# ALGEBRA

Algebra is a branch of Mathematics which uses letters of the alphabet to find unknown numbers. These letters are called variables. The values that are known in the given expression such as numbers are called constants. Algebra plays a vital role in the mathematics section of  SSC CGL and CHSL. It is important because it helps you not just in solving questions in the exams, but also in answering many of your day-to-day life problems. Algebra is an important topic that carries a lot of marks. In SSC CGL tier-1 and CHSL tier-1 the weightage of Algebra is 2-3 questions and, in tier-2 exams, the weightage is 10 - 12 questions. Let us discuss some important models in Algebra.

Model Questions

1. If a, b are the roots of the Quadratic Equation x2 − 7x + 9 = 0, then find the value of .

2. A root of an equation ax2 + bx + c = 0 (where a, b, c are rational numbers) is  What is the value of

3. If then the value of   is .....

A) 0    B) 6    C) 2    D) 4

4. If   and are the roots of equation x2 − 4x + 5 = 0 then which equation will have roots

and ?

A)  x− 4x + 125 = 0    B) x2 + 4x − 125 = 0

C) x2 + 2x + 110 = 0    D) x2 − 4x = 0

5. If are the roots of cubic equation x3 − 21x2 + 140x − 45 = 0, then find

A) 140 B) 45 C) ¾¾ D) 1

6.

and are the roots of the quadratic equation. If and then which of the following equations will have roots a2 and b2.
A) x2 − 7x + 8 = 0    B) x2 − 14x + 15 = 0
C) x2 − 14x + 1 = 0    D) x2 + 4x + 5 = 0

7. If K and L are the roots of the quadratic equation Kx2 − K2x + KL = 0, then what is the  value of K3 + L3?
A) 0    B) 8    C) 1    D) 16

8. If the sum and product of the roots of tx2 − 15x + 10t are equal. Then the value of 't' is ......

9. The quadratic equation 4x2 −17x + 20 = 0 has roots a and b. Find the value of .

10. If the sum and product of roots of a  quadratic equation are and  respectively, then the equation is ......

A) 15x2 − 41x + 145 = 0
B) 35x2 + 98x + 100 = 0
C) 10x2 + 16x + 32 = 0
D) 42x2 + 37x + 100 = 0

11. 152x + 98y = 100; 98x + 152y = 150, then find the value of x + y.

A) 0    B) −5    C) 1    D) 7

12. If 7x + 4y + 6z = 50; 28x + 11y + 24z = 175, then find the value of y.

A) 3    B) 2    C) 4    D) 5

13. Cost of 5 Red balls, 4 Blue balls and 6 Yellow balls is Rs.400. Cost of 4 Red balls, 5 Blue balls and 7 Yellow balls is Rs.250. What is the cost in (Rs.) of 6 Red balls, 3 Blue balls and 5 Yellow balls?

A) 460    B) 555    C) 550    D) 625

14. If p = 245, q = 246, r = 247, find the value of p2 + q2 + r2 − pq − qr − rp.

A) 0    B) 5    C) 3    D) 10

15. If a − b = 6; b − c = 5; c − a = 3, then find a2 + b2 + c2 − ab − bc − ca.

A) 35    B) 43    C) 30    D) 18

16. If then find the value of 10ab.

A) 10    B) 15    C) 1    D)

17. Find the value of

A) 246    B) 245    C)     D) 1

18. If x = 49, find the value of x(x + 2x).

A) 2999    B) 2501   C) 2643    D) 2499

19. If find the value of 10(a2 + b2).

A) 150    B) 110    C) 70    D) 90

20. If x + y = 5, then find the value of x3 + y3 + 15xy.

A) 105    B) 64    C) 125    D) 140

Key

1-D   2-B   3-C   4-A   5-A   6-C   7-C   8-B   9-D  10-B   11-C   12-D   13-C   14-C  15-A   16-B   17-A   18-D  19-B   20-C

Explanations

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A) 5   B) 4   C) 3   D) 1

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3. ax + by = 6; bx − ay = 2 and a2 + b2 = 4. Find x2 + y2.

A) 5  B) 1   C) 0   D) 10

4. If x = 11, find the value of x4 − 12x3 + 12x2 − 12x + 15.

A) 1   B) 3   C) 4   D) 10

5. If , find the value of x6 + x3 + 1.

A) 0   B) 1   C) −1   D) 2

6. If

, find the value of

A) 1   B) 2     C) 3   D) 4

7. If

, then find
A) 0   B) 1  C) 2   D) 3

8. If  then find x82 + x72 − x70 − x60.

A) 0   B) 1   C) 2   D) 3

9. If x is a real number, find the minimum value of .

A) 0    B) 1    C) 2    D) 3

10. If

then find .

A) 0    B) 1    C) 2    D) 3

11. If x is a real number

,find .

12. If

find the value of

13. If find

14. If find

A) 123     B) 121     C) 125     D) 144

15. If

, find

16. If , find

A) 198   B) 204   C) 216   D) 225

17. If x2 + 1 = 2x, find the value of

A) −2     B) −1     C) 0     D) 1

18. If , find

A) 1   B) 4   C) 7   D) 10

19. If  then

A) 1   B) 4   C) 10   D) 0

20.  then

A) 0   B) 1   C) −1    D) −2

21. Iffind

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22. If

find

A) 1000     B) 999    C) 970     D) 1331

23. If

then the value of p6 + q6 is?

A) 0 B) 1 C) 2 D) 3

Key

1-A   2-B   3-D   4-C   5-B   6-B  7-C   8-A   9-C   10-C    11-B    12-A    13-D    14-A   15-A    16-B    17-A   18-C    19-A    20-D    21-B    22-C    23-A

Explanations

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Model Questions

Algebra for SSC CGL and SSC CHSL has a good weightage so if you are preparing for these exams algebra plays the vital role in the scoring. Mathematics is a very wide subject that has different parts, including geometry, theory, analysis concepts. Algebra is one of the most appreciated parts of applied mathematical subjects. Applications of algebra are wide spread, whether it is solving complicated equations, handling data, or finding out the volume of water in a tank, calculating the velocity of your car. It is used everywhere and in various situations in day to day life. It is vital subject when it comes to competitive exams so being very sure about your knowledge and concepts are all clear.

Algebraic Expression

An algebraic expression is a combination of constants and variables combined together with the help of the four fundamental signs.
Example: x3 + 4x2 − 8x + 1

x3 + 4x2 − 8x + 1 is an algebraic expression containing four terms.
​​​​​​​ The variable of this expression is x, coefficient of x2 is 4, the coefficient of x is −8 and the constant is 1.

Degree of the Polynomial

In a polynomial of one variable, the highest power of the variable is called the degree of the polynomial.

In case of a polynomial of more than one variable, the sum of the powers of the variables in each term is considered and the highest sum
so obtained is called the degree of the polynomial.
Example: Find the degree of each term for the following polynomial and also find the degree of the polynomial 6ab8 + 5a2b3c2 − 7ab + 4b2c + 3
Given polynomial is
6ab8 + 5a2b3c2 − 7ab + 4b​​​​​​​2c + 3

Degree of each of the terms:
6ab8 has degree (1 + 8) = 9
5a2b3c2 has degree (2 + 3 + 2) = 7
7ab has degree (1 + 1) = 2
4b​​​​​​​2c has degree (2 + 1) = 3
The constant term 3 is always regarded as having degree Zero.
The degree of the polynomial
6ab8 + 5a2b3c2 − 7ab + 4b​​​​​​​2c + 3 = the largest exponent in the polynomial = 9

The general form of quadratic equation
x2 − (sum of the roots)x + product of the roots = 0

H For a quadratic equation ax2 + bx + c where the roots will be given by the equation as

Remainder Theorem

​​​​​​​ If a polynomial p(x) of degree greater than or equal to one is divided by a linear polynomial (x − a) then the remainder is p(a), where a is any real number.
​​​​​​​ If p(x) is divided by (x + a), then the remainder is p(− a).
Example: Find the remainder when
p(x) = x3 + 3x2 + 3x + 1 is divided by x + 1?
p(x) = x3 + 3x​​​​​​​2 + 3x + 1
p(−1) = (−1)3 + 3(−1)2 + 3(−1) + 1
= −1 + 3 − 3 + 1 = 0
Hence, the remainder is 0.

Factor Theorem

​​​​​​​  If p(x) is a polynomial of degree n >1 and a is any real number, then
i) (x − a) is a factor of p(x), if p(a) = 0, and
ii) p(a) = 0, if (x − a) is a factor of p(x).
Example: Show that (x + 2) is a factor of
x3 − 4x2 − 2x + 20
p(x) = x3 − 4x2 − 2x + 20
p(−2) = (−2)3 − 4(−2)2 − 2(−2) + 20 = 0
p(−2) = 0
(x + 2) is a factor of x3 − 4x2 − 2x + 20

Posted Date : 13-07-2022

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.