# CBSE Class 10 Maths Question Paper | 2018

###### Class 10 Board Paper Solution | 4 Mark Questions

Section D contains 8 questions of 4 marks each. Scroll down for explanatory answer and video solution to the 4-mark questions that appeared in Class 10 Maths CBSE board exam in 2018. Questions appeared from the following chapter : Quadratic Equations, Arithmetic Progressions, Triangles, Construction, Trigonometry, Surface Areas & Volumes, Applications of Trigonometry, and Statistics.

1. A motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

How to solve this Quadratic Equations word problem?

Step 1: Assign variable for the speed of the stream. Let it be s km/hr.
Step 2: Compute upstream speed. Frame an equation to compute time taken to travel 24 km upstream.
Step 3: Compute downstream speed. Frame an equation to compute time taken to travel 24 km downstream.
Step 4: Difference between expression in step 2 and step 3 is 1 hour. The equation thus obtained will be a quadratic equation.
Step 5: Solve the quadratic equation obtained in step 4 by factorization and find the speed of the stream.

OR

A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed?

Approach to solve this Quadratic Equations problem

Step 1: Assign variables for the speed and time for the first part of journey.
Step 2: Compute speed and time for second part of journey in terms of the variables assigned in step 1.
Step 3: Express time in terms of speed from equation derived in step 1.
Step 4: Substitute expression derived for time in terms of speed in the equation obtained in step 2. You will get a quadratic equation.
Step 5: Factorize and solve the quadratic equation to find the answer.

2. The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15. Find the numbers.

How to solve this Arithmetic Progressions question?

Step 1: It is very important how the four terms in AP are represented. Let the four consecutive terms in AP be (x - 3d), (x - d), (x + d), and (x + 3d).
Note: The first term is (x - 3d) and the common difference is 2d.
Step 2: Frame an equation for the sum of the four terms, solve the equation to find the value of x.
Step 3: Frame equations - one for the product of the first and last term and another for the product of the 2 middle terms.
Step 4: Find the ratio of the two equations framed in step 3 to the ratio 7 : 15 given in the question. Solve for 'd'
Step 5: Now that we have values for x and d, find the 4 terms.

3. In an equilateral ∆ABC, D is a point on side BC such BD = $$frac{1}{3}$BC. Prove that 9$AD)2 = 7(AB)2

Hint to solve this Equilateral Triangle problem

Step 1: Draw equilateral triangle ABC such that AD = (1/3) BC. Draw altitude AH to side BC of equilateral triangle ABC.
Step 2: By RHS congruence, triangles ABH and ACH are congruent. So, BH = HC = a/2, where a is the side of the equilateral triangle.
Step 3: Using Pythagoras theorem on right triangle ABH, find AH2 in terms of 'a'
Step 4: Using Pythagoras theorem on right triangle ADH, find AD2 in terms of 'a' to get the required proof.

OR

Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the square on the other two sides.

Approach to prove Pythagoras Theorem

Step 1: Draw right triangle ABC, right angled at B. Draw altitude BD to hypotenuse AC of the triangle.
Step 2: Establish similarity between triangles ABC and ADB and write proportions.
Step 3: Establish similarity between triangles ABC and BDC and write proportions.
Step 4: Adding equations obtained in steps 2 and 3 will provide the proof of Pythagoras Theorem.

4. Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are $\frac{3}{4}$th of the corresponding sides of the ΔABC.

5. Prove that: $\frac{sin A - 2 sin^3 A}{2cos^3 A - cos A}$ = tan A

Hint to solve this Trigonometry problem

Step 1: Take sin A common in the numerator and cos A common in the denominator of the LHS.
Step 2: Express 1 as sin2 A + cos2 A in both the numerator and denominator of the expression obtained in step 1.
Step 3: Cancel common terms in both the numerator and the denominator to get the RHS.

6. The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find the area of the metal sheet used to make the bucket.

Hint to solve this Surface Areas & Volume problem

Step 1: Compute slant height of the frustum of the cone using data about the height of the bucket and the radii of the lower and upper ends of the bucket.
Step 2: Compute curved surface area of the frustum of the cone.
Step 3: Compute the area of the circular base of the bucket.
Step 4: Add the values computed in steps 2 and 3 to find the area of metal sheet used to make the bucket.

7. As observed from the top of a 100 m high light house from the sea–level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Use $\sqrt{3}\\$ = 1.732]

Hint to solve this Heights & Distances problem

Step 1: Represent the given information as two right triangles.
Step 2: Angle of elevation is the same as the angle of depression. So, angles of elevation to the top of the light house from the two ships will be 30° and 45°.
Step 3: Compute tan of the two angles of elevation to compute the distance of the two ships from the base of the lighthouse
Step 4: The difference between the distance of the two ships from the base of the lighthouse is the distance between the two ships.

8. The mean of the following distribution is 18. Find the frequency f of the class 19 – 21.

Class 11 - 13 13 - 15 15 - 17 17 - 19 19 - 21 21 - 23 23 - 25
Frequency 3 6 9 13 f 5 4
Hint to solve this Statistics problem

Step 1: Compute fixi for all the class intervals for the data given in the table.
Step 2: Compute the sum of fixi and the sum of all the frequencies. Both sum will include the variable 'f'.
Step 3: Equate the mean given in the question to the mean computed by dividing the sum of fixi by the sum of the frequencies of the distribution to find the value of 'f' - the frequency of the class interval 19 to 21.

OR

The following distribution gives the daily income of 50 workers of a factory:

Daily Income 100 - 120 120 - 140 140 - 160 160 - 180 180 - 200
Number of Workers 12 14 8 6 10

Convert distribution above to a less than type cumulative frequency distribution and draw its ogive.

Hint to solve this Statistics problem

Step 1: Convert the distribution to a less than type cumulative frequency distribution.
Step 2: Draw the ogive on the x-y plane. x-axis is the upper limit of the class interval and y-axis is the cumulative frequency corresponding to each class interval.

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