Maximum Marks : 80                                                              Time Taken : 3 Hours

 

General Instructions:

(i) All the questions are compulsory.

(ii) The question paper consists of 40 questions divided into 4 sections A, B, C, and D.

(iii) Section A comprises of 20 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 8 questions of 3 marks each. Section D comprises of 6 questions of 4 marks each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark each, two questions of 2 marks each, three questions of 3 marks each, and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.

(v) Use of calculators is not permitted.

 

SECTION - A

(Q1- Q10) Multiple Choice Questions

Question 1:

A bag contains fifteen cards bearing number 11, 12, 13, 14 ..., 25. If a card is chosen from the bag then the probability that it bears a square number is

(a)

(b)

(c)

(d)

 

Question 2:

The value of a for which the given system of equations 2x + 3y = 7, (a - 1)x + (a + 1)y = 3a - 1  has infinitely many solutions

(a) 5

(b) 7

(c) 2

(d) 7

 

Question 3:

If ABC is a right angle triangle at C then cos(A + B) is

(a) 0

(b) 1

(c)

(d)

 

Question 4:

The value of cos² 17° - sin² 73° is 

(a) 1

(b) -1

(c) 0

(d)

 

Question 5:

If the LCM of a and 18 is 36 and the HCF of a and 18 is 2, then a is equal to  

(a) 1

(b) 2

(c) 3

(d) 4

 

Question 6:

If the arithmetic mean of 7, 8, x, 11, 14 is x, then x = 

(a) 4

(b) 7

(c) 10

(d) 13

 

Question 7:

In a triangle, ΔABC ˜ ΔDEF, ar(ΔABC) = 9 cm², ar(ΔDEF) = 16 cm². If BC = 2.1 cm, then the measure of EF is 

(a) 1.8 cm

(b) 2.8 cm

(c) 3.8 cm

(d) 4.8 cm

 

Question 8:

If (x, 2), (-3, -4) and (7,-5) are collinear, then what is the value of x? 

(a) -40

(b) -51

(c) -63

(d) -74

 

Question 9:

The ratio in which the x-axis divides the segment joining (3, 6) and (12, -3) is 

(a) 1 : 2

(b) 2 : 1

(c) 1 : 3

(d) 3 : 1

 

Question 10:

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is 

(a) x² + 4 = 0 

(b) x² - 4 = 0 

(c) x² + 2 = 0 

(d) x² - 2 = 0 

 

(Q 11- Q 15) Fill in the blanks 

Question 11:

If α, β are the zeros of the polynomial f (x) = x² + x + 1, then  = _____.

 

Question 12:

The number of terms of the A.P. 3, 7, 11, 15,...... to be taken so that the sum is 406 is ______.

 

Question 13:

The number of bricks, each measuring 25 cm * 11.25 cm * 6 cm, will be needed to build a wall of 8 m * 6 m * 22.5 cm, is ______.

 

Question 14:

In the given figure, if radius of the circle is 10 then the area of the shaded region is _______.

                                                                                                                                  

 

Question 15:

The length of the hypotenuse of an isosceles right triangle whose one side is 4√2 cm is ______.

 

(Q 16 - Q 20) Answer the following:

Question 16:

In triangle ABC, AD bisects angle BAC and AD = DC. If angle BDA = 70°, then find the angle ACD.

 

Question 17:

If n = 2³ * 3⁴ * 4⁴ * 7, then find the number of consecutive zeros in n, where n is a natural number.

 

Question 18:

If the points A (5, p), B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then what is the value of p?

 

Question 19:

In the given figure, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then find the value of OR.

 

 

Question 20:

If mean of a set of observations is X, then evaluate Σ(xi - X). 

 

SECTION - B

Question 21:

If the HCF of 657 and 963 is expressible in the form of 657x + 963 * (-15), then find the value of x.

 

Question 22:

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is

(i) red ?                                        (ii) not red ?

 

Question 23:

What is the angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height?

 

Question 24:

Prove that the tangents at any point of a circle are perpendicular to the radius through the point of contact.

 

Question 25:

The decimal expansion of the rational number  will terminate after how many places of decimal?

 

Question 26:

A square of side 60 m and a rectangular field of length 80 m have the same perimeter. Which field has a larger area?

 

SECTION – C

Question 27:

If √(ab) is an irrational number then prove that √a + √b is also an irrational number.

 

Question 28:

The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

 

Question 29:

If α and β are the zeroes of the polynomial f(x) = x² + px + q then form a polynomial whose zeroes are (α + β)² and (α - β)².

 

Question 30:

Solve the following system of equations: 

 

 

 

Question 31:

If A(-2, 5) , B(1, -3) and C(a, b) form an isosceles triangle with CB = CA, then show that 6a - 16b + 19 = 0

 

Question 32:

If sin θ + 2 cos θ = 1 then prove that 2 sin θ - cos θ = ±2

 

Question 33:

The shape of a garden is rectangular in the middle and semi-circular at the ends as shown in the diagram. Find the area and the perimeter of this garden.

                                                                                                 

 

Question 34:

In a triangle ABC, DEF trisect BC prove that 8AE² = 3AC² + 5AD²

 

SECTION - D

Question 35:

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q. Give the justification of the construction.

 

Question 36:

A plane left 30 min later than the scheduled time and in order to reach the destination 1500 km away in time it has to increase the speed by 250  from  from the usual speed. Find its usual speed.

 

Question 37:

A ladder rests against a wall at an angle α to the horizontal its foot is pulled away from the wall through the distance A. So, that it slides a distance B down the wall making an angle β with the horizontal.

Show that

 

Question 38:

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

 

Question 39:

The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

      

Question 40:

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

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