Board Paper of Class 10 2018 Maths (SET 3)  Solutions
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.
(iii) Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and 3 questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.
(v) Use of calculated is not permitted.
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.
(iii) Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and 3 questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.
(v) Use of calculated is not permitted.
 Question 1
What is the value of (cos^{2} 67° – sin^{2}^{ }23°)? VIEW SOLUTION
 Question 2
In an AP, if the common difference (d) = –4, and the seventh term (a_{7}) is 4, then find the first term. VIEW SOLUTION
 Question 3
Given ∆ABC ~ ∆PQR, if $\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{1}{3},$ then find $\frac{\mathrm{ar}\u2206\mathrm{ABC}}{\mathrm{ar}\u2206\mathrm{PQR}}.$ VIEW SOLUTION
 Question 4
What is the HCF of smallest prime number and the smallest composite number? VIEW SOLUTION
 Question 5
Find the distance of a point P(x, y) from the origin. VIEW SOLUTION
 Question 6
If x = 3 is one root of the quadratic equation x^{2} – 2kx – 6 = 0, then find the value of k. VIEW SOLUTION
 Question 7
Two different dice are tossed together. Find the probability :
(i) of getting a doublet
(ii) of getting a sum 10, of the numbers on the two dice. VIEW SOLUTION
 Question 8
Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m. VIEW SOLUTION
 Question 9
An integer is chosen at random between 1 and 100. Find the probability that it is:
(i) divisible by 8.
(ii) not divisible by 8. VIEW SOLUTION
 Question 10
 Question 11
Find the sum of first 8 multiples of 3. VIEW SOLUTION
 Question 12
Given that $\sqrt{2}$ is irrational, prove that $\left(5+3\sqrt{2}\right)$ is an irrational number. VIEW SOLUTION
 Question 13
If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.
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. VIEW SOLUTION
 Question 14
Find all zeroes of the polynomial $\left(2{x}^{4}9{x}^{3}+5{x}^{2}+3x1\right)$ if two of its zeroes are $\left(2+\sqrt{3}\right)\mathrm{and}\left(2\sqrt{3}\right)$. VIEW SOLUTION
 Question 15
Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers. VIEW SOLUTION
 Question 16
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION
 Question 17
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 diagonal.
OR
If the area of two similar triangles are equal, prove that they are congruent. VIEW SOLUTION
 Question 18
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. VIEW SOLUTION
 Question 19
The table below shows the salaries of 280 persons :
Salary (In thousand Rs) 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
Calculate the median salary of the data. VIEW SOLUTION
 Question 20
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 2. 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.
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? VIEW SOLUTION
 Question 21
Find the area of the shaded region in Fig. 3, where arcs drawn with centres A, B, C and D intersect in pairs at midpoints P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14]
VIEW SOLUTION
 Question 22
If 4 tan θ = 3, evaluate $\left(\frac{4\mathrm{sin}\mathrm{\theta}\mathrm{cos}\mathrm{\theta}+1}{4\mathrm{sin}\mathrm{\theta}+\mathrm{cos}\mathrm{\theta}1}\right)$
ORIf tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A. VIEW SOLUTION
 Question 23
As observed from the top of a 100 m high light house from the sealevel, 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. $\left[\mathrm{Use}\sqrt{3}=1.732\right]$ VIEW SOLUTION
 Question 24
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:
(i) The area of the metal sheet used to make the bucket.
(ii) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14] VIEW SOLUTION
 Question 25
Prove that: $\frac{\mathrm{sin}\mathrm{A}2{\mathrm{sin}}^{3}\mathrm{A}}{2{\mathrm{cos}}^{3}\mathrm{A}\mathrm{cos}\mathrm{A}}=\mathrm{tan}\mathrm{A}$. VIEW SOLUTION
 Question 26
The mean of the following distribution is 18. Find the frequency f of the class 19 – 21.
Class 1113 1315 1517 1719 1921 2123 2325 Frequency 3 6 9 13 f 5 4
OR
The following distribution gives the daily income of 50 workers of a factory :
Daily Income (in Rs) 100120 120140 140160 160180 180200 Number of workers 12 14 8 6 10
 Question 27
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.
OR
A train travels at a certain average speed for a distance of 63 km and then travels at 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? VIEW SOLUTION
 Question 28
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. VIEW SOLUTION
 Question 29
Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the ∆ABC. VIEW SOLUTION
 Question 30
In an equilateral ∆ ABC, D is a point on side BC such that BD = $\frac{1}{3}$ BC. Prove that 9(AD)^{2} = 7(AB)^{2}
OR
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. VIEW SOLUTION
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