Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Application of Derivatives

Name: Zigya - For The Curious Learner
Rating: 4.4 (740 reviews)

Long Answer Type

351.
Find the largest possible area of a right-angled triangle whose hypotenuse is 5 cm long.

Let x cm and y cm be two sides of a right angled triangle with hypotenuse 5 cm.

therefore space space space straight x squared plus straight y squared space equals space 25 space space space rightwards double arrow space space space straight y squared space equals space 25 minus straight x squared therefore space space space space straight y space equals space square root of 25 minus straight x squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis

Let A be area of triangle

therefore space space space space straight A space equals space 1 half xy space equals space 1 half straight x square root of 25 minus straight x squared end root space space space space space space open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets therefore space space space space space dA over dx space equals 1 half open square brackets straight x. space fraction numerator negative 2 straight x over denominator 2 square root of 25 minus straight x squared end root end fraction plus square root of 25 minus straight x squared end root.1 close square brackets space space space space space space space space space space space space space space space space space space space equals 1 half open square brackets negative fraction numerator straight x squared over denominator square root of 25 minus straight x squared end root end fraction plus fraction numerator square root of 25 minus straight x squared end root over denominator 1 end fraction close square brackets

equals space 1 half open square brackets fraction numerator negative straight x squared plus 25 minus straight x squared over denominator square root of 25 minus straight x squared end root end fraction close square brackets space equals space 1 half open parentheses fraction numerator 25 minus 2 straight x squared over denominator square root of 25 minus straight x squared end root end fraction close parentheses

For maximum or minimum values,

dA over dx space equals space 0

therefore space space space space 1 half open parentheses fraction numerator 25 minus 2 straight x squared over denominator square root of 25 minus straight x squared end root end fraction close parentheses space equals space 0 space space space space space space space space space space space space rightwards double arrow space space space 25 minus 2 straight x squared space equals space 0 rightwards double arrow space space space space space space straight x squared space equals space 25 over 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction as space straight x space cannot space be space negative fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space 1 half open square brackets fraction numerator square root of 25 minus straight x squared end root. space left parenthesis negative 4 straight x right parenthesis space minus space left parenthesis 25 minus 2 straight x right parenthesis squared. space begin display style fraction numerator negative 2 straight x over denominator 2 square root of 25 minus straight x squared end root end fraction end style over denominator 25 minus straight x squared end fraction close square brackets space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets negative 4 straight x square root of 25 minus straight x squared end root plus fraction numerator straight x left parenthesis 25 minus 2 straight x squared right parenthesis over denominator square root of 25 minus straight x squared end root end fraction close square brackets space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets fraction numerator negative 4 straight x left parenthesis 25 minus straight x squared right parenthesis space plus space straight x left parenthesis 25 minus 2 straight x squared right parenthesis over denominator square root of 25 minus straight x squared end root end fraction close square brackets space space space space space space space space space space space space

equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets fraction numerator negative 100 straight x plus 4 straight x cubed plus 25 straight x minus 2 straight x cubed over denominator square root of 25 minus straight x squared end root end fraction close square brackets equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space left parenthesis 2 straight x cubed minus 75 straight x right parenthesis

When space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction. fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space fraction numerator 1 over denominator 2 open parentheses 25 minus begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets fraction numerator 2 cross times 125 over denominator 2 square root of 2 end fraction minus 75 cross times fraction numerator 5 over denominator square root of 2 end fraction close square brackets space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 open parentheses begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets fraction numerator 125 over denominator square root of 2 end fraction minus fraction numerator 375 over denominator square root of 2 end fraction close square brackets space equals space fraction numerator 1 over denominator 2 open parentheses begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets negative fraction numerator 250 over denominator square root of 2 end fraction close square brackets less than 0

therefore space space space straight A space is space maximum space when space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction therefore space space space maximum space area space space equals space 1 half fraction numerator 5 over denominator square root of 2 end fraction square root of 25 minus 25 over 2 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 5 over denominator 2 square root of 2 end fraction cross times square root of 25 over 2 end root space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction cross times fraction numerator 5 over denominator square root of 2 end fraction space equals space 25 over 4 space sq. space units. space

131 Views

352. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is

open parentheses straight a to the power of 2 over 3 end exponent plus straight b to the power of 2 over 3 end exponent close parentheses to the power of 3 over 2 end exponent.

80 Views

353. Find the maximum-area of an isosceles triangle inscribed in the ellipse

straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1

with its vertex at one end of the major axis.

770 Views

Short Answer Type

354.

Let AP and BQ be two vertical poles at points A and B, respectively. If AP = 16 m. BQ = 22 m and AB = 20 m, then find the distance of a point R on AB from the point A such that RP² + RQ² is minimum.

80 Views

Long Answer Type

355.

If length of three sides of a trapezium other than base are equal to 10 cm. then find the area of the trapezium when it is maximum.

172 Views

Short Answer Type

356.

Use differentials to approximate $square root of 25.3 end root.$

86 Views

357.

Use differentials to approximate $square root of 0.037 end root$ .

73 Views

358.

Use differentials to approximate $square root of 36.6 end root$

76 Views

359.

Use differentials to approximate:
$square root of 49.5 end root$

79 Views

360.

Use differentials to approximate:
$square root of 0.6 end root$

78 Views

Previous Year Papers

Test Series

Test Yourself

Long Answer Type

351. Find the largest possible area of a right-angled triangle whose hypotenuse is 5 cm long.

Short Answer Type

Long Answer Type

Short Answer Type

351.
Find the largest possible area of a right-angled triangle whose hypotenuse is 5 cm long.