(Each question 4 marks) - MATHS STD 11 Science Questions - Vidyadip

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18 questions · self-marked practice — reveal the answer and mark yourself.

Question 14 Marks

IQ of a person is given by the formula
$\mathrm{IQ}=\frac{\mathrm{MA}}{\mathrm{CA}} \times 100$
where MA is mental age and CA is chronological age. If 80 $\le$ IQ $\le$ 140 for a group of 12 years old children, find the range of their mental age.

Answer

It is given that $80 \leqslant Q \leqslant 140$ and CA = 12
We have $IQ = \frac{{MA}}{{CA}} \times 100$
$\therefore 80 \leqslant \frac{{MA}}{{12}} \times 100 \leqslant 140$
$ \Rightarrow 960 \leqslant MA \times 100 \leqslant 1680$
$ \Rightarrow 9.6 \leqslant MA \leqslant 16.8$
Thus minimum MA is 9.6 and maximum 16.8

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Question 24 Marks

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Answer

Let x litres of water be added to 1125 litres of 45% acid solution.
Then total quantity of mixture = (1125 +x ) litres
$\frac{{45}}{{100}} \times 1125 + 0 \times \frac{x}{{100}} > $$\frac{{25}}{{100}} \times (1125 + x)$
and $\frac{{45}}{{100}} \times 1125 + 0 \times \frac{x}{{100}} < \frac{{30}}{{100}}$$ \times (1125 + x)$
Combining the above inequations, we get
$\frac{{25}}{{100}} \times 100 \leqslant \frac{{2025 \times 100}}{{4(1125 + x)}} \leqslant \frac{{30}}{{100}} \times 100$
$ \Rightarrow 25 \leqslant \frac{{50625}}{{1125 + x}} \leqslant 30$
$ \Rightarrow 25 \leqslant \frac{{50625}}{{1125 + x}}$ and $\frac{{50625}}{{1125 + x}} \leqslant 30$
$ \Rightarrow 28125 + 25x \leqslant 50625$ and $50625 \leqslant 33750 + 30x$
$ \Rightarrow 25x \leqslant 22500$ and $30x \geqslant 1687.5$
$ \Rightarrow x \leqslant 900$ and $x \geqslant 562.5$
$ \Rightarrow 562.5 \leqslant x \leqslant 900$
Thus minimum 562.5 litres and maximum 900 litres of water need to be added.

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Question 34 Marks

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

Answer

Let x litre of 2% boric acid solution be added to 640 litres of 8% boric acid solution. Then
Total quality of mixture = (640 + x) litres
Total boric acid in (640 + x) litres of mixture $ = \frac{{2x}}{{100}} + \frac{8}{{100}} \times 640$
$ = \frac{x}{{50}} + \frac{{256}}{5}$
It is given that the resulting mixture must be more than 4% but less than 6% boric acid.
$\therefore \frac{4}{{100}}(640 + x) < \frac{x}{{50}} + \frac{{256}}{5}$$ < \frac{6}{{100}}(640 + x)$
$ \Rightarrow \frac{{640 + x}}{{25}} < \frac{{x + 2560}}{{50}} < \frac{{1920 + 3x}}{{50}}$
$ \Rightarrow $ 1280 + 2x < x + 2560 < 1920 + 3x
$ \Rightarrow $ 1280 + 2x < x + 2560 and x + 2560 < 1920 + 3x
$ \Rightarrow $ x < 1280 and
$ \Rightarrow $ x < 1280 and x > 320
$ \Rightarrow $ 320 < x < 1280
Thus 2% boric acid solution must be more than 320 litres but less than 1280 litres.

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Question 44 Marks

Solve the system of inequality graphically: 5x + 4y $\le$ 20, x $\ge$ 1, y $\ge$ 2

Answer

Given 5x + 4y $\leq$ 20,
let 5x + 4y = 20 .......(i)
Putting value of x = 0 and y = 0 in equation (i) one by one, we get value of
y = 5 and x = 4
The required points are (0, 5) and (4, 0) for 5x+4y=20
Checking If the origin lies in the solution area (0, 0)
0 $\leq$ 20
Which is true, hence the origin would lie in the solution area. The required area of the line`s graph is on the left side of the graph.
x $\ge$ 1
let x = 1 ......(ii)
for all the values of y, x would be 1,
The required points would be (1, 0), (1, 2) and so on.
Checking for origin (0, 0)
0 $\ge$ 1, which is not true
So the origin would not lie in the required area. The required area on the graph will be on the right side of the line graph.
y $\ge$ 2
let y = 2 ......(iii)
Similarly, for all the values of x, y would be 2.
The required points would be ( 0, 2), (1, 2) and so on.
Checking for origin (0, 0)
0 $\ge$ 2, this is no true
Hence the required area would be on the right side of the line`s graph.
The shaded area on the graph shows the required solution to the given inequalities.

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Question 54 Marks

Solve the system of inequality graphically: x + y $\le$ 9, y > x, x $\ge$ 0

Answer

The given inequality is $x + y \leqslant 9$
Draw the graph of the line x + y = 9

Table of values satisfying the equation x + y = 9

X	5	4
Y	4	5

Putting (0, 0) in the given inequation, we have
$0 + 0 \leqslant 9 \Rightarrow 0 \leqslant 9$, which is true.
$\therefore $ Half plane of $x + y \leqslant 9$ is towards origin.
Also the given inequality is x - y < 0
Draw the graph of the line x - y = 0
Table of values satisfying the equation x - y = 0

X	1	2
Y	1	2

Putting (0, 3) in the given inequation we have
$0 - 3 < 0 \Rightarrow - 3 < 0$, which is true.
$\therefore $ Half plane of x - y < 0 containing the point (0, 3).

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Question 64 Marks

Solve the system of inequality graphically: 2x + y $\ge$ 8, x + 2y $\ge$ 10

Answer

The given inequality is $2x + y \geqslant 8$.
Draw the graph of the line 2x + y = 8.

Table of values satisfying the equation 2x + y = 8

X	3	4
Y	2	0

Putting (0, 0) in the given inequation, we have
$2 \times 0 + 0 \geqslant 8 \Rightarrow 0 \geqslant 8$, which is false
$\therefore $ Half plane of $2x + y \geqslant 8$ is away from origin.
Also the given inequality is $x + 2y \geqslant 10$
Draw the graph of the line x + 2 y = 10
Table of the values satisfying the equation x + 2y = 10

X	2	4
Y	4	3

Putting (0, 0) in the given inequation, we have
$0 + 2 \times 0 \geqslant 10 \Rightarrow 0 \leqslant 10$, which is false.
$\therefore $ Half plane of $x + 2y \geqslant 10$ is away from origin.

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Question 74 Marks

Solve the system of inequality graphically: x + y $\le$ 6, x + y $\ge$ 4

Answer

The given in equality is $x + y \leqslant 6$
Draw the graph of the line x + y = 6.
Table of values satisfying the equation x + y = 6

X	3	4
Y	3	2

Putting (0, 0) in the given inequation, we have
$0 + 0 \leqslant 6 \Rightarrow 0 \leqslant 6$, which is true.
$\therefore $ Half plane of $x + y \leqslant 6$ is towards origin.

Also the given inequality is $x + y \geqslant 4$
Draw the graph of the line x + y = 4
Table of values satisfying the equation x + y = 4

X	2	1
Y	2	3

Putting (0, 0) in the given inequation, we have
$0 + 0 \geqslant 4 \Rightarrow 0 \geqslant 4$, which is false
$\therefore $ Half plane of $x + y \geqslant 4$ is away from origin.

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Question 84 Marks

Solve the system of inequality graphically: 2x – y >1, x – 2y < –1

Answer

The given inequality is 2x - y > 1
Draw the graph of the line 2x - y = 1
Table of values satisfying the equation
2x - y = 1

X	1	2
Y	1	3

Putting (0, 0) in the given inequation, we have
$2 \times 0 - 0 > 1 \Rightarrow 0 > 1$, which is false.
$\therefore $ Half plane of 2x - y > 1 is away from origin.
Also the given inequality is x - 2y < -1
Draw the graph of the line x - 2y = -1
Table of values satisfying the equation x - 2y = -1

X	1	2
Y	1	2

Putting (0, 0) in the given inequation, we have
$0 - 2 \times 0 < - 1 \Rightarrow 0 < - 1$ which is false
$\therefore $ Half plane of x - 2y < - 1 is away from origin.

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Question 94 Marks

Solve the system of inequality graphically: x + y $\ge$ 4, 2x – y < 0

Answer

The given inequality is $x + y \geqslant 4$
Draw the graph of the line x + y = 4.
Table of values satisfying the equation x + y = 4.

X	3	2
Y	1	2

Putting (0, 0) in the given inequation, we have
$0 + 0 \geqslant 4 \Rightarrow 0 \geqslant 4$, which is false.
$\therefore $ Half plane of $x + y \geqslant 4$ is away from origin.
Also the given inequality is 2x - y < 0
Draw the graph of the line 2x - y = 0
Table of values satisfying the equation 2x - y = 0

X	1	2
Y	2	4

Putting (3, 0) in the given inequation, we have
$2 \times 3 - 0 <0 \Rightarrow 6 <0$, which is false.
$\therefore $ Half plane of 2x - y = 0 does not contain (3, 0)

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Question 104 Marks

Solve the system of inequality graphically: 2x + y $\ge$ 6, 3x + 4y $\le$ 12

Answer

The given inequality is $2x + y \geqslant 6$
Draw the graph of the line 2x + y = 6
Table of values satisfying the equation 2x + y = 6

X	3	2
Y	0	2

Putting (0, 0) in the given in equation, we have
$2 \times 0 + 0 \geqslant 6 \Rightarrow 0 \geqslant 6$, which is false.
$\therefore $ Half plane of $2x + y \geqslant 6$ is away from origin.
Also the given inequality is $3x + 4y \leqslant 12$.
Draw the graph of the line 3x + 4y = 12
Table of values satisfying the equation
3 x + 4y = 12

X	0	4
Y	3	0

Putting (0, 0) in the given inequation, we have
$3 \times 0 + 4 \times 0 \geqslant 12 \Rightarrow 0 \leqslant 12$, which is true.
$\therefore $ Half plane of $3x + 4y \leqslant 12$ is towards origin.

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Question 114 Marks

Solve the system of inequality graphically: 3x + 2y $\le$ 12, x $\ge$ 1, y $\ge$ 2

Answer

The given inequality is
$3x + 2y \leqslant 12$,
Draw the graph of the line
3x + 2y = 12
Table of values satisfying the equation 3x + 2y = 12

X	2	4
Y	3	0

Putting (0, 0) in the given in equation, we have
$3 \times 0 + 2 \times 0 \leqslant 12 \Rightarrow 0 \leqslant 12$ which is true.
$\therefore $ Half plane of $3x + 2y \leqslant 12$, is towards origin.
Also the given inequality is $x \geqslant 1$.
Draw the graph of the line x = 1.
Putting (0, 0) in the given inequation, we have $0 \geqslant 1$ which is false.
$\therefore $ Half plane of $x \geqslant 1$ is away from origin.
The given inequality is $y \geqslant 2$.
Draw the graph of the line y = 2.
Putting (0, 0) in the given inequation, we have $0 \geqslant 2$ which is false.
$\therefore $ Half plane of $y \geqslant 2$ is away from origin.

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Question 124 Marks

Solve the system of inequality graphically: x + 2y $\le$ 10, x + y $\ge$ 1, x – y $\le$ 0, x $\ge$ 0, y $\ge$ 0

Answer

The given inequality is $x + 2y \leqslant 10$.
Draw the graph of the line x + 2y = 10.
Table of values satisfying the equation x + 2 y = 10

X	2	4
Y	4	3

Putting (0, 0) in the given inequation, we have
$0 + 2 \times 0 \leqslant 10 \Rightarrow 0 \leqslant 10,$ which is true.

$\therefore $ Half plane of $x + 2y \leqslant 10$is towards origin.
Also the given inequality is $x + y \geqslant 1$.
Draw the graph of the line x + y = 1
Table of values satisfying the equation x + y = 1.

X	0	1
Y	1	0

Putting (0, 0) in the given inequation we have
$0 + 0 \geqslant 1 \Rightarrow 0 \geqslant 1,$ which is false.
$\therefore $ Half plane of $x + y \geqslant 1$ is away from origin.
Also the given inequality is $x - y \leqslant 0$
Draw the graph of the line x - y = 0
Table of values satisfying the equation x - y = 0

X	1	2
Y	1	2

Putting (2, 0) in the given inequation, we have
$2 - 0 \leqslant 0 \Rightarrow 2 \leqslant 0$ which is false.
$\therefore $ Half plane of $x - y \leqslant 0$ is away from origin.

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Question 134 Marks

Solve the system of inequality graphically: 3x + 2y $\le$ 150, x + 4y $\le$ 80, x $\le$ 15, y $\ge$ 0, x $\ge$ 0

Answer

The given inequality is 3x + 2y $\le$ 150,
Draw the graph of the line 3x + 2y = 150
Table of values satisfying the equation 3x + 2y = 150

X	30	40
Y	30	15

Putting (0, 0) in the given inequation, we have
3 $\times$ 0 + 2 $\times$ 0 $\le$ 150
$\Rightarrow$ 0 $\le$ 150, which is true.

$\therefore $ Half plane of 3x + 2y $\le$ 150 is towards the origin.
Also, the given inequality is x + 4y $\le$ 80
Draw the graph of the line x + 4y = 80
Table of values satisfying the equation x + 4y = 80

X	0	40
Y	20	10

Putting (0, 0) in the given inequation, we have
0 + 4 $\times$ 0 $\le$ 80
$\Rightarrow$ 0 $\le$ 80 which is true.
$\therefore $ Half plane of x + 4y $\le$ 80 is towards origin.
The given inequality is x $\le$ 15
Draw the graph of the line x = 15.
Putting (0, 0) in the given inequation, we have
0 $\le$ 15, which is true.
$\therefore $ Half plane of x $\le$ 15 is towards origin.

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Question 144 Marks

Solve the system of inequality graphically: 4x + 3y $\le$ 60, y $\ge$ 2x, x $\ge$ 3, x, y $\ge$ 0

Answer

The given inequality is $4x + 3y \leqslant 60$
Draw the graph of the line 4x + 3y = 60.
Table of values satisfying the equation
4x + 3y = 60

X	15	0
Y	0	20

Putting (0, 0) in the given inequation, we have
$4 \times 0 + 3 \times 0 \leqslant 60 \Rightarrow 0 \leqslant 60,$which is true.
$\therefore $ Half plane of $4x + 3y \leqslant 60$ is towards origin.
Also the given inequality is $2x - y \leqslant 0$.
Draw the graph of the line 2x - y = 0.
Table of values satisfying the equation 2x - y = 0

X	5	10
Y	10	20

Putting (10, 0) in the given inequation, we have
$2 \times 10 - 0 \leqslant 0 \Rightarrow 20 \leqslant 0,$which is false.
$\therefore $Half plane of $2x - y \leqslant 0$ does not contain (10, 0).
The given inequality is $x \geqslant 3$.
Draw the graph of the line x = 3.
Putting (0, 0) in the given inequation, we have
$0 \geqslant 3,$ which is false.
$\therefore $ Half plane of $x \geqslant 3$ is away from origin.

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Question 154 Marks

Solve the system of inequality graphically: x – 2y $\le$ 3, 3x + 4y $\ge$ 12, x $\ge$ 0 , y $\ge$ 1

Answer

The given inequality is $x - 2y \leqslant 3$
Draw the graph of the line x - 2y = 3
Table of values satisfying the equation x - 2y = 3

X	1	5
Y	-1	1

Putting (0, 0) in the given inequation, we have
$0 - 2 \times 0 \leqslant 3 \Rightarrow 0 \leqslant 3$, which is true.
$\therefore $ Half plane of $x - 2y \leqslant 3$ is towards origin.
Also the given inequality is $3x + 4y \geqslant 12$
Draw the graph of the line 3x + 4y = 12
Table of values satisfying the equation 3x + 4y = 12

X	4	0
Y	0	3

Putting (0, 0) in the given inequation, we have
$3 \times 0 + 4 \times 0 \geqslant 12 \Rightarrow 0 \geqslant 12$, which is false.
$\therefore $ Half plane of $3x + 4y \geqslant 2$ is away from origin.
The given inequality is $y \geqslant 1$.
Draw the graph of the line y = 1.
Putting (0, 0) in the given inequation, we have
$0 \geqslant 1$, which is false.
$\therefore $ Half plane of $y \geqslant 1$ is away from origin.

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Question 164 Marks

Solve the system of inequality graphically: 2x + y $\geq$ 4, x + y $\le$ 3, 2x – 3y $\le$ 6

Answer

The given inequality is 2x + y $\geq$ 4
Draw the graph of the line 2x + y = 4
Table of values satisfying the equation 2x + y = 4

X	2	1
Y	0	2

Putting (0, 0) in the given inequation, we have
$\Rightarrow$ 2 $\times$ 0 + 0 $\Rightarrow$ 0 $\geq$ 4, which is false.
$\therefore $ Half plane of 2x + $\geq$ 4 is away from origin.
Also, the given inequality is $x + y \leq 3$
Draw the graph of the line x + y = 3
Table of values satisfying the equation x + y = 3

X	2	1
Y	1	2

Putting (0, 0) in the given inequation, we have
0 + 0 $\leq$ 3 $\Rightarrow$ 0 $\leq$ 3, which is true
$\therefore $ Half plane of x + y $\leq$ 3 is towards origin.
The given inequality is 2x - 3y $\leq$ 6
Draw the graph of the line 2x - 3y = 6
Table of values satisfying the equation 2x - 3y = 6

X	0	3
Y	-2	0

Putting (0, 0) in the given inequation, we have
2 $\times$ 0 - 3 $\times$ 0 $\leq$ 6
$\Rightarrow$ 0 $\leq$ 6, which is true,
$\therefore $ Half plane of 2x - 3y $\leq$ 6 is towards origin.

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Question 174 Marks

Solve the system of inequality graphically: 3x + 4y $\le$ 60, x +3y $\le$ 30, x $\ge$ 0, y $\ge$ 0

Answer

The given inequality is $3x + 4y \leqslant 60$
Draw the graph of the line 3x + 4y = 60
Table of values satisfying the equation 3x + 4y = 60

X	8	12
Y	9	6

Putting (0, 0) in the given in equation, we have
$3 \times 0 + 4 \times 0 \leqslant 60 \Rightarrow 0 \leqslant 60$, which is true.
$\therefore $ Half plane of $3x + 4y \leqslant 60$ is towards origin.
Also the given inequality is $x + 3y \leqslant 30$,
Draw the graph of the line x + 3y = 30
Table of values satisfying the equation x + 3y = 30

X	0	9
Y	10	7

Putting (0, 0) in the given inequation, we have
$0 + 3 \times 0 \leqslant 30 \Rightarrow 0 \leqslant 30$, which is true.
$\therefore $ Half plane of $x + 3y \leqslant 30$ is towards origin.

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Question 184 Marks

Solve the system of inequality graphically: x $\ge$ 3, y $\ge$ 2

Answer

The given inequality is $x \geqslant 3$.
Draw the graph of the line x = 3.
Putting (0, 0) in the given in equation, we have $0 \geqslant 3$ which is false.
$\therefore $ Half plane of $x \geqslant 3$ is away from origin.
Also the given inequality is $y \geqslant 2$
Draw the graph of the line y = 3.
Putting (0, 0) in the given inequation, we have $0 \geqslant 2$ which is false.

$\therefore $ Half plane of $y \geqslant 2$ is away from origin.

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