The curve $$y=$$ $$ax3+$$ $$bx2+cx$$ $$+$$ $$5$$ touches the $$x-axis$$ at $$P(-2$$, $$0)$$ and cuts the $$y-axis$$ at a point Q where its gradient is $$3$$, then the value of $$-10$$ $$a-$$ $$100b+$$ $$1000$$ c must be

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Infinity Learn · Accepted Answer

∵$$dydx=3ax2+2bx+c----iSince$$, the curve y $$=$$ $$ax3$$ $$+$$ $$bx2$$ $$+$$ cx $$+5$$ touches $$X-axis$$ at $$P(-2$$ $$0)$$ then $$X-axis$$ is the tangent at $$(-2$$, $$0)$$ the curvemeets $$Y-axis$$ in (0$$, $$5).∴ $$dydx(0$$,$$5)=0+0+c=3$$ (given) $$----ii$$ and $$dydx($$−$$2$$,$$0)=0$$⇒$$12a$$−$$4b+c=0$$⇒$$12a$$−$$4b+3=0$$  [from Eq. (ii)] ..........(iii)and$$ $$(-2$$, $$0) lies on the curve, $$then0=$$−$$8a+4b$$−$$2c+5$$⇒ −$$8a+4b$$−$$1=0$$⇒ $$8a$$−$$4b+1=0----ivFrom$$ Egs. (ii) and (iv) we $$geta=$$−$$12$$,$$b=$$−$$34$$∴−$$10a$$−$$100b+1000c=$$−$$10$$×−$$12$$−$$100$$×−$$34+1000$$×$$3=5+75+3000=3080$$

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The curve y= ax³+ bx²+cx + 5 touches the x-axis at P(-2, 0) and cuts the y-axis at a point Q where its gradient is 3, then the value of -10 a- 100b+ 1000 c must be

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Tangents and normals

Remember concepts with our Masterclasses.

The curve y= ax3+ bx2+cx + 5 touches the x-axis at P(-2, 0) and cuts the y-axis at a point Q where its gradient is 3, then the value of -10 a- 100b+ 1000 c must be

Ready to Test Your Skills?

free study material

The curve y= ax³+ bx²+cx + 5 touches the x-axis at P(-2, 0) and cuts the y-axis at a point Q where its gradient is 3, then the value of -10 a- 100b+ 1000 c must be