# 10th Maths Paper Solutions Set 2 : CBSE All India Previous Year 2009

General Instructions:
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.
Q1 :

Find the [HCF × LCM] for the numbers 105 and 120.

For two numbers a and b, HCF × LCM = a × b

∴ For the given numbers 105 and 120:

HCF × LCM = 105 × 120

HCF × LCM = 12600

Q2 :

Find the number of solutions of the following pair of linear equations:

x + 2y − 8 = 0

2x + 4y = 16

The given pair of linear equations is

x + 2y − 8 = 0

2x + 4y − 16 = 0

On comparing with general equations

Hence, the given pair of linear equations has infinitely many solutions.

Q3 :

If, a, and 2 are three consecutive terms of an A.P., then find the value of a.

If three terms a, b, and c are in A.P., then we have ba = cb

⇒ 2b = a + c

∴ Ifare three consecutive terms of an A.P., then

Thus, the value of a is.

Q4 :

Two coins are tossed simultaneously. Find the probability of getting exactly one head.

Q5 :

In figure 1, ΔABC is circumscribing a circle. Find the length of BC.

Q6 :

If the diameter of a semicircular protractor is 14 cm, then find its perimeter.

Q7 :

If 1 is a zero of the polynomial p(x) = ax2 − 3(a − 1) x − 1, then find the value of a.

Q8 :

In ΔLMN, ∠L = 50° and ∠N = 60°. If ΔLMN ∼ ΔPQR, then find ∠Q.

Q9 :

If sec2θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.

Q10 :

Find the discriminant of the quadratic equation

Q11 :

Find all the zeroes of the polynomial 2x3 + x2 − 6x − 3, if two of its zeroes are and.

Q12 :

If , then evaluate

OR

Find the value of tan 60°, geometrically.

Q13 :

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.

Q14 :

Which term of the A.P. 3, 15, 27, 39… will be 120 more than its 21st term?

Q15 :

In Figure 2, ΔABD is a right triangle, right-angled at A and AC BD. Prove that AB2= BC . BD.

Q16 :

Prove that is an irrational number.

Q17 :

In Figure, 3, AD ⊥ BC and BD CD. Prove that 2CA2 = 2AB2 + BC2.

OR

In Figure 4, M is mid-point 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.

Q18 :

The area of an equilateral triangle is cm2. 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=]

Q19 :

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?

Q20 :

The sum of first six terms of an arithmetic progression is 42. The ratio of its 10thterm to its 30thterm is 1 : 3. Calculate the first and the thirteenth term of the A.P.

Q21 :

Evaluate:

Q22 :

Find the ratio in which the point (x, 2) divides the line segment joining the points (−3, −4) and (3, 5). Also find the value of x.

Q23 :

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, − 1), (2, 1) and (0, 3).

Q24 :

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

Q25 :

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.

Q26 :

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]

Q27 :

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, prove that the angle opposite to the first side is a right angle.

Use the above theorem to find the measure of PKR in the below figure.

Q28 :

From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30° and 60° respectively. Find

(i) the horizontal distance between the building and the lamp post

(ii) The height of the lamp post.

[Take ]

Q29 :

During the medical check-up 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.

Q30 :

Solve the following equation for x:

9x2− 9(a+ b)x+ (2a2+ 5ab + 2b2) = 0

OR

If (−5) is a root of the quadratic equation 2x2+ px − 15 = 0 and the quadratic equation p(x2+ x) + k= 0 has equal roots, then find the values of pand k.