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) Determine the value of c for which the following system of linear equation has no solution:
cx + 3y = 3; 12x + cy = 6  (Marks 2)

Q2) The GCD and LCM of two polynomials P(x) and Q(x) are x(x + a) and 12x²(x + a)(x² + a²) respectively. If P(x) = 4(x + a)², find Q(x)  (Marks 2)

Q3) If q is the mean proportional between p and r, show that pqr(p + q + r)³ = (pq + qr + pr)³  (Marks 2)

Q4) Find the value of such that the quadratic equation ( - 12)x² + 2(  - 12)x + 2 = 0 has equal roots.  (Marks 2)

Q5) In Fig 1, OPQR is a rhombus, three of whose vertices lie on a circle with centre O. If the area of the rhombus is 323 cm², find the radius of the circle.  (Marks 2)

Q6) The sales price of a television, inclusive of sales tax, is Rs. 13,500. If sales tax is charged at the rate of 8% of the list price, find the list price of the television.  (Marks 2)

Q7) Without using trigonometric tables evaluate : 2(cos 67^o/sin 23^o) - (tan 40^o/cot 50^o) - sin90^o  (Marks 2)

Q8) If cos /cos = m and cos /sin = n, show that
(m² + n²)cos² = n² (Marks 2)

Q9) In Fig 2, ABCD is a parallelogram in which DC = 10 cm and BC = 43 cm. AP is perpendicular to DC. If ADC = 60^o, find the length of AP.  (Marks 2)

Q10) ABC is right angled at B. On the side AC, a point D is taken such that AD = DC and AB = BD. Find the measure of CAB.  (Marks 2)

Q11) In Fig. 3, ABCD is a quadrilateral in which AD = BC and ADC = BCD. Show that the points A, B, C and D lie on a circle.  (Marks 2)

Q12) D is a point on the side BC of ABC such that ADC and BAC are equal. Prove that CA² = DC x CB.  (Marks 2)

Q13) If the mean of n observations
x₁, x₂, x₃,  ..., x_n is , prove that 
(x₁ - ) + (x₂ - ) + (x₃ - ) + ... + (x_n - ) = 0  (Marks 2)

Q14) Flow Chart - Omitted being out of Syllabus.  (Marks 2)

Q15) The median of the following observations arranged in ascending order is 24. Find x.
11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41  (Marks 2)

Section - B

Q16) Solve graphically the following system of linear equations : 2x - y = 2; 4x - y = 8.
Also, find the co-ordinates of the points where the lines meet the axis of x.  (Marks 4)

Q17) Factorise. Omitted, being out of Syllabus.  (Marks 4)

Q18) Students of a class are made to stand in rows. If 4 students are extra in a row, there would be 2 rows less. If 4 students are less in a row, there would be 4 more rows. Find the number of students in the class.  (Marks 4)

Q19) In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 100 km/hour and time increased by 30 minutes. Find the original duration of flight.  (Marks 4)

Q20) A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 19 cm and the diameter of the cylinder is 7cm. Find the volume and surface area of the solid.  (Marks 4)

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