section - 2 algebra 1. linear inequations 2....

SECTION - 2

Algebra1. Linear inequations2. Quadratic Equations3. Ratio and Proportion4. Remainder and Factor theorem5. Matrices

Submission Date : / /

175

2003

1. Solve using the quadratic formula : x2 – 4x + 1 = 0.

22. If3a 4b 3a 4b3c 4d 3c 4d

, prove that

a cb d

3. Find the value of a, if (x – a) is a factor of x3 – a2x + x + 2.

4. The work done by (x – 3) men in (2x + 1) days and the work done by (2x + 1)men in (x + 4) days are in the ratio of 3 : 10. Find the values of x.

5. In an auditorium, seats were arranged in rows and columns. The number ofrows was equal to the number of seats in each row. When the number of rowswas doubled and the number of seats in each row was reduced by 10, thetotal number of seats increased by 300. Find :(i) The number of rows in the original arrangement.(ii) The number of seats in the auditorium after re-arrangement.

6. Find x and y if :-

3 2 2 4 22 4

1 4 1 5

xy

7. Solve 2 < 2x – 3 < 5, x R and mark it on a number line.

2004

1. Given A =2 1 3 2 1 0

,B and C2 0 4 0 0 2

find the matrix X, such that A + X = 2B + C.

2. Use the factor theorem to factorise completely x3 + x2 – 4x – 4.

3. What number should be subtracted from each of the following numbers, 23,30, 57 and 78 so that the remainders are in proportion?

4. Solve the equation 3x2 – x – 7 = 0 and give your answer correct to two decimalplaces.

5. Given that x I, solve the inequation and graph the solution on the numberline:

43 22 3

x x

AlgebraAssignment Sheet

176

MT EDUCARE LTD.X - ICSE (Maths)

6. P and Q are centres of circle of radius 9 cm and 2 cm respectively. PQ = 17 cm.R is the centre of a circle of radius x cms, which touches the above circlesexternally. Given that PRQ = 90º, write an equation in x and solve it.

2005

1. (x – 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression isdivided by (x – 3), it leaves the remainder 3. Find the value of a and b.

2. What number must be added to each of numbers 6, 15, 20 and 43 to makethem porportional?

3. A = {x : 11x – 5 > 7x + 3, x R} andB = {x : 18x – 9 > 15 + 2x, x R}.Find the range of Set A B and represent it on a number line.

4. Solve the following equation and give your answer up to two decimal places :x2 – 5x – 10 = 0.

5. Find the value of x, given that A2 = B

A =2 120 1

, B =40 1

x.

6. By increasing the speed of a car by 10 km/hr, the time of journey for a distanceof 72 km is reduced by 36 minutes. Find the original speed of the car.

2006

1. If3 5 73 5 3

x yx y , find x : y.

2. Show that 2x + 7 is a factor of 2x3 + 5x2 – 11x – 14. Hence, factorise the givenexpression completely, using the factor theorem.

3. Given that x R, solve the following inequality and graph the solution on thenumber line: –1 < 3 + 4x < 23

4. If A =4 26 3

, B =

0 21 1

and C =2 3

1 1

Find A2 – A + BC.

5. Solve the equation 2x –1x = 7. Write your answer correct to two decimal

places.

6. A shopkeeper buys a certain number of books for ` 720. If the cost per bookwas ` 5 less, the number of books that could be bought for ` 720 would be 2more. Taking the original cost of each book to be ` x. Write an equation inx and solve it.

MT EDUCARE LTD. X - ICSE (Maths)

177

2007

1. If x =a 3b a – 3ba 3b a 3b

, prove that 3bx2 – 2ax + 3b = 0

2. If 23 4 1 7 05 0 1 10 5

yx

find the values of x and y.

3. If A =1 02 1

B =2 31 0

. Find A2 + AB + B2.

4. Show that (x – 1) is a factor of x³ – 7x2 + 14x – 8. Hence, completely factorisethe above expression.

5. Solve the following quadratic equation for x and give your answer correct totwo decimal places :x2 – 3x – 9 = 0

6. Five years ago, a woman’s age was the square of her son’s age. Ten yearshence her age will be twice that of her son’s age. Find :(i) The age of the son five years ago.(ii) The present age of the woman.

7. Solve the following inequation and graph the solution on the number line.

2 1 1–2 33 3 3 x ; Rx .

2008

1. Some students planned a picnic. The budget for the food was ` 480. As eightof them failed to join the party, the cost of the food for each member increasedby ` 10. Find how many students went for the picnic.

2. If (x – 2) is a factor of 2x3 – x2 – px – 2.With the value of p, factorize the above expression completely.

3. Solve the given inequation and graph the solution on the number line.2y – 3 < y + 1 < 4y + 7; y R.

4. If1 4

2 3

+ 2M = 33 20 3

.

Find the matrix M.

178


5. If8a 5b 8a 5b8c 5d 8c 5d

’ prove that

a cb d

6. Solve the following quadratic equation for x and give your answer to twodecimal places:5x (x + 2) = 3

7. Given A =p 00 2

, B =0 q 2 2

, C1 2 2 2

and BA = C2.

Find p and q.

2009

1. The speed of an express train is x km/h and the speed of an ordinary train is12 km/h less than that of the express train. If the ordinary train takes onehour longer than the express train to cover a distance of 240 km, find the speedof the express train

2. Solve the following quadratic equation and give the answer correct to twosignificant figures.

4x2 – 7x + 2 = 0

3. Solve the inequation and represent the solution set on the number line.

– 3 + x <83x

+ 2 <143

+ 2x, where x I.

4. Given that x + 2 and x + 3 are factors of 2x3 + ax² + 7x – b. Determine thevalues of a and b.

5. What least number must be added to each of the numbers, 5, 11, 19 and 37 sothat they are in proportion?

6. Find x and y, if2 3 16

3 2 9

x xy y .

7. Given that3 2

3 2

a 3ab 63b 3a b 62

find a : b.

2010

1. Given x =2 2 2 2

2 2 2 2

a b a b

a b a b

use componendo and dividendo to prove that b2 =2

2

2a1

xx


179

2. Solve the following inequation and represent the solution set on the numberline.

– 3 < –12

–23x

<56

, x R.

3. Without solving the following quadratic equation, find the value of ‘p’ for whichthe roots are equal.px2 – 4x + 3 = 0.

4. When divided by x – 3 the polynomials x3 – px2 + x + 6 and2x3 – x2 – (p + 3)x – 6 leave the same remainder. Find the value of ‘p’.

5. Evaluate4sin30º 2cos60º 4 5sin90º 2cos0º 5 4

6. Given A =3 21 4

, B =

61

, C =45

and D =22

. Find AB + 2C – 4D

7. If x, y, z are in continued proportion, prove that2

2

( )( )

x y xy z z

8. A positive number is divided into two parts such that the sum of the squaresof the two parts is 208. The square of the larger part is 8 times the smallerpart. Taking x as the smaller part, find the number.

9. Use the Remainder Theorem to factorise the following expression :2x3 + x² – 13x + 6

2011

1. Find the value of ‘k’ if (x – 2) is a factor of : x3 + 2x2 – kx + 10.Hence determine whether (x + 5) is also a factor.

2. If A =3 54 2

and B =24

, is the product AB possible ? Give a reason. If yes,

find AB.

3. Solve the following inequation and represent the solution set on the numberline 2x – 5 < 5x + 4 < 11, where x I

4. Using componendo and dividendo, find the value of x .

3 4 3 53 4 3 5

x xx x = 9

180


5. ` 480 is divided equally among ‘x’ children. If the number of children were 20more then each would have got ` 12 less. Find ‘x’ .

6. If A =2 51 3

, B =4 21 3

and I is the identity matrix of the same order and

‘At’ is the transpose of matrix A, find At B + BI.

7. Solve the following equation : x –18x = 6. Give your answer correct to two

significant figures.

8. 6 is the mean proportional between two numbers x and y and 48 is the thirdproportional of x and y. Find the numbers.

2012

1. If A =3 11 2

and I =1 00 1

, find A2 – 5A + 7 I.

2. A car covers a distance of 400 km at a certain speed. Had the speed been 12km/h more, the time taken for the journey would have been 1 hour 40 minutesless. Find the original speed of the car.

3. Solve the following question and give your answer correct to 3 significantfigures:5x² – 3x – 4 = 0

4. If x =a 1 a 1a 1 a 1

, using properties of proportion show that x2 – 2ax + 1 = 0

5. Solve the following inequation and represent the solution set on the numberline :

4x – 19 <35x

– 2 <2

5

+ x, x R

6. Given2 13 4

X =76

. Write :

(i) the order of the matrix X(ii) the matrix X.

7. Using the Remainder Theorem, factorise completely the following polynomial.3x3 + 2x2 – 19x + 6

8. The monthly pocket money of Ravi and Sanjeev are in the ratio 5 : 7. Theirexpenditures are in the ratio 3 : 5. If each saves ` 80 every month, find theirmonthly pocket money.


181

9. Without solving the following quadratic equation, find the value of ‘m’ forwhich the given equation has real and equal roots.x2 + 2 (m – 1)x + (m + 5) = 0.

2013

1. Given A =2 62 0

, B =3 24 0

, C =4 00 2

.

Find the matrix X such that A + 2X = 2B + C.

2. What number must be added to each of the numbers 6, 15, 20 and 43 to makethem proportional?

3. If (x – 2) is a factor of the expression 2x3 + ax2 + bx – 14 and when the expressionis divided by (x – 3), if leaves a remainder 52; find the values of a and b.

4. Solve the following equation and calculate the answer correct to two decimalplaces :-x2 – 5x – 10 = 0.

5. Solve the following inequation, write the solution set and represent it on the

number line :1 1

1 ,3 2 3 6

x xx R

6. Find x and y, if34

x xy y

21

=5

12

7. Without solving the following quadratic equation, find the value of ‘p’ for whichthe given equation has real and equal roots :x2 + (p – 3) x + p = 0

8. Using the properties of proportion solve for x, given4

2

1 172 8

xx .

9. A shopkeeper purchases a certain number of books for ` 960. If the cost perbook was ` 8 less, the number of books that could be purchased for` 960 would be 4 more. Write an equation, taking the original cost of eachbook to be ` x, and solve it to find the original cost of the books.

2014

1. Find the values of x, which satisfy the inequation

–256

<12

–x232, x W. Graph the solution set on the number line.

182


2. Find x, y if

–2 03 1 x

–12 + 3

–21 = 2

y 3

3. Using the Remainder and Factor Theorem, factorise the following polynomial:x3 + 10x2 – 37x + 26

4. The numbers 6, 8, 10, 12, 13, and x are arranged in an ascending order.If the mean of the observations is equal to the median, find the value of x.

5. If2 2

2 2

x yx y =

178

, then find the value of :

(i) x : y

(ii)3 3

3 3

x yx y

6. Let A =

2 10 –2 , B =

4 1–3 –2 , C =

–3 2–1 4 .

Find A2 + AC – 5B.

7. If (x – 9) : (3x + 6) is the duplicate ratio of 4 : 9, find the value of x.

8. Solve for x using the quadratic formula. Write your answer correct totwo significant figures. (x – 1)2 – 3x + 4 = 0.

9. A two digit positive number is such that the product of its digits is 6. If 9 isadded to the number, the digits interchange their places. Find the number.

2015

1. Solve the following inequation and write the solution set :13x – 5 < 15x + 4 < 7x + 12, x R

Represent the solution on a real number line.

2. If A =30 1

x, B =

9 160 – y , find x and y where A2 = B.

3. Find the value of ‘k’ for which x = 3 is a solution of the quadratic equation,(k + 2)x2 – kx + 6 = 0.Thus find the other root of the equation.

4. If a, b, c are in continued proportion, prove that(a + b + c) (a – b + c) = a2 + b2 + c2.


183

5. Find ‘a’ if the two polynomials ax3 + 3x2 – 9 and 2x3 + 4x + a, leave the sameremainder when divided by x + 3.

6. If A =3 72 4

, B =0 25 3

and C =1 –5–4 6

Find AB – 5C.

7. Sum of two natural numbers is 8 and the difference of their reciprocal is2

15 .

Find the numbers.

8. Given3 3

2 2

12 276 8 9 27

x x y yx y . Using componendo and dividendo find x : y.

2016

1. Using remainder theorem, find the value of k if on dividing 2x3 + 3x2 - kx + 5by x – 2, leaves a remainder 7.

2. Given2 0 1 0

A = and I =-1 7 0 1

and A2 = 9A + mI. Find m.

3. Solve the following inequation, write the solution set and represent it onthe number line.

-3(x – 7) > 15 – 7x >x +1

3, x R

4. If (3a + 2b): (5a + 3b) = 18: 29, find a : b.

5. Solve the quadratic equation x2 – 3(x + 3) = 0; Give your answer correct to twosignificant figures.

6. Given matrix A =o o

o o4sin30 cos0cos0 4sin30

and B =45

If AX = B(i) Write the order of matrix X.(ii) Find the matrix ‘X’.

7.3 3 3x y z x y z 3xyz

If = = show that + + =3 3 3a b c abca b c

184


8. A bus covers a distance of 240 km at a uniform speed. Due to heavy rain itsspeed gets reduced by 10 km/h and as such it takes two hrs longer to coverthe total distance. Assuming the uniform speed to be ‘x’ km/h, form anequation and solve it to evaluate ‘x’.

2017

1. If b is the mean proportion between a and c, show that :4 2 2 4

4 2 2 4

a a b bb b c c

=2

2

ac

2. Solve the equation 4x2 – 5x – 3 = 0 and give your answer correct to twodecimal places.

3. If A =1 33 4

and B =–2 1–3 2

and A2 – 5B2 = 5C. Find matrix C where C is a

2 by 2 matrix.

4. What must be subtracted from 16x3 – 8x2 + 4x + 7 so that the resultingexpression has 2x + 1 as a factor ?

5. Solve the following inequation and represent the solution set on a numberline.

–812

< –12

– 4x 712

, x I

6. Given matrix B =1 18 3

. Find the matrix X if, X = B2 – 4B.

Hence solve for a and b given Xab

=550

.

7. If7 27 – 2m nm n

=53

, use properties of proportion to find

(i) m : n

(ii)2 2

2 2–m nm n

8. The sum of the ages of Vivek and his younger brother Amit is 47 years. Theproduct of their ages in years is 550. Find their ages.

section - 2 algebra 1. linear inequations 2....

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