From the chapter Basic Algebra, some of the important questions are covered in this question paper. The questions are covers from the book back and the previous year questions.

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Questions:

Multiple Choice Question

60 x 1 = 60

Which whole number is not a natural number?

(a)

(b)

(c)

(d)

If \(x={1\over 2+\sqrt{3}}\) then the value of x³ - x² - 11x + 3 is

(a)

(b)

(c)

(d)

The roots of the equation \(x+{1\over x}=3{1\over 3},x\ne 0\) are

(a)

1, 3

(b)

\({1\over 3},3\)

(c)

\(3,{-1\over 3}\)

(d)

\(1,{1\over 3}\)

Zero of the polynomial p(x) = x² - 4x + 4

(a)

(b)

(c)

-2

(d)

-1

Let \(\alpha\) and \(\beta\) are the roots of a quadratic equation px² + qx + r = 0 then ___________

(a)

\(\alpha +\beta=-{p\over r}\)

(b)

\(\alpha\beta={p\over r}\)

(c)

\(\alpha +\beta={-q\over p}\)

(d)

\(\alpha \beta =r\)

For the below figure of ax² + bx + c = 0

(a)

a < 0, D > 0

(b)

a > 0, D > 0

(c)

a < 0, D < 0

(d)

a > 0, D = 0

If a and b are roots of x² + x + 1 = 0 then the value of a² + b² = ___________

(a)

(b)

-1

(c)

cannot be determined

(d)

The number of real solutions of the equation |x²| - 3|x| + 2 = 0 is ___________

(a)

(b)

(c)

(d)

If one root of the quadratic equation ax² + bx + c = 0 is the reciprocal of the other then ___________

(a)

a = b

(b)

a = c

(c)

ac = 1

(d)

b = c

10.

The condition for one root of the quadratic equation ax² + bx + c = 0 to be double the other ___________

(a)

b² = 3ac

(b)

b² = 4ac

(c)

2b² = 9ac

(d)

c² = ac - b²

11.

The value of log_a^x+ log_1/a^x is ___________

(a)

(b)

(c)

2 log_a^x

(d)

2 log_a^x

12.

If P(x) = x³ + 3x² + 2x + 1, then the remainder on dividing p(x) by (x - 1) is ___________

(a)

(b)

(c)

(d)

13.

If (x + 1) and (x - 3) are factors of x³ - 4x² + x + 6 then other linear factor is ___________

(a)

x + 2

(b)

x - 2

(c)

x - 1

(d)

x + 3

14.

The value of \({3^{-3}\times6^4\times 12^{-3}\over 9^{-4}\times 2^{-2}}\) is ___________

(a)

3⁵

(b)

3⁶

(c)

3⁴

(d)

15.

The value of 2 log₁₀3 + log₁₀ 16 - 2 log₁₀ \(6\over 5\) is ___________

(a)

(b)

(c)

(d)

16.

The value of log₂3 . log₂₇32 ___________

(a)

\(\frac{5}{2}\)

(b)

\(\frac{2}{5}\)

(c)

\(\frac{5}{3}\)

(d)

\(\frac{3}{5}\)

17.

Logarithm of 144 to the base 2\(\sqrt{3}\) is ___________

(a)

(b)

(c)

(d)

18.

\(\sqrt [ 4 ]{ { \left( -2 \right) }^{ 4 } } \times { \left( -1000 \right) }^{ \frac { 1 }{ 3 } }\) is ___________

(a)

(b)

-20

(c)

2^-10

(d)

100

19.

If \(\frac{1}{\sqrt{3}\times\sqrt{2}}=\sqrt{3}+a\) then a is ___________

(a)

\(\sqrt{2}\)

(b)

-\(\sqrt{2}\)

(c)

\(\sqrt{\frac{3}{2}}\)

(d)

\(\sqrt{\frac{2}{3}}\)

20.

If \(\frac{x}{x^2-5x+6}=\frac{A}{x-2}+\frac{B}{x-3}\) then value of A is _____________

(a)

(b)

(c)

(d)

-2

21.

Find the other root of x²- 4x + 1 = 0 given that 2 +\(\sqrt{3}\) is a root ___________

(a)

\(\sqrt{3}\) + 2

(b)

-\(\sqrt{3}\) -\(\sqrt{2}\)

(c)

2 - \(\sqrt{3}\)

(d)

\(\sqrt{3}\) - 2

22.

The value of a when x³- 2x²+ 3x + a is divided by (x - 1), the remainder is 1, is ___________

(a)

-1

(b)

(c)

(d)

-2

23.

The zero of the polynomial function f(x) = 9x²-16 are ____________

(a)

(9, 16)

(b)

(3, 4)

(c)

\((\frac{4}{3},-\frac{4}{3})\)

(d)

\((\frac{3}{4},-\frac{3}{4})\)

24.

Solve 3x² + 5x - 2≤0

(a)

(2,\(\frac{1}{3}\))

(b)

[2,\(\frac{1}{3}\)]

(c)

(-2,\(\frac{1}{3}\))

(d)

(-2,\(\frac{-1}{3}\))

25.

Solve \(\sqrt{7+6x-x^2}=x+1\)

(a)

(1, -3)

(b)

(3, -1)

(c)

(1, -1)

(d)

(3, -3)

26.

The condition that the equation ax² + bx + c = 0 may have one root is the double the other is ___________

(a)

2b² = 9ac

(b)

b²= ac

(c)

b² = 4ac

(d)

9b² = 2ac

27.

The minimum point of y = x² - 4x - 5 is ___________

(a)

(2, -9)

(b)

(-2, -9)

(c)

(-2, 9)

(d)

(4, 5)

28.

If \(\alpha\) and \(\beta\) are the roots of 2x²+ 4x + 5 = 0 the equation where roots are 2\(\alpha\) and 2\(\beta\) is ___________

(a)

4x²+ 4x + 5 = 0

(b)

2x² + 4x + 50 = 0

(c)

x²+ 4x + 5 = 0

(d)

x²+ 4x + 10 = 0

29.

If \(\alpha\) and \(\beta\) are the roots of 2x² - 3x - 4 = 0 find the value of \(\alpha^2+\beta^2\)

(a)

\(\frac{41}{4}\)

(b)

\(\frac{\sqrt{14}}{2}\)

(c)

(d)

none of these

30.

Given \(|\frac{3}{x-4}|<1\) then ___________

(a)

x∈(∞,3)

(b)

x∈(4, ∞)

(c)

x∈(1, 7)

(d)

x∈(1, 4)U(4, 7)

31.

The factors of the polynomial \(6\sqrt { { 3x }^{ 2 } } -47x+5\sqrt { 3 } \) are ___________

(a)

\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

(b)

\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x+1)\)

(c)

\((2x+5\sqrt { 3 } )(3\sqrt { 3 } x+1)\)

(d)

\((2x+5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

32.

(x²-2x+2)(x²+2x+2) are the factors of the polynomial ___________

(a)

(x²-2x)²

(b)

x⁴-4

(c)

x⁴+4

(d)

(x²-2x+2)²

33.

The value of log₁₀⁸+ log₁₀⁵- log₁₀⁴= ___________

(a)

\({ log }_{ 10 }^{ 9 }\)

(b)

\({ log }_{ 10 }^{ 36 }\)

(c)

(d)

-1

34.

The Value of \({ log }_{ 3/4 }^{ (4/3) }\) is ___________

(a)

-2

(b)

(c)

(d)

-1

35.

The logarithmic form of 5²= 25 is ___________

(a)

\({ log }_{ 5 }^{ 2 }=25\)

(b)

\({ log }_{ 2 }^{ 5 }=25\)

(c)

\({ log }_{ 2 }^{ 25 }=2\)

(d)

\({ log }_{ 25 }^{ 5 }=2\)

36.

If the roots of x²-bx + c = 0 are two consecutive integer,then b²- 4c is ___________

(a)

(b)

(c)

(d)

none of these

37.

If x is real and k = \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then ___________

(a)

\(k\epsilon \left[ \frac { 1 }{ 3 } ,3 \right] \)

(b)

k ≥ 3

(c)

\(k\le \frac { 1 }{ 3 } \)

(d)

none of these

38.

The number of real solution of |2x - x²- 3| = 1 is ___________

(a)

(b)

(c)

(d)

39.

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

(a)

(b)

(c)

(d)

40.

The rationalising factor of \(\frac { 5 }{ \sqrt [ 3 ]{ 3 } } \) is

(a)

\(\sqrt [ 3 ]{ 6 } \)

(b)

\(\sqrt [ 3 ]{ 3 } \)

(c)

\(\sqrt [ 3 ]{ 9 } \)

(d)

\(\sqrt [ 3 ]{ 27 } \)

41.

\(\sqrt [ 4 ]{ 11 } \) is equal to ___________

(a)

\(\sqrt [ 8 ]{ 11^{ 2 } } \)

(b)

\(\sqrt [ 8 ]{ 11^{ 4 } } \)

(c)

\(\sqrt [ 8 ]{ 11^{ 8 } } \)

(d)

\(\sqrt [ 8 ]{ 11^{ 6 } } \)

42.

If |x + 3| ≥ 10 then ___________

(a)

x ∊ (-13, 7]

(b)

x ∊ [-13, 7)

(c)

x ∊ (-∞, -13] \(\cup\) [7, ∞)

(d)

x ∊ (-∞, -13] \(\cup\) [7, ∞)

43.

If x is a real number and |x| < 5 then ___________

(a)

x ≥ 5

(b)

-5 < x < 5

(c)

x ≤ -5

(d)

-5 ≤ x ≤ 5

44.

If - 3x + 17 < -13 then ___________

(a)

x ∈ (10, ∞)

(b)

x ∈ [10, ∞)

(c)

x ∈ (-∞, 10]

(d)

x ∈ [10, 10)

45.

If x < 7, then ___________

(a)

-x < -7

(b)

- x ≤ -7

(c)

-x > -7

(d)

-x ≥ -7

46.

The value of log₃ 11.log₁₁ 13.log₁₃ 15.log₁₅ 27.log₂₇ 81 is

(a)

(b)

(c)

(d)

47.

The number of roots of (x + 3)⁴+ (x + 5)⁴= 16 is

(a)

(b)

(c)

(d)

48.

If \(\frac { 1-2x }{ 3+2x-{ x }^{ 2 } } =\frac { A }{ 3-x } +\frac { B }{ x+1 } \), then the value of A + B is

(a)

\(\frac { -1 }{ 2 } \)

(b)

\(\frac { -2 }{ 3 } \)

(c)

\(\frac { 1 }{ 2 }\)

(d)

\(\frac { 2 }{ 3 } \)

49.

If \(\frac { kx }{ (x+2)(x-1) } =\frac { 2 }{ x+2 } +\frac { 1 }{ x-1 } \), then the value of k is

(a)

(b)

(c)

(d)

50.

If a and b are the real roots of the equation x²- kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

(a)

\(\sqrt { { k }^{ 2 }-4c } \)

(b)

\(\sqrt { { 4k }^{ 2 }-c } \)

(c)

\(\sqrt { 4c-{ k }^{ 2 } } \)

(d)

\(\sqrt { k-8c } \)

51.

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x²+ dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

(a)

1, 2

(b)

-1, 1

(c)

9, 1

(d)

-1, 2

52.

The equation whose roots are numerically equal but opposite in sign to the roots 3x²- 5x -7 = 0 is

(a)

3x²- 5x - 7 = 0

(b)

3x²+ 5x - 7 = 0

(c)

3x²- 5x + 7 = 0

(d)

3x²+ x - 7

53.

The number of solution of x²+ |x - 1| = 1 is

(a)

(b)

(c)

(d)

54.

If a and b are the roots of the equation x²- kx + 16 = 0 and a²+ b²= 32, then the value of k is

(a)

(b)

-8

(c)

-8, 8

(d)

55.

Find a so that the sum and product of the roots of the equation 2x²+ (a - 3) x + 3a - 5 = 0 are equal is

(a)

(b)

(c)

(d)

56.

If 3 is the logarithm of 343, then the base is

(a)

(b)

(c)

(d)

57.

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

(a)

0.5

(b)

2.5

(c)

1.5

(d)

1.25

58.

The value of \({ log }_{ 3 }\frac { 1 }{ 81 } \) is

(a)

-2

(b)

-8

(c)

-4

(d)

-9

59.

Given that x, y and b are real numbers x < y, b > 0, then

(a)

xb < yb

(b)

xb > yb

(c)

xb ≤ yb

(d)

\(\frac { x }{ b } \ge \frac { y }{ b } \)

60.

If |x+2| \(\le\) 9, then x belongs to

(a)

\((-\infty ,-7)\)

(b)

[-11, 7]

(c)

\((-\infty ,-7)\cup (11,\infty)\)

(d)

(-11, 7)

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Answers Key:

Multiple Choice Question

60 x 1 = 60

(d)
0

(a)
0

(b)
\({1\over 3},3\)

(b)
2

(b)
a > 0, D > 0

(a)
1

(d)
4

(b)
a = c

10.

11.

(b)
0

12.

(a)
7

13.

(b)
x - 2

14.

(b)
3⁶

15.

16.

17.

18.

(b)
-20

19.

(b)
-\(\sqrt{2}\)

20.

(d)
-2

21.

22.

(a)
-1

23.

24.

(d)
(-2,\(\frac{-1}{3}\))

25.

(b)
(3, -1)

26.

(a)
2b² = 9ac

27.

(a)
(2, -9)

28.

(d)
x²+ 4x + 10 = 0

29.

(b)
\(\frac{\sqrt{14}}{2}\)

30.

(d)
x∈(1, 4)U(4, 7)

31.

(a)
\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

32.

33.

34.

(d)
-1

35.

36.

(b)
1

37.

(a)
\(k\epsilon \left[ \frac { 1 }{ 3 } ,3 \right] \)

38.

(b)
2

39.

(a)
1

40.

41.

(a)
\(\sqrt [ 8 ]{ 11^{ 2 } } \)

42.

(d)
x ∊ (-∞, -13] \(\cup\) [7, ∞)

43.

(b)
-5 < x < 5

44.

(a)
x ∈ (10, ∞)

45.

46.

The value of
\(\log _{3} 11 \log _{11} 13 \log _{13} 15 \log _{15} 27 \log _{27} 81=\log _{3} 81 \)
\(=\log _{3} 3^{4}=4 \log _{3} 3=4 \)

47.

(a)
4

48.

\(\frac{1-2 x}{3+2 x-x^{2}}= \frac{\mathrm{A}}{3-x}+\frac{\mathrm{B}}{x+1} \)
\( 1-2 x=\mathrm{A}(x+1)+\mathrm{B}(3-x) \)
\( \text { Put } x=3,-5=\mathrm{A}(4) \Rightarrow \mathrm{A}=\frac{-5}{4} \)
\( \text { Put } x=-1,3=\mathrm{B}(4) \Rightarrow \mathrm{B}=\frac{3}{4} \)
\(\mathrm{~A}+\mathrm{B}= \frac{-5}{4}+\frac{3}{4}=\frac{-2}{4}=\frac{-1}{2} \)

49.

\(\frac{2}{x+2}+\frac{1}{x-1}=\frac{2 x-2+x+2}{(x+2)(x-1)}=\frac{3 x}{(x+2)(x-1)}= k = 3\)

50.

\(x^{2}-\mathrm{k} x+\mathrm{c}=0\)
a and b are the roots
\(\therefore a+b=k, a b=c\)
To find
\(\sqrt{(a-b)^{2}+0^{2}}=a-b=\sqrt{(a+b)^{2}-4 a b}=\sqrt{k^{2}-4 c}\)

51.

\(x^{2}+a x+c=0 \)
\(x^{2}+d x+b=0 \)
\(8 \& 2 \text { are the roots }\)\(\text { 3. } 3 \text { are the roots }\)
\(\therefore a=-10 ; c=16 \quad d=-6, \quad b=9\)
\(x^{2}+a x+b =0 \)
\(x^{2}-10 x+9 =0 \)
\(\Rightarrow(x-1)(x-9) =0 \)
\(\therefore x =1 \text { (or) } 9 \)

52.

\(3 x^{2}-5 x-7=0\)
\(\text { Let the roots be } \alpha, \beta\)
\(\Rightarrow \operatorname{Sum}: \alpha+\beta=\frac{5}{3}\)
\(\text { Product : } \alpha \beta=\frac{-7}{3}\)
\(\text { Now take the roots are }-\alpha,-\beta\)
\(\Rightarrow \text { Sum : }-\alpha-\beta \quad-(\alpha+\beta)=-\frac{5}{3}\)
\(\text { Product : }(-\alpha)(-\beta)=\alpha \beta=\frac{-7}{3}\)
\(\text { The reouired equation }\)
\(x^{2}+\frac{5}{3} x-\frac{7}{3}=0 \)
\(3 x^{2}+5 x-7=0 \)

53.

\(x^{2}+|x-1| =1\)
\(|x-1| =1-x^{2} \)
\(1-x^{2} =x-1 \text { (or) } 1-x^{2} =-x+1 \)
\(x^{2}+x-2 =0 \ x^{2}-x =0 \)
\((x+2)(x-1) =0 \ x(x-1) =0 \)
\(x =-2 \text { (or) } 1 \ x=0 ; x =1 \)
The roots are -2, 0, 1. Since -2 does not satisfy. The roots are 0 and 1.
.'. Number of solution is 2.

54.

\(x^{2}-k x+16=0, a^{} \& b \text { are roots }\)
\(a+b =k ; a b=16 \)
\(a^{2}+b^{2} =(a+b)^{2}-2 a b \)
\(32 =k^{2}-32 \)
\(k^{2} =64 \)
\(k =\pm 8 \Rightarrow k=-8,8 \)

55.

\(2 x^{2}+(a-3) x+3 a-5=0\)
\(\text {Sum }=\frac{-(a-3)}{2} ; \quad \text { Product }=\frac{3 a-5}{2}\)
\(\text {Given they are equal, } \frac{-(a-3)}{2}=\frac{3 a-5}{2}\)
\(-4 a =-8 \)
\(a =2 \)

56.