Janardan Bhagat Shikshan Prasarak Sanstha's

Ramsheth Thakur Public School,

Practice Worksheet
(2025-2026)
Subject: APPLIED MATHEMATICS (241)
STD & DIV.: XII- D

2
d y dy 1
1. If y= √ x +1− √ x−1 ,prove that ( x −1 )
2
2
+ x − y=0.
dx dx 4

( ) ( )
2
1 1 2d y dy
2. If x=a t+ , y=a t− show that y 2
+ x − y=0.
t t dx dx
2
cm
3. The area of a circle of radius r increases at the rate of 5 , find the rate at which the radius
sec
increases. Also find the value of this rate when the circumference is 10cm.
4. A particle moves along the curve 6 y=x 3 +2. Find the points on the curve at which y-coordinate is
changing 8 times as fast as the x-coordinate.
5. Find the least value of a so that the function f ( x )=x 2 +ax +1 is strictly increasing on [1,2].
6. Find the interval in which the function f ( x )=2 x 3−15 x 2+ 36 x+1 is strictly increasing or decreasing.
7. Find the absolute maximum and minimum values of f ( x )=2 x 3−9 x 2 +12 x−5 in [0,3].
8. Find all the points of local maxima and local minima of the function
−3 4 3 45 2
f ( x )= x −8 x − x +105.
4 2
9. A company is selling a certain product. The demand function for the product is linear. The company
can sell 2000 units when the price is Rs 8 oer unit and it can sell 3000 units when the price is Rs 4
per unit. Determine : demand function and total revenue function.

50
10. For a monopolist product, the demand function is p= and average cost function AC=0.5+2000/x
√x
. Find the profit maximizing level of output. At this level, show that the marginal revenue and
marginal cost are equal.
1
11. Evaluate: 1) ∫ dx
√ 2 x +3+ √2 x−3
3
x
2) ∫ 2 3
dx
( x +1)
2 x−3
3) ∫ 2 dx
( x −1)(2 x+ 3)
4) ∫ √ 4 x + 9 dx
2

5) ∫ x ¿ ¿ ¿
1 −x
e
6) ∫ x
dx
0 1+e

12. The demand function p for maximizing a profit monopolist is given by p=274−x 2 while the marginal
cost is 4+3x, for x units of the commodity. Using integrals, find the consumer surplus.
 2

13. Evaluate ∫ x dx and hence show the region on the graph where area it represents.
2

14. Find the order and degree of equation y ' ' ' + y 2 +e y ' =0

15. Form the DE for the family of circles passing through origin and have centres on x-axis.

dy x+ y 2 y
16. Solve using variable separable method: =e + x e .
dx

17. Using determinants , find the value of k so that points (k,7) , (1,-5) and (-4,5) are collinear.

[ ]
3 −1 1
18. Find the inverse of the matrix A= −15 6 −5 and hence show that A−1 A=I .
5 −2 2

19. A couple went to a restaurant for dinner. Husband ordered 2 small pizzas where 1 had veggies and
the other pizza had double extra cheese. Along with pizza, he ordered a fruit bowl. This was a
combo which cost him 200 Rs. Wife ordered from a combo which had 1 veggie pizza and 3 fruit
bowls. She decided to take two fruit bowls as a parcel. This combo cost 150 Rs. Can you
determine the cost of pizza and fruit bowl. Can this problem be solved using Cramer’s rule?
20. An amount of Rs. 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per
annum respectively. The total annual income is Rs. 358. If the combined income from the first two
investments is Rs 70 more than the income from the third, find the amount of each investment by
matrix method.

21. Find the value of k if M = [ ]

22. If A is a square matrix such that A2=I , then find the simplified value of ( A−I )3 +( A + I )3−7 A .

23. Cost of a pen and a notebook are Rs. 12 and Rs. 27 respectively. On a given day shopkeeper P
sells 5 pens and 7 notebooks whereas shopkeeper Q sells 6 pens and 4 notebooks. Find the
money received by both the booksellers using matrix algebra.

24. Solve the following inequality and graph the solution set on the number line:
2 y−3< y+ 2≤ 3 y+ 5
25. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the
resulting mixtures will contain more than 25% but less than 30% acid content?