ALGEBRAIC IDENTITIES
1) The square root of a²+ 1/a²+ 2 is
a) a+ 1/a b) a- 1/a c) a²+ 1/a² d) a²- 1/a²
2) The square root of a+ 1/a - 2 is
a) a -1/a b) √a+ 1/√a c) ±(√a- 1/√a) d) a+ 1/a
3) The value of (a+ b)²/{(b -c)(c - a)} + (b + c)²/{(a - b)(c - a) + (c + a)²/{(a - b)(b - c)} is
a) -1 b) 0 c) 1 d) 2
4) The square root of the expression (1/abc) (a²+ b²+ c²) +2(1/a + 1/b+ 1/c) is
a) (a+ b+ c)/abc b) √a + √b + √c c) √(bc/a) + √(ca/b) + √(ab/c) d) √(a/bc)+ √(b/ca) + √c/ab)
5) The square root of x²/9 + 9/4x² - x/3 - 3/2x + 5/4 is
a) 2x/3 + 3/2x - 1/2 b) x/3 + 3/2x +1 c) 3/x + 2/3x - 1/2 d) x/3 + 3/2x - 1/2
6) The square root of the expression is (xy + xz - yz)² - 4xyz(x - y) is
a) xy + yz - 2xyz b) x + y - 2xyz c) xy + z - y d) xy + yz - xz
7) The square root of a²/4 + 1/a² - 1/a + a/2 - 3/4 is
a) a/2 - 1/a + 1/2 b) a/2 + 2/a - 1 c) a/2 + 1/a - 1/2 d) a/2 - 2/a - 1/2
8) The expression (4a + 5b+ 5c)² - (5a + 4b+ 4c)² + 9a² is a perfect square of the expression
a) √3(b + c) b) 3(a+ b + c) c) 3(b+ c) d) 3(-b + c - a)
9) The expression (3a + 2b+ 3c)² - (2a + 3b+ 2c)² + 5b² is a perfect square of the expression √(a + b+ c) b) √((a + b) c) √5(a +c) d) √5(a - b+ c)
10) If a/b + b/a =2, then (a/b)¹⁰ - (b/a)¹⁰ is equal to
a) (2¹⁰-1)/2¹⁰ b) 2 c) 0 d) (2²⁰+1)/2¹⁰
11) If abc = 6 and a + b + c = 6, then 1/ac + 1/ab + 1/bc =
a) 2 b) 1 c) 3 d) 0
12) √{(a + b+ c)²+ (a + b- c)²+ 2(c² -b²- a²- 2ab) is equal to
a) 2c b) 2a c) 2b d) a + b+ c
13) If a/b + b/a = -1, then a³- b³=
a) 1 b) -1 c) 1/2 d) 0
14) If a+ b=8 and ab= 12, then a³+ b³=
a) 244 b) 224 c) 144 d) 284
15) If (a + 1/a +2)²=4, then a²+ 1/a²=
a) 12 b) 13 c) 14 d) -14
16) If x+ 1/x =7, then x³- 1/x³=
a) 9√5 b) 144√5 c) 135√5 d) √5
17) {(a - b)³ - (a + b)³}/2 + a(a²+ 3b²)=
a) a³- b³ b) (a + b)³ c) a³+ b³ d) (a - b)³
18) If x+ 1/x =5, then x²+ 1/x²=
a) 25 b) 10 c) 23 d) 27
19) If x+ 1/x =2, then x³ + 1/x³=
a) 64 b) 14 c) 8 d) 2
20) If x+ 1/x =4, then x⁴ + 1/x⁴=
a) 196 b) 194 c) 192 d) 190
21) If x+ 1/x =3, then x⁶ + 1/x⁶=
a) 927 b) 414 c) 364 d) 322
22) If x² + 1/x² =102, then x- 1/x=
a) 8 b) 10 c) 12 d) 13
23) If x³+ 1/x³ =110, then x + 1/x =
a) 5 b) 10 c) 15 d) none
24) If x³- 1/x³ =14, then x- 1/x=
a) 5 b) 4 c) 3 d) 2
25) If a+ b+ c= 9 and ab+ bc+ ca= 23, then a²+ b²+ c²=
a) 35 b) 58 c) 127 d) none
26) (a - b)³+ (b - c)³+ (c - a)³=
a) (a+ b+ c)(a²+ b²+ c²- ab - bc - ca)
b) (a - b)(b - c)(c - a)
c) 3(a - b)(b - c)(c - a) d) none
27) a+ b= 3 and ab = 2, then a³+ b³=
a) 6 b) 4 c) 9 d) 12
28) If a- b =-8 ab = -12, then a³- b³=
a) -244 b) -240 c) -224 d) -260
29) if the volume of a cuboid is 3x²- 27, then its possible dimensions are
a) 3, x, -27x b) 3, x -3, x+3 c) 3, x², 27x d) 3,3,3
30) 75 x 75 +2 x 75 x 25+25 x 25 is equal to
a) 10000 b) 6250 c) 7500 d) 3750
31) (x - y)(x+ y)(x²+ y²)(x⁴+ y⁴) is equal to
a) x¹⁶- y¹⁶ b) x⁸- y⁸ c) x⁸+ y⁸ d) x¹⁶+ y¹⁶
32) If x⁴+ 1/x⁴ =623, then x + 1/x=
a) 27 b) 25 c) 3√3 d) -3√3
33) If x- 1/x = 15/4, then x + 1/x =
a) 4 b) 17/4 c) 13/4 d) 1/4
34) If 3x+ 2/x = 7, then 9x² - 4/x² =
a) 25 b) 35 c) 49 d) 30
35) If a²+ b²+ c²- ab - bc - ca = 0, then
a) a+ b = c b) b + c = a c) c + a= b d) a= b= c
36) If a+ b + c = 0, then a²/bc + b²/ca + c²/ab is
a) 0 b) 1 c) -1 d) 3
37) If a¹⁾³ + b¹⁾³ + c¹⁾³= 0, then
a) a+ b+ c= 0 b) (a+ b + c)³= 27abc c) a+ b + c = 3abc d) a³+ b³+ c³= 0
38) If a+ b + c = 9, then ab+ bc + ca =23, then a³+ b³+ c³- 3abc=
a) 108 b) 207 c) 669 d) 729
39) {(a² - b²)³+ (b²- c²)³+ (c²- a²)³}/{(a - b)+ (b - c)+(c - a)}=
a) 3(a + b) (b +c)(c +a) b) 3(a - b)(b - c)(c - a)} c) (a - b) (b - c)(c - a) d) (a + b) (b +c)(c+ a)
40) The product (a + b)(a - b)(a²- ab+ b²)(a²+ ab+ b²) =
a) a⁶+ b⁶ b) a⁶- b⁶ c) a³- b³ d) a³+ b³
41) The product (x²-1)(x⁴+ x²+1) is equal to
a) x⁸-1 b) x⁸+1 c) x⁶-1 d) x⁶+1
42) If a/b + b/a = 1, then a³+ b³=
a) 1 b) -1 c) 1/2 d) 0
43) If 49a²- b = (7a + 1/2)(7a - 1/2), then the value of b is
a) 0 b) 1/4 c) 1/√2 d) 1/2
44) One of the factors of (5x +1)² -(5x -1)² is
a) 5 + x b) 5- x c) 5x -1 d) 20x
45) If 9x² - b =(3x + 1/2)(3x - 1/2), then the value of b is
a) 0 b) 1/√2 c) 1/4 d) 1/2
46) The Coefficient of x in (x +3)³ is
a) 1 b) 9 c) 18 d) 27
47) The value of 249²- 248² is
a) 1 b) 477 c) 487 d) 497
48) Which of the following is a factor of (x + y)³-(x³+ y³)?
a) x²+ 2xy + y² b) x² - xy + y² c) xy² d) 3xy
49) If x/y + y/x = -1 (x,y ≠ 0), the value of x³- y³ is
a) 1 b) -1 c) 0 d) 1/2
50) If x + y=2 and xy = 1, then x⁴+ y⁴=
a) 6 b) 4 c) 8 d) 2
51) If x² + y²+ xy =1 and x + y = 2, then xy=
a) -3 b) 3 c) -3/2 d) 0
52) If a, b, c are natural numbers such that a²+ b²+ c²= 29 and ab + bc + ca = 26, and a+ b + c=
a) 9 b) 6 c) 7 d) 10
53) If 2x + y/3= 12 and xy = 30, then 8x³+ y³/27=
a) 1008 b) 168 c) 106 d) none
