2011 ICSE (Class 10) Board Paper Solution: Mathematics

MATHEMATICS (ICSE – Class X Board Paper 2011)

Two and Half Hour. Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper.

The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions form 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 the loss of marks.

The intended marks for questions or parts of questions are given in brackets [ ].

Mathematical tables are provided.

SECTION A [40 Marks]

(Answer all questions from this Section.)

Question 1:

(a) Find the value of $\displaystyle k$ if $\displaystyle (x-2)$ is a factor of $\displaystyle x^3+2x^2-kx+10$ . Hence determine whether $\displaystyle (x+5)$ is also a factor. [3]

$\displaystyle \text{(b) If } A = \begin{bmatrix} 3 & 5 \\ 4 & -2 \end{bmatrix} \text{ and } B = \begin{bmatrix} 4 \\ 2 \end{bmatrix} , \text{ is the product } AB \text{ possible ? } \\ \text{Give a reason. If yes, find } AB . \hspace{5.0cm} [3]$

(c) Mr. Kumar borrowed Rs. 15000 for two years. The rate of interest for the two successive years are 8% and 10% respectively. If the repays Rs. 6200 at the end of the first year, find the outstanding amount at the end of the second year. [4]

Answer:

(a) Let $\displaystyle f(x) = x^3+2x^2-kx+10$ .

Since given that $\displaystyle (x-2)$ is a factor $\displaystyle f(2) = 0$

Substituting the value of $\displaystyle x =2$ in the above function we get:

$\displaystyle f(2) = 0$

$\displaystyle f(2) = 8+8-2k+10=0$

$\displaystyle \Rightarrow k=13$

For $\displaystyle (x + 5)$ to be a factor $\displaystyle f(-5) = 0$

Substituting the value of $\displaystyle x =-5$ in the above function we get:

$\displaystyle f(-5) = (-5)^3+2(-5)^2-k(-5)+10 = -125+50+65+10=0$

Hence $\displaystyle (x+5$ ) is a factor of $\displaystyle x^3+2x^2-kx+10$

(b) The order of matrix $\displaystyle A = 2 \times 2$ and the order of matrix $\displaystyle B \text{ is } 2 \times 1 .$

Since the number of columns in $\displaystyle A$ is equal to the number of rows in $\displaystyle B$ , the product $\displaystyle AB$ is possible.

$\displaystyle AB = \begin{bmatrix} 3 & 5 \\ 4 & -2 \end{bmatrix} . \begin{bmatrix} 4 \\ 2 \end{bmatrix} = \begin{bmatrix} 6+20 \\ 8-8 \end{bmatrix} = \begin{bmatrix} 26 \\ 0 \end{bmatrix}$

The rate of interest for the two successive years are $\displaystyle 8\%$ and $\displaystyle 10\%$ respectively.

$\displaystyle \text{Formula: } A = P(1+\frac{r}{100})^n$

$\displaystyle \text{Therefore Amount after } 1^{st} \text{ year } = 15000 \times \Big(1+\frac{8}{100} \Big)^1 = 16200 \text{ Rs. }$

Principal at the start of $\displaystyle 2^{nd}$ year after repayment $\displaystyle = 16200 - 6200 = 10000 \text{ Rs. }$

$\displaystyle \text{Amount outstanding at the end of second year } = 10000 \times (1+\frac{10}{100})^1 = 11000 \text{ Rs. }$

$\displaystyle \\$

Question 2:

(a) From a pack of 52 playing cards all cards whose numbers are multiples of 3 are removed. A card is now drawn at random. What is the probability that the card drawn is;

(i) a face card (King, Jack, or Queen)

(i) an even number red card [3]

(b) Solve the following equation: $\displaystyle x - \frac{18}{x} = 6$ Give your answer correct to two significant figures. [3]

2011-1 (c) In the given figure $\displaystyle O$ is the center of the circle Tangent of $\displaystyle A$ and $\displaystyle B$ meet at $\displaystyle C$ if $\displaystyle \angle AOC = 30^o$ , find (i) $\displaystyle \angle BCO$ (ii) $\displaystyle \angle AOB$ (iii) $\displaystyle \angle APB$ [4]

Answer:

(a) Total number of cards $\displaystyle = 52$

Number of cards which are multiples of $\displaystyle 3 = 12$

Total number of cards left $\displaystyle = 52 - 12 = 40$

(i) Number of face cards $\displaystyle = 12$

$\displaystyle \text{Probability (of a face card) } = \frac{12}{40} = 0.3$

(ii) Even numbered red cards $\displaystyle = 10$

$\displaystyle \text{Probability (of a even number red card) } = \frac{10}{40} = 0.25$

$\displaystyle \text{(b) Given } x - \frac{18}{x} = 6$

Simplifying: $\displaystyle x^2 - 6x - 18 = 0$

Compare with equation $\displaystyle ax^2 + bx + c = 0$ , we get $\displaystyle a =1 b = -6$ and $\displaystyle c = -18$

$\displaystyle \text{We know, } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$\displaystyle = \frac{6 \pm \sqrt{36 + 72 }}{2}$

$\displaystyle = \frac{6 \pm 6\sqrt{3}}{2} = 3 \pm 3 \sqrt{3}$

Therefore $\displaystyle x = 8.196 \text{ or } -2.196$

Answer correct to two significant figures: $\displaystyle x = 8.2 \text{ or } -2.2$

$\displaystyle \angle OAC = \angle OBC = 90^{\circ}$

$\displaystyle OC$ is common

$\displaystyle AC = BC$ (two tangents drawn from a point on a circle are of equal lengths)

Therefore $\displaystyle \triangle AOC \cong \triangle BOC$ (RHS postulate)

(i) $\displaystyle \angle ACO = \angle BCO = 30^{\circ}$

(ii) $\displaystyle \angle AOC = 180^{\circ} - 90^{\circ} - 30^{\circ} = 60^{\circ}$

$\displaystyle \angle BOC = 180^{\circ} - 90^{\circ} - 30^{\circ} = 60^{\circ}$

$\displaystyle \therefore \angle AOB = \angle AOC + \angle BOC = 120^{\circ}$

(iii) $\displaystyle \angle AOB = 2 \angle APB$ (chord subtends twice the angle at the center than that it subtends on the circumference)

$\displaystyle \Rightarrow \angle AOB = 2 \angle APB$

$\displaystyle \Rightarrow \angle APB = 60^{\circ}$

$\displaystyle \\$

Question 3:

(a) Ahmed has a recurring deposit account in a bank. He deposits Rs. 2500 per month for 2 years. If he gets Rs. 66250 at the time of maturity, find;

(i) The interest paid by the bank

(ii) The rate of interest [3]

2011-2 (b) Calculate the area of the shaded region, if the diameter of the semi circle is $\displaystyle \text{equal to} 14\text{ cm } . \text{(Take } \pi = \frac{22}{7} )$ [3]

(c) $\displaystyle ABC$ is a triangle and $\displaystyle G(4,3)$ is the central of the triangle. If $\displaystyle A=(1,3), B=(4,b)$ and $\displaystyle C=(a,1)$ find $\displaystyle 'a'$ and $\displaystyle 'b'$ find the length of side $\displaystyle BC .$ [4]

Answer:

$\displaystyle \text{(a) } P = \text{ Rs. } 2500, \text{ no of months} = 24, rate = r\% \\ \\ \text{ Maturity Amount } = \text{ Rs. } 66250$

$\displaystyle \text{ Maturity Value }= P \times n + P \times \frac{n(n+1)}{2 \times 12} \times \frac{r}{100}$

$\displaystyle 66250 =2500 \times 24 +2500 \times \frac{24(24+1)}{2 \times 12} \times \frac{r}{100} \Rightarrow r=10\%$

$\displaystyle \text{ Interest }= P \times \frac{n(n+1)}{2 \times 12} \times \frac{r}{100}$

$\displaystyle = 2500 \times \frac{24(24+1)}{2 \times 12} \times \frac{10}{100} = 6250$

(b) Area of shaded portion = Total area – area of the two quadrants

$\displaystyle = (\text{ Area of ACDE + Area of semi circle EFD}) - \\ (\text{ Area of Quadrant ABE + Area of Quadrant BCD})$

$\displaystyle = \Big(14 \times 7 + \frac{1}{2} \pi \times 7^2) - (\frac{1}{4} \times \pi \times 7^2 + \frac{1}{4} \times \pi \times 7^2 \Big)$

$\displaystyle = 98\text{ cm}^2$

$\displaystyle 4 = \frac{1+4+a}{3} \Rightarrow a=7$

$\displaystyle 3= \frac{3+b+1}{3} \Rightarrow b = 5$

Therefore $\displaystyle B(4, 5) \text{ and } C(7, 1)$

Therefore $\displaystyle BC=\sqrt{(7-4)^2+(1-5)^2} = \sqrt{25} = 5$ units.

$\displaystyle \\$

Question 4:

(a) Solve the following in equation and represent the solution set on the number line:

$\displaystyle 2x-5 \le 5x + 4 \le 11 \text{ where } x \in I$ [3]

(b) Evaluate without using trigonometric tables:

$\displaystyle 2 \Big(\frac{\tan 35^{\circ}}{\cot 55^{\circ}} \Big)^2 + \Big(\frac{\cot 55^{\circ}}{\tan 35^{\circ}} \Big)^2 - 3 \Big(\frac{\sec 40^{\circ}}{\mathrm{cosec} 50^{\circ}} \Big)$ [3]

Marks	50-60	60-70	70-80	80-90	90-100
No. of students	4	8	14	19	5

Draw a histogram for the above data using a graph paper and locate the mode. [4]

Answers:

(a) $\displaystyle 2x-5 \leq 5x+4 < 11$

$\displaystyle 2x-5 \leq 5x+4 \text{ or } -9 \leq 3x \text{ or } -3 \leq x$

$\displaystyle 5x+4 < 11 \text{ or } 5x < 7 \text{ or } x < \frac{7}{5}$

$\displaystyle -3 \leq x <\frac{7}{5}$

Therefore $\displaystyle x \in \{-3, -2, -1, 0, 1 \}$

(b) $\displaystyle 2 \Big(\frac{\tan 35^o}{\cot 55^o} \Big)^2 + \Big(\frac{\cot 55^o}{\tan 35^o} \Big)^2 - 3 \Big(\frac{\sec 40^o}{\mathrm{cosec} 50^o} \Big)$

$\displaystyle = 2 \Big(\frac{\tan 35^o}{\cot (90^o - 35^o)} \Big)^2 + \Big(\frac{\cot 55^o}{\tan (90^o - 55^o)} \Big)^2 - 3 \Big(\frac{\sec 40^o}{\mathrm{cosec} (90^o - 40^o)}\Big)$

$\displaystyle = 2 \Big(\frac{\tan 35^o}{\tan 35^o} \Big)^2 + \Big(\frac{\cot 55^o}{\cot 55^o} \Big)^2 - 3 \Big(\frac{\sec 40^o}{\sec 40^o} \Big)$

$\displaystyle = 2 + 1 - 3 = 0$

(c)

$\displaystyle \\$

SECTION B [40 Marks]

(Answer any four questions in this Section.)

Question 5:

(a) A manufacturer sells a washing machine to a wholesaler for Rs. 15000. The wholesaler sells it to a trader at a profit of Rs. 1200 and the trader in turn sells it to a consumer at a profit of Rs. 1800. If the rate of VAT is 8% find:

(i) The amount of VAT received by the state government on the sale of this machine from the manufacture and the wholesaler.

(ii) The amount that the consumer pays for the machine. [3]

(b) A solid cone of radius 5 cm and height 8 cm is melted and made into small spheres of radius 0.5 cm . Find the number of sphere formed. [3]

(c) $\displaystyle ABCD$ is a parallelogram where $\displaystyle A(x, y), B (5, 8), C (4, 7)$ and $\displaystyle D (2, -4)$ Find

(i) Coordinates of $\displaystyle A$

(ii) Equation of diagonal $\displaystyle BD$ [4]

Answer:

(a)

$\displaystyle \text{(i) Tax received by the manufacturer } = \frac{8}{100} \times 1500 = \text{ Rs. } 1200$

For the trader the price $\displaystyle = 15000+1200= \text{ Rs. } 16200$

$\displaystyle \text{Tax paid by the trader } = \frac{8}{100} \times 16200 = \text{ Rs. } 1296$

Therefore VAT received from wholesaler $\displaystyle = 1296-1200 = \text{ Rs. } 96$

Price for the consumer $\displaystyle = 16200+1800 = \text{ Rs. } 18000$

$\displaystyle \text{(ii) Tax paid by the consumer } = \frac{8}{100} \times 18000 = \text{ Rs. } 1440$

Hence the total price paid by the consumer $\displaystyle = 18000+1440 = \text{ Rs. } 19440$

(b) Cone: Radius : $\displaystyle 5\text{ cm }$ and Height $\displaystyle 8\text{ cm }$

Sphere: $\displaystyle 0.5\text{ cm }$

$\displaystyle \text{Number of sphere } = \frac{\text{Volume of the cone}}{\text{Volume of the small sphere}}$

$\displaystyle = \frac{\frac{1}{3} \pi (5)^2 \times 8}{\frac{4}{3} \pi (0.5)^2}$

$\displaystyle = \frac{25 \times 8}{4 \times 0.125} = 400$

(c)

$\displaystyle \text{(i) Mid point of } BD = (\frac{5+2}{2}, \frac{-4+8}{2})= (\frac{7}{2}, 2)$

$\displaystyle \text{Therefore we have } A(x, y), O(\frac{7}{2}, 2) \text{ and } C(4, 7)$

$\displaystyle O$ is the mid point of $\displaystyle AC$ as well (diagonals of a parallelogram bisect each other)

$\displaystyle \text{Hence } \frac{x+4}{2} = \frac{7}{2} \Rightarrow x = 3$

$\displaystyle \text{and } \frac{y+7}{2} = 2 \Rightarrow y = -3$

Hence $\displaystyle A ( 3, -3)$

(ii) Equation of $\displaystyle BD$

$\displaystyle y - 8 = \frac{-4-8}{2-5} (x-5)$

$\displaystyle y-8 = 4(x-5)$

$\displaystyle y - 8 = 4x - 20$

$\displaystyle y + 12 = 4x$

$\displaystyle \\$

Question 6:

(a) Use a graph paper to answer the following (Take $\displaystyle 1\text{ cm }=1 \text{ unit }$ on both axes)

(i) Plot $\displaystyle A(4,4), B(4,-6)$ and $\displaystyle C(8,0)$ the vertices of a $\displaystyle \triangle ABC$

(ii) Reflect $\displaystyle ABC$ on the $\displaystyle y-axis$ and name it as $\displaystyle A'B'C'$

(iii) Write the coordinates of the image $\displaystyle A'B'$ and $\displaystyle C'$

(iv) Give a geometrical name for the figure $\displaystyle AA' C' B' BC$

(v) Identify the line of symmetry of $\displaystyle AA' C' B' BC'$ [5]

(b) Choudhury opened a Saving account at State Bank of India an 1st April 2007, The entries of one year as shows in his pass book are given below:the bank pays interest at the rate of 5% per annum, find the interest paid on 1st April 2008. Give your answer correct to the nearest rupees. [5]

Date	Particulars	Withdrawals (Rs)	Deposit (Rs.)	Balance (in Rs.)
1st April 2007	By cash	–	8550	8550
12th April 2007	To Self	1200	–	7350
24th April 2007	By cash	–	4550	11900
8th July 2007	By cheque	–	1500	13400
10th Sep.2007	By cheque	–	3500	16900
17th Sep. 2007	By cheque	2500	–	14400
11th Oct.2007	By cash	–	800	15200
6th Jan 2008	To Self	2000	–	13200
9th March 2008	By cheque	–	950	14150

Answer:

(a)

(i) & (ii) Shown in the graph below.

(iii) $\displaystyle A'(-4, 4), C'(-8, 0)$ and $\displaystyle B'(-4, -6)$

(iv) The shape is that of a hexagon.

(v) Line of symmetry : $\displaystyle y-axis$

(b) Qualifying principal for various months:

Month	Principal (Rs.)
April	7350
May	11900
June	11900
July	13400
August	13400
September	14400
October	14400
November	15200
December	15200
January	13200
February	13200
March	14150
Total	157700

$\displaystyle P = \text{ Rs. } 157700 R = 5\% \text{ and } T= \frac{1}{12}$

$\displaystyle I = P \times R \times T = 157700 \times \frac{5}{100} \times \frac{1}{12} = \text{ Rs. } 657.08 \text{ or } \text{ Rs. } 657$

$\displaystyle \\$

Question 7:

(a) Using component and dividend, find the value of x:

$\displaystyle \frac{\sqrt{3x+4}+ \sqrt{3x-5}}{\sqrt{3x+4}- \sqrt{3x-5}} = 9 \hspace{10.0cm} [3]$

$\displaystyle \text{(b) If } A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} , B = \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix} \text{ and } I \text{ is the identity matrix of the same order and } \\ \\ A^t \text{ is transpose of matrix } A , \text{ find } A^t B +BI . \hspace{7.0cm} [3]$

sm21 (c) In the adjoining figure $\displaystyle ABC$ is a right angled triangle with $\displaystyle \angle BAC = 90^o$ .

(i) Prove $\displaystyle \triangle ADB \sim \triangle CDA$

(ii) If $\displaystyle BD=18 \text{ cm }, CD=8 \text{ cm, find } AD$

(iii) Find the ratio of the area of $\displaystyle \triangle ADB$ is to the area of $\displaystyle \triangle CDA$ [4]

Answer:

$\displaystyle \text{(a) } \frac{\sqrt{3x+4}+ \sqrt{3x-5}}{\sqrt{3x+4}- \sqrt{3x-5}} = 9$

Applying componendo and dividendo

$\displaystyle \frac{(\sqrt{3x+4}+\sqrt{3x-5})+(\sqrt{3x+4}-\sqrt{3x-5})}{(\sqrt{3x+4}+\sqrt{3x-5})-(\sqrt{3x+4}-\sqrt{3x-5})}=\frac{9+1}{9-1}$

$\displaystyle \frac{2\sqrt{3x+4}}{2\sqrt{3x-5}}=\frac{10}{8}$

Simplifying

$\displaystyle \frac{\sqrt{3x+4}}{\sqrt{3x-5}}=\frac{5}{4}$

Square both sides

$\displaystyle \frac{3x+4}{3x-5} = \frac{25}{14}$

$\displaystyle 42x+56 = 75x-125$

Simplifying we get $\displaystyle x = 7$

$\displaystyle \text{(b) } A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$

$\displaystyle A^t = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$

$\displaystyle A^t.B+BI$

$\displaystyle = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} . \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}+ \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix} . \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 7 & -1 \\ 17 & -1 \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 11 & -3 \\ 16 & 2 \end{bmatrix}$

(c)

(i) Let $\displaystyle \angle DAB = \theta$

Therefore $\displaystyle \angle DAC = 90^o - \theta$

$\displaystyle \angle DBA = 90^o - \theta$

$\displaystyle \angle DCA = \theta$

Therefore $\displaystyle \triangle ADB \sim \triangle CDA$ (AAA postulate)

$\displaystyle \text{(ii) } \frac{CD}{AD}=\frac{AD}{BD}$

$\displaystyle \Rightarrow AD^2 = CD \times BD = 8 \times 18 = 144$

Therefore $\displaystyle AD = \sqrt{144} = 12$

$\displaystyle \text{(iii) } \frac{Area of \triangle ADB}{ Area of \triangle CDA} = \frac{\frac{1}{2} AD \times BD} {\frac{1}{2} AD \times CD} = \frac{BD}{CD} = \frac{18}{8}= \frac{9}{4}$

$\displaystyle \\$

Question 8:

(a) Using step division method, calculate the mean marks of the following distribution: State the modal class: [5]

Class Interval	50-55	55-60	60-65	65-70	70-75	75-80	80-85	85-90
Frequency	5	20	10	10	9	6	12	8

(b) Marks obtained by 200 students in an examination 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	11	10	20	28	37	40	29	14	6

Draw an ogive for the given distribution taking 2 cm =10 marks on one axis and 2 cm =20 students on the other axis. Using the graph, determine.

(i) The median marks

(ii) The number of students who failed if minimum marks required to pass is 40.

(iii) If scoring 85 and more marks is considered as grade one, find the number of students who secured grade on in the examination; [5]

Answers:

(a)

$\displaystyle \left| \begin{array}{c | c | c | c | c | c } \hline \text{C.I} \hspace{1.0cm} & f \hspace{1.0cm} & x \hspace{1.0cm} & d=x-67.5 \hspace{1.0cm} & u \hspace{1.0cm} & f \cdot u \hspace{1.0cm} \\ \hline 50-55 \hspace{1.0cm} & 5 \hspace{1.0cm} & 52.5 \hspace{1.0cm} & -15 \hspace{1.0cm} & -3 \hspace{1.0cm} & -15 \hspace{1.0cm} \\ \hline 55-60 \hspace{1.0cm} & 20 \hspace{1.0cm} & 57.5 \hspace{1.0cm} & -10 \hspace{1.0cm} & -2 \hspace{1.0cm} & -40 \hspace{1.0cm} \\ \hline 60-65 \hspace{1.0cm} & 10 \hspace{1.0cm} & 62.5 \hspace{1.0cm} & -5 \hspace{1.0cm} & -1 \hspace{1.0cm} & -10 \hspace{1.0cm} \\ \hline 65-70 \hspace{1.0cm} & 10 \hspace{1.0cm} & 67.5 \hspace{1.0cm} & 0 \hspace{1.0cm} & 0 \hspace{1.0cm} & 0 \hspace{1.0cm} \\ \hline 70-75 \hspace{1.0cm} & 9 \hspace{1.0cm} & 72.5 \hspace{1.0cm} & 5 \hspace{1.0cm} & 1 \hspace{1.0cm} & 9 \hspace{1.0cm} \\ \hline 75-80 \hspace{1.0cm} & 6\hspace{1.0cm} & 77.5 \hspace{1.0cm} & 10 \hspace{1.0cm} & 2 \hspace{1.0cm} & 12 \hspace{1.0cm} \\ \hline 80-85 \hspace{1.0cm} & 12 \hspace{1.0cm} & 82.5 \hspace{1.0cm} & 15 \hspace{1.0cm} & 3 \hspace{1.0cm} & 36 \hspace{1.0cm} \\ \hline 85-90 \hspace{1.0cm} & 8 \hspace{1.0cm} & 87.5 \hspace{1.0cm} & 20 \hspace{1.0cm} & 4 \hspace{1.0cm} & 32 \hspace{1.0cm} \\ \hline \hspace{1.0cm} & \Sigma f = 80 \hspace{1.0cm} & \hspace{1.0cm} & \hspace{1.0cm} & \hspace{1.0cm} & \Sigma fu = 24 \hspace{1.0cm} \\ \hline \end{array} \right|$

$\displaystyle A.M = 67.5$

$\displaystyle \text{(i) } \bar{x} = A.M. + \frac{\Sigma fu}{\Sigma f} \times i = 67.5 + \frac{24}{80} \times 5 = 69$

(ii) Modal class is $\displaystyle 55-60$ (class with highest freq.)

(b)

Class Interval	Frequency	Cumulative Frequency
0-10	5	5
10-20	11	16
20-30	10	20
30-40	20	46
40-50	28	74
50-60	37	111
60-70	40	151
70-80	29	180
80-90	14	194
90-100	6	200

(i) $\displaystyle n = 200$ (even)

$\displaystyle \text{Median} = (\frac{n}{2})^{th} \text{observation} = (\frac{200}{2})^{th} \text{observation} = 100^{th} \text{observation} = 57$

(ii) Number of student who failed $\displaystyle = 46$

(iii) Number of students who secured grade one $\displaystyle = 200 - 1888 = 12$

$\displaystyle \\$

Question 9:

(a) Parekh invested Rs. 52000 on 100 shares at a discount of Rs. 20 paying 8% dividend. At the end of one year he sells the shares at a premium of Rs. 20. Find;

(i) The annual dividend

(ii) The profit earned including hid dividend; [3]

(b) Draw a circle of radius 3.5 cm. Mark a point $\displaystyle P$ outside the circle at a distance of 6 cm from the center. Construct two tangent from $\displaystyle P$ to the given circle. Measure and write down the length of one tangent. [3]

Answer:

(a) Nominal Value of the share $\displaystyle = 100 \text{ Rs. }$

Market Value of the share $\displaystyle = 80 \text{ Rs. }$

$\displaystyle \text{Number of shares bought } = \frac{52000}{80} = 650$

$\displaystyle \text{Dividend earned } = 650 \times 100 \times \frac{8}{100} = 5200 \text{ Rs. }$

Sale proceeds $\displaystyle = 650 \times 120 = 78000 \text{ Rs. }$

Profit $\displaystyle = (78000-52000)+5200 = 31200 \text{ Rs. }$

(b) 2011-91

Steps of construction (diagram not to scale):

With the help of a ruler measure 3.5 cm in your compass and draw a circle of radius 3.5 cm.
The draw a point P, 6 cm away from the circle.
The next step is to bisect PP’
The with this as the center, O, draw a circle that goes through point Q and R on the original circles.
Join PQ and PR. These are the tangents.
Length of the tangents is 4.9 cm

$\displaystyle \text{LHS } = (\mathrm{cosec} A- \sin A) (\sec A-\cos A) \sec^2 A$

$\displaystyle = \frac{1 - \sin^2 A}{\sin A} . \frac{1 - \cos^2 A}{\cos A} . \frac{1}{\cos^2 A}$

$\displaystyle = \frac{\cos^2 A}{\sin A}. \frac{\sin^2 A}{\cos A} . \frac{1}{\cos^2 A}$

$\displaystyle = \frac{\sin A}{\cos A} = \tan A = \text{RHS}$

Hence Proved.

$\displaystyle \\$

Question 10:

(a) 6 is the mean proportion two numbers $\displaystyle x$ and $\displaystyle y$ and 48 is the third proportion of $\displaystyle x$ and $\displaystyle y$ . Find the numbers. [3]

(b) In what period of time will Rs. 12000 yield Rs. 3972 as compound interest at 10% per annum, if compounded on an yearly basis? [3]

(c) A man observed the angle of elevation of the top of a building to be $\displaystyle 30^{\circ}$ . He walks towards it in a horizontal line through its base. On covering 60 m the angle of evaluation changes to $\displaystyle 60^{\circ}$ . Find the height of the building correct to the nearest meter. [4]

Answer:

(a) Given 6 is the mean proportion between two numbers $\displaystyle x \text{ and } y$

$\displaystyle \text{Therefore } \frac{x}{6}={6}{y} \Rightarrow xy=36 \Rightarrow x = \frac{36}{y} \text{... ... ... ... ... ... i)}$

Also given 48 is the third proportion to $\displaystyle x \text{ and } y$

$\displaystyle \text{Therefore } \frac{x}{y}=\frac{y}{48} \Rightarrow y^2=48x \text{ ... ... ... ... ... ... ii)}$

Solving i) and ii)

$\displaystyle y^2 = 48 \frac{36}{y}$

$\displaystyle y^3 = 2^3 \times 6^3 \Rightarrow y = 12$

$\displaystyle \text{v} x = \frac{36}{12} = 3$

Hence the numbers are $\displaystyle 3 \text{ and } 12 .$

$\displaystyle \text{(b) Given } P=12000 \text{Rs.}; \hspace{0.3cm} A= (12000+3972)= 15972 \text{ Rs.} ; \hspace{0.1cm} r=10\%; n=n$

$\displaystyle A=P(1+\frac{r}{100})^n \Rightarrow 15972 = 12000(1+\frac{10}{100})^n \Rightarrow n = 3 \text{ years }$

$\displaystyle \frac{h}{x} = \tan 60^o$

$\displaystyle \Rightarrow x = \frac{h}{\sqrt{3}}$

Similarly, from $\displaystyle \triangle ADB$

$\displaystyle \frac{h}{60+x} = \tan 30^o$

$\displaystyle 60+x = h \sqrt{3}$

Equating for $\displaystyle x$ we get

$\displaystyle h \sqrt{3} - 60 = \frac{h}{\sqrt{3}}$

$\displaystyle \Rightarrow 3 h - 60 \sqrt{3} = h$

$\displaystyle \Rightarrow 2h = 60 \sqrt{3} \Rightarrow h = 30 \sqrt{3} = 51.96 \text{ m}$

$\displaystyle \\$

Question 11:

(a) $\displaystyle \triangle ABC \text{ with } AB=10 \text{ cm }, BC=8\text{ cm } \text{ and } AC=6\text{ cm }$ (not drawn to scale) Three circle are drawn touching each other with the vertices as their centers. Find the radii of the three circles. [3]

2011-6

(b) Rs. 480 is divided equally among $\displaystyle x$ children. If the number of children were $\displaystyle 20$ more than each would have got Rs. 12 less. Find $\displaystyle x.$ [3]