MATHEMATICA

Saturday, 12 January 2013

10 IMPORTANT QUESTIONS FROM EACH CHAPTER FOR CLASS X SA2

Coordinate Geometry - x
Short answer questions
1. Find the distance of the point A (4, - 3) from the Y-axis.
2. Find perimeter of a triangle with vertices (0, 5), (0, 0) and (4, 0).
3. If the points P (1, 2), O (0, 0) and Q (a, b).
4. Find area of triangle PQR whose vertices are P (-6, 4), Q (-5, 5) and R (-3, 9).
5. In what ratio does the x-axis divide line segment joining the points (-4, -6) and (-1, 7)? Find the coordinates of the point of division.
6. Find the area of a rhombus PQRS if its vertices are P (3, 0), Q (4, 5),R (-1, 4) and S (-2, -1).
7. The line joining the points (2, -1) and (-5, -6) is bisected at Z. If Z lies on the line 2x + 4y + k = 0, find the value of k.
8. Find area of quadrilateral ABCD whose vertices are A (1, 0), B (5, 3),
C (2, 7) and D (-2, 4).
9. If the points (r, 0) (0, s) and (1, 1) are collinear. Prove that 1/r + 1/s = 1
10. Using coordinate geometry prove midpoint of hypotenuse.

Arithmetic Progressions - x
short answer questions
1. Find 11th term from the end of the AP 14, 19, 24, 29, ... 249
2. Is 450 a term of the 4, 7, 10, 13, 16,
3. If the 8th term of the AP is 31 and the 15th term is 16 more than the 11th term, find the A.P.
4. Which term of the AP of the given AP is the first negative term? 55, 50, 45,.....................
5. Determine the general term of AP whose 7th term is 47 and 13th term is 83
6. 5 times the 5th term of an AP is equal to 8 times its 8th term. Find its 13th term.
7. How many 3 digit numbers are divisible by 11.
8. Find 3 terms in AP whose sum is – 3 and product is 8.
9. The sum of first six terms of an AP is 42. If the ratio of its 10th term to its 30th term is 1:3. Find its 20th term.
10. The sum of the third and seventh terms of AP is 6 and their product is 8.Find the sum of first sixteen terms of the AP.

Some Applications of Trigonometry - x
1. A vertical pole is 10m high and the length of its shadow is 10 3 m. find the angle of elevation at that instant?
2. The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.
3. If the height of a tower and the distance of the point of observation from its foot, both, are increased by 20%, then the angle of elevation of its top remains unchanged.
4. A round balloon of radius r subtends an angle a at the eye of the observer while the angle of elevation of its centre is β. Prove that the height of the centre of the balloon is r sin β cosec a/2
5. From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower are a and β (β > a ) Find the distance between the two objects.
6. At the foot of a mountain, the angle of elevation of its summit is 45°.After ascending 1km towards the mountain up at an inclination of 30°,the angle of elevation changes to 60°. Find the height of the mountain.
7. The angles of elevation of the top of a tower from two points at a distance of a m and b m from the base of the tower and in the same straight line with it are complementary.Prove that the height of the tower is (ab)1/2 metre.

Area Related to Circles - x
1. Find area of the largest triangle that can be inscribed in a semi-circle of radius 2r units.
2. A circle of radius 11cm is inscribed in a square find the area of the region enclosed between the square and the circle.
3. Triangle ABC is inscribed in a circle of radius 5cm, AB being the diameter and BC = 8cm. find the area of the region enclosed between the circle and the triangle.
4. A piece of wire 40cm long is bent into the form of an arc of a circle subtending an angle of 60° at its centre. Find the radius of the circle.
5. Arcs with radii 7cm each and with centers A, B and C have been drawn at the vertices of triangle ABC. Find area of triangle enclosed between the arcs.
6. The diameter of front and rear wheels of a tractor are 80cm and 2m respectively. Find the number of revolutions that rear wheel will make in covering a distance in which the front wheel make 700 revolutions.
7. The area of an equilateral triangle ABC is 49 (3)1/2 sq cm. With each vertex of the triangle as center, a circle is drawn with radius equal to half the length of the side of the triangle. Find area of triangle enclosed between the arcs of three circles.
8. A race track is in the form of a ring whose inner circumference is 440m and the outer circumference is 506m. Find the width of the track and also the area.
9. a playground has the shape of a rectangle, with two semi-circles on its smaller sides as diameters, added to its outside. If the sides of the rectangle are 36m and 24.5m, find the area of the playground.
10. Two circles with centres A and B touch each other at the point C. If AC = 8cm and AB = 3cm, find the area of the shaded region enclosed between the two circles.

Probability - x
1. Two coins are tossed simultaneously. Find probability of getting at most 1 tail.
2. Find probability of 52 Thursdays in a leap year.
3. A card is drawn from a well shuffled deck of playing cards. Find probability of getting a king or a red queen.
4. A card is drawn from a well shuffled deck of playing cards. Find probability of getting a king or spade.
5. Three coins are tossed simultaneously. Find probability of getting exactly 2 tails or exactly 2 heads.
6. Cards are numbered 1 to 100. Find probability of getting an even number or an odd number.
7. Two dice are thrown once. Find probability of getting same number on both dice.
8. A card is drawn from a well shuffled deck of playing cards. Find probability of getting a non face card.
9. A bag contains 36 balls out of which z are red. If 18 more red balls are put in the bag, the probability of drawing a red ball will double. Find z.
10. Two dice are thrown once. Find probability of getting 7 as sum on numbers on both dice.

Quadratic Equations - x
1. Determine the set of values of p for which the quadratic equation px2 + 6x +1=0 has real roots.
2. Solve by factorisation 4x2 – 4ax + (a2 – b2) = 0
3. Find the roots of the equations r2s2 – 3rsx + 2s2 = 0 by the method of completing the square.
4. Find the value of k for which kx2 – 6x – k = 0 has real roots.
5. Find the value of k for which the equation x2 + 4kx + 25 = 0 has no real roots.
6. If the sum and product of the roots of the equation rx2 + 6x + 4k = 0 are equal, then find r.
7. Find the value of r for which the given quadratic equation x2 + 4rx + 25= 0 has real and distinct roots.
8. Using quadratic formula solve 9x2 – 9 (a + b) x + (2a2 + 5ab + 2b2) = 0
9. Prove the both the roots of the equation (x – r) (x – s) + (x – s) (x – t)+ (x – t) (x – r) = 0 are real but they are equal only when r = s = t
10.The difference of square of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the two numbers.

Surface Area & Volume - x
1. A cylindrical pencil sharpened at one edge is combination of ___ and ___
2. A metallic spherical shell of internal and external diameters 6cm and 12cm respectively is melted and recast into the form a cone of base diameter 6cm. Find the height of the cone.
3. Three metal cubes with edges in ratio 3:4:5 are melted and converted into a single cube whose diagonal is36(3)1/2 . Find edges of the given cubes
4. The height of a cone is 90cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27of the volume of the given cone,then find the height above the base at which the section has been made.
5. Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius 2a.
6. A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water over flows?
7. A solid toy is in the form of a hemisphere surmounted by a right circular cone. the height of the cone is 2cm and diameter of the base is 4cm.Determine the volume of the toy.
8. A golf ball has diameter equal to 4.1cm. Its surface has 150 hemispherical dimples each of radius 2mm. Calculate total surface area which is exposed to the surrounding.
9. A hemisphere, cone and a cylinder stand on the same base and have equal heights. Find the ratio of their volumes.
10. A hemispherical sheet of metal of radius 63cm is joined along the radii to form a cone. Find the volume of the cone.

Sunday, 10 June 2012

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 30^th 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 a_n = 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 bg_n = (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)

(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.

Wednesday, 6 June 2012

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).

Saturday, 12 May 2012

POLOYNOMIALS (X)

POLOYNOMIALS (X)

1. Show that x² – 3 is a factor of 2x⁴ + 3x³ -2x² -9x – 12

2. Divide: 4x³ + 2x² + 5x - 6 by 2x² + 3x + 1

3. Find other zeroes of the polynomial p(x) = 2x⁴ + 7x³ – 19x² – 14x + 30 if two of its zeroes are √2 and -√2

4. Find all the zeroes of the polynomial 3x⁴ + 6x³ - 2x² – 10x – 5, if two of its zeroes are √5/3 and -√5/3

5. Find all the zeroes of 2x⁴ – 3x³ – 3x² + 6x – 2, if it is known that two of its zeroes are √2 and -√2

6. Find all the zeroes of 2x⁴ - 9x³ + 5x² +3x – 1, if two of its zeroes are 2 + √3 and 2 - √3

7. Find all the zeroes of polynomial 4x⁴ – 20x³ + 23x² + 5x – 6 if two of its zeroes are 2 and 3

8. If the polynomial f(x) = x⁴ - 6x³ +16x² - 25x + 10, is divided by another polynomial x² - 2x + k the remainder

Comes out to be x + a, find k and a

9. On dividing x³ – 3x²+ x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and -2x +4, respectively

Find g(x)

10. If the polynomial 6x⁴ + 8x³ – 5x² + ax + b is exactly divisible by the polynomial 2x² – 5, then find the values of

a and b

11. Find the values of m and n so that x⁴ + mx³ + nx² – 3x + n is divisible by x² – 1

12. What must be subtracted from 2x⁴ – 11x³ + 29 x² – 40x + 29, so that the resulting polynomial is exactly divisible

By x²-3x + 4

13. Find the polynomial, whose zeroes are 2 + √3 and 2 - √3 14.Form a quadratic polynomial, one of whose zero is 2 + √5 and the sum of zeroes is 4

15. If α and β are zeroes of the polynomial x²– 2x – 15, then form a quadratic polynomial whose zeroes are 2α and 2β 16.Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2

17. Find the zeroes of the polynomial and verify the relationship between the zeroes and the coefficient

a) 4x² – 4x + 1 b) x² – 3 c) √3x² – 8x + 4√3

18. If α and β are the zeroes of the polynomial 2y² + 7y + 5, write the value of α +β + αβ

19. If one root of the polynomial 5x³ + 13x + k is reciprocal of the other, then find the value of k?

20. If one zero of the polynomial (a² + 9) x² +13x + 6a is reciprocal of the other. Find the value of a

21. If the zeroes of the polynomial x³ – 3x² + x+1 are a – b, a, a + b, find a and b

22. If α and β are the zeroes of the polynomial f(x) = 6x² + x -2, find the value of 1 + 1 - αβ

α β

23.If α and β are the zeroes of the quadratic polynomial 2x²+ 3x - 5, find the value of 1 + 1

α β

24. If α and β are the zeroes of the polynomial f(x) = x² – 5x + k such that α – β = 1, find k

25. If α, β are the zeroes of a polynomial, such that α + β = 6 and α β = 4, then writes the polynomial

26. If the product of zeroes of the polynomial ax² – 6x – 6 is 4, find the value of a

27.If α, β are the zeroes of quadratic polynomial 2x² + 5x + k, find the value of k such that (α + β)² – α β = 24

28. If α and β are zeroes of x² + 5x + 5, find the value of α^-1 + β^-1

29. α, β are the zeroes of the quadratic polynomial x² – (k+6)x +2 (2k – 1). Find the value of k if α + β = ½ α β

30. if α, β are the zeroes of the quadratic polynomial x² – 7x + 10, find the value of α³ + β³

31. Find the sum and the product of the zeroes of cubic polynomial 2x³ -5x² – 14x + 8

32. Find the sum and product of the zeroes of quadratic polynomial x² – 3

33. If 1 is a zero of polynomial ax² – 3(a-1) -1, then find the value of a

34. If α, β are zeroes of quadratic polynomial x² – (k + 6)x + 2(2k-1).Find k if α + β = 1/2αβ

35. Divide (6 + 19x + x² – 6x³) by (2 +5x – 3x²) and verify the division algorithm