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 trigonometric 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) Reduce to its lowest term:
(2x² - 2x)/(x³ + x² + x)  (Marks 2)

Q3) Find whether the number 4, 7, 6 and 10 are in proportional or not. If not what number be added to each number so that they become proportional ? (Marks 2)

Q4) For what value of x, is the mode of the following data 9 ?
5, 8, 9, 3, 9, 8, 7, 6, 8, 9, x, 4  (Marks 2)

Q5) The mean weight of 25 students of a class is 60 kg. If the mean weight of the first 13 students of the class is 57 kg and that of the last 13 students is 63 kg, find the weight of the 13th student. (Marks 2)

Q6) Ahmed purchased a washing machine, with a market price of Rs. 8500 at a discount of 4%. If the sales tax is charged at 7%, find the amount Ahmed had to pay for purchasing the washing machine.  (Marks 2)

Q7) 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 X takes food for 25 days, he has to pay Rs. 1750 as hostel charges where as a student Y, who takes food for 28 days, pays Rs. 1900 as hostel charges. Find the fixed charge and the cost of food per day.  (Marks 4)

Q8) Find the value of a and b so that the polynomials p(x) and q(x) have (x + 1)(x - 4) as their HCF.
p(x) = (x² + 3x + 2)(x² - 7x + a)
q(x) = (x² - 2x - 8)(x² - 2x + b)  (Marks 4)

Q9) Show that cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A
  (Marks 4)

Q10) Solve for x :
3^{4x + 1} - 2 x 3^{2x + 2} = 81  (Marks 6)

Q11) Find whether the number 4, 11, 12 and 27 are in proportional or not. If not what number be added to each number so that they become proportional ?  (Marks 2)

Q12) Reduce the following rational expression in to lowest form:
(x⁴ - 13x² + 36 )/(x³ - x² - 6x)  (Marks 2)

Q13) Sarla purchased an almirah, with a market price of Rs. 5600 at a discount of 5%. If the sales tax is charged at 10%, find the amount Sarla had to pay for purchasing the almirah.  (Marks 2)

Q14) The mean of 23 observations is 34. If the mean of the first 12 observations is 32 and that of the last 12 observations is 38, find the 12th observation.  (Marks 2)

Q15) For what value of x, is the mode of the following data 11 ?
8, 9, 11, 10, 12, 14, 11, 14, 15, 14, x, 11  (Marks 2)

Q16) Find the value of a and b so that the polynomials p(x) and q(x) have (x - 1)(x + 4) as their HCF.
p(x) = (x² - 3x + 2)(x² + 7x + a), q(x) = (x² + 5x + 4)(x² - 5x + b)  (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 22 days, he has to pay Rs. 1380 as hostel charges where as a student B, who takes food for 28 days, pays Rs. 1680 as hostel charges. Find the fixed charge and the cost of food per day.  (Marks 4)

Q18) Solve for x:
5^4x - 3 x 5^{2x + 1} = 250  (Marks 6)