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:

The number 3825 expressed as a product of its prime factor is

(A) 3² * 5³ * 17

(B) 3³ * 5² * 17

(C) 3² * 5² * 17

(D) 3³ * 5³ * 17

 

Question 2:

The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. The model class of this data is

 

(A) 50 - 55

(B) 40 - 45

(C) 30 - 35

(D) 20 - 25

 

Question 3:

If a = 2 * 3, b = 2 * 3 * 5, c = 3ⁿ *5 and the LCM (a, b, c) = 2 * 3 * 3 * 5, then the value of n is

(A) 2

(B) 3

(C) 5

(D) 7

 

Question 4:

The given pair of linear equation 2x + y = 5, 3x + 2y = 8 has

(A) unique solution

(B) no solution

(C) infinite solution

(D) none of these

 

Question 5:

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is

(A) 2 : 1                       

(B) 1 : 2                        

(C) 4 : 1                             

(D) 1 : 4

 

Question 6:

If tan A +  = 2, then the value of tan² A +  is

(A) 1                       

(B) 2                        

(C)                              

(D)

 

Question 7:

In triangle ABC, right-angled at B, if A = , find the value of

sin A cos C + cos A sin C

(A) 1                       

(B)                         

(C)                              

(D)  

 

Question 8:

The co-ordinates of a point which divides the join of (2, –1) and (–3, 4) in the ratio of 2 : 3 internally is

(A) (1, 0)                       

(B) (1, 1)                        

(C) (0, 0)                             

(D) (0, 1)

 

Question 9:

If the perimeter and area of a circle are numerically equal, then the radius of the circle is                                                      

(a) 2 units                      

(b) π units                     

(c) 4 units                        

(d) 7 units

 

Question 10:

If A and B are the points (0, 6) and (8, 0) respectively, then the value of AB is

(A) 5                        

(B) 10           

(C) 12                

(D) 15

 

(Q 11- Q 15) Fill in the blanks 

Question 11:

If the sum of the roots of the equation x² - (k + 6)x + 2(2k - 1) = 0 is equal to half of their product, then the value of k is ______.

 

Question 12:

The common difference of an AP in which T₂₇ - T₂₃ = 64 is _______.

 

Question 13:

If a family has 3 children then the probability of having at least one boy is ______.

 

Question 14:

If in Δ ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm then the angle B is ______.

 

Question 15:

A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord is ______.

 

(Q 16 – Q 20) Answer the following:

Question 16:

Find the sum of the exponents of the prime factors in the prime factorization of number 196.

 

Question 17:

In the figure, AP is a tangent to the circle with centre O such that OP = 4 cm and ∠OPA = 30^o, then find the value of AP. 

                                                                     

 

Question 18:

If the first term of an A.P. is a and n^th term is b, then what is the common difference of AP?

 

Question 19:

If one root of the equation 2x² + kx + 4 = 0 is 2, then find the other root.

 

Question 20:

If ABC and DEF are similar triangles such that ∠A = 47 degree and ∠E = 83 degree, then what is the value of ∠C?

 

SECTION - B

Question 21:

If a square is inscribed in a circle, find the ratio of the areas of the circle and square.

 

Question 22:

The horizontal distance between two towers is 80 m. The angle of depression of the top of the first tower when seen from the top of the second tower is 30^o. If the height of the second tower is 160 m, find the height of the first tower.

 

Question 23:

A number is selected from first 50 natural numbers. What is the probability that it is a multiple of 3 or 5? 

 

Question 24:

Two cubes each of 10 cm edge are joined end to end. Find the surface area of the resulting cuboid.

 

Question 25:

A Circular field has a circumference of 360 km. Three Cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?

 

Question 26:

If AD and PM are medians of triangles ABC and PQR, respectively where

Δ ABC ~ Δ PQR, prove that .

 

SECTION - C

Question 27:

If the sum of the LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then what is the product of these numbers?

 

Question 28:

Let S_n denotes the sum of the first n terms of an A.P. If S_2n = 3S_n then find the ratio of S_3n : S_n.

 

Question 29:

Solve the system of linear equation:

ax + by - (a - b) = 0

bx – ay - (a + b) = 0

 

Question 30:

If the polynomial x⁴ – 6x³ + 16x² – 25x + 10 is divided by another polynomial

x² – 2x + k, the remainder comes out to be x + a. Find the value of k and a.

 

Question 31:

Prove that:

 

Question 32:

ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

 

Question 33:

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs 500 per m². (Note that the base of the tent will not be covered with canvas.)

 

Question 34:

During the medical check-up of 35 students of a class, their weights were recorded as follows:

        

(a) Draw a less than type ogive for the given data.

(b) Find the median weight from the graph and verify the result by using the formula.

 

SECTION - D

Question 35:

Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are 34 of the corresponding sides of the triangle ABC. Give the justification of the construction.

 

Question 36:

The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them saves Rs 200 per month, find their monthly incomes.

 

Question 37:

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 kmh, in how much time will the tank be filled?

 

Question 38:

The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.

 

 

                     

 

Question 39:

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

 

Question 40:

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

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