10th Maths Paper Solutions Set 1 : CBSE Abroad Previous Year 2015

General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections - A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
Q1 :

Find the 25th term of the A.P.

Thus, the 25th term of the given AP is 55.

Q2 :

A pole casts a shadow of length $2\sqrt{3}$ m on the ground, when the sun's elevation is 60°. Find the height of the pole.

Let AB be the height of the pole and BC be the shadow of the pole.
Given:
BC = $2\sqrt{3}$ m
$\angle$ACB = 60°
In ΔABC,

Hence, the height of the pole is 6 m.

Q3 :

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and these are equally likely outcomes. Find the probability that the arrow will point at any factor of 8.

Total number of outcomes = 8
Let E be the event that the arrow will point at any factor of 8. So, the outcomes in favour of the event E are 1, 2, 4 and 8.
Total number of favourable outcomes = 4

Thus, the probability that the arrow will point at any factor of 8 is $\frac{1}{2}$.

Q4 :

Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

Q5 :

In figure 1, O is the centre of a circle. PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70°, find ∠TRQ.

Q6 :

In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the lengths of TP and TQ.

Q7 :

Solve for x :

Q8 :

The fourth term of an A.P. is 11. The sum of the fifth and seventh terms of the A.P. is 34. Find its common difference.

Q9 :

Show that the points (a, a), (–a, –a) and are the vertices of an equilateral triangle.

Q10 :

For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?

Q11 :

Point A lies on the line segment PQ joining P(6, –6) and Q(–4, –1) in such a way that $\frac{\mathrm{PA}}{\mathrm{PQ}}=\frac{2}{5}.$ If point P also lies on the line 3x + k (y + 1) = 0, find the value of k.

Q12 :

Solve for x :
x2 + 5x − (a2 + a − 6) = 0

Q13 :

In an A.P., if the 12th term is −13 and the sum of its first four terms is 24, find the sum of its first ten terms.

Q14 :

A bag contains 18 balls out of which x balls are red.
(i) If one ball is drawn at random from the bag, what is the probability that it is not red?
(ii) If 2 more red balls are put in the bag, the probability of drawing a red ball will be $\frac{9}{8}$ times the probability of drawing a red ball in the first case. Find the value of x.

Q15 :

From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pole are 30° and 45° respectively. Find
(i) how far the pole is from the bottom of a tower,
(ii) the height of the pole. (Use $\sqrt{3}=1.732$)

Q16 :

The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14)

Q17 :

Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.

Q18 :

A metallic cylinder has radius 3 cm and height 5 cm. To reduce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of $\frac{3}{2}$ cm and its depth is $\frac{8}{9}$ cm. Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape.

Q19 :

In Figure 3, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. If arcs of equal radii 7 cm have been drawn, with centres A,B, C  and D, then find the area of the shaded region.

Figure 3

Q20 :

A solid right-circular cone of height 60 cm and radius 30 cm is dropped in a right-circular cylinder full of water of height 180 cm and radius 60 cm. Find the volume of water left in the cylinder, in cubic metres.

Q21 :

If x = −2 is a root of the equation 3x2 + 7x + p = 0, find the values of k so that the roots of the equation x2 + k(4x + k − 1) + p = 0 are equal.

Q22 :

Find the middle term of the sequence formed by all three-digit numbers which leave a remainder 3, when divided by 4. Also find the sum of all numbers on both sides of the middle term separately.

Q23 :

The total cost of a certain length of a piece of cloth is Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre ?

Q24 :

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Q25 :

In Figure 4, O is the centre of the circle and TP is the tangent to the circle from an external point T. If ∠PBT = 30°, prove that BA : AT = 2 : 1.

 Figure 4

Q26 :

Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre draw two tangents to the circle. Measure the length of each tangent.

Q27 :

Two poles of equal heights are standing opposite to each other on either side of the road which is 80 m wide. From a point P between them on the road, the angle of elevation of the top of a pole is 60° and the angle of depression from the top of another pole at point P is 30°. Find the heights of the poles and the distances of the point P from the poles.

Q28 :

A box contains cards bearing numbers from 6 to 70. If one card is drawn at random from the box, find the probability that it bears

(i) a one digit number.

(ii) a number divisible by 5.

(iii) an odd number less than 30.

(iv) a composite number between 50 and 70.

Q29 :

The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the mid-point of the base. Find the coordinates of the points A and B. Also find the coordinates of another point D such that BACD is a rhombus.

Q30 :

A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. One-fourth of the water flows out of the vessel. Find the radius of a spherical ball.