class 9 Pearson - number systems

Which of the following fractions lie between $\frac{1}{5}$ and $\frac{1}{4}$ ?
A. $\frac{7}{33}$
B. $\frac{4}{11}$
C. $\frac{13}{57}$
D. $\frac{7}{17}$
1. A and B
2. A and C
3. B, C and D
4. A, B and D

Express $0.34$ + $0.34$ as a single decimal.
1. $0.6788$
2. $0.689$
3. $0.6878$
4. $0 . \overset{⇀}{687}$

If $\sqrt{5^{n}}$ =125, then $5^{\sqrt[n]{64}}$ =_______
1. 25
2. $\frac{1}{125}$
3. 625
4. $\frac{1}{25}$

If $x^{4} + 1 = 1297$ and $y^{4} - 1 = 2400$ then $y^{2} - x^{2}$ =______
1. 10
2. 25
3. 13
4. 43

What is the value of $4^{2 x - 2}$ , if ${(16)}^{2 x + 3}$ = ${(64)}^{x + 3}$ ?
1. 64
2. 256
3. 32
4. 512

Which of the following pairs is having two equal values? (where x $\in$ R)________.
1. $9^{x / 2}, 24^{x / 3}$
2. ${(256)}^{4 / x}, {(4^{3})}^{4 / x}$
3. ${(343)}^{x / 3}, {(7^{4})}^{x / 12}$
4. ${(36^{2})}^{2 / 7}, {(6^{3})}^{2 / 7}$

The expression $(\sqrt{5} - \sqrt{3}) (\sqrt{7} - \sqrt{2})$ when simplified becomes a
1. simple surd
2. mixed surd
3. compound surd
4. binomial surd

If m and n are positive integers, then for a positive number a, ${\{\sqrt[m]{(\sqrt[n]{a})}\}}^{m n}$ =_______
1. $a^{m n}$
2. a
3. $a^{m / n}$
4. 1

If $2^{- m} \times \frac{1}{2^{m}} = \frac{1}{4}$ , then $\frac{1}{14} [{(4^{m})}^{1 / 2} + {(\frac{1}{5^{m}})}^{- 1}]$ =_______
1. $\frac{1}{2}$
2. 2
3. 4
4. $\frac{- 1}{4}$

10.

The surds $\sqrt{2}, \sqrt[3]{3}$ and $\sqrt[5]{5}$ , in their descending order are
1. $\sqrt[3]{3}, \sqrt[5]{5}, \sqrt{2}$
2. $\sqrt{2}, \sqrt[3]{3}, \sqrt[5]{5}$
3. $\sqrt{2}, \sqrt[5]{5}, \sqrt[3]{3}$
4. $\sqrt[3]{3}, \sqrt{2}, \sqrt[5]{5}$

11.

$2 {[{(16 - 15)}^{- 1} + 25 {(13 - 8)}^{- 2}]}^{- 1} + {(1024)}^{0}$ =______
1. 2
2. 3
3. 1
4. 5

12.

If x=2 and y=4, then ${(\frac{x}{y})}^{x - y} + {(\frac{y}{x})}^{y - x}$ =_________
1. 4
2. 8
3. 12
4. 2

13.

In which of the following pairs of surds are the given two surds similar?
1. $\sqrt{5}, 7 \sqrt{5}$
2. $\sqrt[3]{7}, \sqrt[2]{7}$
3. $\sqrt{7}, \sqrt{28}$
4. Both 1 and 3

14.

Which of the following is the greatest?
1. $7^{2}$
2. ${(49)}^{3 / 2}$
3. ${(\frac{1}{343})}^{- 1 / 3}$
4. ${(2401)}^{- 1 / 4}$

15.

$(\sqrt[6]{5}) (\sqrt[3]{2}) (\sqrt{3}) (\sqrt[12]{6})$ =___________
1. $\sqrt[12]{1749600}$
2. $\sqrt[3]{2} \times \sqrt[12]{109350}$
3. $\sqrt[12]{177960}$
4. Both 1 and 2

16.

If p=3 and q=2, then ${(3 p - 4 q)}^{q - p} \div {(4 p - 3 q)}^{2 q - p}$ =________
1. 1
2. 6
3. $\frac{1}{6}$
4. $\frac{2}{3}$

17.

$[\frac{{(32)}^{0.2} + {(81)}^{0.25}}{{(256)}^{0.5} - {(121)}^{0.5}}]$ =_______
1. 2
2. 5
3. 1
4. 11

18.

Which of the following surd is the smallest?
$\sqrt{10} - \sqrt{5}, \sqrt{19} - \sqrt{14}, \sqrt{22} - \sqrt{17}$ and $\sqrt{8} - \sqrt{3}$
1. $\sqrt{10} - \sqrt{5}$
2. $\sqrt{19} - \sqrt{14}$
3. $\sqrt{22} - \sqrt{17}$
4. $\sqrt{8} - \sqrt{3}$

19.

If $\sqrt{m} = \sqrt{a} + \sqrt{c}$ and $\sqrt{m}$ , $\sqrt{a}$ and $\sqrt{c}$ are three surds, then
1. $\sqrt{m}$ is dissimilar to $\sqrt{a}$ and $\sqrt{c}$
2. $\sqrt{a}$ and $\sqrt{c}$ are similar to $\sqrt{m}$
3. only $\sqrt{a}$ is similar to $\sqrt{m}$
4. None of these

20.

The surd obtained after rationalizing the numerator of $\frac{4 - \sqrt{25 - a}}{a - 9}$ is equal to
1. $\frac{a - 9}{4 - \sqrt{25 - a}}$
2. $\frac{1}{4 - \sqrt{25 - a}}$
3. $\frac{1}{(a - 9) (4 + \sqrt{25 - a})}$
4. $\frac{1}{4 + \sqrt{25 - a}}$

21.

If $\sqrt{13 - x \sqrt{10}} = \sqrt{8} + \sqrt{5}$ , then what is the value of x?
1. -5
2. -6
3. -4
4. -2

22.

If the surds $\sqrt[4]{4}$ , $\sqrt[6]{5}$ , $\sqrt[8]{6}$ and $\sqrt[12]{8}$ are arranged in descending order from left to right, then the third surd from the left is
1. $\sqrt[12]{8}$
2. $\sqrt[4]{4}$
3. $\sqrt[8]{6}$
4. $\sqrt[6]{5}$

23.

$\sqrt{11 \sqrt{11 \sqrt{11 . . . . . . . 4 t e r m s}}} =$
1. $\sqrt[16]{11^{5}}$
2. $\sqrt[16]{11}$
3. $\sqrt[16]{11^{14}}$
4. $\sqrt[16]{11^{15}}$

24.

If $\frac{5 - \sqrt{3}}{2 + \sqrt{3}} = x + y \sqrt{3}$ , then (x, y) is
1. (13, -7)
2. (-13, 7)
3. (-13, -7)
4. (13, 7)

25.

The simplified form of $\sqrt{125} + \sqrt{125} - \sqrt{845}$ is
1. $\sqrt{15}$
2. $2 \sqrt{5}$
3. $- \sqrt{5}$
4. $- 2 \sqrt{5}$

26.

Which of the following statements is true?
I. If x is a conjugate surd of y, then x can be a RF of y.
II. If x is a RF of y, then x need not be the conjugate of y.
1. Only I
2. Only II
3. Both I and II
4. Neither I nor II

27.

If $\frac{3 - 2 \sqrt{5}}{6 - \sqrt{5}} = a + b \sqrt{5}$ hwere and a and b are rational numbers, then what are the values of a and b?
1. $\frac{8}{35}, \frac{- 9}{35}$
2. $\frac{8}{31}, \frac{- 9}{31}$
3. $\frac{- 8}{31}, \frac{9}{31}$
4. $\frac{- 8}{35}, \frac{9}{35}$

28.

If $\frac{3^{5 x} \times {(81)}^{2} \times 6561}{3^{2 x}} = 3^{7}$ , then x=________
1. 3
2. -3
3. $\frac{1}{3}$
4. $\frac{- 1}{3}$

29.

If $\sqrt{2^{n}} = 1024$ , then $3^{(\frac{n}{4} - 4)}$ =________
1. 3
2. -3
3. 27
4. 81

30.

If ${[{\{{(\frac{1}{7^{2}})}^{- 2}\}}^{- 1 / 3}]}^{1 / 4}$ $= 7^{m}$ , then m =________
1. $\frac{- 1}{3}$
2. $\frac{1}{4}$
3. -3
4. 2

31.

${[{\{{(\frac{1}{x^{a^{2} - b^{2}}})}^{\frac{1}{a - b}}\}}^{a + b}]}^{\frac{1}{{(a + b)}^{2}}}$ =____________
1. $x^{2}$
2. $\frac{1}{x}$
3. $7^{3}$
4. $\frac{1}{x^{2}}$

32.

If $\frac{2^{m + n}}{2^{m - n}} = 16$ and $a = 2^{\frac{1}{10}}$ then $\frac{{(a^{2 m + n - p})}^{2}}{{(a^{m - 2 n + 2 p})}^{- 1}} =$ _____________
1. 2
2. $\frac{1}{4}$
3. 9
4. $\frac{1}{8}$

33.

Simplify
$[(p^{- 1} + q^{- 1}) (p^{- 1} - q^{- 1}) \div (\frac{1}{p^{- 1}} - \frac{1}{q^{- 1}}) (\frac{1}{p^{- 1}} + \frac{1}{q^{- 1}})] {(p q)}^{2}$
1. ${(p q)}^{2}$
2. -1
3. $- {(p q)}^{- 2}$
4. 1

34.

If x= $\frac{2}{\sqrt{10} - \sqrt{8}}, y = \frac{2}{\sqrt{10} + 2 \sqrt{2}}$ , then ${(x - y)}^{2}$ =
1. $4 \sqrt{2}$
2. 32
3. $8 \sqrt{2}$
4. 64

35.

If $a = \sqrt{6} - \sqrt{3}, b = \sqrt{3} - \sqrt{2}$ and $c = \sqrt{2} - \sqrt{6}$ , then find the value of $a^{3} + b^{3} + c^{3} - 2 a b c$ .
1. $3 \sqrt{2} - 5 \sqrt{3} - \sqrt{6}$
2. $3 \sqrt{2} - 5 \sqrt{3} - \sqrt{6}$
3. $3 \sqrt{2} - 4 \sqrt{3} + \sqrt{6}$
4. $3 \sqrt{2} + 4 \sqrt{3} + \sqrt{6}$

36.

$\sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64} \sqrt{\frac{81}{64}}}} . . . . \infty}$ =
1. $\frac{81}{64}$
2. $\frac{9}{8}$
3. $\frac{3}{2}$
4. $\frac{3}{2 \sqrt{2}}$

37.

If $a^{p} = b^{q} = c^{r} = a b c$ , then pqr=_____
1. $p^{2} q + q^{2} r$
2. pq+qr+pr
3. ${(p q + q r + r p)}^{2}$
4. pq(qr+rp)

38.

The value of ${[{(23 + 2^{2})}^{2 / 3} + {(140 - 29)}^{1 / 2}]}^{2}$ is _______
1. 196
2. 289
3. 324
4. 400

39.

If $x = \sqrt{6} + \sqrt{5}$ , then $x^{2} + \frac{1}{x^{2}} - 2 =$
1. $2 \sqrt{6}$
2. $2 \sqrt{5}$
3. 24
4. 20

40.

$\sqrt{6 + \sqrt{6 + \sqrt{6 + . . . . . \infty}}}$ is equal to______
1. -3
2. 3
3. 6
4. 2

41.

Simplify
$\frac{1}{\sqrt{19 - \sqrt{360}}} - \frac{1}{\sqrt{21 - \sqrt{440}}} + \frac{2}{\sqrt{20 + \sqrt{396}}}$ =___________
1. 1
2. 2
3. 0
4. None of these

42.

If $a = \sqrt{17} - \sqrt{16}$ and b= $\sqrt{16} - \sqrt{15}$ then
1. a<b
2. a>b
3. a=b
4. None of these

43.

$(\sqrt[6]{15 - 2 \sqrt{56}}) . (\sqrt[3]{\sqrt{7} + 2 \sqrt{2}})$ =_________
1. 0
2. 1
3. -1
4. 2

44.

$\sqrt{\sqrt{63} + \sqrt{56}} =$
1. $\sqrt[4]{7} (\sqrt{3} + \sqrt{5})$
2. $\sqrt[4]{7} (\sqrt{3} + 1)$
3. $\sqrt[4]{7} (\sqrt{3} + \sqrt{5})$
4. $\sqrt[4]{7} (\sqrt{2} + 1)$

45.

If $\frac{\sqrt{7} + 2 \sqrt{3}}{2 \sqrt{7} - \sqrt{5}} = \frac{c + \sqrt{p} + \sqrt{q} + \sqrt{r}}{23}$ (p<q<r), where p, q, r are rational numbers, then q+r-p=
1. 361
2. 302
3. 418
4. 426

46.

The following are the steps involved in finding the value of x - y from $\frac{8 - \sqrt{5}}{8 + \sqrt{5}} = x - y \sqrt{50}$ . Arrange them in sequential order.
(A) $\frac{13 - 2 \sqrt{40}}{8 - 5} = x - y \sqrt{40}$
(B) $\frac{{(\sqrt{8})}^{2} + {(\sqrt{5})}^{2} - 2 (\sqrt{8}) (\sqrt{5})}{{(\sqrt{8})}^{2} - {(\sqrt{5})}^{2}} = x - y \sqrt{40}$
(C) $x - y = \frac{11}{3}$
(D) $x = \frac{13}{3} a n d y = \frac{2}{3}$
(E) $\frac{(\sqrt{8} - \sqrt{5}) (\sqrt{8} - \sqrt{5})}{(\sqrt{8} + \sqrt{5}) (\sqrt{8} - \sqrt{5})} = x - y \sqrt{40}$
1. EABDC
2. EBADC
3. ABDEC
4. DEBAC

47.

The following are the steps involved in finding the least among $\sqrt{3}, \sqrt[3]{4} a n d \sqrt[6]{15} .$ Arrange them in sequential order.
(A) $∴ \sqrt[6]{15} is the smallest$
(B) $∴ 3^{\frac{1}{2}} = 3^{\frac{3}{6}}, 4^{\frac{1}{3}} = 4^{\frac{2}{6}}, 15^{\frac{1}{6}} = 15^{\frac{1}{6}}$
(C) The LCM of the denominators of the exponents is 6.
(D) $\sqrt{3} = 3^{\frac{1}{2}}, \sqrt[3]{4} = 4^{\frac{1}{3}}, \sqrt[6]{15} = 15^{\frac{1}{6}}$
(E) $∴ \sqrt{3} = \sqrt[6]{27}, \sqrt[3]{4} = \sqrt[6]{16}, \sqrt[6]{15} = \sqrt[6]{15}$
1. DCABE
2. DABEB
3. DCBEA
4. DBCAE

48.

If $y = 3 - \sqrt{8}$ , then ${(y - \frac{1}{y})}^{2} =_________$
1. 9
2. 81
3. 4
4. 32

49.

The following are the steps involved in finding the value of a + b from $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} = a + b \sqrt{3}$ . Arrange them in sequential order.
(A) $\frac{2^{2} + {(\sqrt{3})}^{2} + 2 x 2 x \sqrt{3}}{2^{2} - {(\sqrt{3})}^{2}} = a + b \sqrt{3}$
(B) a + b = 7 + 4 = 11
(C) $\frac{(2 + \sqrt{3}) (2 + \sqrt{3})}{(2 - \sqrt{3}) (2 + \sqrt{3})} = a + b \sqrt{3}$
(D) $\frac{7 + 4 \sqrt{3}}{4 - 3} = a + b \sqrt{3}$
(E) a = 7 and b = 4
1. CDAEB
2. CAEBD
3. CADEB
4. CEDAB

50.

The following are the steps involved in finding the greatest among $\sqrt[3]{2}, \sqrt[6]{3} a n d \sqrt{6}$ . Arrange them in sequential order.
$(A) T h e L C M o f t h e d e n o m i n a t o r s o f t h e e x p o n e n t s i s 6 . (B) ∴ \sqrt[6]{216} i . e ., \sqrt{6} . i s t h e g r e a t e s t . (C) \sqrt[3]{2} = 2^{\frac{1}{3}}, \sqrt[6]{3} = 3^{\frac{1}{6}}, \sqrt{6} = 6^{\frac{1}{2}} (D) 2^{\frac{1}{3}} = 2^{\frac{2}{6}}, 3^{\frac{1}{6}} = 3^{\frac{1}{6}}, 6^{\frac{1}{2}} = 6^{\frac{3}{6}} (E) \sqrt[3]{2} = \sqrt[6]{4}, \sqrt[6]{3} = \sqrt[6]{3}, \sqrt{6} = \sqrt[6]{216}$
1. CDAEB
2. CDABE
3. DCAEB
4. DACBE

51.

If $x = \frac{1}{\sqrt{3} + 2}$ , then ${(x + \frac{1}{x})}^{2} =____________.$
1. 16
2. 3
3. 12
4. 6

52.

If $\sqrt[x]{3} x \sqrt[y]{5} = 10125, t h e n 12 x y =__________$ .
1. 1
2. $\frac{1}{3}$
3. 2
4. $\frac{1}{2}$

53.

If $x = \frac{1}{5 + 2 \sqrt{6}}, t h e n x^{2} - 10 x + 1 =_________$
1. 1
2. -1
3. 0
4. 6

54.

If $x = \frac{2}{\sqrt{3} - \sqrt{5}} a n d y = \frac{2}{\sqrt{3} + \sqrt{5}}, t h e n x + y =__________$ .
1. 3
2. 4 $\sqrt{3}$
3. -2 $\sqrt{3}$
4. 6

55.

$\frac{3}{7}$ lies between the fraction _________ .
1.   $\frac{4}{9}, \frac{5}{9}$
2.   $\frac{43}{99}, \frac{4}{9}$
3.   $\frac{42}{99}, \frac{4}{9}$
4.   $\frac{41}{99}, \frac{42}{99}$

56.

If $\sum_{k = 4}^{143} \frac{1}{\sqrt{k} + \sqrt{k + 1}} = a - \sqrt{b},$ then a and b respectively are
1. 10 and 0
2. -10 and 4
3. 10 and 4
4. -10 and 0

57.

The surd $\frac{12}{3 + \sqrt{5} + 2 \sqrt{2}},$ after rationalizing the denominator becomes
1.   $\sqrt{5} - \sqrt{2} + \sqrt{10} + 1$
2.   $\sqrt{5} + \sqrt{10} + \sqrt{2} + 1$
3. $\sqrt{10} + \sqrt{2} + \sqrt{5} + 1$
4.   $\sqrt{5} - \sqrt{10} - \sqrt{2} - 1$

58.

If $A^{\frac{1}{A}} = B^{\frac{1}{B}} = C^{\frac{1}{C}}, A^{B C} + B^{A C} + C^{A B} = 729 .$ Which of the following equals $A^{\frac{1}{A}}$ ?
1.   $\sqrt[A B C]{81}$
2.   $\sqrt{2}$
3.   $\sqrt[A B C]{27}$
4. $\sqrt[A B C]{9}$

59.

If $x = \frac{1}{2 - \sqrt{3}},$ then the value of $x^{2} - 2 x^{2} - 7 x + 10$ is equal to
1.   $2 + \sqrt{3}$
2.   $10$
3.   $7 + 2 \sqrt{3}$
4. 8

60.

If x = $1 + 5^{\frac{1}{3}} + 5^{\frac{2}{3}},$ then find the value of $x^{3} - 3 x^{2} - 12 x + 6 .$
1. 22
2. 20
3. 16
4. 14

61.

$\frac{4}{\sqrt{10 - 2 \sqrt{21}}} =________.$
1. $\frac{1}{4} (\sqrt{7} + \sqrt{3})$
2.   $\frac{1}{4} (\sqrt{7} - \sqrt{3})$
3.   $\sqrt{7} + \sqrt{3}$
4.   $\sqrt{7} - \sqrt{3}$

62.

If $y = 3^{\frac{1}{3}} + 3, t h e n y^{3} - 9 y^{2} + 27 y =________.$
1. 27
2. -27
3. -30
4. 30

63.

$\frac{1}{\sqrt{8 + 2 + \sqrt{15}}} =_________.$
1.   $\frac{1}{2} (\sqrt{5} + \sqrt{3})$
2.   $\frac{1}{2} (\sqrt{5} - \sqrt{3})$
3.   $\frac{1}{2} (\sqrt{5} + 1)$
4.   $\frac{1}{2} (\sqrt{5} - 1)$

64.

If $x = 2^{\frac{1}{3}} - 2, t h e n x^{3} + 6 x^{2} + 12 x =________.$
1. 6
2. -6
3. 8
4. -8

65.

$\frac{3}{\sqrt{19 - 2 \sqrt{88}}} - \frac{8}{\sqrt{14 + 2 \sqrt{33}}} =__________.$

1.   $\sqrt{19 + 2 \sqrt{33}}$
2.   $\sqrt{14 - 2 \sqrt{88}}$
3.   $\sqrt{11 + 2 \sqrt{24}}$
4.   $\sqrt{11 - 2 \sqrt{55}}$

66.

$\sqrt{\sqrt[x]{2^{x} \sqrt[x^{2}]{3^{x^{3}} \sqrt[x^{3}]{6^{x^{6}} \sqrt[x^{4}]{9^{x^{10}}}}}}} =$
1. 18
2. 54
3. 24
4. 36

67.

$\sqrt{7 + 2 \sqrt{6}} + \sqrt{7 - 2 \sqrt{6}} =_________.$
1. 14
2. $\sqrt{6}$
3. $2 \sqrt{6}$
4. 7

68.

$\sqrt{3^{2} \sqrt{9^{2} \sqrt{{(81)}^{2} \sqrt{16^{16}}}}} =___________.$

1.   $6 x 2^{4}$
2.   $3^{3} x 2$
3.   $6^{3} x 2^{3}$
4.   $6^{3} x 2$

69.

$\sqrt[6]{15 - 2 \sqrt{56}} \sqrt[3]{\sqrt{7} + 2 \sqrt{2}} =$
1. 0
2. $\sqrt{2}$
3. 1
4. $\sqrt[6]{2}$

70.

If p = $7 - 4 \sqrt{3}, t h e n \frac{p^{2} + 1}{7_{p}} =_________.$
1. 2
2. 1
3. 7
4. $\sqrt{3}$

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