PT and PS are tangents drawn to a circle, with cantre C, from a point P. If ∠TPS = 50° , then the measure of ΔTCS is

1

(a)

150^o

(b)

120^o

(c)

100^o

(d)

130^o

The length of shadow of a tower on the plane ground is √3 times the height of the tower. The angle of elevation of sun is :

1

(a)

90^o

(b)

60^o

(c)

30^o

(d)

45^o

If the radius of the base of a right circular cylinder is halved keeping the height same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is

1

(a)

1:2

(b)

2:1

(c)

4:1

(d)

1:4

There is a circular path around a sport field. Sonia takes 18min to drive one around of the field, while Ravi takes 12min for the same. Suppose they both start at the same point at the same time, and go in the same direction, after how minutes will they meet again at the starting point?

1

(a)

3

(b)

36

(c)

6

(d)

18

1. The difference of any two sides of a triangle is always __________ than the third side.

   1

   (a)

   Less

2. Find the distance of the point (-4, -7) from the y-axis.

   1

   (a)

   Points are (-4, -7) and (0, -7)
   Distance = \(=\sqrt { { \left( 0+4 \right) }^{ 2 }+{ \left( -7+7 \right) }^{ 2 } } =\sqrt { { 4 }^{ 2 }+0 } =\sqrt { 16 } \) = 4 units

1. What is the sum of five positive integers divisible by 6.

   1

   (a)

   Here, a = 6, d = 6, n = 5
   \(\because\) S = n/2[2a+(n-l)d]
   S₅ = 5/2[2x6+(5-1)(6)]
   = 5/2[12+4x6]
   = 5/2[12+24] = 5/2[36]
   = 5x18 = 90

2. Two tangents making an angle of 60° between them are drawn to a circle of radius \(\sqrt{3}\) cm, then find the length of each tangent.

   1

   (a)

   \(\tan { { 30 }^{ ° } } =\frac { OA }{ AP } \)
   \(\Rightarrow \quad \frac { 1 }{ \sqrt { 3 } } =\frac { \sqrt { 3 } }{ AP } \)
   \(\Rightarrow \quad AP=\sqrt { 3 } \times \sqrt { 3 } =\) 3 cm

If ax²+bx+c=0 has equal roots, what is the value of c?

2

(a)

Since the quadratic equation has equal roots, then
b²-4ac=0
4ac=b²
c=\(b^2\over 4a\)

For what value of p, are 2p + 1, 13, 5p - 3 three consecutive terms of an AP?

2

(a)

If terms are in AP, then 13 - (2p + 1) = (5p - 3) - 13
\(\Rightarrow\)   13 - 2p -1 = 5p - 3 - 13
\(\Rightarrow\) 28 = 7p
\(\Rightarrow\) p = 4

Write the name of the common point of the tangent to a circle and the circle.

2

(a)

The common point of the tangent to a circle and the circle is called the point of contact.

The probability of getting a bad egg from a lot of 400 eggs is 0.035.Find the number of bad eggs in the lot.

2

(a)

Number of bad eggs\(=400\times0.035=400\times{35\over1000}\)
= 14

1. Draw a right triangle in which the sides containing the right angle are 5 cm and  4cm.  Construct a similar triangle whose sides are \(5\over 3\) times the sides of the given triangle.

   2

   (a)

2. Draw a circle of radius 4 cm. Take a point P outside the circle. Without using the centre of the circle, draw two tangents to the circle from P.

   2

   (a)

1. Find the fourth vertex D of a parallelogram ABCD whose three vertices are A(-2,3) B(6,7) and C(8,3).

   2

   (a)

2. Check whether 301 is a term of the list of numbers 5, 11, 17, 23,......

   2

   (a)

A tower stands vertically on the ground. From a point on the ground which is 60m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60^o. Find the height of the tower.

3

(a)

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer. (-1, -2), (1, 0), (-1, 2), (-3, 0)

3

(a)

A 20m deep well with diameter 7m is dug and the earth from digging is evenly spread out to form a platform 22m by 14m.Find the height of the platform.

3

(a)

Find the value of k, if - 1 is a zero of the polynomial p(x) = kx² - 4x + k.

3

(a)

If \(\sqrt { 2 } \sin { \theta } =1\), find the value of \(\sec ^{ 2 }{ \theta } -cosec^{ 2 }\theta \)

3

(a)

1. If the point A(0,2) is equidistant from the points B(3,p) and C(p,5), find p. Also find the length of AB.

   3

   (a)

2. The shaded area, in the figure between the circumference of two concentric circles is 346.5 cm² . The circumference of the inner circle is 88 cm. Calculate the radius of the outer circle.
   
   

   3

   (a)

1. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point 'A' on the ground is 60^o and the angle of depression of the point 'A' from the top of the tower is 45^o . Find the height of the tower. \((\sqrt { 3 } =1.73)\)  

   3

   (a)

2. A box contain 100red cards, 200 yellow cards and 50blue cards.If a card is drawn at random from the box, the find the probability that it will be:
   (i)A blue card
   (ii)Not a yellow card
   (iii)Neither yellow nor a blue card.

   3

   (a)

1. AB is a diameter of a circle and AC is its chord such that \(\angle BAC=30°\). If the tangent at C intersects AB extended at D, then prove that BC = BD.

   3

   (a)

2. The ages of employees in a factory are as follows:

   Age (in years)	17-23	23-29	29-35	35-41	41-47	47-53
   Number of employees	2	5	6	4	2	1

   Find the median age of the employees. 

   3

   (a)

1. The sum of first n, 2n and 3n terms of an AP are S₁, S₂ and S₃, respectively. Prove that S₃ = 3(S₂ - S₁).

   3

   (a)

If in \(\triangle\)ABC, AD is median and AE \(\bot\) BC, then prove that AB² + AC² = 2AD² + \(\frac {1}{2}\)BC².

4

(a)

Check whether -3 is a solution of the equation \(3x^{ 2 }+5x+2=0\)

4

(a)

The angle of elevation of the top of a tower from two distinct points s and t from its foot are complementary.Prove that the height of the tower is \(\sqrt{st}\)

4

(a)

All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting
(i) a spade

4

(a)

Find the cost of fencing a circular field of area 9856 m² at the rate of Rs. 20 per metre.

4

(a)

Prove that \(\cot ^{ 2 }{ A } \left[ \frac { \sec { A } -1 }{ 1+\sin { A } } \right] +\sec ^{ 2 }{ A } \left[ \frac { \sin { A } -1 }{ 1+\sec { A } } \right] =0\)

4

(a)

To find out the concentration of SO₂ in the air (in parts per million, i.e, ppm), the data was collected for 30 localities in a certain city and its presented below:

Concentration of SO2 (in ppm)	Frequency
0.00-0.04	4
0.04-0.08	9
0.08-0.12	9
0.12-0.16	2
0.16-0.20	4
0.20-0.24	2

Find the mean concentration of SO₂ in the air.

4

(a)

Find the zeroes of the following quadratic polynomials and verify the relationship betwee the zeros and its coefficients.
(i) x²-2x-8
(ii) 4s²-4s+1
(iii) 6x²-3-7x
(iv) 4u²+8u
(v) t²-15
(vi) 3x²-x-4

4

(a)

Form the pair of linear equations in the following problems and find their solutions graphically:
10 students of class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

4

(a)

Water is flowing at the rate of 15 km/hour through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?

4

(a)

Two sets of English and Social Science Books containing 336 and 96 books respectively in a library, have to be stacked in such a way that all the books are stored topicwise and the height of each stack is the same. Assuming that the books are of the same thickness, determine the total number of stacks. What are the characteristics of library?
[Number of stacks = \(\frac { 336+96 }{ HCF\left( 336,96 \right) } \)]

4

(a)