Advanced Math MCQ Quiz in मराठी - Objective Question with Answer for Advanced Math - मोफत PDF डाउनलोड करा

To find the number of gadgets that minimizes the production cost, we look for the vertex of the quadratic function \(C = 4x^2 - 12x + 20\). The vertex \(x\)-coordinate is found using \(x = -\frac{b}{2a}\), where \(a = 4\) and \(b = -12\). Thus, \(x = -\frac{-12}{2(4)} = \frac{12}{8} = 1.5\). Therefore, the production cost is minimized at \(x = 1.5\), or \(1.5\) gadgets. Option \(3\) is the correct choice. Options \(0\), \(1\), and \(3\) do not yield the minimum cost as they do not correspond to the vertex of the parabola.

The height of the ball when it reaches the ground is 0. Therefore, we set \(-4.9t^2 + 20t + 1.5 = 0\) and solve for \(t\). This is a quadratic equation in standard form, \(at^2 + bt + c = 0\), where \(a = -4.9\), \(b = 20\), and \(c = 1.5\). Solving this using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we have: \(t = \frac{-20 \pm \sqrt{20^2 - 4(-4.9)(1.5)}}{2(-4.9)} = \frac{-20 \pm \sqrt{400 + 29.4}}{-9.8}\). Simplifying, \(t = \frac{-20 \pm \sqrt{429.4}}{-9.8}\). The positive solution is approximately \(t = 4.1\). Therefore, the correct option is 3. Option 1 (1.5) and Option 2 (3.5) are too early for the ball to reach the ground, and Option 4 (4.5) is slightly beyond the correct time.

A quadratic function \(ax^2 + bx + c\) has only one real root when its discriminant \(b^2 - 4ac\) is zero. For \(f(x) = x^2 - 6x + k\), the discriminant is \((-6)^2 - 4(1)(k) = 36 - 4k\). Set the discriminant to zero for one real root: \(36 - 4k = 0\). Solving for \(k\), we have \(4k = 36\) and \(k = 9\). Therefore, the value of \(k\) is 9.

To simplify \(4m^2 - 12m^2 + 3m^2\), combine the coefficients of the like terms, \(m^2\). This gives us \((4 - 12 + 3)m^2 = -5m^2\). Therefore, option 4 is the correct answer. Option 1 incorrectly adds the exponents. Option 2 incorrectly combines coefficients. Option 3 incorrectly calculates the result.

The function \(g(x)\) halves in value for each increase in \(x\) by \(1\). Therefore, the function can be expressed as \(g(x) = a(0.5)^x\). Given \(g(1) = 10\), we can substitute \(10\) for \(g(1)\) to find \(a\). Hence, \(g(x) = 10(0.5)^{x-1}\) represents the function correctly. Option 1 correctly uses the initial condition at \(x = 1\). Option 2 and Option 3 misrepresent the decay process, while Option 4 uses incorrect scaling.

The function \(h(x)\) decreases by \(30\%\) with every unit increase in \(x\). This indicates an exponential decay model \(h(x) = a(1 - 0.3)^x\). Given \(h(0) = 20\), \(a = 20\). Therefore, \(h(x) = 20(0.7)^x\). Option 1 is correct. Option 2 suggests growth rather than decay. Option 3 incorrectly models the decay rate, and Option 4 misrepresents the exponential factor.

To find \( V(100) \), substitute \( x = 100 \) into the function: \( V(100) = 1.05(100)^2 + 20 \). First, calculate \( (100)^2 = 10000 \). Then \( 1.05(10000) = 10500 \). Add \( 20 \): \( 10500 + 20 = 10520 \). Thus, \( V(100) = 10520 \). Option 3 is correct, as it reflects the correct computation of the investment value.

Profit is calculated as sales minus expenses. Therefore, the expression for profit is \( (9x^3 - 7) - (5x^3 + 3) \). Simplifying this, we have: \( 9x^3 - 5x^3 = 4x^3 \) for the \( x^3 \) terms, and \( -7 - 3 = -10 \) for the constants. Thus, the profit expression is \( 4x^3 - 10 \), making option 1 the correct answer.

The quadratic \(25x^2 + bx + 36 = 0\) requires the discriminant \(b^2 - 4ac\) to be positive for two distinct real solutions. Here, \(a = 25\) and \(c = 36\). Calculating, \(b^2 - 4(25)(36) > 0\) simplifies to \(b^2 - 3600 > 0\), or \(b^2 > 3600\). Solving gives \(b > 60\) or \(b < -60\). Therefore, the value \(100\) satisfies \(b > 60\), ensuring two distinct real solutions. Thus, \(100\) is the correct option.

To find the value of \(k\) where the line \(y = 2x + k\) intersects the parabola \(y = x^2 - 3x + 4\) at one point, equate the equations: \(2x + k = x^2 - 3x + 4\). Rearrange to form a quadratic: \(x^2 - 5x + 4 - k = 0\). For the quadratic to have exactly one solution, the discriminant must be zero. The discriminant is \((-5)^2 - 4(1)(4 - k) = 25 - 16 + 4k = 0\). Simplifying gives \(9 + 4k = 0\), solving for \(k\) gives \(k = 6.25\). Therefore, the value of \(k\) is \(6.25\).