Math's Class X (CBSE)
Sample Paper Set - I

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) Find the value of k for which the system of equations has a unique solution :
x - ky = 2
3x + 2y = - 5  (Marks 2)

Q2) Evaluate : 2(tan 53^o/cot 37^o) - (cot 80^o/tan 10^o)  (Marks 2)

Q3) Determine the values of K such that the equation Kx² - 5x + K = 0 has equal roots.  (Marks 2)

Q4) A spherical cannon ball, 28 cm in diameter is melted and cast into a right circular conical mould, the base of which is 35 cm in diameter. Find the height of the cone, correct to one place of decimal.  (Marks 4)

Q5) Five years hence, father's age will be three times the age of his son. Five years ago father was seven times as old as his son. Find there present ages. (Marks 4)

Q6) Find the mean of the following data:   (Marks 4)

Q7) Solve for x :
4(x² + 1/x²) - 8(x + 1/x) + 3 = 0  (Marks 6)

Q8) Find the g.c.d. of the following polynomials :
p(x) = 8(x⁴ - 16); q(x) = 12(x³ - 8)  (Marks 2)

Q9) If b is the mean proportional between a and c, prove that
(a² - b² + c²)/(a^-2 - b^-2 + c^-2) = b⁴  (Marks 2)

Q10) Find the values of k so that the quadratic equation
x² - 2x(1 - 3k) + 7(3 + 2k) = 0
has equal roots.  (Marks 2)

Q11) Flow Chart. Omitted being out of Syllabus.  (Marks 2)

Q12) Fill in the blanks in the following table and find the Crude Death Rate (CDR) for the data:  (Marks 2)

Section - B

Q13) Solve graphically the following system of linear equations:
x + y = 3, 2x + 5y = 12  (Marks 4)

Q14) Factorise. Omitted, being out of Syllabus.  (Marks 4)

Q15) A part of monthly expenses of a family is constant and the remaining varies with the price of wheat. When the rate of wheat is Rs. 250 a quintal, the total monthly expenses of the family are Rs. 1,000 and when it is Rs. 240 a quintal, the total monthly expenses of the family are Rs. 980. Find the total monthly expenses of the family when the cost of wheat is Rs. 350 a quintal.  (Marks 4)

Q16) A piece of cloth costs Rs. 200. If the piece were 5 m longer, and each metre of cloth costed Rs. 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre ?  (Marks 4)

Q17) A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel ?  (Marks 4)

Q18) P is a point in the interior of rectangle ABCD. If P is joined to each of the vertices of the rectangle and the lengths of PA, PB and PC are 3 cm, 4 cm and 5 cm respectively, find the length of PD. (Marks 4)

Q19) Flow Chart. Omitted, being out of Syllabus. (Marks 4)

Q20) The data on number of patients attending a hospital in a month are given below. Find the average number of patients attending the hospital in a day.  (Marks 4)

Q21) Algorithm. Omitted, being out of Syllabus.  (Marks 4)

Section - C

Q22) If PAB is a secant to a circle, intersecting the circle at A and B, and PT is a tangent to the circle at T, then PA x PB = PT² - Prove.
Use the above result to prove the following:
" In Fig 3, ABC is an isosceles triangle in which AB = AC. A circle through B touches the side AC at D and intersect the side AB at P. If D is the mid point of side AC, then AB = 4AP."  (Marks 6)

Q23) The annual income of Ramesh (excluding HRA) is Rs. 1,65,000. He contributes Rs. 2000 per month in his Provident Fund and pays annual premium of Rs. 8,000 towards his Life Insurance Policy. Calculate the income tax paid by Ramesh in the last month of the year if his earlier deductions for 11 months for income tax were at the rate of Rs. 800 per month.  (Marks 6)

Assume the following for calculating income tax :

Note: Question done according to the latest syllabus Annual income taken as Rs.165000 instead of Rs.125000.