Board Paper of Class 10 2020 Math - Solutions

(a) i. Out of the following which is the Pythagorean triplet?
(A) (1, 5, 10) (B) (3, 4, 5) (C) (2, 2, 2) (D) (5, 5, 2)
ii. Two circles of radii 5.5 cm and 3.3 cm respectively touch each other externally. What is the distance between their centres?
(A) 4.4 cm (B) 2.2 cm (C) 8.8 cm (D) 8.9 cm
iii. Distance of point (–3, 4) from the origin is _______. (A) 7 (B) 1 (C) –5 (D) 5
iv. Find the volume of a cube of side 3 cm:
(A) 27 cm³ (B) 9 cm³ (C) 81 cm³ (D) 3 cm³

(b) i. The ratio of corresponding sides of similar triangles is 3 : 5, then find the ratio of their areas.
ii. Find the diagonal of a square whose side is 10 cm.
iii. ABCD is cyclic. If $\angle \mathrm{B}=110°$, then find measure of $\angle \mathrm{D}$.
iv. Find the slope of the line passing through the points A(2, 3) and B(4, 7) VIEW SOLUTION

(a) In the figure given above, ‘O’ is the centre of the circle, seg PS is a tangent segment and S is the point of contact. Line PR is a secant. If PQ = 3.6, QR = 6.4, find PS

ii. If $\sec \theta =\frac{25}{7}$ , find the value of $\tan \theta$. 
iii.
 
In the figure given above, O is the centre of the circle. Using given information complete the following table: 

Type of arc	Name of the arc	Measure of the arc
Minor arc
Major arc

(b) i. 

In $△\mathrm{PQR}$, $\mathrm{NM}\parallel \mathrm{RQ}$. If PM = 15, MQ = 10, NR = 8, then find PN. 
ii. 

In $△\mathrm{MNP}$, $\angle \mathrm{MNP}=90°$, $\mathrm{Seg} \mathrm{NQ}\perp \mathrm{seg} \mathrm{MP}$. If MQ = 9, QP = 4, then find NQ. 
iii. 

In the figure given above, M is the centre of the circle and seg KL is a tangent segment. L is a point of contact. If $\mathrm{MK}=12, \mathrm{KL}=6\sqrt{3}$, then find the radius of the circle. 
iv. Find the co-ordinates of midpoint of the segment joining the points (22, 20) and (0, 16). 
v. A person is standing at a distance of 80 metres from a Church and looking at its top. The angle of elevation is of $45°$. Find the height of the Church. 

  VIEW SOLUTION

(a) i. 

In the given figure, X is any point in the interior of the triangle. Point X is joined to the vertices of triangle. $\mathrm{seg} \mathrm{PQ}\parallel \mathrm{seg} \mathrm{DE}$, $\mathrm{seg} \mathrm{QR}\parallel \mathrm{seg} \mathrm{EF}$Complete the activity and prove that $\mathrm{seg} \mathrm{PR}\parallel \mathrm{seg} \mathrm{DF}$. 
ii. If A(6, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of $\square \mathrm{ABCD}$, show that $\square \mathrm{ABCD}$ is a parallelogram.

(b) i. If $△\mathrm{PQR}$, point S is the mid-point of side QR. If PQ = 11, PR = 17, PS = 13, find QR.   
ii. Prove that, tangent segments drawn from an external point to the circle are congruent. 
iii. Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.   
iv. A metal cuboid of measures $16 \mathrm{cm}\times 11 \mathrm{cm}\times 10 \mathrm{cm}$ was melted to make coins. How many coins were made, if the thickness and diameter of each coin was 2 mm and 2 cm respectively? $\left(\pi =3.14\right)$ VIEW SOLUTION

(a) In $△\mathrm{ABC}$, PQ is a line segment intersecting AB at P and AC at Q such that $\mathrm{seg} \mathrm{PQ}\parallel \mathrm{seg} \mathrm{BC}$. If PQ divides $△\mathrm{ABC}$ into two equal parts having equal areas, find $\frac{\mathrm{BP}}{\mathrm{AB}}$ . 
  

(b) Draw a circle of radius 2.7 cm and draw a chord PQ of length 4.5 cm. Draw tangents at points P and Q without using centre. 

(c)
In the figure given above $\square \mathrm{ABCD}$ is a square of side 50 m. Points P, Q, R, S are midpoints of side AB, side BC, side CD, side AD respectively. Find area of shaded region.  VIEW SOLUTION

(a) Circles with centres A, B and C touch each other externally. If AB = 3 cm, BC = 3 cm,  CA = 4 cm, then find the radii of each circle. 

(b) If  $\sin \theta +{\sin }^2\theta =1$
Show that: ${\cos }^2\theta +{\cos }^4\theta =1$ VIEW SOLUTION