BRAIN INTERNATIONAL SCHOOL

Term I Class XII 2018-19

SUB: MATHEMATICS REVISION WORKSHEET

Chapter-2 Inverse Trigonometry

Chapter-3&4 Matrices and Determinants

1. Show that x=2 is a root of the determinant |𝑥 −6 −12 −3𝑥 𝑥 − 3

−3 −2𝑥 𝑥 + 2| =0 and solve it completely.

2. Find the inverse of the following matrix by using elementary operations: [2 −3 53 2 −41 1 −2

] .Hence

solve the following system of linear equations:

2x+3y+z = 11, -3x + 2y +z = 4, 5x-4y -2z = -9.

3. Using properties of determinants, prove the following :

|𝑎2 𝑏𝑐 𝑎𝑐 + 𝑐2

𝑎2 + 𝑎𝑏 𝑏2 𝑎𝑐𝑎𝑏 𝑏2 + 𝑏𝑐 𝑐2

| = 4a2b2c2.

4. If p ≠ 0, q ≠ 0 and |

𝑝 𝑞 𝑝 ∝ + 𝑞𝑞 𝑟 𝑞 ∝ + 𝑟

𝑝 ∝ +𝑞 𝑞 ∝ +𝑟 0| = 0, then using properties of

determinants, prove that at least one of the following statements is true.

(a) p, q, rare in G.P. (b) ∝ is a root of the equation px2 + 2qx + r = 0

5. Find the inverse of the matrix

2 0 1

1 2 3

2 2 1

by using elementary transformation. Hence solve the

following system of equations, 2x – z = 4, x + 2y + 3z = 0, 2x + 2y –z = 2

6. If A is any square matrix of order 3 x 3 and IAI=5. Find IA.Adj(A)I

7. The management committee of a residential colony decided to award some of its members (say x)for

honesty, some (say y) for helping others (say z) for supervising the workers to keep the colony neat and

clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and

supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix

method, find the number of awardees of each category. Apart from these values, namely, honesty,

cooperation and supervision, suggest one more value which the management of the colony must include for awards.

8. Use elementary column operation C2 → C2 + 2C1 in the following matrix equation:

[2 12 0

] = [3 12 0

] [1 0

−1 1]

9. If B is a skew-symmetric matrix , write whether the matrix ( ABA’) is symmetric or Skew-symmetric.

10. Using properties of determinants prove the following:

(a) 𝑎 𝑏 𝑐

𝑎2 𝑏2 𝑐2

𝑎3 𝑏3 𝑐3 = abc(a-b)(b-c)(c-a)

(b) 𝑎 + 𝑏 + 2𝑐 𝑎 𝑏

𝑐 𝑏 + 𝑐 + 2𝑎 𝑏𝑐 𝑎 𝑐 + 𝑎 + 2𝑏

= 2(a + b + c)3

11. Show that the pts A(a, b+c), B(b, c+a) and C (c, a+b) are collinear.

12. Solve the system of linear equations:

x + y + z = 4

2x – y + z = -1

2x + y – z = -9

Chapter-5 Continuity and Differentiability

Chapter-6 Apps Of Derivatives

1. Find the approximate value of f(5.001) ,wheref(x)=x3-7x2+15.

2. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate

error in calculating its surfacearea.

3. The length of the sides of an isosceles triangle are 9+x²,9+x² and 18-2x² units. Calculate the value

of x which makes the area maximum. Also find the maximum area of thetriangle.

4. A window has the shape of a rectangle surmounted by an equilateral triangle . If the perimeter of

the window is 12m, find the dimensions of the rectangle will produce the largest area of thewindow.

5. An Apache helicopter of enemy is flying along the curve given by y = x2 + 7.A soldier

placed at ( 3 , 7 ) wants to shoot down the helicopter when it is nearest tohim.

6. Find the points on the curve 9y2= x3 where the normal to curve makes equal intercepts with theaxes.

7. Sand is pouring from a pipe at the rate of 12cm3/sec. The falling sand forms a cone on the ground

in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is

the height of the sand-cone increasing when the height is4cm?

8. Water is dripping out from a conical funnel at a uniform rate of 4cm3/sec through a tiny hole at the

vertex in the bottom. When the slant height of the water is 3cm, find the rate of decrease of the slant

height of the water cone .Given that the vertical angle of the funnel is1200.

Chapter-8 Apps Of Integration

1. Find the area bounded by the curves y = 2x − x2and line y = −x

2. Find the area bounded by curves y = 6x − x2and y = x2 – 2x. 3. Find the area bounded by the triangle whose vertices are A(2,0), B(4,5), & C(6,3).

4. Find the area bounded by the lines 2y + x = 4, 3y – 2x = 6 and y – 3x + 5 = 0. 5. Find the area bounded by the curves y = 2 + |x + 2| ; x = – 4 and x = 4 and y = 0.

6. Find the area bounded by the curves x2 + y2 = 1 and (x – 1)2 + y2 = 1.

7. Find the area bounded by the curves (𝑥 – 3)2 + 𝑦2 ≥ 9 and 𝑥2 + 𝑦2 ≤ 9.

Chapter-9 Differential Equations

1. Solve the following differentialequation:

( x2 − y2) dx + 2xy dy = 0 given that y = 1 when x = 1

2. Find the particular solution, satisfying the given condition, for the

followingdifferential equation:

3. Find the particular solution of the differential equation satisfying the givenconditions:

4. Find the general solution of the differentialequation,

5. Solve the following differentialequation:

6. Find the particular solution of the following differentialequation:

7. Find the particular solution of the differentialequation

given that y =0 when x =0.

8. Find theparticular solution ofthe differential equation x2dy=(2xy+y2) dx, given thaty=1,

when x = 1.

9. Find the particular solution of the differentialequation

, given that y =1 when x = 0.

Chapter-12 Linear Programming 1. A merchant plans to sell two types of personal computers –a desktop model and a portable model

that will cost Rs 25000 and Rs40000 respectively. He estimates that the total monthly demand of computers will not exceed 250units. Determine the number of units of each type of computers

which the merchant should stock to get maximum profit If he does not want to invest more than

Rs70 lakhs and if his profit on the desk top model is Rs4500 and on portable model is Rs5000

2. A diet is to contain atleast 80units of vitaminA and 100 units of minerals. Two foods F1 and F2

are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6units of

vitamin A and 3units of minerals. Formulate this as a linear programming problem. Find the

minimum cost for diet that consist s of mixture of these two foods and also meets the minimal

nutritional requirements.

3. There are two types of fertilizers F1and F2.F1consists of 10% nitrogen and 6% phosphoric acid

and F2consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a

farmer finds that she needs at least14kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costs Rs 5/kg, determine how much of each type of fertiliser should be

used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

4. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each

executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, atleast 4times as many passengers prefer

to travel by economy class than by the executive class. Determine how many tickets of each type

must be soldinordertomaximisetheprofitfortheairline.Whatisthemaximumprofit?

5. A toy company manufactures two types of dolls, A and B. Market tests and available resources have

indicated that the combined production level should not exceed 1200 dolls per week and the demand

for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of

type Acan exceed three times the production of dolls of other type by atmost 600 units. If the company makes profit of Rs12 and Rs16 per doll respectively on dollsA and B, how many of each

should be produced weekly in order to maximize the profit?

INTEGRATION

Chapter -7 Integration