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32. Prove that $\frac{1}{\sqrt{2}}$ is irrational.

Assume, for the sake of contradiction, that $\frac{1}{\sqrt{2}}$ is a rational number.
If $\frac{1}{\sqrt{2}}$ is rational, then it can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are coprime integers and $b \neq 0$ .
$\frac{1}{\sqrt{2}} = \frac{a}{b}$
Rearrange the equation to isolate $\sqrt{2}$ .
$b = a\sqrt{2}$ $\sqrt{2} = \frac{b}{a}$
Analyze the result.
Since $a$ and $b$ are integers and $a \neq 0$ , the ratio $\frac{b}{a}$ is a rational number. This implies that $\sqrt{2}$ is a rational number.
State the contradiction and conclusion.
However, it is a known fact that $\sqrt{2}$ is an irrational number. This contradicts our initial assumption. Therefore, our assumption that $\frac{1}{\sqrt{2}}$ is rational must be false.
Hence, $\frac{1}{\sqrt{2}}$ is irrational.

33. Find the nature of the roots of the quadratic equation $2x^2 - 6x + 3 = 0$ .
Identify the coefficients $a, b, c$ from the quadratic equation $ax^2 + bx + c = 0$ .
For $2x^2 - 6x + 3 = 0$ , we have $a=2$ , $b=-6$ , and $c=3$ .
Calculate the discriminant $D = b^2 - 4ac$ .
$D = (-6)^2 - 4(2)(3)$ $D = 36 - 24$ $D = 12$
Determine the nature of the roots based on the value of $D$ .
Since $D = 12 > 0$ , the roots are real and distinct.
The nature of the roots is real and distinct.

34. Find a quadratic polynomial whose sum and product of zeroes are $\sqrt{2}$ and $\frac{1}{3}$ respectively.
Identify the sum of zeroes ( $S$ ) and product of zeroes ( $P$ ).
Given sum of zeroes, $S = \sqrt{2}$ .
Given product of zeroes, $P = \frac{1}{3}$ .
Use the general form of a quadratic polynomial
$k(x^2 - Sx + P)$ .
Substitute the values of $S$ and $P$ :
$P(x) = k\left(x^2 - \sqrt{2}x + \frac{1}{3}\right)$
Choose a suitable value for $k$ to simplify the polynomial (e.g., to clear fractions).
Let $k=3$ .
$P(x) = 3\left(x^2 - \sqrt{2}x + \frac{1}{3}\right)$ $P(x) = 3x^2 - 3\sqrt{2}x + 3 \times \frac{1}{3}$ $P(x) = 3x^2 - 3\sqrt{2}x + 1$
A quadratic polynomial is $\boxed{3x^2 - 3\sqrt{2}x + 1}$ .

35. Evaluate: $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}$ .
Substitute the value of $\tan 30^\circ$ .
We know that $\tan 30^\circ = \frac{1}{\sqrt{3}}$ .
$\frac{2 \left(\frac{1}{\sqrt{3}}\right)}{1 + \left(\frac{1}{\sqrt{3}}\right)^2}$
Simplify the expression.
$= \frac{\frac{2}{\sqrt{3}}}{1 + \frac{1}{3}}$ $= \frac{\frac{2}{\sqrt{3}}}{\frac{3+1}{3}}$ $= \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}}$ $= \frac{2}{\sqrt{3}} \times \frac{3}{4}$ $= \frac{6}{4\sqrt{3}}$ $= \frac{3}{2\sqrt{3}}$
Rationalize the denominator.
$= \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$ $= \frac{3\sqrt{3}}{2 \times 3}$ $= \frac{\sqrt{3}}{2}$
Alternatively, recognize the identity $\frac{2 \tan A}{1 + \tan^2 A} = \sin 2A$ .
Here $A = 30^\circ$ , so the expression is $\sin (2 \times 30^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}$ .
The value is $\boxed{\frac{\sqrt{3}}{2}}$ .

36. Prove that: $\sec A (1 - \sin A) (\sec A + \tan A) = 1$ .
Start with the Left Hand Side (LHS) and express $\sec A$ and $\tan A$ in terms of $\sin A$ and $\cos A$ .
$LHS = \sec A (1 - \sin A) (\sec A + \tan A)$ $LHS = \frac{1}{\cos A} (1 - \sin A) \left(\frac{1}{\cos A} + \frac{\sin A}{\cos A}\right)$
Combine the terms in the second parenthesis.
$LHS = \frac{1}{\cos A} (1 - \sin A) \left(\frac{1 + \sin A}{\cos A}\right)$
Multiply the terms.
$LHS = \frac{(1 - \sin A)(1 + \sin A)}{\cos A \cdot \cos A}$ $LHS = \frac{1^2 - \sin^2 A}{\cos^2 A}$
Use the trigonometric identity $\sin^2 A + \cos^2 A = 1$ , which implies $1 - \sin^2 A = \cos^2 A$ .
$LHS = \frac{\cos^2 A}{\cos^2 A}$ $LHS = 1$
Conclude that LHS = RHS.
Since LHS = 1 and RHS = 1, the identity is proven.
$\boxed{\sec A (1 - \sin A) (\sec A + \tan A) = 1}$

37. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box. Find the probability that it bears a perfect square number.
Determine the total number of possible outcomes.
Total number of discs = 90.
Identify the perfect square numbers between 1 and 90.
Perfect squares are numbers obtained by squaring an integer.
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 16$
$5^2 = 25$
$6^2 = 36$
$7^2 = 49$
$8^2 = 64$
$9^2 = 81$
$10^2 = 100$ (This is greater than 90, so it's not included).
The perfect square numbers are $\{1, 4, 9, 16, 25, 36, 49, 64, 81\}$ .
Count the number of favorable outcomes.
Number of perfect square numbers = 9.
Calculate the probability.
Probability (Perfect Square) = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
$P(\text{Perfect Square}) = \frac{9}{90}$ $P(\text{Perfect Square}) = \frac{1}{10}$
The probability that the disc bears a perfect square number is $\boxed{\frac{1}{10}}$ .

38. E and F are points on the sides PQ and PR respectively of a $\triangle PQR$ . If PE = 4cm, EQ = 4.5cm, PF = 8cm and RF = 9cm. State whether EF || QR.
Calculate the ratio of the segments on side PQ.
$\frac{PE}{EQ} = \frac{4 \text{ cm}}{4.5 \text{ cm}} = \frac{40}{45} = \frac{8}{9}$
Calculate the ratio of the segments on side PR.
$\frac{PF}{RF} = \frac{8 \text{ cm}}{9 \text{ cm}}$
Compare the two ratios.
We have $\frac{PE}{EQ} = \frac{8}{9}$ and $\frac{PF}{RF} = \frac{8}{9}$ .
Since $\frac{PE}{EQ} = \frac{PF}{RF}$ , the line segment EF divides the sides PQ and PR in the same ratio.
Apply the Converse of Basic Proportionality Theorem (BPT).
According to the Converse of BPT, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Therefore, EF || QR.

Write in english please

Step-by-step solution

Assume, for the sake of contradiction, that $\frac{1}{\sqrt{2}}$ is a rational number.

Rearrange the equation to isolate $\sqrt{2}$ .

Analyze the result.

State the contradiction and conclusion.

Identify the coefficients $a, b, c$ from the quadratic equation $ax^2 + bx + c = 0$ .

Calculate the discriminant $D = b^2 - 4ac$ .

Determine the nature of the roots based on the value of $D$ .

Identify the sum of zeroes ( $S$ ) and product of zeroes ( $P$ ).

Use the general form of a quadratic polynomial

Choose a suitable value for $k$ to simplify the polynomial (e.g., to clear fractions).

Substitute the value of $\tan 30^\circ$ .

Simplify the expression.

Rationalize the denominator.

Start with the Left Hand Side (LHS) and express $\sec A$ and $\tan A$ in terms of $\sin A$ and $\cos A$ .

Combine the terms in the second parenthesis.

Multiply the terms.

Use the trigonometric identity $\sin^2 A + \cos^2 A = 1$ , which implies $1 - \sin^2 A = \cos^2 A$ .

Conclude that LHS = RHS.

Determine the total number of possible outcomes.

Identify the perfect square numbers between 1 and 90.

Count the number of favorable outcomes.

Calculate the probability.

Calculate the ratio of the segments on side PQ.

Calculate the ratio of the segments on side PR.

Compare the two ratios.

Apply the Converse of Basic Proportionality Theorem (BPT).

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More English Questions

Write in english please

Step-by-step solution

Assume, for the sake of contradiction, that 12\frac{1}{\sqrt{2}}2​1​ is a rational number.

Rearrange the equation to isolate 2\sqrt{2}2​.

Analyze the result.

State the contradiction and conclusion.

Identify the coefficients a,b,ca, b, ca,b,c from the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0.

Calculate the discriminant D=b2−4acD = b^2 - 4acD=b2−4ac.

Determine the nature of the roots based on the value of DDD.

Identify the sum of zeroes (SSS) and product of zeroes (PPP).

Use the general form of a quadratic polynomial

Choose a suitable value for kkk to simplify the polynomial (e.g., to clear fractions).

Substitute the value of tan⁡30∘\tan 30^\circtan30∘.

Simplify the expression.

Rationalize the denominator.

Start with the Left Hand Side (LHS) and express sec⁡A\sec AsecA and tan⁡A\tan AtanA in terms of sin⁡A\sin AsinA and cos⁡A\cos AcosA.

Combine the terms in the second parenthesis.

Multiply the terms.

Use the trigonometric identity sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1sin2A+cos2A=1, which implies 1−sin⁡2A=cos⁡2A1 - \sin^2 A = \cos^2 A1−sin2A=cos2A.

Conclude that LHS = RHS.

Determine the total number of possible outcomes.

Identify the perfect square numbers between 1 and 90.

Count the number of favorable outcomes.

Calculate the probability.

Calculate the ratio of the segments on side PQ.

Calculate the ratio of the segments on side PR.

Compare the two ratios.

Apply the Converse of Basic Proportionality Theorem (BPT).

Still stuck on this one?

More English Questions

Assume, for the sake of contradiction, that $\frac{1}{\sqrt{2}}$ is a rational number.

Rearrange the equation to isolate $\sqrt{2}$ .

Identify the coefficients $a, b, c$ from the quadratic equation $ax^2 + bx + c = 0$ .

Calculate the discriminant $D = b^2 - 4ac$ .

Determine the nature of the roots based on the value of $D$ .

Identify the sum of zeroes ( $S$ ) and product of zeroes ( $P$ ).

Choose a suitable value for $k$ to simplify the polynomial (e.g., to clear fractions).

Substitute the value of $\tan 30^\circ$ .

Start with the Left Hand Side (LHS) and express $\sec A$ and $\tan A$ in terms of $\sin A$ and $\cos A$ .

Use the trigonometric identity $\sin^2 A + \cos^2 A = 1$ , which implies $1 - \sin^2 A = \cos^2 A$ .