# Board Paper of Class 10 2020 Math - Solutions

Board Paper of Class 10 2020 Math MeritNation - Solutions

- Question 1
**(a) **Solve the following Quadratic Equation:

${x}^{2}-7x+3=0$
Give your answer correct to two decimal places.

**(b) **Given

$\mathrm{A}=\left[\begin{array}{cc}x& 3\\ y& 3\end{array}\right]$
If

${A}^{2}=3I$, where I is the identity matrix of order 2, find

*x* and

*y*.

**(c) **Using ruler and compass construct a triangle ABC where AB = 3 cm, BC = 4 cm and

$\angle \mathrm{ABC}=90\xb0$. Hence construct a circle circumscribing the triangle ABC. Measure and write down the radius of the circle.

VIEW SOLUTION

- Question 2
**(a) **Use factor theorem to factories

$6{x}^{3}+17{x}^{2}+4x-12$ completely.

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

$\frac{3x}{5}+2<x+4\le \frac{x}{2}+5,x\in \mathbf{R}$**(c) **Draw a Histogram for the give data, using a graph paper:

Weekly Wages(in Rs) |
Number of People |

3000-4000 |
4 |

4000-5000 |
9 |

5000-6000 |
18 |

6000-7000 |
6 |

7000-8000 |
7 |

8000-9000 |
2 |

9000-10000 |
4 |

VIEW SOLUTION

- Question 3
**(a) **In what ratio is the line joining P(5, 3) and Q(-5, 3) divided by the y-axis? Also find coordinates of the point of intersection.

**(b) **Prove that:

$\frac{\mathrm{sin}A}{1+\mathrm{cot}A}-\frac{\mathrm{cos}A}{1+\mathrm{tan}A}=\mathrm{sin}A-\mathrm{cos}A$**(c) **In the figure given bellow,

$\mathrm{O}$ is the center of the circle and

$\mathrm{AB}$ is a diameter.

If

$\mathrm{AC}=\mathrm{BD}$ and

$\angle \mathrm{AOC}=72\xb0$ Find:

(i) ∠ABC (ii) ∠BAD (iii) ∠ABD

VIEW SOLUTION

- Question 4
**(a) **A solid spherical ball of radius 6 cm is melted and recast into 64 identical spherical marbles. Find the radius of each marble,

**(b) **Each of the letters of the word `AUTHORIZES’ is written on identical circular discs and put in a bag. They are well shuffled. If a disc is drawn at random from the bag, what is the probability that the letter is:

(i) a vowel (ii) one of the first 9 letters of the English alphabet which appears in the given word (iii) one of the last 9 letters of the English alphabet which appears in the in word?

**(c) **Mr. Bedi visits the market and buys the following articles:

Medicines costing ₹ 950, GST @ 5% A pair of shoes costing ₹ 3000, GST @ 18% A Laptop bag costing ₹ 1000 with a discount of 30%, GST @, 18%, (i) Calculate the total amount of GST paid. (ii) The total bill amount including GST paid by Mr. Bedi,

VIEW SOLUTION

- Question 5
**(a) **A company with 500 shares of nominal value 120 declares an annual dividend ut 131 15%. Calculate:

(i) the total amount of dividend paid by the company.

(ii) annual income of Mr. Sharma who holds 80 shares of the company If the return percent of Mr. Sharma from his shares is 10%, find the Market value of each share.

**(b) **The mean of the following data is 16 Calculate the value of f

Marks |
5 |
10 |
15 |
20 |
25 |

Number of Students |
3 |
7 |
*f* |
9 |
6 |

**(c) **The 4th, 6th and the last term of a geometric progression are 10, 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio number of terms of the series.

VIEW SOLUTION

- Question 6
**(a) **From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be

$45\xb0$and

$60\xb0$ respectively. If the height of the tower is 20 m. Find: (i) the height of the cliff (ii) the distance between the cliff and the tower.

**(b) **In the given figure AB = 9cm, PA = 7.5 and PC = 5Cm. Chords AD and BC intersect at P.

(i) Prove that ∆ PAB ~ ∆PCD

(ii) Find the length of CD

(iii) Find area of ∆PAB: area of ∆PCD

**(c) **If

$A=\left[\begin{array}{cc}3& 0\\ 5& 1\end{array}\right],B=\left[\begin{array}{cc}-4& 2\\ 1& 0\end{array}\right]$
Find

${A}^{2}-2AB+{B}^{2}$ VIEW SOLUTION

- Question 7
**(a) **
In the given figure TP and TQ are two tangents to the circle with center O, touching at A and C respectively, If

$\angle \mathrm{BCQ}=55\xb0$ and

$\angle \mathrm{BAP}=60\xb0$, find: (i)

$\angle \mathrm{OBA}$ and

$\angle \mathrm{OBC}$
(ii)

$\angle \mathrm{AOC}$
(iii)

$\angle \mathrm{ATC}$
**(b) **Using properties of find

*x* :

*y*, given:

$\frac{{x}^{2}+2x}{2x+4}=\frac{{y}^{2}+3y}{3y+9}$**(c) ** Find the value of '

*p*' if the lines,

$5x-3y+2=0$ and

$6x-py+7=0$ are perpendicular to each other. Hence find the equation of a line passing through

$\left(-2,-1\right)$ and parallel to

$6x-py+7=0$.

VIEW SOLUTION

- Question 8
**(a) **What must be added to the polynomial

$2{x}^{3}-3{x}^{2}-8x$, so that it leaves a remainder 10 when divided by 2

*x* + 1?

**(b) **Mr. Sonu has a recurring deposit account and deposits ₹ 750 per month for 2 years If he gets ₹ 19125 at the time of maturity, find the rate of interest.

**(c) **Use graph paper for this question. Take 1 cm = 1 unit on both

* x* and

*y* axes

(i) Plot the following points on your graph sheets: A(-4, 0), B(-3, 2), C(0, 4) D(4, 1) and E(7, 3)

(ii) Reflected the point B,C,D and E on the x-axis and name then as B,C,D and E respectively.

(iii) Join the points A,B,C,D,E,E,D,C,B and A in order.

(iv) Name the close figure farmed

VIEW SOLUTION

- Question 9
**(a) ** If

$x=\frac{\sqrt{2a+1}+\sqrt{2a-1}}{\sqrt{2a+1}-\sqrt{2a-1}}$, prove that

${x}^{2}-4ax+1=0$**(b) **40 student enter for a game of shot-put competition. The distance thrown (in meters) is recorded below:

Distance (In m) |
12-13 |
13-14 |
14-15 |
15-16 |
16-17 |
17-18 |
18-19 |

Number Of Students |
3 |
9 |
12 |
9 |
4 |
2 |
1 |

Use a graph paper to draw an ogive for the above distribution. Use a scale of 2 cm = 1 m on one axis and 2 cm = 5 students on the other axis. Hence using your graph find: (i) the median (ii) Upper Quartile (iii) number of student who cover a distance which is above

$16\frac{1}{2}\mathrm{m}$.

VIEW SOLUTION

- Question 10
**(a) **If the 6th term of an A.P. is equal to four times its first term and the sum of first six terms is 75, find the first term and the common difference.

**(b) ** The difference of two natural numbers is 7 and their product is 450. Find the numbers.

**(c) **Use ruler and compass for this question. Construct a circle of radius 4.5 cm. Draw a chord. AB = 6cm.

(i) Find the locus of points equidistant from A and B. Mark the point where it meets the circle as D.

(ii) Join AD and find the locus of points which are equidistant from AD and AB. Murk the point where it meets the circle as C.

(iii) Join and CD. Measure and write down the length of side CD of the quadrilateral ABCD. VIEW SOLUTION

- Question 11
**(a) ** A model of a high rise building is made to a scale of 1: 50. (i) If the height of the model is 0.8 m, find the height of the actual building. (ii) If the floor area of a flat in the building is 20 m2, find the floor area of that in the model.

**(b) **From a solid wooden cylinder of height 28 cm and diameter 6 cm, two conical cavities are hollowed out. The diameters of the cones are also of 6 cm and height 10.5 cm. Taking 22 7 π= find the volume of the remaining solid.

**(c) **Prove the identity

${\left(\frac{1-\mathrm{tan}\theta}{1-\mathrm{cot}\theta}\right)}^{2}={\mathrm{tan}}^{2}\theta $ VIEW SOLUTION