Speedlabs > ICSE Class 10 Maths Previous Year Question Paper 2018

ICSE – Class 10th

Previous Year – 2018

Mathematics

Time: 2 ½ Hrs Marks:80

General Instructions:-

- 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.

SECTION A (40 Marks) Attempt all questions from this Section. | ||||||||||||||||||

1. | ||||||||||||||||||

a) | Find the value of and ‘y’ if: $2\left[\begin{array}{cc}x & 7 \\ 9 & y-5\end{array}\right]+\left[\begin{array}{rr}6 & -7 \\ 4 & 5\end{array}\right]=\left[\begin{array}{cc}10 & 7 \\ 22 & 15\end{array}\right]$ | 3 | ||||||||||||||||

b) | Sonia had a recurring deposit account in a bank and deposited Rs. 600 per month for $2^{1} / 2$ years. If the rate of interest was $10 \%$ p.a., find the maturity value of this account. | 3 | ||||||||||||||||

c) | Cards bearing numbers $2,4,6,8,10,12,14,16,18$ and 20 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card which is: | 4 | ||||||||||||||||

i. | a prime number. | |||||||||||||||||

ii. | a number divisible by 4. | |||||||||||||||||

iii. | a number that is a multiple of 6. | |||||||||||||||||

iv. | an odd number. | |||||||||||||||||

2. | ||||||||||||||||||

a) | The circumference of the base of a cylindrical vessel is $132 \mathrm{~cm}$ and its height is 25 $\mathrm{cm}$. Find the | 3 | ||||||||||||||||

i. | radius of the cylinder | |||||||||||||||||

ii. | volume of cylinder (use $\pi=\frac{22}{7}$ ) | |||||||||||||||||

b) | If $(\mathrm{k}-3),(2 \mathrm{k}+1)$ and $(4 \mathrm{k}+3)$ are three consecutive terms of an A.P., find the value of $\mathrm{k}$. | 3 | ||||||||||||||||

c) | PQRS is a cyclic quadrilateral. Given $\angle \mathrm{QPS}=73^{\circ}, \angle \mathrm{PQS}=55^{\circ}$ and $\angle \mathrm{PSR}=82^{\circ}$, calculate: | 4 | ||||||||||||||||

i. | $\angle QRS$ | |||||||||||||||||

ii. | $\angle RQS$ | |||||||||||||||||

iii. | $\angle PRQ$ | |||||||||||||||||

3. | ||||||||||||||||||

a) | If $(x+2)$ and $(x+3)$ are factors of $x^{3}+a x+b$, find the values of ‘a’ and ‘b’. | 3 | ||||||||||||||||

b) | Prove that $\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}=\tan \theta+\cot \theta$ | 3 | ||||||||||||||||

c) | Using a graph paper draw a histogram for the given distribution showing the number of runs scored by 50 batsmen. Estimate the mode of the data:
| 4 | ||||||||||||||||

4. | ||||||||||||||||||

a) | Solve the following inequation, write down the solution set and represent it on the real number line: $-2+10 x \leq 13 x+10<24+10 x, x \in Z$ | 3 | ||||||||||||||||

b) | If the straight lines $3 x-5 y=7$ and $4 x+a y+9=0$ are perpendicular to one another, find the value of a. | 3 | ||||||||||||||||

c) | Solve $x^{2}+7 x=7$ and give your answer correct to two decimal places. | 4 | ||||||||||||||||

SECTION B (40 Marks) Attempt any four questions from this Section | ||||||||||||||||||

5. | ||||||||||||||||||

a) | The $4^{\text {th }}$ term of a G.P. is 16 and the $7^{\text {th }}$ term is $128 .$ Find the first term and common ratio of the series. | 3 | ||||||||||||||||

b) | A man invests Rs. 22,500 in Rs. 50 shares available at $10 \%$ discount. If the dividend paid by the company is $12 \%$, calculate: | 3 | ||||||||||||||||

i. | The number of shares purchased | |||||||||||||||||

ii. | The annual dividend received. | |||||||||||||||||

iii. | The rate of return he gets on his investment. Give your answer correct to the nearest whole number. | |||||||||||||||||

c) | Use graph paper for this question (Take $2 \mathrm{~cm}=1$ unit along both $x$ and y axis). $A B C D$ is a quadrilateral whose vertices are $\mathrm{A}(2,2), \mathrm{B}(2,-2), \mathrm{C}(0,-1)$ and $\mathrm{D}(0,1)$. | 4 | ||||||||||||||||

i. | Reflect quadrilateral $\mathrm{ABCD}$ on the $\mathrm{y}$ -axis and name it as A’B’CD. | |||||||||||||||||

ii. | Write down the coordinates of $\mathrm{A}^{\prime}$ and $\mathrm{B}^{\prime}$. | |||||||||||||||||

iii. | Name two points which are invariant under the above reflection. | |||||||||||||||||

iv. | Name the polygon $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{CD}$. | |||||||||||||||||

6. | ||||||||||||||||||

a) | Using properties of proportion, solve for $x$. Given that $x$ is positive: $\frac{2 x+\sqrt{4 x^{2}-1}}{2 x-\sqrt{4 x^{2}-1}}=4$ | 3 | ||||||||||||||||

b) | if $A=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right], B=\left[\begin{array}{rr}0 & 4 \\ -1 & 7\end{array}\right]$ and $c=\left[\begin{array}{rr}1 & 0 \\ -1 & 4\end{array}\right]$, find $A C+B^{2}-10 C .$ | 3 | ||||||||||||||||

c) | Prove that $(1+\cot \theta-\operatorname{cosec} \theta)(1+\tan \theta+\sec \theta)=2$ | 4 | ||||||||||||||||

7. | ||||||||||||||||||

a) | Find the value of $\mathrm{k}$ for which the following equation has equal roots. $x^{2}+4 k x+\left(k^{2}-k+2\right)=0$ | 3 | ||||||||||||||||

b) | On a map drawn to a scale of $1: 50,000$, a rectangular plot of land ABCD has the following dimensions. $\mathrm{AB}=6 \mathrm{~cm} ; \mathrm{BC}=8 \mathrm{~cm}$ and all angles are right angles. Find: $[3]$ | 3 | ||||||||||||||||

i. | the actual length of the diagonal distance AC of the plot in $\mathrm{km}$. | |||||||||||||||||

ii. | the actual area of the plot in sq. $\mathrm{km}$. | |||||||||||||||||

c) | $\mathrm{A}(2,5), \mathrm{B}(-1,2)$ and $\mathrm{C}(5,8)$ are the vertices of a triangle $\mathrm{ABC},{ }^{\prime} \mathrm{M}^{\prime}$ is a point on $\mathrm{AB}$ such that $\mathrm{AM}: \mathrm{MB}=1: 2$. Find the co-ordinates of ‘ $\mathrm{M}$ ‘. Hence find the equation of the line passing through the points $\mathrm{C}$ and $\mathrm{M}$. | 4 | ||||||||||||||||

8. | ||||||||||||||||||

a) | Rs. 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received Rs. 100 more. Find the original number of children. | 3 | ||||||||||||||||

b) | If the mean of the following distribution is 24 , find the value of ‘ $a$ ‘
| 3 | ||||||||||||||||

c) | Using ruler and compass only, construct a $\triangle \mathrm{ABC}$ such that $\mathrm{BC}=5 \mathrm{~cm}$ and $\mathrm{AB}=6.5$ $\mathrm{cm}$ and $\angle \mathrm{ABC}=120^{\circ}$ | 4 | ||||||||||||||||

i. | Construct a circum-circle of $\Delta \mathrm{ABC}$ | |||||||||||||||||

ii. | Construct a cyclic quadrilateral $A B C D$, such that $D$ is equidistant from $A B$ and $\mathrm{BC}$. | |||||||||||||||||

9. | ||||||||||||||||||

a) | Priyanka has a recurring deposit account of Rs. 1000 per month at $10 \%$ per annum. If she gets Rs. 5550 as interest at the time of maturity, find the total time for which the account was held. | 3 | ||||||||||||||||

b) | In $\Delta \mathrm{PQR}, \mathrm{MN}$ is parrallel to $\mathrm{QR}$ and $\frac{\mathrm{PM}}{\mathrm{MQ}}=\frac{2}{3}$ | 3 | ||||||||||||||||

i. | Find $\frac{\mathrm{MN}}{\mathrm{QR}}$ | |||||||||||||||||

ii. | Prove that $\Delta 0 M N$ and $\Delta O R Q$ are similar. | |||||||||||||||||

iii. | Find, Area of $\Delta 0 \mathrm{MN}:$ Area of $\Delta \mathrm{ORQ}$ | |||||||||||||||||

c) | The following figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. Their common radius is $7 \mathrm{~cm}$. The height of the cylinder and cone are each of $4 \mathrm{~cm}$. Find the volume of the solid. | 4 | ||||||||||||||||

10. | ||||||||||||||||||

a) | Use Remainder theorem to factorize the following polynomial: $2 x^{3}+3 x^{2}-9 x-10$ | 3 | ||||||||||||||||

b) | In the figure given below ‘ 0 ‘ is the centre of the circle. If $Q R=0 P$ and $\angle O R P=20^{\circ}$. Find the value of ‘x’ giving reasons. | 3 | ||||||||||||||||

c) | The angle of elevation from a point $\mathrm{P}$ of the top of a tower $\mathrm{QR}, 50 \mathrm{~m}$ high is $60^{\circ}$ and that of the tower PT from a point $Q$ is $30^{\circ} .$ Find the height of the tower $P T$, correct to the nearest metre. | 4 | ||||||||||||||||

11. | ||||||||||||||||||

a) | The $4^{\text {th }}$ term of an A.P. is 22 and $15^{\text {th }}$ term is 66 . Find the first terns and the common difference. Hence find the sum of the series to 8 terms. | 4 | ||||||||||||||||

b) | Use Graph paper for this question. A survey regarding height (in $\mathrm{cm}$ ) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded:
Taking $2 \mathrm{~cm}=$ height of $10 \mathrm{~cm}$ along one axis and $2 \mathrm{~cm}=10$ boys along the other axis draw an ogive of the above distribution. Use the graph to estimate the following: | 6 | ||||||||||||||||

i. | the median | |||||||||||||||||

ii. | lower Quartile | |||||||||||||||||

iii. | if above $158 \mathrm{~cm}$ is considered as the tall boys of the class. Find the number of boys in the class who are tall. |

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