# Board Paper of Class 10 2018 Maths - Solutions

(i) Solve all questions. Draw diagrams wherever necessary

(ii) Use of calculator is not allowed

(iii) Diagram is essential for writing the proof of the theorem.

(iv) Marks of constructions should be distinct. They should not be rubbed off.

(ii) Use of calculator is not allowed

(iii) Diagram is essential for writing the proof of the theorem.

(iv) Marks of constructions should be distinct. They should not be rubbed off.

- Question 1
1. **Attempt any five sub-questions from the following.**[5] (i) $\u2206\mathrm{DEF}~\u2206\mathrm{MNK}$. If DE = 5 and MN = 6, then find the value of $\frac{\mathrm{A}\left(\u2206\mathrm{DEF}\right)}{\mathrm{A}\left(\u2206\mathrm{MNK}\right)}$. (ii) If two circles with radii 8 cm and 3 cm respectively touch externally, then find the distance between their centres. (iii) Find the length of the altitude of an equilateral triangle with side 6 cm. (iv) If θ = 45 ^{o}, then find tan θ.(v) Slope of a line is 3 and *y*intercept is –4. Write the equation of a line.(vi) Using Euler’s formula, find V, if E = 30, F = 12.

- Question 2
2. **Attempt any four sub-questions from the following:**[8] (i) The ratio of the areas of two triangles with the common base is

10 : 7. Height of the larger triangle is 15 cm, then find the corresponding height of the smaller triangle.(ii) In the following figure, point ‘A’ is the centre of the circle. Line MN is tangent at point M. If AN= 16 cm and MN = 8 cm. Determine the radius of the circle.

(iii) Draw $\angle $XYZ of measure 50 ^{o}and bisect it.(iv) If $\mathrm{cos}\mathrm{\theta}=\frac{24}{25}$, where θ is an acute angle. Find the value of

sin θ.(v) The volume of a cube is 216 cm ^{3}. Find its side.(vi) The radius and slant height of a cone are 10 cm and 30 cm respectively. Find the curved surface area of that cone (π = 3.14)

- Question 3
3. **Attempt any three sub-questions from the following:**[9] (i)

In the following figure, seg DH$\perp $ seg EF and seg GK $\perp $seg EF. If DH = 12 cm, GK = 20 cm and $\u2206(\u2206\mathrm{DEF})$ = 300 cm ^{2}, then find:

(a) EF

(b) A($\u2206$GEF)

(c) A($\square $DFGE)(ii)

In the following figure, ray PA is tangent to the circle at A and PBC is a secant. If AP = 18, BP = 10, then find BC. (iii)

Draw the circle with centre C and radius 3.3 cm. Take a point B at a distance 6.6 cm from the centre C. Draw tangents to the circle from the point B. (iv) Show that: $\sqrt{\frac{1-\mathrm{cos}\mathrm{A}}{1+\mathrm{cos}\mathrm{A}}}=\mathrm{cosec}\mathrm{A}-\mathrm{cot}\mathrm{A}$. (v)

Write the equation of the line passing through A (–2, –3) and B (–4, 7) in the form of ax + by + c = 0.

- Question 4
4. **Attempt any two sub-questions from the following:**[8] (i) Prove that, “the lengths of the two tangent segments to a circle drawn from an external point are equal”. (ii) A tree is broken by the wind. The top of that tree struck the ground at an angle of 30 ^{o}and at a distance of 30 m from the root. Find the height of the whole tree. ( $\sqrt{3}=$1.73)(iii) A (5, 4), B (–3, –2) and C (1, –8) are the vertices of triangle ABC. Find the equation of median AD.

- Question 5
5. **Attempt any two sub-questions from the following:**[10] (i) Prove that, in a right-angle triangle, the square of hypotenuse is equal to the sum of the square of remaining two sides. (ii) ∆SHR ∼ ∆SVU, in ∆SHR, SH = 4.5 cm, HR = 5.2 cm, SR = 5.8 cm and $\frac{\mathrm{SH}}{\mathrm{SV}}=\frac{3}{5}$

Construct ∆SVU.(iii) If ‘V’ is the volume of a cuboid of dimensions a × b × c and ‘S’ is its surface area,

then prove that: $\frac{1}{\mathrm{V}}=\frac{2}{\mathrm{S}}\left[\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}\right]$