http://www.cbseguess.com/------------------------------------------------------------------------------------------------------------------------www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com HALF YEARLY EXAMINATION -2006 SAMPLE PAPER MATHEMATICS CLASS -X TIME -3 HOURS M.M.80 INSTRUCTIONS: (1) Answer all questions. (2) Work should be neat and tidy. (3) As far as possible attempt all questions in order. SECTION -A (Each question carries 2 marks.) 1. The 7th term of an A.P. is 34 and 15 th is 74.Find 40th term. 2. A sum of Rs.8, 400/-is borrowed at 10% per annum compound interest, to be paid back in 2 years by two equal annual installments. Find the value of each installment. 3. Determine the value (s) of p for which the quadratic equation 2x2 + p x+8 = 0, have real roots. 4. Evaluate: tan1º . tan 20. tan 30…………….tan 870. tan 880. tan 890 5. Find 104th term of the following A.P. 35, 29, 23, 17, 11,…………………… Also find the sum of 104 terms? 6. D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Prove that CDCA = CACB. 7. If AB and CD are the chords of a circle, which intersect each other at a point P out side the circle. If AB = 7cm, CD = 10 cm, PD = 8 cm. Find BP? 8. The diameter of a metallic sphere is 12 cm. The sphere is melted and drawn into a wire of uniform circular cross-section. If the length of the wire is 72 m. find its radius. 9. Find the value of k for which the H.C.F. of x2 + (k-2) x -3k + 6 and x2 + 3x – 3k is x +6 . http://www.cbseguess.com/------------------------------------------------------------------------------------------------------------------------www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com -2-10. A radio is available for Rs. 1500 cash or Rs 360 as cash down payment followed by three equal monthly instalments of Rs. 390. Find the rate of interest charged under instalment scheme. SECTION -B (each question carries 3 marks.) 11. Solve graphically: 4x + y = 16 6y – 4x = 12 Determine the vertices of the triangle formed by the lines representing these equations and the x-axis? 12. In an equilateral triangle ABC, D is a point on BC such that BD= ⅓ BC. Prove that 9AD2 = 7AB2 . 13. Prove: 1tan1tanSecAAASecA = AAcossin1 14. From a point on the ground 60 m away from the foot of a tower, the angle of elevation of the top of the tower is 45º. The angle of elevation to the top of a water tank on the top of the tower is 60º Find the height of the tower and the depth of the tank. 15. Find the sum of first 50 terms of A.P. whose third term is 11 and the seventh term is 31. 16. Find the values of p and q , if the H.C.F. of the polynomials F(x) = (x+3) ( 2x2 -3x + p) G(x) = (x-2) ( 3x2 + 10x -q) is (x+3) (x-2) 17. Simplify: 11x -11x -122x -144x 18. An aeroplane left 30 minutes later than its scheduled time; and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hour from its usual speed. Determine its usual speed. 19. Prove:AAsin1sin1 = sec A + tan A 20. In figure ABCD is a quadrilateral with AB=AD. AE and AF are respectively Bisectors of angle BAC and angle DAC. Prove that EF//BD. http://www.cbseguess.com/------------------------------------------------------------------------------------------------------------------------www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com SECTION -C (Each question carries 6 marks) 21. Prove alternate segment theorem. In the adjacent figure two circles are intersecting each other at the points A and B. If PQ is the common tangent of the circle. Using the above theorem, Prove, ∠ PAQ + ∠ PBQ = 180º 22. A container made up of a metal sheet is in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 7cm and 14 cm respectively. Find the cost of milk which can completely fill the container at the rate of Rs. 15 per litre and the cost of the metal sheet used, if it cost 50 paise per square cm. 23. State and prove “ Basic Proportionality Theorem”. Using this theorem prove the diagonals of a trapezium divides each other in the same ratio. 24. Two ships are sailing in the sea on either side of a lighthouse. The angles of depression of the two ships are observed as 600 and 450 respectively. If the distance between the two ships is Find the height of the lighthouse. 25.. Mr Gupta monthly income is Rs. 30,000. He contributes Rs. 2000 per month towards is C.P.F. account, he invested Rs. 80,000 in N.S.C. He also invested Rs. 20,000 in infrastructure bonds. If he pays Rs. 4000 per month as tax for 11 months. Calculate the tax http://www.cbseguess.com/------------------------------------------------------------------------------------------------------------------------www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com paid by him in the last month of the year. INCOME TAX SLAB 1. (A) For men (below 65 years) Taxable income Rate Up to Rs. 1,00,000 Nil Rs. 1,00,001-1,50,000 10% of amount exceeding rs. 1,00,000 Rs 1,50,001-2,50,000 Rs.5000+20% of the amount exceeding Rs.1,50,000 Exceeding Rs 2,50,000 Rs. 25,000+30% of the amount exceeding Rs 2,50,000 2. Women (below 65 years) Taxable income Rate Up to Rs. 1,35,000 Nil Rs 1,35,001-Rs 1,50,000 10% of the amount exceeding Rs 1,35,000 Rs 1,50,001-Rs 2,50,000 Rs 1,500+20% of them amount exceeding Rs 1,50,000 Exceeding Rs 2,50,000 Rs 21,500+30% of the amount exceeding Rs 2,50,000 3. (C) For senior citizens (65 years or more) Taxable income Rate Up to Rs 1,85,000 Nil Rs 1,85,001-2,50,000 20% of the amount exceeding Rs 1,85,000 Above Rs 2,50,000 Rs 13,000+30% of amount exceeding Rs 2,50,000 4. Surcharge: 10% of the amount of tax payable if the taxable income exceeds rs 10,00,000 5. Educational cess: 2% of the amount of tax payable. 6. Concession for saving: notified savings (PF, LIC, PPF, mutual fund etc) up to a maximum of rs 1, 00,000 are exempted from payment of income tax. ALL THE BEST