" to compute ratio of areas of the two triangles.

OR

If the area of two similar triangles are equal, prove that they are congruent.

Hint to solve this Similar & Congruent Triangles problem

Step 1: Use theorem "Ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides".
Step 2: Because their areas are equal, the ratio of their sides, which is the square root of the ratio of the areas will also be equal.

6. Prove that the lengths of tangents drawn from an external point to a circle are equal.

Hint to solve this Circles & Trangents problem

Step 1: The two tangents, corresponding radii drawn from the tangents where they meet the circle and the line joining the centre of the circle and the point where the tangents meet will form two congruent right triangles by RHS test of congruence.
Step 2: By CPCT, we can prove that the tangents are equal in length.

7. If 4 tanθ = 3, evaluate $$frac{\text{4sinθ - cosθ + 1}}{\text{4sinθ + cosθ - 1}}$. Hint to solve this Trigonometry problem Step 1: Compute value of tan θ. Step 2: Compute hypotenuse of the right triangle. Step 3: Compute values of sin θ and cos θ. Step 4: Compute value of the expression given in the question. OR If tan 2A = cot$A - 18°), where 2A is an acute angle, find the value of A.

Hint to solve this Trigonometry - Complementary Angles problem

Step 1: By trigonometric ratios of complementary angles, tan 2A can be written as cot (90° - 2A).
Step 2: Establish that (90° - 2A) and (A - 18°) are acute angles.
Step 3: Because (90° - 2A) and (A - 18°) are acute angles, both will be positive as the angles are in the first quadrant.
Step 4: Equate (90° - 2A) and (A - 18°) to find the value of A.

8. Find the area of the shaded region in figure given below, where areas drawn with centres A, B, C, and D intersect in pairs at mid – points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm.

Hint to solve this Areas Related to Circles & Squares problem

Step 1: Area of shaded region is the difference between the area of the square and the sum of the areas of the 4 sectors.
Step 2: Compute area of the given square.
Step 3: The sectors subtend an angle of 90°. The radii of all 4 sectors are equal. So, they are 4 equal quarter circles. Sum of their area equals area of 1 circle. Compute the area of the resultant circle.
Step 4: Difference between value obtained in step 2 and step 3 is the answer to the question.

9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the given figure. If the height of the cylinder is 10 cm and its base is of radius 3. 5 cm. Find the total surface area of the article.

Hint to solve this Surface Areas & Volumes problem

Step 1: Total Surface Area of wooden article is the sum of the CSA of the cylinder and the CSA of the two hemispheres scooped from the top and bottom of the cylinder.
Step 2: Compute Curved Surface Area (CSA) of the cylinder.
Step 3: Compute CSA of the two equal hemispheres.
Step 4: Compute Total Surface Area of the wooden article by adding values obtained in step 2 and 2 times the value obtained in step 3.

OR

A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap ?

Hint to solve this Surface Areas & Volumes problem

Step 1: Using base radius and height compute volume of cone.
Step 2: Compute slant height using base radius and height of the cone.
Step 3: Compute curved surface area (CSA) of cone to find area of canvas required to cover the heap.

10. The table below shows the salaries of 280 persons. Calculate the median salary of the data.

Salary (In thousand) No. of Persons
5 - 10 49
10 - 15 133
15 - 20 63
20 - 25 15
25 - 30 6
30 - 35 7
35 - 40 4
40 - 45 2
45 - 50 1
Hint to solve this Statistics problem

Step 1: Compute cumulative frequency for the given data.
Step 2: Identify the class interval in which the median lies.
Step 3: Use the formula to compute median for the given data.

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