MathematicsAdvancedHigh School (11–12)Functions and Calculus › Application of DerivativesJEE Advanced
Created by StemPy LabLet be a real constant, and consider the equation As changes, the whole curve moves up or down without changing its shape, so the number of real roots changes only when a turning point of the curve passes through the -axis. A root that is repeated counts once here: we count distinct real roots only. Pay special attention to the values of at which a turning point lies exactly on the axis. How many integer values of make the equation have at least three distinct real roots?
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← The figure shows the curve y² = x³, which has a sharp point (a cusp) at the origin and two branches, one above the x-axis and one below it, mirror images of each other. A convenient way to describe its points is x = t², y = t³, where t is any real number. The point P(4, 8) lies on the upper branch, where t = 2. The tangent to the curve at P crosses the curve once more, and the normal at P is the line through P perpendicular to that tangent. Which of the following statements are true?A student has to find the indefinite integral ∫ (x² - 1)/(x⁴ + 5x² + 1) dx, x > 0. The denominator does not factor into real quadratics with rational coefficients, so partial fractions would be slow. There is a faster way. Divide the numerator and the denominator by x², and look for a new variable whose derivative is exactly the new numerator. Then write the new denominator in terms of that variable; take care with the square of the new variable, because it does not give back x² + 1/x² on its own. In each option below, C is an arbitrary constant. Which of the following is the integral? →
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