**Time Allowed : 3 to 3 ^{1}/ ₂ hours [Maximum Marks: 80 General Instructions :**

**(i)All questions are compulsory.**

**(ii) The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 10 questions of 1 mark each, Section B comprises of8 questions of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D comprises of 6 questions of 4 marks each.**

**(iii) Question numbers 1 to 10 in Section A are multiple choice questions where you are to select one correct option out of the given four.**

**(iv) There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of four marks each. You have to attempt only one of the alternatives in all such questions.**

**(v) Use of calculators is not permitted. –**

**(vi) An additional 15 minutes time has been allotted to read this question paper only**

**Section ‘A’**

**The value of P for which the polynomial x³+4x² -px +8 is exactly divisible by (x-2) is**

**(a) 0 (b)3**

**(c)5 (d)16**

**2.If HCF and LCM of two numbers are 4 and 9696 then the product of the number is**

**(a)9696 (b)24242**

**(c)38784 (d)4848**

**3.The pair of linear equations 2x +7y = k, kx +21y = 18 has infinitely many solution if**

**(a)3 (b)6**

**(c)9 (d)18**

**4. If cosecθ=4 and cotθ=√3k then k =**

**(a)1 (b)2**

**(c)√5 (d)5**

**Section B**

**5. Can two numbers have 18 as their HCF and 380 as their LCM? Give reason.**

**Class 10 maths ncert solutions chapter 1: Real numbers**

**6. The cost of 2 kg of apples and 1 kg of grapes on a day was found to be Rs. 160. Also, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent this situation algebraically.**

**7. Write three rational numbers between √2 and √3.**

**8.Express 43 in the form of 7q + 8.**

**9. Use Euclid’s division algorithm to find HCF of 456 and 84.**

**10-If one zero of the polynomial p(x) = (a²+9)x²+45x +6a is reciprocal of other, find the value of a.**

**SectionC**

**11. For which value of p does the equations 4x +py +8 = 0 and x + y +1=0 have unique solution.**

**12. If 7sin²θ + 3cos²θ = 4 then prove that**

**13. What must be subtracted from the polynomial 4x³+ 5x²–10x +7 so that resulting polynomial is exactly divisible by 2x +5.**

**14. Find the value of a and b if the following pair of the linear equation has infinite number of solutions.**

**4x–(a +2)y =b+2, 6x + (1 – 3a)y = b**

**15. If the sum of the square of the zeroes of the polynomial is 6x² + x +k is 25/36.find the value of k.**

**16.The altitude AD of ΔABC is drown from the vertex A on BC where ∠A is obtuse . If AD = 10cm, BD =10 cm and CD =10√3cm, find ∠A.**

**17. State and prove Pythagoras theorem. Also, prove thatΔ ABC is an isosceles triangle with AC =BC, if AB² = 2AC² then ABC is a right triangle.**

**18. If A,B and C are interior angles of a triangle ABC,prove that**

**19. Find the mean of the following distribution.**

Class | Frequency |

0─20 | 5 |

20─40 | 8 |

40─60 | 10 |

60─80 | 12 |

80─100 | 7 |

100─120 | 8 |

**20. Find the mode of the following data.**

Class Interval | Frequency |

10─20 | 7 |

20─30 | 12 |

30─40 | 20 |

40─50 | 11 |

50─60 | 8 |

**Section D**

**21. Find the largest positive integer that will divide 100,245 and 343 leaving reminders 4,5 and 7 respectively.**

**22. Solve the following equations graphically also find the points where the lines meet the x-axis .**

**x +2y =5**

**2x + 3y = 4**

**23.Find all the zeroes of the polynomial 4x ^{4}–2oxᶾ +23x² +5x –6, if two of its zeroes are 2 and 3.**

**24.In an equilateral ABC, AD is the altitude drawn from A on side BC, Prove that 3AB² = 4AD².**

**25. For the f0llowing frequency distribution draw a more than type ogive and then determine the median.**

Measurements | 0─10 | 10─20 | 20─30 | 30─40 | 40─50 | 50─60 |

Frequency | 8 | 12 | 21 | 30 | 22 | 7 |

**26. If the median of the following frequency distribution is 28.5, find values of x and y.**

Measurements | 0─10 | 10─20 | 20─30 | 30─40 | 40─50 | 50─60 | Total |

Frequency | 5 | x | 20 | 15 | y | 5 | 60 |

**Q27.Sum of the areas of two squares is 468 sq.m, if the difference of their **perimeter is 24 m, find the sides of two square.

**Q28. If in Δ ABC AD is median and AM ⊥BC, then prove that AB² +AC² =2AD²+1/2BC².**

**Q30. Prove that:**

**Q31. Prove that √3 is an irrational number, Hence show that 7 + 2√3 is also an irrational number.**

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