Maximum Marks : 80                                                                                                                                         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:

The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

(a) 9000                

(b) 9400

(c) 9600

(d) 9800                                                                                                             

 

Question 2:

If the median of the data : 6, 7, x - 2, x, 17, 20 written in ascending order, is 16, then x =                 

(a) 15                

(b) 16

(c) 17

(d) 18                                                                                                                 

 

Question 3:

If n any natural number, then 6ⁿ - 5ⁿ always ends with

(a) 0                

(b) 1

(c) 2

(d) any natural number                                                                                         

 

Question 4:

The value of p so that the pair of equations 4x + py + 8 = 0, 2x + 2y + 2 = 0 has unique solution, is

(a) p = 2                

(b) p ≠ 2

(c) p = 4

(d) p ≠ 4                                                                                                               

 

Question 5:

If θ and 2θ - 45^o are acute angles such that sin θ = cos (2θ - 45^o), then tan θ is equal to

(a) 1                

(b)

(c)

(d)                                                                                                                

 

Question 6:

The value of cosec 31⁰ – sec 59⁰ is

(a) 0                

(b) 1

(c)

(d)                                                                                                                 

 

Question 7:

The value of (sec A + tan A) (1 – sin A) is

(a) sec A              

(b) sin A             

(c) cosec A                     

(d) cos A                                                                                                                                                  

 

Question 8:

The area of the triangle whose vertices are (2, 3), (–1, 0) and (2, – 4), is

(a) 5.5 square units              

(b) 10.5 square units             

(c) 15.5 square units                     

(d) 20.5 square units                                                                                                                            

 

Question 9:

If A and B are (–2, –2) and (2, –4), respectively, then the coordinates of P such that  and P lies on the line segment AB, is

(a)               

(b)              

(c)                      

(d)

                                                                                                                                            

Question 10:

If the points A(3, 1), B(5, 1), C(a, b), D(4, 3) are vertices of a parallelogram ABCD then the value of a and b are

(a) a = 3, b = 6              

(b) a = 6, b = 3             

(c) a = 6, b = 6                     

(d) a = 3, b = 3                                                                                                                                     

 

(Q 11- Q 15) Fill in the blanks 

Question 11:

2 cubes each of volume 64 cm³ are joined end to end. Then the surface area of the resulting cuboid is _____.                                                                                                                  

 

Question 12:

If one root of the quadratic equation x² - 12x + a = 0, is thrice the other then the value of a is ______.                                                                                      

 

Question 13:

In the given figure, ABC is an isosceles triangle right angled at C with AC = 4 cm. Then the length of AB is                                                                                                                                          

 

Question 14:

The 10^th term from the end of the AP -5, -10, -15, ......., -1000 is _____.   

 

Question 15:

Two dice are tossed. The probability that the total score is a prime number is _____.                                                                                                                               

 

(Q 16- Q 20) Answer the following:

Question 16:

In the figure, PS = 3 cm, QS = 4 cm, ∠ PRQ = θ, ∠ PSQ = 90⁰, PQ ⊥ RQ and RQ = 9 cm. Evaluate tan θ.                                                                                                                                         

 

Question 17:

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.                                                                                                     

 

Question 18:

Find the quadratic equation whose roots are the reciprocals of the roots of 2x² + 5x + 3 = 0?                                                                                                                                                                                                                                                                                                                           

Question 19:

If a, b, c, d, e are in AP then find the value of a - 4b + 6c - 4d + e.                  

 

Question 20:

Express 0.6666....... in the form of .                                                                 

 

SECTION - B

Question 21:

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

 

Question 22:

The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30⁰ than when it was 60⁰. Find the height of the tower. (Given √3 = 1·732)                                                                                          

 

Question 23:

A piggy bank contains hundred 50 p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin:

(i) Will be a 50 p coin?                                 

(ii) Will not be a Rs.5 coin?                                                                                                                        

 

Question 24:

A cone of height 24 cm and radius of base 6 cm made up modelling clay a child reshapes it in the form of a sphere. Find the radius of the sphere.        

 

Question 25:

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.                                  

 

Question 26:

Find the smallest number which is divisible by both 306 and 657.            

 

SECTION - C

Question 27:

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.                                                                                                                                       

 

Question 28:

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the AP.                                                                                        

 

Question 29:

Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2CD, find the ratio of the areas of triangles AOB and COD.                                                       

 

Question 30:

If α and β are the zeroes of the quadratic polynomial p(s) = 3s² – 6s + 4 then find the value of .                                                 

 

Question 31:

(a) In the given figure, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then find the length of QR (in cm).

(b) In the figure, ABCD is a rectangle. Find the value of x and y.

Question 32:

If 16cot x = 12, then find the value of sinx – cosxsinx + cosx.              

 

Question 33:

A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8 gm mass. (Use π = 3.14)                                                                                             

 

Question 34:

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:             

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.                                                          

 

SECTION - D

Question 35:

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60⁰. Give the justification of the construction.                                                                

 

Question 36:

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.              

 

Question 37:

In the figure, ABC is a triangle in which ∠ ABC > 90° and AD ⊥ CB produced. Prove that AC² = AB² + BC² - 2 BC . BD

 

Question 38:

Tw0 equal circles with centres O and O' touch each other at X. OO' is produced to meet the circle with centre O' at A. AC is a tangent to the circle with centre O, at the point C. If O'D is perpendicular to AC, find .                                                                                                       

 

Question 39:

A girl empties a cylindrical bucket full of sand, of base radius 18 cm and height 32 cm on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct to one place of decimal.                                                                                      

 

Question 40:

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30⁰, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60⁰. Find the time taken by the car to reach the foot of the tower from this point.                    

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