Q1 :

Find the 25^{th} term of the A.P. $-5,\frac{-5}{2},0,\frac{5}{2},...$

**Answer :**

$\mathrm{Here},a=-5\phantom{\rule{0ex}{0ex}}d=\frac{-5}{2}-\left(-5\right)=\frac{-5}{2}+5=\frac{5}{2}\phantom{\rule{0ex}{0ex}}\mathrm{Now},{a}_{n}=a+\left(n-1\right)d\phantom{\rule{0ex}{0ex}}\Rightarrow {a}_{25}=-5+\left(25-1\right)\times \frac{5}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {a}_{25}=-5+24\times \frac{5}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {a}_{25}=-5+60\phantom{\rule{0ex}{0ex}}\Rightarrow {a}_{25}=55$

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.

**Answer :**

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,

$\mathrm{tan}60\xb0=\frac{\mathrm{AB}}{\mathrm{BC}}\phantom{\rule{0ex}{0ex}}\Rightarrow \sqrt{3}=\frac{\mathrm{AB}}{2\sqrt{3}}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{AB}=6\mathrm{m}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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.

**Answer :**

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

$\therefore \mathrm{P}\left(E\right)=\frac{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{favourable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}=\frac{4}{8}=\frac{1}{2}$

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.

**Answer :**

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.

**Answer :**

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.

**Answer :**

Q7 :

Solve for *x* :

**Answer :**

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.

**Answer :**

Q9 :

Show that the points (*a*, *a*), (–*a*, –*a*) and $\left(-\sqrt{3}a,\sqrt{3}a\right)$ are the vertices of an equilateral triangle.

**Answer :**

Q10 :

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

**Answer :**

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.

**Answer :**

Q12 :

Solve for *x* :

*x*^{2} + 5*x* −
(*a*^{2} + *a* − 6) = 0

**Answer :**

Q13 :

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

**Answer :**

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*.

**Answer :**

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 $\sqrt{3}=1.732$)

**Answer :**

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)

**Answer :**

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.

**Answer :**

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.

**Answer :**

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

**Answer :**

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.

$\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$

**Answer :**

Q21 :

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

**Answer :**

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.

**Answer :**

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 ?

**Answer :**

Q24 :

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

**Answer :**

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 |

**Answer :**

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.

**Answer :**

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.

**Answer :**

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.

**Answer :**

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.

**Answer :**

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.

**Answer :**

Q31 :

Milk in a container, which is in the form of a frustum of a cone of height 30 cm and the radii of whose lower and upper circular ends are 20 cm and 40 cm respectively, is to be distributed in a camp for flood victims. If this milk is available at the rate of Rs 35 per litre and 880 litres of milk is needed daily for a camp, find how many such containers of milk are needed for a camp and what cost will it put on the donor agency for this. What value is indicated through this by the donor agency ?

**Answer :**

- 10th Maths Paper Solutions Set 2 : CBSE Delhi Previous Year 2015
- 10th Maths Paper Solutions Set 3 : CBSE Delhi Previous Year 2015
- 10th Maths Paper Solutions Set 1 : CBSE Delhi Previous Year 2015
- 10th Maths Paper Solutions Set 1 : CBSE All India Previous Year 2015

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