Time allowed : 3 hours
Maximum Marks : 100

General Instructions :

(i) Question number 1 to 15 carry 2 marks each.
(ii) Question number 16 to 25 carry 4 marks each.
(iii) Question number 26 to 30 carry 6 marks each.
(iv) Write the serial number of the question before attempting it.
(v) Use of logarithmic and trignometric tables is permitted. Use of calculator is not permitted.

Section - A

Q1) For what value of k will the following system of linear equations have an infinite number of solutions :
2x + 3y = 2; (k + 2)x + (2k + 1)y = 2(k - 1) ?  (Marks 2)

Q2) Find whether the number 6, 10, 14 and 22 are in proportional or not. If not what number be added to each number so that they become proportional ?  (Marks 2)

Q3) Reduce the following relation expression into lowest form:
(x⁴ - 10x² + 9)/(x³ + 4x² + 3x)  (Marks 2)

Q4) Solve for x and y
ax + by = a - b  - (i)
bx - ay = a + b  - (ii)  (Marks 2)

Q5) Find the value of k such that sum of the roots of the quadratic equation
3x² + (2k + 1)x - (k + 5) = 0 is equal to the product of its roots.  (Marks 2)

Q6) Find two consecutive numbers, whose square have sum 85.  (Marks 2)

Q7) Without using trignometric table, show that:
tan 7^o . tan 23^o . tan 60^o . tan 67^o . tan 83^o = 3  (Marks 2)

Q8) Show that (sin - 2sin³ )/(2cos³ - cos ) = tan .  (Marks 2)

Q9) In the given figure chord PQ and RS of a circle intersect at T. If RS = 18 cm, ST = 6 cm and PT = 18 cm, find the length of TQ.  (Marks 2)

Q10) In the given figure DE || BC and AD : DB = 5 : 4
Find ar(DFE)/ar(CFB).  (Marks 2)

Q11) In the figure BAC = 30^o. Show that BC is equal to the radius of the circum-circle of ABC whose centre is O.  (Marks 2)

Q12) Rita purchased a car, with a market price of Rs. 2,10,000 at a discount of 5%. If the sales tax charged at 10%, find the amount Rita had to pay for purchasing the car.  (Marks 2)

Q13) The mean weight of 21 students of a class is 52 kg. If the mean weight of first 11 students of the class is 50 kg and that of last 11 students is 54 kg, find the weight of the 11th student.  (Marks 2)

Q14) The following data has been arranged in ascending order: 12, 14, 17, 21, x, 26, 28, 32, 36. If the median of the data is 23, find x. If 32 is changed to 23, find the new median.  (Marks 2)

Q15) For what value of x, in the mode of the following data 5 ?
2, 4, 3, 5, 4, 5, 6, 4, x, 7, 5.  (Marks 2)

Section - B

Q16) Determine graphically the co-ordinates of the vertices of the triangle, the equation of whose sides are:
y = x, 3y = x, x + y = 8.  (Marks 4)

Q17) A part of monthly hostel charges in a college are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, he has to pay Rs. 1000 as hostel charges where as a student B, who takes food for 26 days, pays Rs. 1180 as hostel charges. Find the fixed charge and the cost of food per day.  (Marks 4)

Q18) Find the value of a and b so that the polynomials p(x) and q(x) have (x + 1)(x - 2) as their HCF.
p(x) = (x² + 3x + 2)(x² + x + a)
q(x) = (x² - 3x + 2)(x² - 3x + b)  (Marks 4)

Q19) A page of pass book of Ved is given below :-

Find the interest Ved gets for the period March, 98 to Dec.' 98 at 5% per annum simple interest.  (Marks 4)

Q20) The annual income of Seema (excluding HRA) is Rs. 1,60,000. She contributes Rs. 5000 per month to her provident fund and pays a half yearly insurance premium of Rs. 5000. Calculate the income tax along with surcharge Seema has to pay in the last month of the year if her earlier deductions as income tax for the first 11 months were at the rate of Rs. 400 per month.  (Marks 4)
Assume the following for calculating income tax :

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