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REAL NUMBERS

90318 The HCF and LCM of two numbers are 9 and 90 respectively. If one number is 18, then the other number is:

1 )63

2 )36

3 )45

4 )54

Explanation:

Exp: C
45
Using the result,
HCF × LCM = product of two natural numbers,
$\Rightarrow The other number = \frac{9 \times 90}{18} = 45$

REAL NUMBERS

90320 If 9 $^{x}$ + 2 = 240 + 9 $^{x}$ , then the value of x is:

1 )0.3

2 )0.2

3 )0.1

4 )0.5

Explanation:

Exp: D
0.5
$9^{x + 2} = 240 + 9 x$
$\Rightarrow 9^{x} . 9^{2} = 240 + 9 x$
$\Rightarrow 9^{x} (81 - 1) = 240$
$\Rightarrow 9^{x} = 3$
$\Rightarrow 9^{x} = 9 \frac{1}{2}$
$\Rightarrow x = \frac{1}{2} = 0.5$

REAL NUMBERS

90321 The smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is

1 )1002

2 )1001

3 )1000

4 )1080

Explanation:

Exp: D
1080
LCM (12, 15, 18, 27) = 540
Now, smallest four-digit number = 1000
$∴$ 1000 ÷ 540 = 1 × 540 + 460 (Remainder = 460)
Therefore, the smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is 1000 + (540 - 460) = 1000 + 80 = 1080

REAL NUMBERS

90323 If two positive integers m and n can be expressed as m = x $^{2}$ y $^{5}$ and n = x $^{3}$ y $^{2}$ , where x and y are prime numbers, then HCF(m, n) =

1 )x

^{2}

y

^{3}

2 )x

^{2}

y

^{2}

3 )x

^{3}

y

^{2}

4 )x

^{3}

y

^{3}

Explanation:

Exp: B
x $^{2}$ y $^{2}$
x $^{2}$ y $^{5}$ = y $^{3}$ (x $^{2}$ y $^{2}$ )
x $^{3}$ y $^{3}$ = x(x $^{2}$ y $^{2}$ )
Therefore HCF (m, n) is x $^{2}$ y $^{2}$

REAL NUMBERS

90386 If one of the zeroes of the cubic polynomial x $^{3}$ + ax $^{2}$ + bx + c is -1, then the product of other two zeroes is:

1 )b - a + 1

2 )b - a – 1

3 )a - b + 1

4 )a - b - 1

Explanation:

Exp: A
b - a + 1
Let p(x) = x $^{3}$ + ax $^{2}$ + bx + c
Now, -1 is a zero of the polynomial
So, p(0) = 0
$\Rightarrow$ (-1) $^{3}$ + a(-1) $^{2}$ + b(-1) + c = 0
$\Rightarrow$ -1 + a - b + c = 0
$\Rightarrow$ a - b + c = 1
$\Rightarrow$ c = 1 - a + b
Now, if $α, β, γ$ are the zeroes of the cubic polynomial ax $^{3}$ + bx $^{2}$ + cx + d, then product of zeroes is given by
$α β γ = - \frac{d}{a}$
So, for the given polynomial, p(x) = x $^{3}$ + ax $^{2}$ + bx + c
$α β (- 1) = \frac{- c}{1} = \frac{- (1 - a + b)}{1}$
$\Rightarrow α β = 1 - a + b$
Hence, the correct answer is option (a)

REAL NUMBERS

90318 The HCF and LCM of two numbers are 9 and 90 respectively. If one number is 18, then the other number is:

1 )63

2 )36

3 )45

4 )54

Explanation:

Exp: C
45
Using the result,
HCF × LCM = product of two natural numbers,
$\Rightarrow The other number = \frac{9 \times 90}{18} = 45$

REAL NUMBERS

90320 If 9 $^{x}$ + 2 = 240 + 9 $^{x}$ , then the value of x is:

1 )0.3

2 )0.2

3 )0.1

4 )0.5

Explanation:

Exp: D
0.5
$9^{x + 2} = 240 + 9 x$
$\Rightarrow 9^{x} . 9^{2} = 240 + 9 x$
$\Rightarrow 9^{x} (81 - 1) = 240$
$\Rightarrow 9^{x} = 3$
$\Rightarrow 9^{x} = 9 \frac{1}{2}$
$\Rightarrow x = \frac{1}{2} = 0.5$

REAL NUMBERS

90321 The smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is

1 )1002

2 )1001

3 )1000

4 )1080

Explanation:

Exp: D
1080
LCM (12, 15, 18, 27) = 540
Now, smallest four-digit number = 1000
$∴$ 1000 ÷ 540 = 1 × 540 + 460 (Remainder = 460)
Therefore, the smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is 1000 + (540 - 460) = 1000 + 80 = 1080

REAL NUMBERS

90323 If two positive integers m and n can be expressed as m = x $^{2}$ y $^{5}$ and n = x $^{3}$ y $^{2}$ , where x and y are prime numbers, then HCF(m, n) =

1 )x

^{2}

y

^{3}

2 )x

^{2}

y

^{2}

3 )x

^{3}

y

^{2}

4 )x

^{3}

y

^{3}

Explanation:

Exp: B
x $^{2}$ y $^{2}$
x $^{2}$ y $^{5}$ = y $^{3}$ (x $^{2}$ y $^{2}$ )
x $^{3}$ y $^{3}$ = x(x $^{2}$ y $^{2}$ )
Therefore HCF (m, n) is x $^{2}$ y $^{2}$

REAL NUMBERS

90386 If one of the zeroes of the cubic polynomial x $^{3}$ + ax $^{2}$ + bx + c is -1, then the product of other two zeroes is:

1 )b - a + 1

2 )b - a – 1

3 )a - b + 1

4 )a - b - 1

Explanation:

Exp: A
b - a + 1
Let p(x) = x $^{3}$ + ax $^{2}$ + bx + c
Now, -1 is a zero of the polynomial
So, p(0) = 0
$\Rightarrow$ (-1) $^{3}$ + a(-1) $^{2}$ + b(-1) + c = 0
$\Rightarrow$ -1 + a - b + c = 0
$\Rightarrow$ a - b + c = 1
$\Rightarrow$ c = 1 - a + b
Now, if $α, β, γ$ are the zeroes of the cubic polynomial ax $^{3}$ + bx $^{2}$ + cx + d, then product of zeroes is given by
$α β γ = - \frac{d}{a}$
So, for the given polynomial, p(x) = x $^{3}$ + ax $^{2}$ + bx + c
$α β (- 1) = \frac{- c}{1} = \frac{- (1 - a + b)}{1}$
$\Rightarrow α β = 1 - a + b$
Hence, the correct answer is option (a)

REAL NUMBERS

90318 The HCF and LCM of two numbers are 9 and 90 respectively. If one number is 18, then the other number is:

1 )63

2 )36

3 )45

4 )54

Explanation:

Exp: C
45
Using the result,
HCF × LCM = product of two natural numbers,
$\Rightarrow The other number = \frac{9 \times 90}{18} = 45$

REAL NUMBERS

90320 If 9 $^{x}$ + 2 = 240 + 9 $^{x}$ , then the value of x is:

1 )0.3

2 )0.2

3 )0.1

4 )0.5

Explanation:

Exp: D
0.5
$9^{x + 2} = 240 + 9 x$
$\Rightarrow 9^{x} . 9^{2} = 240 + 9 x$
$\Rightarrow 9^{x} (81 - 1) = 240$
$\Rightarrow 9^{x} = 3$
$\Rightarrow 9^{x} = 9 \frac{1}{2}$
$\Rightarrow x = \frac{1}{2} = 0.5$

REAL NUMBERS

90321 The smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is

1 )1002

2 )1001

3 )1000

4 )1080

Explanation:

Exp: D
1080
LCM (12, 15, 18, 27) = 540
Now, smallest four-digit number = 1000
$∴$ 1000 ÷ 540 = 1 × 540 + 460 (Remainder = 460)
Therefore, the smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is 1000 + (540 - 460) = 1000 + 80 = 1080

REAL NUMBERS

90323 If two positive integers m and n can be expressed as m = x $^{2}$ y $^{5}$ and n = x $^{3}$ y $^{2}$ , where x and y are prime numbers, then HCF(m, n) =

1 )x

^{2}

y

^{3}

2 )x

^{2}

y

^{2}

3 )x

^{3}

y

^{2}

4 )x

^{3}

y

^{3}

Explanation:

Exp: B
x $^{2}$ y $^{2}$
x $^{2}$ y $^{5}$ = y $^{3}$ (x $^{2}$ y $^{2}$ )
x $^{3}$ y $^{3}$ = x(x $^{2}$ y $^{2}$ )
Therefore HCF (m, n) is x $^{2}$ y $^{2}$

REAL NUMBERS

90386 If one of the zeroes of the cubic polynomial x $^{3}$ + ax $^{2}$ + bx + c is -1, then the product of other two zeroes is:

1 )b - a + 1

2 )b - a – 1

3 )a - b + 1

4 )a - b - 1

Explanation:

Exp: A
b - a + 1
Let p(x) = x $^{3}$ + ax $^{2}$ + bx + c
Now, -1 is a zero of the polynomial
So, p(0) = 0
$\Rightarrow$ (-1) $^{3}$ + a(-1) $^{2}$ + b(-1) + c = 0
$\Rightarrow$ -1 + a - b + c = 0
$\Rightarrow$ a - b + c = 1
$\Rightarrow$ c = 1 - a + b
Now, if $α, β, γ$ are the zeroes of the cubic polynomial ax $^{3}$ + bx $^{2}$ + cx + d, then product of zeroes is given by
$α β γ = - \frac{d}{a}$
So, for the given polynomial, p(x) = x $^{3}$ + ax $^{2}$ + bx + c
$α β (- 1) = \frac{- c}{1} = \frac{- (1 - a + b)}{1}$
$\Rightarrow α β = 1 - a + b$
Hence, the correct answer is option (a)

REAL NUMBERS

90318 The HCF and LCM of two numbers are 9 and 90 respectively. If one number is 18, then the other number is:

1 )63

2 )36

3 )45

4 )54

Explanation:

Exp: C
45
Using the result,
HCF × LCM = product of two natural numbers,
$\Rightarrow The other number = \frac{9 \times 90}{18} = 45$

REAL NUMBERS

90320 If 9 $^{x}$ + 2 = 240 + 9 $^{x}$ , then the value of x is:

1 )0.3

2 )0.2

3 )0.1

4 )0.5

Explanation:

Exp: D
0.5
$9^{x + 2} = 240 + 9 x$
$\Rightarrow 9^{x} . 9^{2} = 240 + 9 x$
$\Rightarrow 9^{x} (81 - 1) = 240$
$\Rightarrow 9^{x} = 3$
$\Rightarrow 9^{x} = 9 \frac{1}{2}$
$\Rightarrow x = \frac{1}{2} = 0.5$

REAL NUMBERS

90321 The smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is

1 )1002

2 )1001

3 )1000

4 )1080

Explanation:

Exp: D
1080
LCM (12, 15, 18, 27) = 540
Now, smallest four-digit number = 1000
$∴$ 1000 ÷ 540 = 1 × 540 + 460 (Remainder = 460)
Therefore, the smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is 1000 + (540 - 460) = 1000 + 80 = 1080

REAL NUMBERS

90323 If two positive integers m and n can be expressed as m = x $^{2}$ y $^{5}$ and n = x $^{3}$ y $^{2}$ , where x and y are prime numbers, then HCF(m, n) =

1 )x

^{2}

y

^{3}

2 )x

^{2}

y

^{2}

3 )x

^{3}

y

^{2}

4 )x

^{3}

y

^{3}

Explanation:

Exp: B
x $^{2}$ y $^{2}$
x $^{2}$ y $^{5}$ = y $^{3}$ (x $^{2}$ y $^{2}$ )
x $^{3}$ y $^{3}$ = x(x $^{2}$ y $^{2}$ )
Therefore HCF (m, n) is x $^{2}$ y $^{2}$

REAL NUMBERS

90386 If one of the zeroes of the cubic polynomial x $^{3}$ + ax $^{2}$ + bx + c is -1, then the product of other two zeroes is:

1 )b - a + 1

2 )b - a – 1

3 )a - b + 1

4 )a - b - 1

Explanation:

Exp: A
b - a + 1
Let p(x) = x $^{3}$ + ax $^{2}$ + bx + c
Now, -1 is a zero of the polynomial
So, p(0) = 0
$\Rightarrow$ (-1) $^{3}$ + a(-1) $^{2}$ + b(-1) + c = 0
$\Rightarrow$ -1 + a - b + c = 0
$\Rightarrow$ a - b + c = 1
$\Rightarrow$ c = 1 - a + b
Now, if $α, β, γ$ are the zeroes of the cubic polynomial ax $^{3}$ + bx $^{2}$ + cx + d, then product of zeroes is given by
$α β γ = - \frac{d}{a}$
So, for the given polynomial, p(x) = x $^{3}$ + ax $^{2}$ + bx + c
$α β (- 1) = \frac{- c}{1} = \frac{- (1 - a + b)}{1}$
$\Rightarrow α β = 1 - a + b$
Hence, the correct answer is option (a)

REAL NUMBERS

90318 The HCF and LCM of two numbers are 9 and 90 respectively. If one number is 18, then the other number is:

1 )63

2 )36

3 )45

4 )54

Explanation:

Exp: C
45
Using the result,
HCF × LCM = product of two natural numbers,
$\Rightarrow The other number = \frac{9 \times 90}{18} = 45$

REAL NUMBERS

90320 If 9 $^{x}$ + 2 = 240 + 9 $^{x}$ , then the value of x is:

1 )0.3

2 )0.2

3 )0.1

4 )0.5

Explanation:

Exp: D
0.5
$9^{x + 2} = 240 + 9 x$
$\Rightarrow 9^{x} . 9^{2} = 240 + 9 x$
$\Rightarrow 9^{x} (81 - 1) = 240$
$\Rightarrow 9^{x} = 3$
$\Rightarrow 9^{x} = 9 \frac{1}{2}$
$\Rightarrow x = \frac{1}{2} = 0.5$

REAL NUMBERS

90321 The smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is

1 )1002

2 )1001

3 )1000

4 )1080

Explanation:

Exp: D
1080
LCM (12, 15, 18, 27) = 540
Now, smallest four-digit number = 1000
$∴$ 1000 ÷ 540 = 1 × 540 + 460 (Remainder = 460)
Therefore, the smallest number of 4 digits exactly divisible by 12, 15, 18 and 27 is 1000 + (540 - 460) = 1000 + 80 = 1080

REAL NUMBERS

90323 If two positive integers m and n can be expressed as m = x $^{2}$ y $^{5}$ and n = x $^{3}$ y $^{2}$ , where x and y are prime numbers, then HCF(m, n) =

1 )x

^{2}

y

^{3}

2 )x

^{2}

y

^{2}

3 )x

^{3}

y

^{2}

4 )x

^{3}

y

^{3}

Explanation:

Exp: B
x $^{2}$ y $^{2}$
x $^{2}$ y $^{5}$ = y $^{3}$ (x $^{2}$ y $^{2}$ )
x $^{3}$ y $^{3}$ = x(x $^{2}$ y $^{2}$ )
Therefore HCF (m, n) is x $^{2}$ y $^{2}$

REAL NUMBERS

90386 If one of the zeroes of the cubic polynomial x $^{3}$ + ax $^{2}$ + bx + c is -1, then the product of other two zeroes is:

1 )b - a + 1

2 )b - a – 1

3 )a - b + 1

4 )a - b - 1

Explanation:

Exp: A
b - a + 1
Let p(x) = x $^{3}$ + ax $^{2}$ + bx + c
Now, -1 is a zero of the polynomial
So, p(0) = 0
$\Rightarrow$ (-1) $^{3}$ + a(-1) $^{2}$ + b(-1) + c = 0
$\Rightarrow$ -1 + a - b + c = 0
$\Rightarrow$ a - b + c = 1
$\Rightarrow$ c = 1 - a + b
Now, if $α, β, γ$ are the zeroes of the cubic polynomial ax $^{3}$ + bx $^{2}$ + cx + d, then product of zeroes is given by
$α β γ = - \frac{d}{a}$
So, for the given polynomial, p(x) = x $^{3}$ + ax $^{2}$ + bx + c
$α β (- 1) = \frac{- c}{1} = \frac{- (1 - a + b)}{1}$
$\Rightarrow α β = 1 - a + b$
Hence, the correct answer is option (a)

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