Board Paper of Class 10 2009 Maths (SET 1)  Solutions
1. All questions are compulsory.
2. The question paper consists of 30 questions divided into four sections – A, B, C and D. Section A comprises of ten questions of 1 mark each, Section B comprises of five questions of 2marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6marks each.
3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
4. In question on construction, the drawing should be neat and as per the given measurements.
5. Use of calculators is not permitted.
 Question 1
Find the [HCF × LCM] for the numbers 100 and 190.
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 Question 2
If 1 is a zero of the polynomial p(x) = ax^{2} − 3(a − 1) x − 1, then find the value of a.
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 Question 3
In ΔLMN, ∠L = 50° and ∠N = 60°. If ΔLMN ∼ ΔPQR, then find ∠Q.
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 Question 4
If sec^{2}θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.
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 Question 5
If the diameter of a semicircular protractor is 14 cm, then find its perimeter.
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 Question 6
Find the number of solutions of the following pair of linear equations:
x + 2y − 8 = 0
2x + 4y = 16
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 Question 7
Find the discriminant of the quadratic equation
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 Question 8
If, a, and 2 are three consecutive terms of an A.P., then find the value of a.
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 Question 9
In figure 1, ΔABC is circumscribing a circle. Find the length of BC.
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 Question 10
Two coins are tossed simultaneously. Find the probability of getting exactly one head.
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 Question 11
Find all the zeroes of the polynomial x^{3} + 3x^{2} − 2x − 6, if two of its zeroes are and.
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 Question 12
Which term of the A.P. 3, 15, 27, 39… will be 120 more than its 21^{st} term?
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 Question 13
In Figure 2, ΔABD is a right triangle, rightangled at A and AC ⊥ BD. Prove that AB^{2} = BC . BD.
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 Question 14
 Question 15
If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2, 3), find the value of x.
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 Question 16
Prove that is an irrational number.
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 Question 17
Solve for x and y:
ax − by = 2ab
OR
The sum of two numbers is 8. Determine the numbers if the sum of their reciprocals is
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 Question 18
The sum of first six terms of an arithmetic progression is 42. The ratio of its 10^{th} term to its 30^{th} term is 1 : 3. Calculate the first and the thirteenth term of the A.P.
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 Question 19
Evaluate:
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 Question 20
Draw a right triangle in which sides (other than hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are times the corresponding sides of the first triangle.
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 Question 21
In Figure, 3, AD ⊥ BC and BD CD. Prove that 2CA^{2} = 2AB^{2} + BC^{2}.
OR
In Figure 4, M is midpoint of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2 BL.
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 Question 22
Find the ratio in which the point (2, y) divides the line segment joining the points A (−2, 2) and B (3, 7). Also find the value of y.
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 Question 23
Find the area of the quadrilateral ABCD whose vertices are A(−4, −2), B(−3, −5), C (3, −2) and D (2, 3).
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 Question 24
The area of an equilateral triangle is cm^{2}. Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of triangle not included in the circles. [Take= 1.73]
OR
Figure 5 shows a decorative block which is made of two solids − a cube and a hemisphere. The base of the block is a cube with edge 5 cm and the hemisphere, fixed on the top, has a diameter of 4.2 cm. Find the total surface area of the block. [Take π =]
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 Question 25
Two dice are thrown simultaneously. What is the probability that
(i) 5 will not come up on either of them?
(ii) 5 will come up on at least one?
(iii) 5 will come up at both dice?
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 Question 26
Solve the following equation for x:
9x^{2} − 9(a + b)x + (2a^{2} + 5ab + 2b^{2}) = 0
OR
If (−5) is a root of the quadratic equation 2x^{2} + px − 15 = 0 and the quadratic equation p(x^{2} + x) + k = 0 has equal roots, then find the values of p and k.
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 Question 27
Prove that the lengths of the tangents drawn from an external point to a circle are equal.
Using the above theorem prove that:
If quadrilateral ABCD is circumscribing a circle, then AB + CD = AD + BC
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 Question 28
An aeroplane when flying at a height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30° and 60° respectively. Find the distance between the two planes at that instant.
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 Question 29
A juice seller serves his customers using a glass as shown in Figure 6. The inner diameter of the cylindrical glass is 5 cm, but the bottom of the glass has a hemispherical portion raised which reduces the capacity of the glass. If the height of the glass is 10 cm, find the apparent capacity of the glass and its actual capacity. (Use π = 3.14)
OR
A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 7 cm and height 6 cm is completely immersed in water. Find the volume of
(i) water displaced out of the cylindrical vessel.
(ii) water left in the cylindrical vessel.
[Take π ]
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 Question 30
During the medical checkup of 35 students of a class their weights were recorded as follows:

Weight (in kg)
Number of students
38 − 40
3
40 − 42
2
42 − 44
4
44 − 46
5
46 − 48
14
48 − 50
4
50 − 52
3
Draw a less than type and a more than type ogive from the given data. Hence obtain the median weight from the graph.
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