Explore topic-wise InterviewSolutions in .

1. bad × good 

2. never × always 

3. negative × positive 

4. cold × hot

I miserably failed in Standard IX. I lost all my hopes. I had decided to discontinue my studies forever. Meanwhile I happened to read a story of King Bruce, the King of Scotland. He fought many battles to free his country from English rule. Lost many battles. Once while hiding in a cave, he saw a spider trying to make his web on the wall of the cave. He slipped and fell again and again. But perseveringly tried again and finally succeeded and made his web.

King Bruce learnt a lesson, got together his army, defeated the English and freed his country. The story impressed me. I was moved by the story and decided to study hard and make an effort to appear for the exam again. The story changed my life.

Let f(x) = x² - 3ax - 2a

Since, x-2 is a factor of f(x) 

so,

f (2) = 0

2²+ 3 a (2) - 2a = 0

4 + 6a -2a = 0

a = -1

Let f(x) be a polynomial and f \((-\frac{1}{2})\) = 0

\(x+\frac{1}{2}=2x+1\) is a factor of f (x)

Let, f (x) = x³ + a

(x +1) is a factor of f(x), so f (-1) = 0

f (-1) =0

(-1)³ + a = 0

-1 + a = 0

a = 1

f (x) = x⁴ + x² - 20 

(x² + 5) (x² - 4) 

Therefore, 

(x² + 5) and (x² - 4) are the factors of f(x)

1. optimistic × pessimistic 

2. drizzle × pouring 

3. lie × truth 

4. new × old 

5. ugly × beautiful 

6. frowned × laughed 

7. hell × paradise 

8. unhappy × happy 

9. disrespect × respect 

10. unimportant × important

Let p(x) = x³ - 2(x²) + ax - b 

q (x) = x² - 2x - 3 

r (x) = x - 6 

Therefore, 

f(x) = p (x) – r (x) 

f(x) = x³ - 2x² + ax - b - x - 6 

= x³- 2x² + (a - 1) x - (b - 6) 

q(x) = x² - 2x - 3 

= (x + 1) (x - 3) 

Thus, 

(x + 1) and (x - 3) are factor of f (x) 

a + b = 4 

f (3) = 0 

3³ - 2 (3)² + (a-1) 3 - b + 6 = 0 

12 + 3a - b = 0 

a = - 2, b = 6

f(x) = x⁴ - 2x² + 3x² - ax - b

Given f(x) is divided by (x-1), then remainder is 6

f (1) = 6

1⁴ - 2 (1)³ - 3 (1)² - a (1) - b = 6

1 - 2 + 3 - a - b = 6

- a - b = 4

a + b = - 4

We have,

f(x) = 2x⁴ - 6x³ + 2x² - x + 2 and g(x) = x + 2

Therefore, by remainder theorem when f (x) is divided by g (x) = x – (-2), the remainder is equal to f (-2)

Now, f(x) = 2x⁴ - 6x³ + 2x² - x + 2

f (-2) = 2 (-2)⁴ – 6 (-2)³ + 2 (-2)² – (-2) + 2

= 2 * 16 + 48 + 8 + 2 + 2

= 32 + 48 + 12

= 92

Hence, required remainder is 92.

Correct option is (C) 260°

\(80^\circ,180^\circ-x\;\&\;180^\circ-y\) are angles of a triangle.

\(\therefore80^\circ+180^\circ-x+180^\circ-y=180^\circ\)

\(\Rightarrow\) \(x+y\) \(=80^\circ+180^\circ=260^\circ\)

Correct option is  C) 260°

We have,

f(x) = 4x³ - 12x² + 14x - 3 and g(x) = 2x - 1

Therefore, by remainder theorem when f (x) is divided by g (x) = 2 (x - \(\frac{1}{2})\), the remainder is equal to f \((\frac{1}{2})\)

Now, f(x) = 4x³ -12x² + 14x - 3

f\((\frac{1}{2})\) = 4 \((\frac{1}{2})^{3}-12(\frac{1}{2})^{2}+14(\frac{1}{2})-3\)

= (4 * \(\frac{1}{8})\) - (12 * \(\frac{1}{4})+7-3\)

= \(\frac{1}{2}-3+7-3\)

= \(\frac{3}{2}\)

Hence, required remainder is \(\frac{3}{2}\)

we have,

f (x) = 2x³ - 3x² + ax + b

Put,

x = 0

f (0) = 2 (0)³ – 3 (0)² + a (0) + b

= 0 – 0 + 0 + b

= b

x = - 1

f (-1) = 2 (-1)³ – 3 (-1)² + a (-1) + b

= - 2 – 3 – a + b

= - 5 – a + b

Since, x = 0 and x = -1 are roots of f (x)

f (0) = 0 and f (-1) = 0

b = 0 and - 5 – a + b = 0

= a – b = - 5

= a – 0 = - 5

= a = - 5

Therefore, 

a = - 5 and b = 0

We have,

f(x) = 4x⁴ - 3x³ - 2x² + x - 7 and g(x) = x - 1

Therefore, by remainder theorem when f (x) is divided by g (x) = x – 1, the remainder is equal to f (+1)

Now, 

f(x) = 4x⁴ - 3x³ - 2x² + x - 7

f (1) = 4 (1)4 – 3 (1)³ – 2 (1)² + 1 – 7

= 4 – 3 – 2 + 1 – 7

= -7

Hence, required remainder is -7.

Correct option is (D) 70°

y is exterior angle whose opposite interior angles in \(\triangle ABC\) are \(30^\circ\;and\;40^\circ.\)

\(\therefore y=30^\circ+40^\circ=70^\circ\)

Correct option is  D) 70°

We have,

f(x) = x³ + 6x² + 11x + 6

Clearly, f (x) is a polynomial with integer coefficient and the coefficient of the highest degree term i.e., the leading coefficient is 1.

Therefore, integer root of f (x) are limited to the integer factors of 6, which are:

+1,+2,+3,+6

We observe that'

f (-1) = (-1)³ + 6 (-1)² + 11 (-1) + 6

= -1 + 6 -11 + 6

= 0 f (-2) = (-2)³ + 6 (-2)² + 11 (-2) + 6 

= -8 + 24 – 22 + 6 

= 0

f (-3) = (-3)³ + 6 (-3)² + 11 (-3) + 6

= -27 + 54 – 33 + 6

= 0

Therefore, 

integral roots of f (x) are -1, -2, -3.

We have,

f(x) = x³ + 4x² - 3x + 10 and g (x) = x + 4

Therefore, by remainder theorem when f (x) is divided by g (x) = x – (-4), the remainder is equal to f (-4)

Now, f(x) = x³ + 4x² - 3x + 10

f (-4) = (-4)³ + 4 (-4)² – 3 (-4) + 10

= -64 + 4 * 16 + 12 + 10

= 22

Hence, required remainder is 22.

Correct option is (A) 8°

\(\because\) \(40^\circ+6x+2^\circ=90^\circ\)

\(\Rightarrow\) \(6x=90^\circ-40^\circ-2^\circ\)

\(=90^\circ-42^\circ=48^\circ\)

\(\Rightarrow\) \(x=\frac{48^\circ}6=8^\circ\)

Correct option is  A) 8°

We have,

f(x) = 2x 3+x 2-7x-6

Clearly, f (x) is a cubic polynomial with integer coefficients. if \(\frac{b}{c}\) is a rational root in lowest term, then the value of b are limited to the factors of 6 which are +1,+2,+3,+6 and values of c are limited to the factors of 2 which are +1,+2.

Hence, 

the possible rational roots of f(x) are:

+1,+2,+3,+6,+\(\frac{1}{2},\)+\(\frac{3}{2}\)

We observe that,

f (-1) = 2 (-1)³ + (-1)² – 7 (-1) – 6

= -2 + 1 + 7 – 6

= 0

f (2) = 2 (2)³ + (2)² – 7 (2) – 6

= 16 + 4 – 14 – 6

= 0

\(f(-\frac{3}{2})\) = 2 \((\frac{-3}{2})^{3}\) + \((\frac{-3}{2})^{2}\) - 7 \((\frac{-3}{2})-6\)

= \(\frac{-27}{4}+\frac{9}{4}+\frac{21}{2}-6\)

= 0

Hence, - 1, 2, \(\frac{-3}{2}\) are the rational roots of f (x).

Correct option is (A) ∠3

\(\angle 2\) and \(\angle 3\) are vertically opposite angles.

\(\therefore\) \(\angle 2\) = \(\angle 3\)

Correct option is  A) ∠3

Correct option is  C) b and c

Correct option is  D) (i) and (iii)

In the given figure, ∠AOB and ∠COD are not adjacent angles. Because, they have common vertex. But they have no common arm.

If the sum of any two angles is 360°, then the angles are called as conjugate angles. 

Conjugate angle of 300° is (360.-300) = 60°

Correct option is D) ∠EOD, ∠DOC

i) Angles between (Door and Wall). (Door and frame) 

ii) Angles between the 3 hands of a clock at a point of time. 

iii) Adjacent angles in the Rangoli. 

iv) Angles between spokes of the window design.

Correct option is (D) 360° – x

The conjugate angle of x is \(360^\circ-x.\)

\((\because\) Sum of two conjugate angles is \(360^\circ)\)

Correct option is  D) 360° – x

Correct option is (C) 45°

\(75^\circ\) is an exterior angle of a triangle whose opposite interior angles are x and \(30^\circ.\)

\(\therefore\) \(x+30^\circ=75^\circ\)

\(\Rightarrow x=75^\circ-30^\circ=45^\circ\)

Correct option is  C) 45°

Correct option is (D) 10

In given figure, AB, AC, AD, AE, BC, BD, BE, CD, CE and DE are line segments.

\(\therefore\) Total line segments in the given figure = 10.

Alternative :-

There are 5 points on line in given figure.

\(\because\) 2 points are required to form a line segment.

\(\therefore\) Total number of line segments \(=\,^5C_2=\frac{5\times4}2=10\)

Correct option is  D) 10

A) Both A and B are true statements

A transversal intersects two straight lines and co-interior angles are supplementary.

∴ By deflations, the lines will be parallel.

Let the monthly income of A and B are Rs.x and Rs.y respectively. 

Then as per the question 

x/y = 5/4 

⇒ y = 4x/5 

Since each save Rs.9,000, so 

Expenditure of A = Rs.(x – 9000) 

Expenditure of B = Rs.(y – 9000) 

The ratio of expenditures of A and B are in the ratio 7:5.

∴ x−9000/y−9000 = 7/5 

⇒7y – 63000 = 5x – 45000 

⇒7y – 5x = 18000 

From (i), substitute y = 4x/5 in (ii) to get 

7 × 4x/5 – 5x = 18000 

⇒28x – 25x = 90000 

⇒3x = 90000 

⇒x = 30000 

Now, putting x = 30000, we get 

y = 4×30000/5 = 4 × 6000 = 24000 

Hence, the monthly incomes of A and B are Rs. 30,000 and Rs.24,000.

Let the cost price of one table and one chair be ₹ x and ₹ y, respectively. 

So, 

The selling price of the table, when it’s sold at a profit of 10% = ₹ x + 10x/100 = ₹ 110x / 100 

The selling price of the chair, when it’s sold at a profit of 25% = ₹ y + 25y/100 = ₹ 125y / 100 

Hence, according to the question 

110x / 100 + 125y / 100 = 1050 … (i) 

Similarly, 

The selling price of the table, when it’s sold at a profit of 25% = ₹ (x + 25x/100) = ₹ 125x/ 100 

The selling price of the chair, when it’s sold at a profit of 10% = ₹ (y + 10y/100) = ₹ 110y / 100 

Hence, again from the question 

125x / 100 + 110y / 100 = 1065 … (ii) 

Re- written (i) and (ii) with their simplest coefficients, 

11x/10 + 5y/4 = 1050…….. (iii) 

5x/4 + 11y/10 = 1065…….. (iv) 

Adding (iii) and (iv), we get 

(11/ 10 + 5/ 4)x + (5/ 4 + 11/ 10)y = 2115 

47/ 20x + 47/ 20y = 2115 

x + y = 2115(20/ 47) = 900 

⇒ x = 900 – y ……. (v) 

Using (v) in (iii), 

11(900 – y)/10 + 5y/4 = 1050 

2(9900 -11y) + 25y = 1050 x 20 [After taking LCM] 

19800 – 22y + 25y = 21000 

3y = 1200 

⇒ y = 400 

Putting y = 400 in (v), we get 

x = 900 – 400 = 500 

Therefore, the cost price of the table is ₹ 500 and that of the chair is ₹ 400.

Correct Option (b) Curve B 

Explanation: 

At lower pressure we can assume that given gas behaves as ideal gas so PV/RT = constant but when pressure increase, the decrease in volume will not take place in same proportion so PV/RT will increases.

When the debentures are redeemed by the company, in annual instalments the serial number of debentures which should be redeemed each year are selected by lottery. This procedure is known as drawings by lots.

Correct option is (A) At the time of liquidation of company

Correct option is (B) Redeemable debenture

No, company can not forfeited the debenture of remaining instalment because debenture is debt of the company.

Correct option is (B) 10 years

Correct option is (C) 30 years

Correct option is (C) Bearer debenture

Correct option is (A) Simple debenture

(a) The public

This statement is False.

On retirement of a partner, his share is acquired by continuing partners in certain proportion and it is nothing but gain for them.

Therefore, on retirement of a partner instead of sacrifice ratio gain ratio is considered.

ADRs are issued in USA.

This statement is True.

When a person retires from the firm due to health issues, financial issues or personal reasons then it is known as person retires from the business and for the business, he is an outgoing partner.

Profit sharing ratio which is acquired by the continuing partners on account of retirement or death of a partner is called Benefit Ratio or Gain Ratio.

Correct option: (b) Debenture holders

Correct option: (b) Debenture holders