2012 ICSE (Class 10) Board Paper Solution: Mathematics

MATHEMATICS (ICSE – Class X Board Paper 2012)

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:

$\displaystyle \text{(a) Given } A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} . \text{ Find } A^2 - 5A + 7I . \hspace{2.0cm} [3]$

(b) The monthly pocket money of Ravi and Sanjeev are in the ratio 5:7. There expenditures are in the ratio 3:5. If each saves Rs. 80 every month, find their monthly pocket money. [3]

Answers:

$\displaystyle \text{(a) Given } A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$\displaystyle A^2 - 5A + 7I$

$\displaystyle = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} . \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}- 5. \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} + 7.\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 9-1 & 3+2 \\ -3-2 & -1+4 \end{bmatrix} - \begin{bmatrix} 15 & 5 \\ -5 & 10 \end{bmatrix} + \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix} - \begin{bmatrix} 15 & 5 \\ -5 & 10 \end{bmatrix} + \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 8-15+7 & 5-5+0 \\ -5-5+0 & 3-10+7 \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$\displaystyle = 0$

(b) Let monthly pocket money be Ravi is $\displaystyle 5x$ and Sanjeev is $\displaystyle 7x .$

They both save $\displaystyle\text{ Rs. } 80$ per month.

Therefore, their expenditure would be $\displaystyle (5x-80) \text{ and } (7x -80)$ respectively.

$\displaystyle \text{Hence } \frac{5x-80}{7x-80} = \frac{3}{5}$

$\displaystyle \Rightarrow 25x - 400 = 21x -240$

$\displaystyle \Rightarrow x = 40\text{ Rs. }$

Ravi’s pocket money $\displaystyle = 5x = 200\text{ Rs. }$

Sanjeev’s pocket money $\displaystyle = 7x = 280\text{ Rs. }$

Try $\displaystyle x = 1, f(1)=3+2-19+6=0$

$\displaystyle f(1)=-3+2+19+6 \neq 0 , \text{ therefore } (x-1)$ is not a factor of the given function.

Try $\displaystyle x = 2, f(2)=24+8-38+6=0$

Therefore $\displaystyle (x-2)$ is a factor of $\displaystyle f(x)$

$\displaystyle \begin{array}{r l l } x-2 ) & \overline {3x^3+ 2x^2-19x+6} & (3x^2+8x-3 \\ (-) & \underline {3x^3-6x^2} & \\ & \hspace{1.0cm} 8x^2-19x+6 & \\ (-) &\hspace{1.0cm} \underline{8x^2-16x} & \\ & \hspace{2.0cm} { -3x+6} & \\ (-) & \hspace{2.0cm} \underline{ -3x+6} & \\ & \hspace{3.0cm} \times & \\ \end{array}$

To factories $\displaystyle 3x ^2+8x-3$

$\displaystyle 3x ^2+8x-3 = 3x ^2+9x-x-3$

$\displaystyle =3x(x+3)-1(x+3)$

$\displaystyle =(3x-1)(x+3)$

Hence, $\displaystyle 3x ^3+ 2x ^2-19x+6=(x-2)(3x-1)(x+3)$

$\displaystyle \\$

Question 2:

(a) On what sum of money will the difference between the compound interest and simple interest for 2 years be equal Rs. 25 if the rate of interest charged for both is 5% p.a.? [3]

(b) $\displaystyle ABC$ is an is isosceles right-angled triangle with $\displaystyle \angle ABC = 90^{\circ} .$ A semi-circle is drawn with $\displaystyle AC$ as the diameter. If $\displaystyle AB=BC=7 \text{ cm }$ find the area of the shaded region. $\displaystyle ( \pi = \frac{22}{7}) .$ [3]

(i) the ratio in which $\displaystyle AB$ is divided by the $\displaystyle y-axis$

(ii) find the coordinates of the point of intersection.

(iii) the length of $\displaystyle AB .$ [4]

Answers:

(a) Let the sum be $\displaystyle x\text{ Rs. }$

$\displaystyle \text{Simple Interest for 2 years } = x \times \frac{5}{100} \times 2 = 0.1x$

Amount Compound Interest

$\displaystyle A=P(1+\frac{r}{100})^n \Rightarrow A = x(1+\frac{5}{100})^2 \Rightarrow A = 1.1025x\text{ Rs. }$

Given difference = 25 Rs.

Therefore $\displaystyle C.I. - S.I. = 25 \Rightarrow (1.1025x-x)-(0.1x) = 25 \Rightarrow x = 10000\text{ Rs. }$

(b) $\displaystyle \triangle ABC$ is a right angled triangle. Therefore

$\displaystyle AC^2 = AB^2+BC^2 = (7)^2 + (7)^2 = 98$

$\displaystyle \Rightarrow AC = 7 \sqrt{2}$

$\displaystyle \text{Area of semi circle } = \frac{1}{2} \times \frac{22}{7} \times (\frac{7 \sqrt{2}}{2})^2 = 38.5 \text{ cm}^2$

$\displaystyle \text{Area of } \triangle ABC = \frac{1}{2} \times 7 \times 7 = 24.5 \text{ cm}^2$

Area of the shaded region = Area of the semi circle – Area of $\displaystyle \triangle ABC = 38.5 - 24.5 = 14 \text{ cm}^2$

(c) Let the required ratio be $\displaystyle k: 1$ and the point of intersection $\displaystyle y-axis$ be $\displaystyle (0, y)$

$\displaystyle \text{Since } x = \frac{kx_2+x_1}{k+1}$

$\displaystyle \Rightarrow 0 = \frac{k \times (8)-4}{k+1}$

$\displaystyle \Rightarrow 8k-4=0$

$\displaystyle \Rightarrow k = \frac{1}{2}$

$\displaystyle \Rightarrow m_1:m_2 = 1:2$

$\displaystyle y = \frac{1 \times (-3)+2 \times (6)}{ 1+2} = 3$

$\displaystyle \text{Therefore the point intersection is } = (0, 3)$

$\displaystyle \text{Length of } AB = \sqrt{(8-(-4))^2+(-3-6)^2} = \sqrt{144+81} = \sqrt{225} = 15 \text{ units} .$

$\displaystyle \\$

Question 3:

(a) In the given figure $\displaystyle O$ is the central of the circle and $\displaystyle AB$ is a tangent at $\displaystyle B .$ If $\displaystyle AB=15 \text{ cm } \text{ and } AC =7.5 \text{ cm } .$ Calculate the radius of the circle. [3]

(b) Evaluate without using trigonometric tables:

$\displaystyle \cos^2 26^{\circ} + \cos 64^{\circ}.\sin 26^{\circ} + \frac{\tan 36^{\circ}}{\cot 54^{\circ}}$ [3]

(c) Marks obtained by $\displaystyle 40$ students in a short assessment is given below, where $\displaystyle a \text{ and } b$ are two missing data:

Marks	5	6	7	8	9
No. of Students	6	A	16	13	B

If the mean of the distribution is 7.2 find $\displaystyle a \text{ and } b .$ [4]

Answers:

(a) Let the radius of the circle $\displaystyle = r$

Here we apply intercept theorem. Therefore:

$\displaystyle AC \times AD = AB^2$

$\displaystyle 7 \times (7.5 + 2r ) = 15^2$

$\displaystyle 2r = 30-7.5 = 22.5 \Rightarrow r = 11.25 \text{ cm }$

$\displaystyle \text{(b) } \cos^2 26^{\circ} + \cos 64^{\circ}.\sin 26^{\circ} + \frac{\tan 36^{\circ}}{\cot 54^{\circ}}$

$\displaystyle = \cos^2 26^{\circ} + \cos (90^{\circ} - 26^{\circ}).\sin 26^{\circ} + \frac{\tan (90^{\circ} - 54^{\circ})}{\cot 54^{\circ}}$

$\displaystyle = \cos^2 26^{\circ} + \sin^2 26^{\circ} + \frac{\cot 54^{\circ}}{\cot 54^{\circ}}$

$\displaystyle = 2$

Therefore $\displaystyle 6 + a+ 16+ 13+ b = 40$

$\displaystyle \Rightarrow a + b = 5 .$ .. … … (i)

$\displaystyle \text{Given mean } (\overline{x}) = \frac {\Sigma fx}{\Sigma f} = 7.2$

Therefore $\displaystyle 7.2 = \frac{5 \times 6 + 6 \times a +7 \times 16 +8 \times 13 +9 \times b}{40}$

$\displaystyle 246 + 6a + 9 b = 288$

$\displaystyle 6a + 9 b = 42$

$\displaystyle 2a + 3b = 14 .$ .. … … (ii)

Solving (i) and (ii) we get $\displaystyle a = 1 \text{ and } b = 4$

$\displaystyle \\$

Question 4:

(a) Kiran deposited Rs. 200 per month for 36 months in a Bank’s recurring deposit account. If the bank pays interest at the rate of 11% per annum, find the amount she gets on maturity. [3]

(b) Two coins are tossed once; Find the probability of getting:

(i) 2 heads

(ii) At 1 tail [3]

(i) Plot the points $\displaystyle A(4,4) \text{ and } B (2,2)$

(ii) Reflect $\displaystyle A \text{ and } B$ in the origin to get the images $\displaystyle A' \text{ and } B'$ respectively.

(iii) Write down the co-ordinates of $\displaystyle A' \text{ and } B' .$

(iv) Give the geometrical name for the figure $\displaystyle ABA'B' .$

(v) Draw and name its lines of symmetry. [4]

Answers:

(a) $\displaystyle P =\text{ Rs. } 200, \text{ no of months } = 36, r = 11\%$

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

$\displaystyle =200 \times 36 +200 \times \frac{36(36+1)}{2 \times 12} \times \frac{11}{100} =\text{ Rs. } 8421$

(b) Let Heads – $\displaystyle H$ and Tails – $\displaystyle T$

If two coins are tossed once, then the total number of possibilities would be as shown: Sample Space $\displaystyle (S) = {HH, HT, TH, TT}$

$\displaystyle n (S) = 4 \text{(i.e. there are 4 possible outcomes)}$

(i) Event: getting two heads $\displaystyle = {HH} \Rightarrow n(E) = 1$

$\displaystyle \text{Hence the probability } P(E) = \frac{n(E)}{n(S)} = \frac{1}{4} = 0.25$

(ii) Events : At least one tail $\displaystyle = {HT, TH, TT} \Rightarrow n(E) = 3$

$\displaystyle \text{Hence the probability } P(E) = \frac{n(E)}{n(S)} = \frac{3}{4} = 0.75$

(c)

(i) Please refer to the graph shown below

(ii) Please refer to the graph shown below

(iii) Reflection of $\displaystyle A \text{ and } B$ in the origin are $\displaystyle A'(4, -4) \text{ and } B'(-2, -2)$ respectively.

(iv) Name of the geometrical figure in the graph show is Rhombus

(v) Two lines of symmetry: Both diagonal $\displaystyle AA' \text{ and } BB'$

$\displaystyle \\$

SECTION B [40 Marks]

(Answer any four questions in this Section.)

Question 5:

c78 (a) In the given figure, $\displaystyle AB$ is the diameter of a circle with center $\displaystyle O .$ $\displaystyle \angle BCD = 130^o .$ Find (i) $\displaystyle \angle DAB$ (ii) $\displaystyle \angle DBA$ [3]

$\displaystyle \text{(b) Given } A = \begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}. X = \begin{bmatrix} 7 \\ 6 \end{bmatrix} \text{Write (i) the order of the matrix } X \text{(ii) the matrix } X . \hspace{0.2cm} [3]$

Date	Particular	Withdrawal	Deposit	Balance
Jan. 1st 2006	B/F	–	–	1,270
Jan. 7th 2006	By Cheque	–	2310	3580
March 9th 2006	To Self	2000	–	1580
March 26th 2006	By Cash		6200	7780
June 10th 2006	To Cheque	4500	–	3280
July 15th 2006	By Clearing	–	2630	5910
October 18th 2006	To Cheque	530	–	5380
October 27th 2006	To Self	2690	–	2690
November 3rd 2006	By Cash	–	1500	4190
December 6th 2006	To Cheque	950	–	3240
December 23rd 2006	By Transfer	–	2920	6160

If he receives Rs. 198 as interest on 1st January, 2007. Find the rate of interest paid by the bank.

Answers:

(a) c78

(i) $\displaystyle \angle DAB = 180^o-\angle DCB = 180^o-130^o = 50^o (ABCD \text{is a cyclic quadrilateral) }$

(ii) In $\displaystyle \triangle ADB$

$\displaystyle \angle DAB + \angle ADB + \angle DBA = 180^o$

$\displaystyle \Rightarrow \angle DBA = 180^o-50^o-90^o = 140^o$

$\displaystyle \text{(b) } \begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix} . X = \begin{bmatrix} 7 \\ 6 \end{bmatrix}$

$\displaystyle A_{2 \times 2} . X_{p \times q} = B_{2 \times 1}$

Therefore $\displaystyle p = 2 \text{ and } q = 1 .$ Hence the order of Matrix is $\displaystyle 2 \times 1$

$\displaystyle \text{Let } X = \begin{bmatrix} x \\ y \end{bmatrix}$

Therefore

$\displaystyle \begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix} . \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 6 \end{bmatrix}$

$\displaystyle \begin{bmatrix} 2x+y \\ -3x+4y \end{bmatrix} = \begin{bmatrix} 7 \\ 6 \end{bmatrix}$

Therefore

$\displaystyle 2x+y = 7$

and $\displaystyle -3x+4y = 6$

Solving we get $\displaystyle x = 2 \text{ and } y = 3$

$\displaystyle \text{Hence } X = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$

Month	Principal (Rs.)
January	3580
February	3580
March	1580
April	7780
May	7780
June	3280
July	3280
August	5910
September	5910
October	2690
November	4190
December	3240
Total	52800

$\displaystyle P =\text{ Rs. } 52800 R = x\% \text{ and } T= \frac{1}{12} I =\text{ Rs. } 198$

$\displaystyle I = P \times R \times T \Rightarrow 52800 \times \frac{x}{100} \times \frac{1}{12} = 198 \Rightarrow x = 4.5 \%$

Therefore Rate $\displaystyle = 4.5\%$

$\displaystyle \\$

Question 6:

(a) The principal price of an article is $\displaystyle\text{ Rs. } 60000 .$ The wholesaler allows a discount of $\displaystyle 20 \%$ to the shopkeeper. The shopkeeper sells the article to the customer at the printed price. Sales tax (under VAT) is charged at the rate of $\displaystyle 6 \%$ at every stage. [4]

Find:

(i) The cost of the shopkeeper inclusive of tax

(ii) VAT paid by the shopkeeper to the Government.

(iii) The cost of the customer inclusive of tax.

(b) Solve the following inequation and represent the solution set on the number line:

$\displaystyle 4x - 19 < \frac{3x}{5} - 2 \le \frac{-2}{5}+ x, x \in R$ [3]

(c) Without solving the following quadratic equation, find the value of $\displaystyle m$ for which the given equation has real and equal roots.

$\displaystyle x^2 + 2 (m-1)x + (m+5) = 0$ [3]

Answers:

(a) Printed Price $\displaystyle =\text{ Rs. } 60000$

$\displaystyle \text{Discounted price } = 60000 \times (1-\frac{20}{100}) =\text{ Rs. } 48000$

$\displaystyle \text{Price charged by the wholesaler }= 48000 + \frac{6}{100} \times 48000 =\text{ Rs. } 50880$

The cost to the shopkeeper inclusive of the tax $\displaystyle =\text{ Rs. } 50880$

VAT paid by the shopkeeper $\displaystyle = \text{Tax Charged - Tax Paid} = \frac{6}{100} \times 60000 - \frac{6}{100} \times 48000 =\text{ Rs. } 720$

The cost to the customer inclusive of the tax $\displaystyle = 60000+ 3600 =\text{ Rs. } 63600$

$\displaystyle \text{(b) } 4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x$

$\displaystyle 4x-19 < \frac{3x}{5}-2 \text{ or } 20x-95 < 3x-10 \text{ or } 17x < 85 \text{ or } x < 5$

$\displaystyle \frac{3x}{5}-2 \leq -\frac{2}{5}+x \text{ or } 3x-10 \leq -2 +5x \text{ or } -8 \leq 2x \text{ or } -4 \leq x$

Therefore $\displaystyle \{x : -4 \leq x < 5, x \in R \}$

$\displaystyle \text{(c) Comparing } x^2+2(m-1)x+(m+5)=0 \text{ with } ax^2+bx+c=0 , \\ \\ \text{ we get } a = 1, b = 2(m-1) and c =(m+5)$

For roots to be equal, we should have $\displaystyle b^2-4ac = 0$

$\displaystyle (2(m-1))^2-4(1)(m+5)=0$

$\displaystyle 4(m^2+1-2m)-4(m+5)=0$

$\displaystyle 4m^2+4-8m-4m-20=0$

$\displaystyle 4m^2-12m-16=0$

$\displaystyle m^2-3m-4=0$

$\displaystyle m^2-4m+m-4=0$

$\displaystyle m(m-4)+(m-4)=0$

$\displaystyle (m-4)(m+1)=0 \Rightarrow m = 4, -1$

$\displaystyle \\$

Question 7:

(a) A hollow sphere of internal and external radii $\displaystyle 6 \text{ cm } \text{ and } 8 \text{ cm }$ respectively is melted and recast into small cones of base radius $\displaystyle 2 \text{ cm }$ and height $\displaystyle 8 \text{ cm } .$ Fins the number of cones. [3]

(b) Solve the following equation and give your answer correct to $\displaystyle 3$ significant figures: $\displaystyle 5x^2-3x-4=0$ [3]

(c) As observed from the top of a 80 m tall lighthouse, the angles of depression of two ships on the same side of the light house in horizontal line with its base are $\displaystyle 30^{\circ} \text{ and } 40^{\circ}$ Find the distance between the two ships. Give your answer correct to the nearest meter. [4]

Answers:

(a) Sphere: Internal radius $\displaystyle = 6 \text{ cm }$ , External radius $\displaystyle = 8 \text{ cm }$

Cone: Radius $\displaystyle = 2 \text{ cm }$ , Height $\displaystyle = 8 \text{ cm }$

$\displaystyle \frac{4}{3} \times \pi \times (8)^3 - \frac{4}{3} \times \pi \times (6)^3 = n \times \frac{1}{3} \times \pi \times (2)^2 \times 8$

$\displaystyle \Rightarrow n = \frac{4 \times (8^3-6^3)}{2^2 \times 8} = 37$

(b) Given : $\displaystyle 5x^2 - 3x - 4 = 0$

Comparing this expression with $\displaystyle ax^2+bx+c = 0$ , we get $\displaystyle a=5, b=-3 \text{ and } C=-4$

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

$\displaystyle x = \frac{3 \pm \sqrt{(-3)^2 - 4(5)(-4)}}{2(5)}$

$\displaystyle x = \frac{3 \pm \sqrt{9 + 80}}{10}$

$\displaystyle x = \frac{3 \pm \sqrt{89}}{10}$

$\displaystyle x = \frac{3 \pm 9.43}{10}$

$\displaystyle \text{Therefore } x = \frac{3 + 9.43}{10} \text{ or } x = \frac{3 - 9.43}{10}$

$\displaystyle x=1.243,\ -0.643$

$\displaystyle \text{(c) In } \triangle ABC, \tan 50^o = \frac{BC}{80}$

$\displaystyle \Rightarrow BC = 80 \times 1.1918$

$\displaystyle \Rightarrow BC = 95.34 \text{ m }$

$\displaystyle \text{In } \triangle ABD, \tan 60^o = \frac{BD}{80}$

$\displaystyle \Rightarrow BD = 80 \sqrt{3} = 138.56 \text{ m }$

Therefore $\displaystyle CD = BD - BC = 138.56 - 95.34 = 43.2 \text{ m }$

$\displaystyle \\$

Question 8:

(a) A man invests Rs. 9,600 on Rs. 100 shares at Rs. 80. If the company pays him 18% dividend. Find:

(i) The number of shares he buys

(ii) His total dividend.

(iii) His percentage return on the shares. [3]

sm101 (b) In the given figure $\displaystyle \triangle ABC \text{ and } \triangle AMP$ are right angled at $\displaystyle B \text{ and } M$ respectively. Given $\displaystyle AC= 10 \text{ cm}, AP = 15 \text{ cm } \text{ and } PM = 12 \text{ cm} .$

(i) Prove $\displaystyle \triangle ABC \sim \triangle AMP$

(ii) Find $\displaystyle AB \text{ and } BC$ [3]

$\displaystyle \text{(c) If } x = \frac{\sqrt{a+1} + \sqrt{a-1}}{\sqrt{a+1} - \sqrt{a-1}} \text{ using properties of proportion show that } \\ \\ x^2-2ax+1=0 \hspace{9.0cm} [4]$

Answers:

(a) Investment $\displaystyle =\text{ Rs. } 9600$ , Dividend $\displaystyle = 18\%$

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

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

$\displaystyle \text{(i) Number of shares } = \frac{\text{Investment}}{\text{Market Value of the share}} = \frac{9600}{80} = 120$

$\displaystyle \text{(ii) Dividend } = 120 \times 100 \times \frac{18}{100} =\text{ Rs. } 2160$

$\displaystyle \text{(iii) } \% Return = \frac{\text{Dividend earned}}{\text{Investment made}} = \frac{2160}{9600} \times 100 = 22.5\%$

(b) In $\displaystyle \triangle ABC \text{ and } \triangle AMP$

$\displaystyle \angle BAC = \angle PAM$ (common angle)

$\displaystyle \angle ABC =\angle PMA$ (right angles)

Therefore $\displaystyle \triangle ABC \sim \triangle AMP$ (AAA postulate)

$\displaystyle AM = \sqrt{AP^2-PM^2} = \sqrt{15^2-12^2} = \sqrt{121} = 11$

Since $\displaystyle \triangle ABC \sim \triangle AMP$

$\displaystyle \frac{AP}{AM}= \frac{BC}{PM}= \frac{AC}{AP}$

Given $\displaystyle AC=10 \text{ cm}, AP = 15 \text{ cm } \text{ and } PM= 12 \text{ cm }$

$\displaystyle \Rightarrow AB = \frac{10}{15} \times 11 = 7.33 \text{ cm }$

$\displaystyle \Rightarrow BC = \frac{10}{15} \times 12 = 8 \text{ cm }$

$\displaystyle \text{(c) Given } x=\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}}$

Applying componendo and dividendo

$\displaystyle \frac{x+1}{x-1}=\frac{(\sqrt{a+1}+\sqrt{a-1})+(\sqrt{a+1}-\sqrt{a-1})}{(\sqrt{a+1}+\sqrt{a-1})-(\sqrt{a+1}-\sqrt{a-1})}$

Simplify

$\displaystyle \frac{x+1}{x-1}=\frac{\sqrt{a+1}}{\sqrt{a-1}}$

Now square both sides

$\displaystyle \frac{x^2+1+2x}{x^2-2x+1}=\frac{a+1}{a-1}$

Simplifying

$\displaystyle x^2+1 = 2ax$

or $\displaystyle x^2-2ax+1 = 0$

$\displaystyle \\$

Question 9:

(a) The line through $\displaystyle A (-2, 3) \text{ and } B (4, b)$ is perpendicular to the line $\displaystyle 2x - 4y = 5 .$ Find the value of $\displaystyle b$ . [3]

$\displaystyle \text{(b) Prove that } \frac{\tan^2 \theta}{(\sec \theta -1)^2} = \frac{1 + \cos \theta}{1 + \cos \theta} \hspace{7.0cm} [3]$

(c) A car covers a distance of 400 km at a certain speed. Had the speed been 12 km per hours more, the line taken for the journey would have been 1 hour and 40 minutes less. Find the original speed of the care. [4]

Answers:

$\displaystyle \text{(a) Slope of } AB = \frac{b-3}{4-(-2)} = \frac{b-3}{6}$

Given equation is $\displaystyle 2x - 4y = 5$

$\displaystyle \Rightarrow 4y=2x-5$

$\displaystyle \Rightarrow y = \frac{1}{2}x-\frac{5}{4}$

$\displaystyle \Rightarrow \text{Slope} (m_2) = \frac{1}{2}$

Since they are perpendicular, $\displaystyle m_1 . m_2 =-1$

$\displaystyle \Rightarrow \frac{b-3}{6}. \frac{1}{2}=-1$

$\displaystyle \Rightarrow b=-9$

$\displaystyle \text{(b) RHS } = \frac{\tan^2 \theta}{(\sec \theta - 1)^2}$

$\displaystyle = \frac{\sin^2 \theta}{\cos^2 \theta} \times \frac{\cos^2 \theta}{(1-\cos \theta)^2}$

$\displaystyle = \frac{(1-\cos \theta)(1 + \cos \theta)}{(1-\cos \theta)^2}$

$\displaystyle = \frac{1+ \cos \theta}{1- \cos \theta} = \text{LHS. Hence proved.}$

$\displaystyle \text{Therefore } \frac{400}{x} = \frac{400}{x+12} + \frac{100}{60}$

$\displaystyle \frac{2400}{x} = \frac{2400}{x+12} +10$

$\displaystyle 2400x+28800=10x^2+2520x$

$\displaystyle 10x^2-120x-28800=0 \Rightarrow x = -60 \text{ or } 48 \text{(ruled out)}$

Therefore speed of the car is 48 km/hr.

$\displaystyle \\$

Question 10:

(a) Construct a $\displaystyle \triangle ABC$ in which base $\displaystyle BC =6 \text{ cm }, AB=5.5 \text{ cm } \text{ and } \angle ABC=120^{\circ}$

(i) Construct a circle circumscribing the $\displaystyle \triangle ABC$

(ii) Draw a cyclic quadrilateral $\displaystyle ABCD$ so that $\displaystyle D$ is equidistant from $\displaystyle B \text{ and } C .$ [4]

(b) The following distribution represents the height of 160 students of a school

Height	140-145	145-150	150-155	155-160	160-165	165-170	170-175	175-180
Students	12	20	30	38	24	16	12	8

Draw a given for the given distribution taking 2 cm =5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine:

(i) The medium height

(ii) The interquartile range

(iii) The number of students whose height is above 172 cm. [6]

Answers:

(a)

(i) Following Step to Constructions:

First, using a ruler, draw a line segment $\displaystyle BC = 6 \text{ cm }$
Construct $\displaystyle \angle CBE=120^{\circ}$ This you can do it by using a compass. Draw an arc with B as the center. The using the same arc and intersection point ‘x”, draw two arcs one at ‘y’ and then one at ‘z’. Craw a line through B and ‘z’. You will get $\displaystyle 120^{\circ}$ angle.
The cut $\displaystyle BA = 5.5 \text{ cm }$ from $\displaystyle BP .$ This you can do by setting the compass for a width of 5.5 cm.
Then join $\displaystyle A \text{ to } C .$ So you get the $\displaystyle \triangle ABC$ .
Now construct perpendicular bisectors of $\displaystyle AB \text{ and } BC$ intersecting at $\displaystyle O .$ Join $\displaystyle AO .$
Taking O as the center and $\displaystyle OA$ as radius draw a circle, passing through $\displaystyle A, B , \text{ and } C$

(ii)

Extend the right bisector of $\displaystyle BC$ intersecting the circle at $\displaystyle D .$
Join $\displaystyle A \text{ to } D \text{ and } C \text{ to } D$
$\displaystyle ABCD$ is required cyclic quadrilateral.

(b) Following table:

Height

c.f.

140-145

145-150

150-155

155-160

160-165

165-170

170-175

175-180

100

124

140

152

160

(i) Mean $\displaystyle = 157.3$

(ii) Interquartile range $\displaystyle = Q_3 - Q_1 = 163.5 - 152 = 11.5$

(iii) No. of students above $\displaystyle 172 \text{ cm } = 160-144 = 16$

$\displaystyle \\$

Question 11:

(a) In $\displaystyle \triangle PQR, PQ = 24 \text{ cm}, QR = 7 \text{ cm } \text{ and } \angle PQR = 90^{\circ}.$ Find the radius of the inscribed circle. [3]

(b) Find the mode and medium of the following frequency distribution: [3]

X	10	11	12	13	14	15
f	1	4	7	5	9	3

l81 (c) The line through $\displaystyle P(5,3)$ intersects $\displaystyle y-axis$ at $\displaystyle Q .$

(i) Write the slope of the line

(ii) Write the equation of the line

(iii) Find the Co-ordinates of $\displaystyle Q$ [4]

Answers:

(a) Given $\displaystyle \triangle PQR$ is a right triangle. Therefore using Pythagoras theorem:

$\displaystyle PR^2 = PQ^2 + QR^2 = (24)^2 + (7)^2 = 625$

$\displaystyle \Rightarrow PR = \sqrt{625} =25$

Draw $\displaystyle OC \text{ and } OB$ as shown. They will be perpendicular to the sides of the triangle.

$\displaystyle AOBQ$ is a square since the angels are $\displaystyle 90^o \text{ and } QA = QB$ (Since the tangent to a circle from an exterior point are equal in length).

Therefore $\displaystyle QA = QB = x$

$\displaystyle AR = 7-x = RC$ (Since the tangent to a circle from an exterior point are equal in length).

$\displaystyle BP = 12-x = PC$ (Since the tangent to a circle from an exterior point are equal in length).

Since $\displaystyle PC +RC = PR$

$\displaystyle \Rightarrow 7-x+12-x = 25$

$\displaystyle \Rightarrow x = 3 \text{ cm }$

(b) Mode is the value of the highest frequency.

Therefore Mode $\displaystyle = 14$

For Median, first write the data in ascending order as follows:

$\displaystyle 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, \\ 13, 13, 13, 13,13, 14, 14, 14,14, 14, 14, 14, 14, 14,15, 15, 15 .$

Since Median is the middle most value.

$\displaystyle \text{Median } = (\frac{N+1}{2})^{th} \text{ observation } = (\frac{29+1}{2})^{th} \text{ observation } = 15^{th} \text{ observation } = 13$