Board Paper of Class 10 2015 Maths - Solutions

Attempt all questions from Section A and any four questions from Section B.
All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the answer.
Omission of essential working will result in loss of marks.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables are provided.

Question 1

(a)	A shopkeeper bought an article for Rs. 3,450. He marks the price of the article 16% above the cost price. The rate of sales tax charged on the article is 10% Find the: (i) market price of the article. (ii) price paid by a customer who buys the article.	[3]
(b)	Solve the following inequation and write the solution set: 13x – 5 < 15x + 4 < 7x + 12, x ∈ R Represent the solution on a real number line.	[4]
(c)	Without using trigonometric tables evaluate: $\frac{\sin 65 °}{\cos 25 °} + \frac{\cos 32 °}{\sin 58 °} - \sin 28 ° . \sec 62 ° + {cosec}^{2} 30 °$	[3]

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Question 2

(a)	If $A = [\begin{matrix} 3 & x \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 9 & 16 \\ 0 & - y \end{matrix}]$ , find x and y where A² = B.	[3]
(b)	The present population of town is 2,00,000. The population is increased by 10% in the first year and 15% in the second year. Find the population of the town at the end of two years.	[3]
(c)	Three vertices of parallelogram ABCD taken in order are A(3, 6), B(5, 10) and C(3, 2) (i) the coordinate of the fourth vertex D (ii) length of diagonal BD (iii) equation of the side AD of the parallelogram ABCD	[4]

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Question 3

(a)

In the given figure, ABCD is the square of side 21 cm. AC and BD are two diagonals of the square. Two semicircles are drawn with AD and BC as diameters. Find the area of the shaded region.

(Take π = \frac{22}{7})

[3]

(b)

The marks obtained by 30 students in a class assignment of 5 marks are given below.

Marks	0	1	2	3	4	5
Number of Students	1	3	6	10	5	5

Calculate the mean, median and mode of the above distribution.

[3]

(c)

In the figure given below, O is the centre of the circle and SP is a tangent.
If ∠SRT = 65°, find the value of x, y and z.

[4]

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Question 4

(a)	Katrina opened a recurring deposit account with a Nationalised Bank for a period of 2 years. If the bank pays interest at the rate 6% per annum and the monthly instalment is Rs. 1,000, find the: (i) Interest earned in 2 years. (ii) Matured value	[3]
(b)	Find the value of ‘k’ for which x = 3 is a solution of the quadratic equation, (k + 2)x² – kx + 6 = 0. Thus find the other root of the equation.	[3]
(c)	Construct a regular hexagon of side 5 cm. Construct a circle circumscribing the hexagon. All traces of construction must be clearly shown.	[4]

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Question 5

(a)

Use a graph paper for this question taking 1 cm = 1 unit along both the x and y axis :
(i) Plot the points A(0, 5), B(2, 5), C(5, 2), D(5, –2), E(2, –5) and F(0, –5).

(ii) Reflect the points B, C, D and E on the y-axis and name them respectively as B’, C’, D’ and E’.

(iii) Write the coordinates of B’, C’, D’ and E’.
(iv) Name the figure formed by B C D E E’ D’ C’ B’.
(v) Name a line of symmetry for the figure formed.

[5]

(b)

Virat opened a Savings Bank account in a bank on 16^thApril 2010.
His pass book shows the following entries :

Date	Particulars	Withdrawal (RS.)	Deposit (Rs.)	Balance (Rs.)
April 16, 2010	By cash	-	2500	2500
April 28^th	By cheque	-	3000	5500
May 9^th	To cheque	850	-	4650
May 15^th	By cash	-	1600	6250
May 24^th	To cash	1000	-	5250
June 4^th	To cash	500	-	4750
June 30^th	To cheque	-	2400	7150
July 3^rd	By cash	-	1800	8950

Calculate the interest Virat earned at the end of 31^st July, 2010 at 4% per annum interest. What sum of money will he receive if he closed the account on 1st August, 2010?

[5]

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Question 6

(a)

If a, b, c are in continued proportion, prove that
(a + b + c) (a – b + c) = a² + b² + c².

[3]

(b)

In the given figure ABC is a triangle and BC is parallel to the y -axis.
AB and AC intersect the y-axis at P and Q respectively.

(i) Write the coordinates of A.
(ii) Find the length of AB and AC.
(iii) Find the ratio in which Q divides AC.
(iv) Find the equation of the line AC

[4]

(c)

Calculate the mean of the following distribution :

Class Interval	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
Frequency	8	5	12	35	24	16

[3]

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Question 7

(a)	Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed.	[3]
(b)	Find 'a' of the two polynomials ax³ + 3x²– 9 and 2x³ + 4x + a, leaves the same remainder when divided by x + 3.	[3]
(c)	Prove that $\frac{\sin θ}{1 - \cot θ} + \frac{\cos θ}{1 - \tan θ} = \cos θ + \sin θ$	[4]

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Question 8

(a)	AB and CD are two chords of a circle intersecting at P. Prove that AP × PB = CP × PD	[3]
(b)	A bag contains 5 white balls, 6 red balls and 9 green balls. A ball is drawn at random from the bag. Find the probability that the ball drawn is: (i) a green ball (ii) a white or a red ball (iii) is neither a green ball nor a white ball.	[3]
(c)	Rohit invested Rs. 9,600 on Rs. 100 shares at Rs. 20 premium paying 8% dividend. Rohit sold the shares when the price rose to Rs. 160. He invested the proceeds (excluding dividend) in 10% Rs. 50 shares at Rs. 40. Find the: (i) original number of shares (ii) sale proceeds (iii) new number of shares. (iv) change in the two dividends.	[4]

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Question 9

(a)

The horizontal distance between two towers is 120 m. The angle of elevation of the top and angle of depression of the bottom of the first tower as observed from the second tower is 30° and 24° respectively.

Find the height of the two towers. Give your answer correct to 3 significant figures.

[4]

(b)

The weight of 50 workers is given below :

Weight in Kg	50 – 60	60 – 70	70 – 80	80 – 90	90 – 100	100 – 110	110 – 120
No. of Workers	4	7	11	14	6	5	3

Draw an ogive of the given distribution using a graph sheet. Take 2 cm = 10 kg on one axis and 2 cm = 5 workers along the other axis. Use a graph to estimate the following:
(i) The upper and lower quartiles.

(ii) If weighing 95 kg and above is considered overweight, find the number of workers who are overweight.

[6]

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Question 10

(a)	A wholesaler buys a TV from the manufacturer for Rs. 25,000. He marks the price of TV 20% above his cost price and sells it to a retailer at a 10% discount on the market price. If the rate of VAT is 8%, find the : (i) Market price (ii) Retailer’s cost price inclusive of tax. (iii) VAT paid by the wholesaler.	[3]
(b)	If $A = [\begin{matrix} 3 & 7 \\ 2 & 4 \end{matrix}], B = [\begin{matrix} 0 & 2 \\ 5 & 3 \end{matrix}] and C = [\begin{matrix} 1 & - 5 \\ - 4 & 6 \end{matrix}]$ . Find AB – 5C.	[3]
(c)	ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that: (i) ∆ADE ∼ ∆ACB (ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD. (iii) Find, Area of ∆ADE : area of quadrilateral BCED.	[4]

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Question 11

(a)	Sum of two natural numbers is 8 and the difference of their reciprocal is $\frac{2}{15}$ . Find the numbers.	[3]
(b)	Given $\frac{x^{3} + 12 x}{6 x^{2} + 8} = \frac{y^{3} + 27 y}{9 y^{2} + 27}$ . Using componendo and dividendo find x : y.	[3]
(c)	Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°. Hence: (i) Construct the locus of points equidistant from BA and BC. (ii) Construct the locus of points equidistant from B and C. (iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.	[4]

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