Q.1. (A) Select the correct alternative answer and write it. 4(i) To draw graph of 4x + 5y = 19, find y when x = 1.(a) 4 (b) 3 (c) 2 (d) –3(ii) √5 m2 – √5 m + √5 = 0 which of the following statement is true for thisgiven equation ?(a) real and unequal roots (b) real and equal roots(c) no real roots (d) three roots(iii) If n(A) = 2, P(A) = 15 , then n(S) = ?(a) 10 (b) 52 (c) 25 (d) 13(iv) Out of the following equations, find the equation having the sum of itsroots –5.(a) 3x2 – 15x + 3 = 0 (b) x2 – 5x + 3 = 0(c) x2 + 3x – 5 = 0 (d) 3x2 + 15x + 3 = 0Q.1. (B) Solve the following. 4(i) Decide whether the following is quadratic equation.(m + 2)(m – 5) = 0(ii) Determine the nature of roots of the following quadratic equation.2y2 – 7y + 2 = 0(iii) If two coins are tossed, find the probability of the following events :(i) Getting atleast one head(ii) Getting no head(iv) Find the values of following determinants.–1 72 4Time : 2 Hours (Pages 5) Max. Marks : 40MT - MATHEMATICS (71) Algebra - SEMI PRELIM - I - PAPER - IMTSeat No.2022 ____ ____ 1100

Sure, here are the answers to your questions:

Q.1. (A) (i) To find y when x = 1 in the equation 4x + 5y = 19, we substitute x = 1 into the equation. This gives us 4(1) + 5y = 19, which simplifies to 4 + 5y = 19. Subtracting 4 from both sides gives us 5y = 15, and dividing both sides by 5 gives us y = 3. So, the correct answer is (b) 3.

(ii) The given equation is √5 m2 – √5 m + √5 = 0. This is a quadratic equation in the form ax^2 + bx + c = 0, where a = √5, b = -√5, and c = √5. The discriminant of a quadratic equation is given by b^2 - 4ac. Substituting the values of a, b, and c into this formula gives us (-√5)^2 - 4*√5*√5 = 5 - 20 = -15. Since the discriminant is less than 0, the roots of the equation are not real. So, the correct answer is (c) no real roots.

(iii) If n(A) = 2 and P(A) = 15, then n(S) = n(A) / P(A) = 2 / 15 = 0.1333. However, this is not one of the options given. There may be a mistake in the question.

(iv) The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a. We are looking for an equation where this sum is -5. Checking the options, we find that for the equation x^2 – 5x + 3 = 0, the sum of the roots is -(-5)/1 = 5, not -5. For the equation 3x^2 – 15x + 3 = 0, the sum of the roots is -(-15)/3 = 5, not -5. For the equation x^2 + 3x – 5 = 0, the sum of the roots is -(3)/1 = -3, not -5. For the equation 3x^2 + 15x + 3 = 0, the sum of the roots is -(15)/3 = -5, which is the correct answer. So, the correct answer is (d) 3x^2 + 15x + 3 = 0.

Q.1. (B) (i) The equation (m + 2)(m – 5) = 0 is a quadratic equation, because it can be rewritten in the form ax^2 + bx + c = 0. If we expand the brackets, we get m^2 - 3m - 10 = 0, which is a quadratic equation.

(ii) The nature of the roots of the quadratic equation 2y^2 – 7y + 2 = 0 can be determined by calculating the discriminant, which is b^2 - 4ac. Substituting the values of a, b, and c into this formula gives us (-7)^2 - 422 = 49 - 16 = 33. Since the discriminant is greater than 0, the roots of the equation are real and unequal.

(iii) If two coins are tossed, there are 4 possible outcomes: HH, HT, TH, TT. The probability of getting at least one head is the number of outcomes with at least one head divided by the total number of outcomes, which is 3/4. The probability of getting no head is the number of outcomes with no head divided by the total number of outcomes, which is 1/4.

(iv) The determinant of a 2x2 matrix [a b; c d] is given by ad - bc. So, the determinant of the matrix [-1 7; 2 4] is (-1)4 - 72 = -4 - 14 = -18.