10类NCERT解决方案-第4章二次方程式-练习4.4

问题1.找到以下二次方程式的根的性质。如果存在真正的根，请找到它们：

(i)2x ² -3x + 5 = 0

^(ⅱ)3×2 +-4√3×4 = 0

(iii)2x ² -6x + 3 = 0

解决方案：

(i) Given: 2x²-3x+5=0

Here a=2,b=-3 and c=5

$\therefore$ Discriminant, D=b²-4ac

= (-3)²– 4 × 2 × 5)

= 9-40 = -31 < 0

Hence, the roots are imaginary.

(ii) Given: 3x²-4√3x + 4 = 0

Here a=3,b=√3 and c=4

$\therefore$ Discriminant, D=b²-4ac

= (-4√3)² – (4 × 3 × 4)

= 48 – 48 = 0

Hence, the roots are real and equal.

Using the formula,

$x=\frac {-b\pm\sqrt{b^2-4ac}} {2a}$, we get 

$x=\frac {-(-4\sqrt3)\pm\sqrt{(-4\sqrt3)^2-4\times3\times4}} {2\times3}$

$= \frac {4\sqrt3\pm\sqrt{48-48}} {6}=\frac {4\sqrt3} {6}=\frac {2} {\sqrt3}$

Hence, the equal roots are $\frac 2 {\sqrt3}$ and $\frac 2 {\sqrt3}$.

(iii) Given: 2x²-6x+3=0

Here, a=2,b=-6 and c=3

$\therefore$ Discriminant, D=b²-4ac

= (-6)² – (4 × 2 × 3)

= 36 – 24 = 12 > 0

Hence, the roots are distinct and real.

Using the formula,

$x=\frac {-b\pm\sqrt{b^2-4ac}} {2a}$,we get

$x=\frac {-(-6)\pm\sqrt{(-6)^2-4\times2\times3}} {2\times2}$

$x=\frac {6\pm\sqrt{36-24}} {4}$

$x=\frac {6\pm\sqrt{12}} {4}$

$x=\frac {6\pm2\sqrt{3}} {4}$

$x=\frac {3\pm\sqrt{3}} {2}$

Hence, the equal roots are $\frac {3+\sqrt{3}} {2}$and $\frac {3-\sqrt{3}} {2}$

问题2。为以下每个二次方程式求k的值，以便它们具有两个相等的根。

(i) 2x ² + kx + 3

(ii)kx(x-2)+ 6 = 0

解决方案：

(i) 2x+kx+3=0

This equation is of the form ax²+bx+x, where a=2, b=k and c=3.

Discriminant, D=b²-4ac

=k²  – 4 × 2 × 3

=k² -24

For equal roots D=0

$\implies$k²-24=0

$\implies$k²=24

$\implies$k² = ±24 = ±2√6​
 

(ii) kx(x-2)+6=0

$\implies$ kx²-2kx+6=0

This equation is of the form ax²+bx+c=0, where a=k, b=-2k and c=6.

Discriminant, D=b²-4ac

=(-2k)² – 4 × k × 6

=4k²-24k

For equal roots D=0

$\implies$ 4k²-24k=0

$\implies$ 4k(k-24)=0

$\implies$k=0 (not possible) or 4k-24=0

$\implies$ k= 24/4=6

问题3.是否可以设计一个长度为宽度的两倍，面积为800 m ²的矩形芒果林?如果是这样，请找到其长度和宽度。

解决方案：

Let the breadth of the rectangular mango grove be x m.

Then, the length of the rectangular mango grove will be 2x m.

The Area of the rectangular mango grove=length × breadth

According to the question, we have

x × 2x= 800

$\implies$ 2x²=800

$\implies$ x²=400

$\implies$ x=20

Hence, the rectangular mango grove is possible to design whose length=40 m and breadth=20 m.

问题4.以下情况是否可能?如果是这样，请确定他们的年龄。两个朋友的年龄之和为20岁。四年前，他们年龄的乘积是48。

解决方案：

Let the present age of one friend be x years.

Then, the present age of other friend be (20-x) years.

4 years ago, one friend’s age was (x-4) years

4 years ago, other friend’s age was (20-x-4)=(16-x) years.

According to the question,

(x-4)(16-x)=48

$\implies$ 16x-64-x²+4x=48

$\implies$ x²-20x+112=0

This equation is of the form ax²+bx+c=0,where a=1, b=-20 and c=112.

Discriminant, D=b²-4ac

= (-20)²-4 × 1 × 112 = -48 < 0

Since, there are no real roots.

So the given situation is not possible.

问题5.是否可以设计一个周长80 m，面积400 m ²的矩形停车库?如果是这样，请找到其长度和宽度。

解决方案：

Let the length of the rectangular park be x.

The perimeter of the rectangular park= 2(length + breadth)

$\implies$ 2(x + breadth)=80

$\implies$ breadth=40-x

The area of rectangular park= length × breadth

$\implies$ x(40-x)=400

\implies 40x-x²=400

\implies x²-40x+400=0

\implies x² -20x-20x+400=0

$\implies$ (x-20)(x-20)=0

$\implies$x=20

Hence, the rectangular park is possible to design. So, the length of the park is 20m and the breadth = 40-20=20m.