# Board Paper of Class 10 2016 Maths - 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) Using remainder theorem, find the value of *k*if on dividing 2*x*^{3}+ 3*x*^{2}–*kx*+ 5 by*x*– 2, leaves a remainder 7.**[3]**(b) Given *A*= $\left[\begin{array}{cc}2& 0\\ -1& 7\end{array}\right]\mathrm{and}\mathrm{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and*A*^{2}= 9*A*+*mI*. Find*m*.**[4]**(c) The mean of following number is 68. Find the value of ‘ *x*’

45, 52, 60,*x*, 69, 70, 26, 81 and 94

Hence estimate the median.**[3]**

- Question 2
(a) The slope of a line joining P (6, *k*) and Q (1 – 3*k*, 3) is $\frac{1}{2}$. Find

(i)*k*

(ii) Midpoint of PQ, using the value of ‘*k*’ found in (i).

**[3]**(b) Without using trigonometrical tables, evaluate:

${\mathrm{cosec}}^{2}57\xb0-{\mathrm{tan}}^{2}33\xb0+\mathrm{cos}44\xb0\mathrm{cosec}46\xb0-\sqrt{2}\mathrm{cos}45\xb0-{\mathrm{tan}}^{2}60\xb0$**[4]**(c) A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number

of cones.**[3]**

- Question 3
(a) Solve the following inequation, write the solution set and represent it on

the number line.

$-3\left(x-7\right)\ge 15-7x>\frac{x+1}{3},x\in \mathrm{R}$**[3]**(b) In the figure given below, AD is a diameter. O is the centre of the circle.

AD is parallel to BC and ∠CBD = 32°. Find:

(i) ∠OBD

(ii) ∠AOB

(iii) ∠BED

**[4]**(c) If (3 *a*+ 2*b*): (5*a*+ 3*b*) = 18 : 29. Find*a*:*b*.**[3]**

- Question 4
(a) A game of numbers has cards marked with 11, 12, 13,..., 40. A card is drawn

at random. Find the Probability that the number on the card drawn is:

(i) A perfect square

(ii) Divisible by 7**[3]**(b) Use graph paper for this question. (Take 2 cm = 1 unit along both *x*and*y*

axis.) Plot the points O (0, 0), A (–4, 4), B (–3, 0) and C (0, –3)

(i) Reflect points A and B on the*y*-axis and name them A' and B' respectively.Write down their coordinates.(ii) Name the figure OABCB'A'.

(iii) State the line of symmetry of this figure**[4]**(c) Mr. Lalit invested Rs. 5000 at a certain rate of interest, compounded annually

for two years. At the end of first year it amounts to Rs. 5325. Calculate

(i) The rate of interest

(ii) The amount at the end of second year, to the nearest rupee.**[3]**

- Question 5
(a) Solve the quadratic equation *x*^{2}– 3(*x*+ 3) = 0; Give your answer correct two significant figures.**[3]**(b) A page from the savings bank account of Mrs. Ravi is given below. **Date****Particulars****Withdrawal**

(Rs.)**Deposit**

(Rs.)**Balance**

(Rs.)April 3rd 2006 B/F 6000 April 7th By cash 2300 8300 April 15th By cheque 3500 11800 May 20th To self 4200 7600 June 10th By cash 5800 13400 June 15th To self 3100 10300 August 13th By cheque 1000 11300 August 25th To self 7400 3900 September 6th 2006 By cash 2000 5900 ^{th}September, 2006. Calculate the interest Mrs.

Ravi earned at the end of 30^{th }September, 2006 at 4.5% per annum interest.

Hence, find the amount she receives on closing the account.

**[4]**(c) In what time will Rs.1500 yield Rs.1996.50 as compound interest at 10% per

annum compounded annually?**[3]**

- Question 6
(a) Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them. **[3]**(b) In the given figure PQRS is a cyclic quadrilateral PQ and SR produced meet at T.

(i) Prove ΔTPS ~ ΔTRQ.

(ii) Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm.

(iii) Find area of quadrilateral PQRS if area of ΔPTS = 27 cm^{2}.

**[4]**(c) Given matrix $\mathrm{A}=\left[\begin{array}{cc}4\mathrm{sin}30\xb0& \mathrm{cos}0\xb0\\ \mathrm{cos}0\xb0& 4\mathrm{sin}30\xb0\end{array}\right]\mathrm{and}\mathrm{B}=\left[\begin{array}{c}4\\ 5\end{array}\right]$

If AX = B

(i) Write the order of matrix X.

(ii) Find the matrix ‘X’.**[3]**

- Question 7
(a) An aeroplane at an altitude of 1500 metres finds that two ships are sailing towards it in the same direction. The angles of depression as observed from the aeroplane are 45° and 30° respectively. Find the distance between the two ships. **[4]**(b) The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution. (Take 2 cm = 10 scores on the X-axis and 2 cm = 20 shooters on the Y-axis). **Scores**0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100 **No.of**

shooters9 13 20 26 30 22 15 10 8 7

(i) The median.

(ii) The interquartile range.

(iii) The number of shooters who obtained a score of more than 85%.**[6]**

- Question 8
(a) If $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$ show that $\frac{{x}^{3}}{{a}^{3}}+\frac{{y}^{3}}{{b}^{3}}+\frac{{z}^{3}}{{c}^{3}}=\frac{3xyz}{abc}$ **[3]**(b) Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using

a ruler and a compass only, construct :

(i) A circle of radius 2.5 cm, passing through A and C.

(ii) Construct two tangents to the circle from the external point B.Measure and record the length of the tangents.**[4]**(c) A line AB meets X-axis at A and Y-axis at B. P (4, –1) divides AB in the ratio 1:2.

(i) Find the coordinates of A and B.

(ii) Find the equation of the line through P and perpendicular to AB.

**[3]**

- Question 9
(a) A dealer buys an article at a discount of 30% from the wholesaler, the marked price being Rs. 6000. The dealer sells it to a shopkeeper at a discount of 10% on the marked price. If the rate of VAT is 6%, find

(i) The price paid by the shopkeeper including the tax.

(ii) The VAT paid by the dealer.**[3]**(b) The given figure represents a kite with a circular and a semicircular motifs stuck on it. The radius of a circle is 2.5 cm and the semicircle is 2 cm. If diagonals AC and BD are of lengths 12 cm and 8 cm respectively, find the area of the:

(i) Shaded part. Give your answer correct to the nearest whole number.

(ii) Unshaded part.

**[4]**(c) A model of a ship is made to a scale 1 : 300

(i) The length of the model of ship is 2 m. Calculate the lengths of the ship.

(ii) The area of the deck ship is 180,000 m^{2}. Calculate the area of the deck of the model.

(iii) The volume of the model in 6.5 m^{3}. Calculate the volume of the ship.**[3]**

- Question 10
(a) Mohan has a recurring deposit account in a bank for 2 years at 6% p.a. simple interest. If he gets Rs. 1200 as interest at the time of maturity, find:

(i) the monthly installment

(ii) the amount of maturity**[3]**(b) (b) The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to :

(i) Frame a frequency distribution table

(ii) To calculate mean

(iii) To determine the Modal class

**[4]**(c) A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be ‘ *x*’ km/h, form an equation and solve it to evaluate ‘*x*’.**[3]**

- Question 11
(a) Prove that $\frac{\mathrm{cos}\mathrm{A}}{1+\mathrm{sin}\mathrm{A}}+\mathrm{tan}\mathrm{A}=\mathrm{sec}\mathrm{A}$. **[3]**(b) Use ruler and compasses only for the following questions.

All constructions lines and arcs must be clearly shown.

(i) Construct a ΔABC in which BC = 6.5 cm, ∠ABC = 60°, AB = 5 cm.

(ii) Construct the locus of points at a distance of 3.5 cm from A.

(iii) Construct the locus of points equidistant from AC and BC.(iv) Mark 2 points X and Y which are a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY.**[4]**(c) Ashok invested Rs. 26,400 on 12%, Rs. 25 shares of a company.

If he receives a dividend of Rs. 2,475. Find the :

(i) number of shares he bought

(ii) Market value of each share**[3]**