Board Paper of Class 10 2015 Maths All India(SET 1)  Solutions
General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
 Question 1
If the quadratic equation ${\mathrm{px}}^{2}2\sqrt{5}\mathrm{px}+15=0$ has two equal roots, then find the value of p. VIEW SOLUTION
 Question 2
In Figure 1, a tower AB is 20 m high and BC, its shadow on the ground, is $20\sqrt{3}$ m long. Find the Sun's altitude.
Figure 1
 Question 3
Two different dice are tossed together. Find the probability that the product of the two number on the top of the dice is 6. VIEW SOLUTION
 Question 4
In Figure 2, PQ is a chord of a circle with centre O and PT is a tangent. If ∠QPT = 60°, find ∠ PRQ.
Figure 2
 Question 5
In Figure 3, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If ∠PRQ = 120°, then prove that OR = PR + RQ.
Figure 3
 Question 6
In figure 4, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments RD and DC are respectively of lengths 6 cm and 9 cm. If the area of ∆ABC is 54 cm^{2}, then find the lengths of sides AB and AC.
Figure 4
 Question 7
Solve the following quadratic equation for x :
4x^{2} + 4bx − (a^{2} − b^{2}) = 0 VIEW SOLUTION
 Question 8
In an AP, if S_{5} + S_{7} = 167 and S_{10} = 235, then find the AP, where S_{n} denotes the sum of its first n terms. VIEW SOLUTION
 Question 9
The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right triangle, rightangled at B find the value of p. VIEW SOLUTION
 Question 10
Find the relation between x and y if the points A(x, y), B(–5, 7) and C(–4, 5) are collinear. VIEW SOLUTION
 Question 11
The 14^{th} term of an AP is twice its 8^{th} term. If its 6^{th} terms is –8, then find the sum of its first 20 terms. VIEW SOLUTION
 Question 12
 Question 13
The angle of elevation of an aeroplane from a point A on the ground 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500$\sqrt{3}$ m, find the speed of the plane in km/hr. VIEW SOLUTION
 Question 14
If the coordinates of points A and B are (–2, –2) and (2, –4) respectively, find the coordinates of P such that $\mathrm{AP}=\frac{3}{7}\mathrm{AB}$, where P lies on the line segment AB. VIEW SOLUTION
 Question 15
The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is $\frac{1}{4}.$ The probability of selecting a blue ball at random from the same jar is $\frac{1}{3}.$ If the jar contains 10 orange balls, find the total number of balls in the jar. VIEW SOLUTION
 Question 16
Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 60°. Also find the area of the corresponding major segment. $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$ VIEW SOLUTION
 Question 17
Due to sudden floods, some welfare associations jointly requested the government to get 100 tents fixed immediately and offered to contribute 50% of the cost. If the lower part of each tent is of the form of a cylinder of diameter 4.2 m and height 4 m with the conical upper part of same diameter but of height 2.8 m, and the canvas to be used costs Rs. 100 per sq. m, find the amount, the associations will have to pay. What values are shown by these associations?$\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$ VIEW SOLUTION
 Question 18
A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into 72 cylindrical bottles of diameter 6 cm. Find the height of the each bottle, if 10% liquid is wasted in this transfer. VIEW SOLUTION
 Question 19
A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have ? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs 5 per 100 sq. cm. [Use π =3.14] VIEW SOLUTION
 Question 20
504 cones, each of diameter 3.5 cm and height 3 cm, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area. [Use $\mathrm{\pi}=\frac{22}{7}$] VIEW SOLUTION
 Question 21
The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field. VIEW SOLUTION
 Question 22
Find the 60^{th} term of the AP 8, 10, 12, ...., if it has a total of 60 terms and hence find the sum of its last 10 terms. VIEW SOLUTION
 Question 23
A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed ? VIEW SOLUTION
 Question 24
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION
 Question 25
Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the end points of the arc. VIEW SOLUTION
 Question 26
Construct a Δ ABC in which AB = 6 cm, ∠ A = 30° and ∠ B = 60°. Construct another Δ AB'C' similar to Δ ABC with base AB' = 8 cm. VIEW SOLUTION
 Question 27
At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A. VIEW SOLUTION
 Question 28
A card is drawn at random from a wellshuffled deck of playing cards.
Find the probability that the card drawn is
(i) a card of spade or an ace
(ii) a black king
(iii) neither a jack nor a king
(iv) either a king or a queen VIEW SOLUTION
 Question 29
Find the values of k so that the area of the triangle with vertices (1, –1), (–4, 2k) and (–k, –5) is 24 sq. units.
OR
Find the ratio in which the point $\mathrm{P}\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points $\mathrm{A}\left(\frac{1}{2},\frac{3}{2}\right)$ and B(2, −5). VIEW SOLUTION
 Question 30
In Figure 5. PQRS is a square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersection of its diagonals. Find the total area of the two flower beds (shaded parts).
Figure 5
 Question 31
From each end of a solid metal cylinder, metal was scooped out in hemispherical form of same diameter. The height of the cylinder is 10 cm and its base is of radius 4.2 cm. The rest of the cylinder is melted and converted into a cylindrical wire of 1.4 cm thickness. Find the length of the wire. $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$ VIEW SOLUTION
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