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Here are the solutions to your questions: 32. Prove that (1)/(sqrt(2)) is irrational. Step 1: Assume, for the sake of contradiction, that (1)/(sqrt(2)) is a rational number. If (1)/(sqrt(2)) is rational, then it can be written in the form (a)/(b), where a and b are coprime integers and b ≠ 0. (1)/(sqrt(2)) = (a)/(b) Step 2: Rearrange the equation to isolate sqrt(2). b = asqrt(2) sqrt(2) = (b)/(a) Step 3: Analyze the result. Since a and b are integers and a ≠ 0, the ratio (b)/(a) is a rational number. This implies that sqrt(2) is a rational number. Step 4: 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 (1)/(sqrt(2)) is rational must be false. Hence, (1)/(sqrt(2)) is irrational. 33. Find the nature of the roots of the quadratic equation 2x^2 - 6x + 3 = 0. Step 1: 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. Step 2: Calculate the discriminant D = b^2 - 4ac. D = (-6)^2 - 4(2)(3) D = 36 - 24 D = 12 Step 3: 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 (1)/(3) respectively. Step 1: Identify the sum of zeroes (S) and product of zeroes (P). Given sum of zeroes, S = sqrt(2). Given product of zeroes, P = (1)/(3). Step 2: Use the general form of a quadratic polynomial: k(x^2 - Sx + P). Substitute the values of S and P: P(x) = k(x^2 - sqrt(2)x + (1)/(3)) Step 3: Choose a suitable value for k to simplify the polynomial (e.g., to clear fractions). Let k=3. P(x) = 3(x^2 - sqrt(2)x + (1)/(3)) P(x) = 3x^2 - 3sqrt(2)x + 3 × (1)/(3) P(x) = 3x^2 - 3sqrt(2)x + 1 A quadratic polynomial is 3x^2 - 3sqrt(2)x + 1. 35. Evaluate: (2 30^)/(1 + ^2 30^). Step 1: Substitute the value of 30^. We know that 30^ = (1)/(sqrt(3)). (2 (1)/(sqrt(3)))1 + ((1)/(sqrt(3)))^2 Step 2: Simplify the expression. = (2)/(sqrt(3))1 + (1)/(3) = (2)/(sqrt(3))(3+1)/(3) = (2)/(sqrt(3))(4)/(3) = (2)/(sqrt(3)) × (3)/(4) = (6)/(4sqrt(3)) = (3)/(2sqrt(3)) Step 3: Rationalize the denominator. = (3)/(2sqrt(3)) × sqrt(3)sqrt(3) = 3sqrt(3)2 × 3 = sqrt(3)2 Alternatively, recognize the identity (2 A)/(1 + ^2 A) = 2A. Here A = 30^, so the expression is (2 × 30^) = 60^ = sqrt(3)2. The value is sqrt(3)2. 36. Prove that: A (1 - A) ( A + A) = 1. Step 1: Start with the Left Hand Side (LHS) and express A and A in terms of A and A. LHS = A (1 - A) ( A + A) LHS = (1)/( A) (1 - A) ((1)/( A) + ( A)/( A)) Step 2: Combine the terms in the second parenthesis. LHS = (1)/( A) (1 - A) ((1 + A)/( A)) Step 3: Multiply the terms. LHS = ((1 - A)(1 + A))/( A · A) LHS = (1^2 - ^2 A)/(^2 A) Step 4: Use the trigonometric identity ^2 A + ^2 A = 1, which implies 1 - ^2 A = ^2 A. LHS = (^2 A)/(^2 A) LHS = 1 Step 5: Conclude that LHS = RHS. Since LHS = 1 and RHS = 1, the identity is proven. A (1 - A) ( A + 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. Step 1: Determine the total number of possible outcomes. Total number of discs = 90. Step 2: 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\. Step 3: Count the number of favorable outcomes. Number of perfect square numbers = 9. Step 4: Calculate the probability. Probability (Perfect Square) = Number of favorable outcomesTotal number of outcomes P(Perfect Square) = (9)/(90) P(Perfect Square) = (1)/(10) The probability that the disc bears a perfect square number is (1)/(10). 38. E and F are points on the sides PQ and PR respectively of a PQR. If PE = 4cm, EQ = 4.5cm, PF = 8cm and RF = 9cm. State whether EF || QR. Step 1: Calculate the ratio of the segments on side PQ. (PE)/(EQ) = 4 cm4.5 cm = (40)/(45) = (8)/(9) Step 2: Calculate the ratio of the segments on side PR. (PF)/(RF) = 8 cm9 cm Step 3: Compare the two ratios. We have (PE)/(EQ) = (8)/(9) and (PF)/(RF) = (8)/(9). Since (PE)/(EQ) = (PF)/(RF), the line segment EF divides the sides PQ and PR in the same ratio. Step 4: 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. Drop the next question.