Board Paper Solutions for ICSE Class 10 MATHS Board Paper 2015 MeritNation Set 1

Question 1

(a)	Given $A = [\begin{matrix} 2 & - 6 \\ 2 & 0 \end{matrix}], B = [\begin{matrix} - 3 & 2 \\ 4 & 0 \end{matrix}], C = [\begin{matrix} 4 & 0 \\ 0 & 2 \end{matrix}]$ Find the matrix X such that A + 2X = 2B + C.	[3]
(b)	At what rate % p.a. will a sum of Rs 4000 yield Rs 1324 as compound interest in 3 years?	[3]
(c)	The median of the following observations 11, 12, 14, (x – 2), (x + 4), (x + 9), 32, 38, 47 arranged in ascending order is 24. Find the value of x and hence find the mean.	[4]

VIEW SOLUTION

Question 2

(a)

What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?

[3]

(b)

If (x – 2) is a factor of the expression 2x³ + ax²+ bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

[3]

(c)

Draw a histogram from the following frequency distribution and find the mode from the graph:

Class	0–5	5–10	10–15	15–20	20–25	25–30
Frequency	2	5	18	14	8	5

[4]

VIEW SOLUTION

Question 3

(a)	Without using tables evaluate: 3 cos 80°. cosec 10° + 2 sin 59° sec 31°.	[3]
(b)	In the given figure, ∠BAD = 65°, ∠ABD = 70°and ∠BDC = 45°. (i) Prove that AC is a diameter of the circle. (ii) Find ∠ACB	[3]
(c)	AB is a diameter of a circle with centre C = (–2, 5). If A = (3, –7). Find (i) the length of radius AC (ii) the coordinates of B.	[4]

VIEW SOLUTION

Question 4

(a)	Solve the following equation and calculate the answer correct to two decimal places: x² – 5x – 10 = 0	[3]
(b)	(b) In the given figure, AB and DE are perpendicular to BC. (i) Prove that ΔABC ~ ΔDEC (ii) If AB = 6 cm; DE = 4 cm and AC = 15 cm. Calculate CD. (iii) Find the ratio of area of ΔABC: area of ΔDEC.	[3]
(c)	Using a graph paper, plot the points A(6, 4) and B(0, 4). (i) Reflect A and B in the origin to get the images A' and B'. (ii) Write the co-ordinates of A' and B'. (iii) State the geometrical name for the figure ABA' B'. (iv) Find its perimeter.	[4]

VIEW SOLUTION

Question 5

(a)	Solve the following inequation, write the solution set and represent it on the number line: $- \frac{x}{3} \leq \frac{x}{2} - 1 \frac{1}{3} < \frac{1}{6}, x \in R$	[3]
(b)	Mr. Britto deposits a certain sum of money each month in a Recurring Deposit Account of a bank. If the rate of interest is 8% per annum and Mr. Britto gets Rs 8088 from the bank after 3 years, find the value of his monthly instalment.	[3]
(c)	Salman buys 50 shares of face value Rs 100 available at Rs 132. (i) What is his investment? (ii) If the dividend is 7.5%, what will be his annual income? (iii) If he wants to increase his annual income by Rs 150, how many extra shares should he buy?	[4]

VIEW SOLUTION

Question 6

(a)

Show that

\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A}

[3]

(b)

In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.

[3]

(c)

Given below are the entries in a Saving Bank A/c pass book.

Date	Particulars	Withdrawls	Deposit	Balance
Feb 8 Feb 18 April 12 June 15 July 8	B/F To self By cash To self By cash	- 4000 - 5000 -	- - 2230 - 6000	8500

Calculate the interest for six months from February to July at 6% p.a.

[4]

VIEW SOLUTION

Question 7

(a)	In ΔABC, A(3, 5), B(7, 8) and C(1, –10). Find the equation of the median through A.	[3]
(b)	A shopkeeper sells an article at the listed price of Rs 1500 and the rate of VAT is 12% at each stage of sale. If the shopkeeper pays a VAT of Rs 36 to the Government, what was the price, inclusive to TAX, at which the shopkeeper purchased the article from the wholesaler?	[3]
(c)	In the figure given, from the top of a building AB = 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to 30° and 60° respectively. Find: (i) The horizontal distance between AB and CD. (ii) The height of the lamp post.	[4]

VIEW SOLUTION

Question 8

(a)	Find x and y if $[\begin{matrix} x & 3 x \\ y & 4 y \end{matrix}] [\begin{matrix} 2 \\ 1 \end{matrix}] = [\begin{matrix} 5 \\ 12 \end{matrix}]$ .	[3]
(b)	A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.	[3]
(c)	Without solving the following quadratic equation, find the value of ‘p' for which the given equation has real and equal roots: x² + (p – 3) x + p = 0.	[4]

VIEW SOLUTION

Question 9

(a)

In the figure alongside, OACB is a quadrant of a circle. The radius OA = 3.5 cm and OD = 2 cm. Calculate the area of the shaded portion.

(Take π = \frac{22}{7})

[3]

(b)

A box contains some black balls and 30 white balls. If the probability of drawing a black ball is two-fifths of a white ball, find the number of black balls in the box.

[3]

(c)

Find the mean of the following distribution by step deviation method:

Class interval	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	70 – 80
Frequency	10	6	8	12	5	9

[4]

VIEW SOLUTION

Question 10

(a)

Using a ruler and compasses only:
(i) Construct a triangle ABC with the following data:

AB = 3.5 cm, BC = 6 cm and ∠ABC = 120°

(ii) In the same diagram, draw a circle with BC as diameter.

Find a point P on thecircumference of the circle which is equidistant from AB and BC.

(iii) Measure ∠BCP.

[4]

(b)

The marks obtained by 120 students in a test are given below:

Marks	0–10	10–20	20–30	30–40	40–50	50–60	60–70	70–80	80–90	90–100
No of students	5	9	16	22	26	18	11	6	4	3

Draw an ogive for the given distribution on a graph sheet.
Use suitable scale for ogive to estimate the following:
(i) The median.
(ii) The number of students who obtained more than 75% marks in the test.
(iii) The number of students who did not pass the test if minimum marks required to pass is 40.

[6]

VIEW SOLUTION

Question 11

(a)	In the figure given below, the line segment AB meets X-axis at A and Y-axis at B. The point P(–3, 4) on AB divides it in the ratio 2: 3. Find the coordinates of A and B.	[3]
(b)	Using the properties of proportion, solve for x, given $\frac{x^{4} + 1}{2 x^{2}} = \frac{17}{8}$ .	[3]
(c)	A shopkeeper purchases a certain number of books for 960. If the cost per book was 8 less, the number of books that could be purchased for 960 would be 4 more. Write an equation, taking the original cost of each book to be x, and solve it to find the original cost of the books.	[4]

VIEW SOLUTION