Board Paper of Class 10 2015 Maths (SET 3) - Solutions

• Question 5
  From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ. VIEW SOLUTION

• Question 6
  Find the middle term of the A.P. 6, 13, 20, ... , 216. VIEW SOLUTION

• Question 7
  In Fig. 2, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.
   

  Figure 2

   

  VIEW SOLUTION

• Question 8
  If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t. VIEW SOLUTION

• Question 9
  Find the ratio in which the point $\mathrm{P} \left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $\mathrm{A} \left(\frac{1}{2}, \frac{3}{2}\right)$ and B(2, −5). VIEW SOLUTION

• Question 10
  Solve the following quadratic equation for x :
  9x² − 6b²x − (a⁴ − b⁴) = 0 VIEW SOLUTION

• Question 11

  In Fig. 3, APB and AQO are semicircles, and AO = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region. $\left[\mathrm{Use} \mathrm{\pi } =\frac{22}{7}\right]$

  Figure 3

  
  
   

  VIEW SOLUTION

• Question 12
  A solid wooden toy is in the form of a hemisphere surrounded by a cone of same radius. The radius of hemisphere is 3.5 cm and the total wood used in the making of toy is $166\frac{5}{6} {\mathrm{cm}}^3$. Find the height of the toy. Also, find the cost of painting the hemispherical part of the toy at the rate of Rs 10 per cm². $\left[\mathrm{Use} \mathrm{\pi } =\frac{22}{7}\right]$ VIEW SOLUTION

• Question 13
  In Fig. 4, from the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid. $\left(\mathrm{Use} \mathrm{\pi }=\frac{22}{7} \mathrm{and} \sqrt{5}=2.236\right)$

  Figure 4

  VIEW SOLUTION

• Question 14
  In Fig. 5, from a cuboidal solid metallic block, of dimensions
  15cm ✕ 10cm ✕ 5cm, a cylindrical hole of diameter 7 cm is drilled out. Find the surface area of the remaining block $\left[\mathrm{Use} \mathrm{\pi }=\frac{22}{7}\right]$

  Figure 5

  VIEW SOLUTION

• Question 15
  The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 45°. If the tower is 30 m high, find the height of the building. VIEW SOLUTION

• Question 16
  In Fig. 6, find the area of the shaded region [Use π = 3.14]
   

  Figure 6

  VIEW SOLUTION

• Question 17
  Find that non-zero value of k, for which the quadratic equation kx² + 1 − 2(k − 1)x + x² = 0 has equal roots. Hence find the roots of the equation. VIEW SOLUTION

• Question 18
  All red face cards are removed from a pack of playing cards. The remaining cards were well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is
  (i) a red card
  (ii) a face card
  (iii) a card of clubs VIEW SOLUTION

• Question 19
  Find the area of the triangle PQR with Q(3,2) and the mid-points of the sides through Q being (2,−1) and (1,2). VIEW SOLUTION

• Question 20
  If S_n denotes the sum of first n terms of an A.P., prove that S₃₀ = 3[S₂₀ − S₁₀] VIEW SOLUTION

• Question 21
  Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION

• Question 22
  In Fig. 7, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.
   

  Figure 7

  VIEW SOLUTION

• Question 23
  From a point P on the ground the angle of elevation of the top of a tower is 30° and that of the top of a flag staff fixed on the top of the tower, is 60°. If the length of the flag staff is 5 m, find the height of the tower. VIEW SOLUTION

• Question 24
  Ramkali required Rs 2,500 after 12 weeks to send her daughter to school. She saved Rs 100 in the first week and increased her weekly saving by Rs 20 every week. Find whether she will be able to send her daughter to school after 12 weeks.

  What value is generated in the above situation?

  VIEW SOLUTION

• Question 25
  The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original fraction is $\frac{29}{20}$. Find the original fraction. VIEW SOLUTION

• Question 26
  Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe. VIEW SOLUTION

• Question 27
  If A(−4, 8), B(−3, −4), C(0, −5) and D(5, 6) are the vertices of a quadrilateral ABCD, find its area. VIEW SOLUTION

• Question 28
  A 21 m deep well with diameter 6 m is dug and the earth from digging is evenly spread to form a platform 27 m ✕ 11 m. Find the height of the platform. $\left[\mathrm{Use} \mathrm{\pi }=\frac{22}{7}\right]$ VIEW SOLUTION

• Question 29
  A bag contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is:
  
  (i) divisible by 3 or 5
  (ii) a perfect square number VIEW SOLUTION

• Question 30
  Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2 cm. Construct tangents to each circle from the centre of the other circle.Steps VIEW SOLUTION

• Question 31
  Solve for x :
  $\frac{3}{x+1}+\frac{4}{x-1}=\frac{29}{4x-1};x\ne 1,-1,\frac{1}{4}$ VIEW SOLUTION