1. Attempt any nine of the following short questions. Each question carries 4 marks. Solve the system of equations x - 4y = - 10 and 2x + 3y = 2 using Cramer's rule. ii. Simplify (sqrt(9x ^ 4 y ^ 2))/(root(27x ^ 6 y ^ 12, 3)) iii. Find the value of x from the equation log(x) = 3.8621 iv. Simplify the rational expression 1/(3x - 6) * (3x + 6)/(2x ^ 2 - 32) v. II 2x-1 is a factor of 24 + kx + 3 find the value of k. x + 1/x and vi. If x = sqrt(15) - 4 find the value of x ^ 2 + 1/(x ^ 2) vii. Find LCM of the polynomials x ^ 2 - 34 + 5x + 2 and x ^ 3 - 3x ^ 2 - x + 3 vii. Solve the radical equation sqrt(5x - 3) - sqrt(3x + 1) = 0 ix. Solve the inequality (3x)/4 + 2 >= (2x - 5)/3 and plot the solution on number line. x. Prove that the points A(1, 2) B(1, 8) and C(5,2) are the vertices of right angled triangle. xi. Prove that in any correspondence of two triangles, if three sides of one triangle are congruent to corresponding three sides of the other, the two triangles are congruent. xii. Prove that the line segment joining the midpoints of two sides of a triangle, is parallel to the third side and is equal to one half of its length. SECTION-C Mark NOTE: Attempt any four of the following questions. Each question carries 6 marks. 2. Prove that any point on the right bisector of a line segment is equidistant from its end points. 1 1+1- 3. Prove that if two sides of a triangle are unequal in length, the longer side has an angle of greater measure opposite to it. 1 + 1 + 4. Prove that if a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the third side. 1 + 1 + 5. Prove that in a right angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the length of the other two sides. 1 + 1 + 6. Prove that parallelogram on the equal bases and having same altitude are equal in area. 1 + 1 + 7. Construct a quadrilateral ABCD such that m overline AB = 5cm m overline BC = 6cm . m angle A = 100 deg m angle B = 60 deg m angle C = 110 deg 1 + 1 +

Question

1. Attempt any nine of the following short questions. Each question carries 4 marks. Solve the system of equations x - 4y = - 10 and 2x + 3y = 2 using Cramer's rule. ii. Simplify (sqrt(9x ^ 4 y ^ 2))/(root(27x ^ 6 y ^ 12, 3)) iii. Find the value of x from the equation log(x) = 3.8621 iv. Simplify the rational expression 1/(3x - 6) * (3x + 6)/(2x ^ 2 - 32) v. II 2x-1 is a factor of 24 + kx + 3 find the value of k. x + 1/x and vi. If x = sqrt(15) - 4 find the value of x ^ 2 + 1/(x ^ 2) vii. Find LCM of the polynomials x ^ 2 - 34 + 5x + 2 and x ^ 3 - 3x ^ 2 - x + 3 vii. Solve the radical equation sqrt(5x - 3) - sqrt(3x + 1) = 0 ix. Solve the inequality (3x)/4 + 2 >= (2x - 5)/3 and plot the solution on number line. x. Prove that the points A(1, 2) B(1, 8) and C(5,2) are the vertices of right angled triangle. xi. Prove that in any correspondence of two triangles, if three sides of one triangle are congruent to corresponding three sides of the other, the two triangles are congruent. xii. Prove that the line segment joining the midpoints of two sides of a triangle, is parallel to the third side and is equal to one half of its length. SECTION-C Mark NOTE: Attempt any four of the following questions. Each question carries 6 marks. 2. Prove that any point on the right bisector of a line segment is equidistant from its end points. 1 1+1- 3. Prove that if two sides of a triangle are unequal in length, the longer side has an angle of greater measure opposite to it. 1 + 1 + 4. Prove that if a line segment intersects the two sides of a triangle in the same ratio then it is parallel to the third side. 1 + 1 + 5. Prove that in a right angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the length of the other two sides. 1 + 1 + 6. Prove that parallelogram on the equal bases and having same altitude are equal in area. 1 + 1 + 7. Construct a quadrilateral ABCD such that m overline AB = 5cm m overline BC = 6cm . m angle A = 100 deg m angle B = 60 deg m angle C = 110 deg 1 + 1 +

Asked on April 10, 2026English

This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.

Accepted Answer · 2026-04-10T04:37:12.213Z

Step 1: Understand the logarithmic equation. The given equation is $\log(x) = 3.8621$. When the base of the logarithm is not explicitly written, it is conventionally assumed to be base 10 (common logarithm). So, the equation is $\log_{10}(x) = 3.8621$. Step 2: Convert the logarithmic equation to an exponential equation. The definition of a logarithm states that if $\log_b(a) = c$, then $b^c = a$. Applying this definition to our equation: $$x = 10^{3.8621}$$ Step 3: Calculate the value of $x$. Using a calculator to evaluate $10^{3.8621}$: $$x \approx 7279.35$$ The value of $x$ is approximately $7279.35$. $$\boxed{x \approx 7279.35}$$ Last free one today — make it count tomorrow, or type /upgrade for unlimited.

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