SEE UR MATHS IQ

    1.     Suppose the statement “If it rains, it pours” is false. Which of the following statements must also be false?
           

	(A)	If it pours, it rains.
	(B)	If it doesn’t rain, it doesn’t pour.
	(C)	If it doesn’t pour, it doesn’t rain.
	(D)	If it doesn’t rain, it pours.
	(D)	If it doesn’t pour, it rains.

  

2. The first term of an arithmetic sequence is –3, and each term thereafter is 3 less than the previous term. What is the value of the 30th term of the sequence? (A) –93 (B) –90 (C) –87 (D) 3 (D) 87

3. The nth term of a sequence is given by the formula an = 6n – 3. What is the sum of the first 100 terms of this sequence? (A) 300 (B) 29,700 (C) 29,850 (D) 30,000 (D) 60,000

4. In a geometric sequence, the nth term is given by the formula bgn = (2n). What is the sum of the first 10 terms of this sequence? (A) 113.3 (B) 170.0 (C) 226.7 (D) 3069 (D) 3072

5. What is the limit of  as x approaches –2? (A) – (B) 0 (C) (D) 2 (D) Impossible to tell

6. What is 3(i5 –i14)? (A) –3 (B) 3i (C) 3 (D) 3i + 3 (D)

ANSWERS

1.      C     

If a statement is false, the contrapositive of the statement will also be false. To find the contrapositive, you need to take the opposite of both parts of the statement and then switch the order. The contrapositive of “If it rains, it pours,” is “If it doesn’t pour, it doesn’t rain.”

2.      B     

The formula for the n^th term of an arithmetic sequence is a_n = a₁ + (n – 1)d, where d is the difference between the terms of an arithmetic sequence.

If the first term of a sequence is –3, and d = –3, then a_n = –3 –3n + 3 = –3n. So, the 30^th term is –3(30) = –90.

3.      D     

This sequence is an arithmetic sequence since the difference between each term is constant. The formula for the sum of the first n terms of an arithmetic sequence is:

To use this formula for this question, first calculate the values of a₁ and a₁₀₀ by plugging n = 1 and n = 100 into the given formula a_n = 6n – 3. So, we find that a₁ = 6 – 3 = 3 and a₁₀₀ = 600 – 3 = 597. The sum of the first 100 terms is therefore:

4.      D     

To answer this question quickly and efficiently, you need to know the formula for the sum of the first n terms of a geometric sequence:

where r is the common ratio of the sequence. In this problem b₁ = ³⁄₂(2¹) = 3 and r = 2, so the formula yields the answer 3069.

5.      A     

This question throws a little curveball at you because function is undefined at x = –2, since –2² – 4 = 0. However, the denominator can be factored into (x – 2)(x + 2). Then (x + 2) can be canceled from the numerator and denominator, leaving ¹⁄_x–2 as the function. Evaluating this function at x = –2, you see the limit is –¹⁄₄.

6.      D     

The powers of i repeat themselves in a cycle of four, that is iⁿ = iⁿ⁺⁴. Since i⁴ = 1, i⁵ must equal i. You can also reduce i¹⁴ by noticing that it equals i¹²  i². Since 12 is a multiple of 4, i¹² equals 1, so

So 3(i – i¹⁴) = 3(i – (–1)) = 3(i + 1) = 3i + 3.

COORDINATE GEOMETRY CLASS 9 (1)

1. In the line segment pictured below, AB + CD = AD, and AB = BC. If AD = 15,what is the distance between the midpoints of AD and BC? Note: Figure may not be drawn to scale. (A) 0.5 (B) 1.5 (C) 3 (D) 4.5 (D) 7.5

2. Which of the following lines is perpendicular to y = 3x + 4 and has 6 for its x-intercept? (A) y = 3x – 6 (B) y = –3x + 6 (C) y = –x + 6 (D) y = –x – 2 (D) y = –x + 2

3. Which of the following inequalities is graphed below? (A) y > –2x – 2 (B) y ≥ –2x – 2 (C) y < –2x – 2 (D) y > x – 2 (D) y > –x – 2

4. The following equation represents which type of graph? 2y = –6x2 + 24x – 12 (A) A parabola that opens downward with vertex (–2 ,6) (B) A parabola that opens upward with vertex (6, 2) (C) A parabola that opens upward with vertex (–2, –6) (D) A parabola that opens downward with vertex (2, 6) (D) A parabola that opens downward with vertex (–6, 2)

POLYNOMIAL TEST PAPER CLASS X

CHAPTER –POLYNOMIAL LEVEL-I (each 1 marks)

1. The zeroes of the polynomial 2x²-3x-2 are

a. 1, 2     b. -1/2,1    c. ½,-2      d. -1/2,2                                                                  

2. If a and b are zeroes of the polynomial 2x²+7x-3, then the value of a² +b² is

a. 49/4     b. 37/4       c. 61/4                d. 61/2                                                         

3. If the polynomial 6x³+16x²+px -5 is exactly divisible by 3x+ 5 , then the value of p is

a. -7      b. -5         c. 5       d. 7                                                                                  

4. If 2 is a zero of the polynomials 3x2+ax-14 and 2x3+bx2+x-2, then the value of 2 - 2b is

a. -1       b. 5         c. 9       d. -9                                                                                  

5. A quadratic polynomial whose product and sum of zeroes are 1/3 and √2 respectively is

(a) 3x² – x +3√ 2 (b) 3x² + x - 3√2 (c) 3x² + 3√2x +1 (d) 3x² – 3√2x +1    

  LEVEL-II (each 2 marks)   

1. If 1 is a zero of the polynomial p(x) = ax² -3(a-1) x -1, then find the value of a.       

2. For what value of k, (-4) is zero of the polynomial x² – x – (2k+2)?                          

3. Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2.

4. Find the zeroes of the quadratic polynomial 2x²-9-3x and verify the relationship between the zeroes and the  coefficients.                                                                                          

5. Write the polynomial whose zeroes are 2 +√3 and 2 - √3.        

   LEVEL – III (each 3 marks)                 

1. Find all the zeroes of the polynomial 2x³+x²-6x-3, if two of its zeroes are -√3 and √3.

2. If the polynomial x⁴+2x³ + 8x²+12x+18 is divided by another polynomial x²+5, the remainder comes out to be  px+q. Find the value of p and q.                                                       

3. If the polynomial 6x⁴+8x³+17x²+21x+7 is divided by another polynomial 3x²+4x+1, the remainder comes out to  be (ax+b), find a and b.                                                             

4. If two zeroes of the polynomial f(x)= x³-4x²-3x+12 are √3 and -√3, then find its third zero.

5. If a, b are zeroes of the polynomial x²-2x-15 then form a quadratic polynomial whose zeroes are (2a) and (2b).       

   LEVEL – IV (each 5 marks)                                        

1. Find other zeroes of the polynomial p(x)=2x⁴ +7x³19x²-14x +30 if two of its zeroes are √2 and -√2.

2. Divide 30x⁴ +11x³-82x²-12x-48 by (3x² +2x-4) and verify the result by division algorithm.

3. If the polynomial 6x⁴ +8x³-5x²+ax+b is exactly divisible by the polynomial 2x²-5, then find the value of a and b.     

4. Obtain all other zeroes of 3x⁴ -15x³+13x²+25x-30, if two of its zeroes are and ±√5/3.

5. If a, b are zeroes of the quadratic polynomial p(x)=kx²+4x+4 such that a² +b²=24, find the value of k.                                                                  

REAL NUMBER CLASS X

10th Real Numbers test paper 2012

1. Express 140 as a product of its prime factors

2. Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.

3. Find the LCM and HCF of 6 and 20 by the prime factorization method.

4. State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.

5. State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.

6. Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.

7. Use Euclid’s division algorithm to find the HCF of 135 and 225

8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m

9. Prove that √3 is irrational.

10. Show that 5 – √3 is irrational

11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

12. An army contingent of 616 members is to march behind an army band of 32 members in a parade.The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

13. Express 156 as a product of its prime factors.

14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.

15. Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.

16. State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating decimal.

17. State whether35/50 will have a terminating decimal expansion or a non-terminating repeatingdecimal.

18. Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.                         

19. Use Euclid’s algorithm to find the HCF of 4052 and 12576.

20. Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.

21. Use Euclid’s divis  lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

22. Prove that 3√2 5 is irrational.

23. Prove that 1/√2 is irrational. (3 marks)

24. In a school there are tow sections- section A and Section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.

25. Express 3825 as a product of its prime factors.

26. Find the LCM and HCF of 8, 9 and 25 by the prime factorization method.

27. Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.

28. State whether 29/343 will have a terminating decimal expansion or a non-terminating repeating decimal.

29. State whether 23/ 23 52 will have a terminating decimal expansion or a non-terminating repeating decimal

30. Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

31. 6. Find the LCM and HCF of 336 and 54 and verify that LCM × HCF = product of the two numbers

32. Use Euclid’s division algorithm to find the HCF of 867 and 255

33. Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

34. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9lm + 1 or 9m + 8.

35. Prove that 7 √5 is irrational.

36. Prove that √5 is irrational.

37. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

38. Express 5005 as a product of its prime factors.

39. Find the LCM and HCF of 24, 36 and 72 by the prime factorization method.

SIMILAR TRIANGLES

                                                SIMILAR TRIANGLES

1. In  ∆ PQR, given that S is a point on PQ such that STIIQR and PS/SQ=3/5 If PR = 5.6 cm, then find PT.

2. In  ABC, AE is the external bisector of <A, meeting BC produced at E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, then find CE.

3. P and Q are points on sides AB and AC respectively, of ∆ ABC. If AP = 3 cm,PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

4. The image of a tree on the film of a camera is of length 35 mm, the distance from the lens to the film is 42 mm and the distance from the lens to the tree is 6 m. How tall is the portion of the tree being photographed?

5. D is the midpoint of the side BC of ∆ABC. If P and Q are points on AB and on AC such that DP bisects <BDA and DQ bisects <ADC, then prove that PQIIBC.

6. If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

7. If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

8. ABCD is a quadrilateral with AB =AD. If AE and AF are internal bisectors of <BAC and <DAC

respectively, then prove that EFIIBD. In a  ∆ABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and <ADE = <DEA. Prove that  ABC is isosceles.

9. In a   ABC, points D, E and F are taken on the sides AB, BC and CA respectively such that  DEIIAC and FEIIAB.

10. The internal bisector of <A of ∆ ABC meets BC at D and the external bisector of <A    meets BC produced at E. Prove that BD/ BE = CD/CE

11. If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the triangles on each side of the perpendicular are similar to the whole triangle.

12. A man sees the top of a tower in a mirror which is at a distance of 87.6 m from the tower. The mirror is on the ground, facing upward. The man is 0.4 m away from the mirror, and the distance of his eye level from the ground is 1.5 m. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line).