**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – Elementary Algebra**

Contents

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A combination of one or more terms including letters in addition to numbers and symbols along with the signs of mathematical operations is termed as an algebraic expression.

It consists of two types of symbols and variables with fundamental arithmetic operations (+,-,÷,x).

**Constants** Constants are fixed, i.e., 0,1, 2, 3, 4, 5, 6, 7, 8, 9

**Variables** Variables are the symbols representing any numeral.

**e.g.,** 3x² + 4xy – 10x, 2x + 5y etc. are algebraic expression.

**Fundamental Operations on Algebraic Expressions**

**I. Method of Addition**

**Step I** Collect different groups of like term.

**Step II** Find the sum of the numerical coefficients of like terms in each group.

**Step III** Write the final terms after addition.

**e.g.,** (x² + 2y) + (3x² – y) =x² + 3x² + 2y-y = 4x² + y

**II. Method of Subtraction**

**Step I** Reverse the sign (from + to – and from – to +) of all the terms of the expression which is to be subtracted.

**Step II** Now, follow the method used in addition.

**e.g.,** (a² + 2b² -ab) – (4a² -b² + 2ab)

= 4a² – a² -b² -2b² + 2ab + ab

=3a² -3b² + 3ab = 3(a² – b² + ab)

**III. Method of Multiplication**

**Step I** Find the product by using the rule that product of two factors with like sign is positive and the product of two factors with unlike sign is negative i.e., (+) x (+) = +, (+) x (-) = -, (-) x (+) = -, (-) x (-) = +

**Step II** Now, from the surds rule, if x is a variable and m, n are positive integer then x^{m} x x^{n}=x.^{m+n}

**e.g.,** Multiply, (x + 2) x (x – 4) =x(x – 4) + 2 (x – 4) = x^{2} – 4x + 2x-8 = x^{2}-2x-8

**IV. Method of Division**

Step I Arrange the terms of both the polynomials in descending order of their highest power.

Step II Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

Step III Multiply all terms of the divisor by the first of the quotient and subtract the result from the dividend.

Step IV Consider the remainder (if any) as a new dividend and proceed as before.

Step V Repeat this process till we obtain a remainder which is either ‘zero’ or polynomial of degree less than, that of

the divisor.

**e.g.,**

**Some Useful Facts/Formulae**

**Reference Corner**

**1. What are the roots of the quadratic equation 21x² – 37x -28 = 0 ? SSC(10 + 2)2017**

(a)-7/3, 4/7

(b) 3/7, -7/4

(c) 7/3,-4/7

(d)-3/7, 7/4

**Answer:**

**2. If x + y = 2a, then the value of a/x-a+a/y-a is SSC(10+ 2)2015**

(a) 2

(b)0

(c) -1

(d) 1

**Answer:**

**3.**

**4. If t² – 4t + 1 = 0, then the value of t³ + 1/t³ is SSC(10 + 2) 2014**

(a) 52

(b) 64

(c) 44

(d) 48

**Answer:**

**5. If x + 1/x = 3, then the value of 3x² – 4x + 3/x² – x + 1 is SSC(10 + 2) 2014**

(a)4/3

(b)3/2

(c)5/2

(d)5/3

**Answer:**

**6. If x = 3 + 2√2, then x ^{6} + x^{4} + x^{2} + 1/x^{3} is equal to SSC(10 + 2)2014**

(a) 216

(b) 192

(c) 198

(d) 204

**Answer:**

**7.**

(a)1/a-b-c

(b)1/a+b-c

(c)1/a-b+c

(d)1/a+b+c

**Answer:**

(d)

**8. If x = p + 1/p and y = p – 1/p then value of x ^{4 }– 2x^{2}y^{2 }+ y^{4} SSC (10 + 2) 2014**

(a) 24

(b) 4

(c) 16

(d) 8

**Answer:**

**9. If xy/x+y=a,xz/x+z=b,yz/y+z=c,where a,b,c are all non-zero numbers, then x equals to SSC (10 + 2) 2013**

(a)2abc/ac + bc- ab

(b)abc/ab + bc + ac

(c)2abc/ab + bc- ac

(d)2abc/ab + ac-bc

**Answer:**

**10. If x + y + z = 13 and x ^{2} + y^{2} + z^{2} = 69, then the value of xy + z (x + y) is equal to SSC (10 +2) 2013**

(a) 50

(b) 60

(c) 70

(d) 40

**Answer:**

**11.**

**12. If 4x – 5z = 16 and xz = 12, the value of 64x ^{3} – 125z^{3} is equal to SSC (10 + 2) 2013**

(a) 15610

(b) 15616

(c) 15618

(d) 15620

**Answer:**

**13. If x + y + z = 15, xy + yz+ zx= 75, then x+4y+z/3z is equal to SSC (10 + 2) 2013**

(a) 1

(b) 0

(c) 2

(d) -1

**Answer:**

**Practice Exercise**

**1.The Solution of the equations p/x+q/y=m and q/x+p/y=n is**

**2.If (x + 1/x) = 4, then (x ^{4} + 1/x^{4}) is equal to**

(a) 190

(b) 180

(c) 193

(d) 194

**3.If x ^{2}**

**+ 5x – 2k is exactly divisible by (x – 1), then the value of k is**

(a)1

(b) 2

(c) 3

(d) 4

**4.If x + 1/x = 4, then the value of x ^{3} + 1/x^{3} is**

(a) 52

(b) 64

(c) 68

(d) 76

**5. If x + y = 8 and xy = 7, then the value of x ^{3} + y^{3} is**

(a) 344

(b) 342

(c) 345

(d) 340

**6. If x ^{100} + 2x^{99} + k, is divisible by (x + 1), then the value of k is**

(a)1

(b) 4

(c) 3

(d)0

**7. The value of k for which x -1 is a factor of 4x ^{3} + 3x^{2} – 4x + k is**

(a) 3

(b) 1

(c) -2

(d) -3

**8. One factor of x ^{4} + x^{2} – 20 is x^{2} + 5. The other factor is**

(a)x

^{2}-4

(b)x-4

(c)x

^{2}-5

(d)x+2

**9. If a + b + c = 0, then a ^{3} + b^{3} + c^{3} is**

(a) 2abc

(b) 3abc

(c) abc

(d) 4abc

**10.The factors of x ^{8} + x^{4} + 1 are**

(a)(x

^{4 }+ 1 – x

^{2}),(x

^{2}+ 1 + x),(x

^{2}+ 1 – x)

(b)(x

^{4 }+ 1 – x

^{2}),(x

^{2}-1 + x),(x

^{2}+ 1 + x)

(c)(x

^{4 }+ 1 + x

^{2}),(x

^{2}-1 + x),(x

^{2}+ 1 + x)

(d)(x

^{4 }– 1 + x

^{2}),(x

^{2}+ 1 – x),(x

^{2}+ 1+ x)

**11.The expression 10xy ^{4} -10x^{4}y can be expressed in factors as**

(a)10xy (x- y) (x

^{2}+ xy + y

^{2})

(b)10xy (y – x) (x

^{2}+ xy + y

^{2})

(c)10xy (y-x) (x

^{2}-xy + y

^{2})

(d) None of these

**12. GCD of (a+b- c) ^{6} and (a+b- c)^{4} is**

(a)(a+b-c)

^{6}

(b)(a+b-c)

^{2}

(c)(a+b-c)

^{10}

(d)(a+b-c)

^{4}

**13. (4x + 3y) ^{2} + (4x – 3y)^{2} is equal to**

(a) 16x

^{2}– 9y

^{2}

(b) 32x

^{2}+ 18y

^{2}

(c) 16x

^{2}+ 9y

^{2}

(d) 32x

^{2}+ 9y

^{2}

**14. If a/b=4/5 and b/c=15/16,then(c ^{2}-a^{2})/(c^{2}+a^{2}) is equal to**

(a)1/7

(b)7/25

(c)3/4

(d)None of these

**15. If a/x + y/b = 1 and b/y + z/c = 1, then (x/a+c/z) is equal to**

(a)0

(b)b/y

(c)1

(d)y/b

**16. If x = (1 – a), y = (2a + 1) and x = y, then a is equal to**

(a) 2

(b) 1/2

(c) 0

(d) -1

**17.**

(a) 0

(b) 1/9

(c) 1/3

(d) 1

**18. If x + y + z = 0, then (x ^{2} + xy + y^{2}) is equal to**

(a)(y

^{2}+ yz + z

^{2})

(b)(y

^{2}-yz + z

^{2})

(c)(z

^{2}-zx+ x

^{2})

(d)(z

^{2}+zx+ x

^{2})

**19.**

(a )[(x+y)(y+z)(z+x)]^{-1}

(b )x+y+z

(c)x-y+z

(d) None of these

**20. If a + b + c = 2s, then [(s – a) ^{2} + (s – b)^{2} + (s – c)^{2} + s^{2}] is equal to**

(a) (s

^{2}– a

^{2}– b

^{2}– c

^{2})

(b) (s

^{2}+ a

^{2}+ b

^{2}+ c

^{2})

(c) (a

^{2}+ b

^{2}+ c

^{2})

(d) (4s

^{2}– a

^{2}– b

^{2}– c

^{2})

**21. If a/3 = b/2, then value of 2a + 3b/3a – 2b is SSC (10 + 2) 2011**

(a)12/5

(b)5/12

(c)1

(d)12/7

**22.**

(a) a

(b) b

(c) 2ab

(d) 2

**23. If x = b + c – 2a, y = c + a – 2b, z = a + b – 2c, then the value of x ^{2} + y^{2} – z^{2} + 2xy is SSC (10 + 2) 2011**

(a) 0

(b) a + b + c

(c) a – b + c

(d) a + b – c

**Answers**

**Hints & Solutions**

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