CUET UG 2025 — Mathematics Calculus

Given that the derivative of $ax^3 + ax^2 + ax + 1$ax3+ax2+ax+1 equals $9x^2 + 6x + 3$9x2+6x+3.

Differentiating $ax^3 + ax^2 + ax + 1$ax3+ax2+ax+1 term by term:

$$\frac{d}{dx}[ax^3] = 3ax^2$$dxd​[ax3]=3ax2

$$\frac{d}{dx}[ax^2] = 2ax$$dxd​[ax2]=2ax

$$\frac{d}{dx}[ax] = a$$dxd​[ax]=a

$$\frac{d}{dx}[1] = 0$$dxd​[1]=0

The derivative is $3ax^2 + 2ax + a$3ax2+2ax+a

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Setting the derivative equal to the given expression:

$$3ax^2 + 2ax + a = 9x^2 + 6x + 3$$3ax2+2ax+a=9x2+6x+3

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For two polynomials to be equal, coefficients of corresponding powers must be equal.

Comparing coefficients of $x^2$x2:

$$3a = 9$$3a=9

$$a = 3$$a=3

Comparing coefficients of $x$x:

$$2a = 6$$2a=6

$$a = 3$$a=3

Comparing constant terms:

$$a = 3$$a=3

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Therefore, $a = 3$a=3