# Board Paper of Class 10 2018 Maths MeritNation(SET 1) - 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) Find the value of 'x' and 'y' if:

$2\left[\begin{array}{cc}x& 7\\ 9& y-5\end{array}\right]$ + $\left[\begin{array}{cc}6& -7\\ 4& 5\end{array}\right]=\left[\begin{array}{cc}10& 7\\ 22& 15\end{array}\right]$**[3]**(b) Sonia had a recurring deposit account in a bank and deposited ₹600 per month for 2$\frac{1}{2}$ 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:

(i) a prime number.

(ii) a number divisible by 4.

(iii) a number that is a multiple of 6.

(iv) an odd number.**[4]**

- Question 2
(a) The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. Find the

(i) radius of the cylinder

(ii) Volume of the cylinder. (use π =$\frac{22}{7}$)**[3]**(b) If (k –3), (2k + 1) and (4k + 3) are three consecutive terms of an A.P., find the value of *k***[3]**(c) PQRS is a cyclic quadrilateral. Given $\angle $QPS = 73 ^{∘}, $\angle $PQS = ${55}^{0}$ and $\angle $PSR =**82**, calculate:^{∘}

(i) $\angle $QRS

(ii) $\angle $RQS

(iii) $\angle $PRQ**[4]**

- Question 3
(a) if (x+2) and (x+3) are factors of x ^{3 }+ax+b, find the value of 'a' and 'b'.[3] (b) prove that $\sqrt{se{c}^{2}\theta +\mathrm{cos}e{c}^{2}\theta}$ = *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: Runs scored 3000- 4000 4000- 5000 5000- 60006000-70007000-

80008000-

90009000-

1000No. of bats man 4 18 9 6 7 2 4 [4]

- Question 4
(a) Solve the following inequation, write down the solution set and represent it on the real number line:

$-2+10x\le 13x+10<24+10x,x\in \mathrm{Z}$**[3]**(b) If the straight lines 3 *x*− 5*y =*7*and 4x+ay+9 =*0 are perpendicular to one another, find the value of a.**[3]**(c) Solve ${x}^{2}$ + $7x$ = 7 and give your answer correct to two decimal places. **[4]**

- Question 5
(a) The 4 ^{th }term of a G.P. is 16 and the 7^{th}term is 128. Find the first term and common ratio of the series.**[3]**(b) A man invests ₹22,500 in ₹50 shares available at 10% discount. If the dividend paid by the company is 12%, calculate: (i) The number of share purchase (ii) The annual divided received (iii) The rate of return he gets on his investment. Give your answer correct to the nearest whole number.

**[3]**(c) Use graph paper for this question (Take 2cm = 1unit along both x and y axis).

ABCD is a quadrilateral whose vertices are A(2,2), B(2,–2), C(0,–1) and D(0,1).

(i) Reflect quadrilateral ABCD on the y-axis and name it as A'B'CD.

(ii) Write down the coordinates of A' and B'.

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

(iv) Name the polygon A'B'CD.**[4]**

- Question 6
(a) Using properties of proportion, solve for x. Given that x is positive:

$\frac{2x+\sqrt{4{x}^{2}-1}}{2x-\sqrt{4{x}^{2}-1}}=4$[3] (b) if $A=\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right],B=\left[\begin{array}{cc}0& 4\\ -1& 7\end{array}\right]andC=\left[\begin{array}{cc}1& 0\\ -1& 4\end{array}\right],$ find AC +B ^{2 }- 10C .[3] (c) Prove that (1+ *cot*$\theta $ -*cosec$\theta $*)(1+*tan*$\theta $+*sec*$\theta $) = 2[4]

- Question 7
(a) Find the value of *k*for which the following equation has equal roots.

${x}^{2}+4kx+({k}^{2}-k+2)=0$**[3]**(b) on a map drawn to a scale of 1:50,000, a rectangale plot of land ABCD has the following dimensions. AB= 6cm; BC=8cm and all angles are right angles. Find:

(i) tha actual length of the diagonal distance AC of the plot in km.

(ii) the actual area of the plot in sq km.**[3]**(c) A(2,5), B(-1,2) and C(5,8) are the vertices of a triangle ABC , 'M' is a point on AB such that AM : MB =1:2 Find the co-ordinates of 'M' . Hence find the equation of the line passing through the points C and M. **[4]**

- Question 8
(a) ₹7500 were divided equally among a certain number of children. Had there been 20 less children , each would have received ₹ 100 more. Find the original number of children.

**[3]**(b) if the mean of the following distribution is 24, find the value of 'a' Marks 0-10 10-20 20-30 30-40 40-50 Number of Student 7 a 8 10 5

**[3]**(c) Using ruler and compass only, constructor a $\u2206ABC$ such that BC = 5 cm and AB = 6.5 cm and $\angle $ABC = 120 ^{∘ }(i) construct a circum-circle of $\u2206$ABC (ii) Construct a cycle quadrilateral ABCD, such that D is equidistant from AB and BC. **[4]**

- Question 9
(a) Priyanka has a recuring deposit account of ₹1000 per month at 10% per annum. if she gets ₹5550 as interest at the time of maturity. find the total time for which the account was held. **[3]**(b) In $\u2206$PQR, *MN*is parallel to QR and

$\frac{PM}{MQ}=\frac{2}{3}$

(i) Find

$\frac{MN}{QR}$(ii) Prove that $\u2206OMN$ and $\u2206ORQ$ are similar (iii) Find, Area of $\u2206OMN$ : Area of $\u2206ORQ$

**[4]**(c) The following figure represent a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at other. Their common radius is 7 cm. The height of the cylinder and cone are each of 4 cm. Find the volume of the solid.

**[3]**

- Question 10
(a) Use Reminder theroem to factorize the following polynomial

$2{x}^{3}+3{x}^{2}-9x-10$**[3]**(b) In the figure given below 'O' is the centre of the circle. If QR = OP and $\angle ORP=20$ ^{∘}. Find the value of 'x' giving reasons.

**[3]**(c) The angle of elevation from a point p of the top of a tower QR, 5m high is 60 ^{∘}and that of the tower PT from a point Q is 30^{∘ }. Find the height of the tower PT, correct to the nearest metre.

**[4]**

- Question 11
(a) The 4th term of an A.P. is 22 and 15th term is 66. Find the first term and the common different. Hence find the sum of the series to 8 terms.

[4](b) Use Graph paper for this question.

A survey regarding height (in cm) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded:Height in cm 135-140 140-145 145-150 150-155 155-160 160-165 165-170 No. of boys 4 8 20 14 7 6 1

Taking 2cm = height of 10 cm along one axis and 2 cm = 10 boys along the other axis draw an ogive of the above distribution. Use the graph to estimate the

following:(i) the median (ii) lower Quartile (iii) if above 158 cm is considered as the tall boys of the class. Find the number of boys in the class who are tall [6]