BUSINESS
MATHEMATICS
HIGHER SECONDARY - SECOND YEAR
CHAPTER-I
MATRIX
WEIGHTAGE
S.NO
QUESTIONS
NO.OF QUESTIONS
1
ONE MARK
5
(1X5=5)
2
SIX MARK
2
(2X6=12)
3
TEN MARK
2
(2X10=20)
TOTAL MARKS
37
SHORT ANSWER IMPORTANT QUESTION:
1 0 𝑎
1) Find the inverse of A = (0 1 𝑏 ) and verify that AA-1=I
0 0 1
4 5 2 2
2) Find the rank of the matrix A = (3 2 1 6)
4 4 8 0
−6 0
3 1
3) Verify (AB)-1 = B-1 A-1 when A = (
) and B = (
)
2 −1
0 9
4) Find k if the equations 2x + 3y -z = 5, 3x -y +4z = 2,
x +7y -6z = k are consistent
1 3 7
5) find the inverse of A =[4 2 3] if it exists.
1 2 1
1
2
3
4
6) Find the rank of the matrix A = ( 2
4
6
8)
−1 −2 − 2 −4
7) Show that the equations x -3y +4z = 3, 2x -5y +7z = 6, 3x -8y +11z = 1 are inconsistent
[mar-07]
1 1 1 1
8) Find the rank of the matrix A = (1 3 − 2 1)
2 0 −3 2
9) Find the value of k, if the equation x +2y -3z = -2, 3x -y -2z = 1,
2x +3y -5z = k are
inconsistent
−1 2 1
10)
Find the inverse of A =( 0 2 3)
1 1 4
1 −1 1
11)
Given A = (2 1
1 ) ,verify that |𝑎𝑑𝑗 𝐴| = |𝐴|²
3 1 −1
LONG ANSWER IMPORTANT QUESTIONS
1) Show that the equations 2x +8y +5z = 5, x +y +z = -2
x +2y -z = 2 are consistent and have unique solution
[june-07]
2) A salesman has the following record of sales during three months for three items A, B and C
which have different rates of commission.
Months
Sales of units
Total
(A)
commission
B
C
January
90
100
20
800
February
130
50
40
900
March
60
100
30
850
Find out the rates of commission on the items A, B and C.
Solve by Cramer’s rule.
[OCT-06]
3) Solve by matrix method the equations
x - 2y +3z = 1, 3x-y +4z = 3, 2x +y - 2z = -1
[MAR-06]
4) Solve by determinant method the equations
2x +2y – z – 1 = 0, x + y - z = 0, 3x +2y - 3z = 1.[ JUNE-06,MAR-07,08]
5) Solve by Cramer’s rule: x + y = 2, y + z = 6, z + x = 4.
[OCT-07]
6) The data below are about an economy of two industries and Q. The values are in lakhs of
rupees.
Producer
User
Final
Demand
Total Output
P
Q
P
16
12
12
40
Q
12
8
4
24
Find the technology matrix and test whether the system is viable as per Hawkins - Simon
conditions.
(7)In an economy there are two industries P and Q and the following table gives the supply and
demand positions in crores of rupees.
Producer
User
Final
Total Output
Demand
P
Q
P
10
25
15
50
Q
20
30
10
60
Determine the outputs when the final demand changes
to 35 for P and 42 for Q.
[JUNE-07, MAR-08]
8) In an economy of two industries P and Q the following table
gives the supply and demand positions in millions of rupees.
Producer
User
Final
Total Output
Demand
P
Q
P
16
8
Q
20
40
4
32
40
80
Find the outputs when the final demand changes to 18 for P
and 44 for Q.
[JUNE, OCT-06]
9) The data below are about an economy of two industries P and
Q. The values are in crores of rupees.
Producer
User
Final
Total Output
Demand
P
Q
P
50
75
75
200
Q
100
50
50
200
Find the outputs when the final demand changes to 300 for P
and 600 for Q.
[MAR-07]
10) Two products P and Q share the market currently with shares 70% and 30% each
respectively. Each week some brand switching takes place. Of those who bought P the previous
week, 80% buy it again whereas 20% switch over to Q. Of those who bought Q the previous
week, 40% buy it again whereas 60% switch over to P. Find their shares after two weeks. If the
price war continues, when is the equilibrium reached? [MAR-06]
(11) Two products A and B currently share the market with shares 60% and 40 % each
respectively. Each week some brand switching takes place. Of those who bought A the previous
week, 70% buy it again whereas 30% switch over to B. Of those who bought B the previous
week, 80% buy it
again whereas 20% switch over to A. Find their shares after one week and after two weeks. If
the price war continues,
when is the equilibrium reached?
[OCT-07]
CHAPTER-II
ANALYTICAL GEOMENTRY
WEIGHTAGE
S.NO
QUESTIONS
NO.OF QUESTIONS
1
ONE MARK
4
(1X4=4)
2
SIX MARK
1
(1X6=6)
3
TEN MARK
1
(1X10=10)
TOTAL MARKS
20
LONG ANSWER IMPORTANT QUESTIONS:
(1) find the equation to the hyperbola which has the lines x+4y-5=0 and 2x-3y+1=0 for its
asymptotes and which passes through the point (1,2)
(2) find the centre,vertices,eccentricity,foci and latus rectum and directrices of the ellipse
:9x²+16y ²+36x-32y-92=0
[MAR 2008]
(3) find the centre,vertices,eccentricity,foci and latus rectum and directrices of the ellipse
:7x²+4y ² -14x+40y+79=0
[OCT2010]
(4) find the centre,vertices,eccentricity,foci and latus rectum and directrices of the ellipse
:3x²+4y ² -6x+8y-5=0
[OCT 2008]
(5) find the equation of the asymptotes of the hyperbola
2x²+5xy+2y ² -11x -7y-4=0
[MAR2009]
(6) Find the equation of the asymptotes of the hyperbola
3x² -5xy -2y ²+17x+y+14=0
[OCT2009]
(7) Find the equation of the asymptotes of the hyperbola
8x²+10xy -3y ² - 2x+4y-2=0
[MAR2010]
(8) find the centre,vertices,eccentricity,foci and latus rectum and directrices
of the
hyperbola :9x² -16y ² -18x - 64y -199=0
[OCT2006]
(9) find the centre,vertices,eccentricity,foci and latus rectum and directrices
of the hyperbola :12x² - 4y ² - 24x+32y -127=0
[MAR2007]
SHORT ANSWER IMPORTANT QUESTION:
(1) Find the equation of the hyperbola whose eccentricity is √3,focus is
(1, 2) and the corresponding directrix is 2x+y=1
(2) Find the equation of the ellipse whose eccentricity is 1/2, one of the
focus is (-1,1) and the corresponding directrix is x-y+3=0
(3) Find the equation of the ellipse whose eccentricity is 1/√2,
focus is
(-1, 1) and the corresponding directrix is 3x-2y+1=0
(4) find the vertex,foci and latus rectum and directrices of the parabola
x² -3y+3 =0
(5) the supply of a commodity is related to the price by the relation
x=5√2𝑝 − 10.show that the supply curve is a parabola .find its vertex and price below
which supply is zero.
(6) Find the equation of the ellipse whose eccentricity is 2/3, whose
focus is (1,2) and the corresponding directrix is 2x-3y+6=0
(7)find the equation of the hyperbola which has 3x-4y+7=0 and 4x+3y+1=0
For asymptotes and which passes through the origin.
(8) find the eccentricity, foci and latus rectum of the ellipse 9x²+16y ²=144
(9) Find the equation of the parabola which has focus (1, 2) and the
directrix is x-y+3=0
CHAPTER-II
DIFFERENTIATION-I
WEIGHTAGE
S.NO
QUESTIONS
NO.OF QUESTIONS
1
ONE MARK
5
(1X5=5)
2
SIX MARK
2
(2X6=12)
3
TEN MARK
1
(1X10=10)
TOTAL MARKS
27
LONG ANSWER IMPORTANT QUESTIONS
1) find the equation of the tangent and normal to the curve
y (x-2)(x-3)-x+7=0 at the point where it cuts the x-axis
[MAR-06]
2) Prove that the curve y = x²-3x+1 and x(y+3)=4 intersects at right angles at the point (2, 1)
[JUN-06]
3) Find the equation of the tangent and normal to the supply curve y=x²+x+2 when x=6
[OCT-06]
4) If AR and MR denote the average marginal revenue at any output level, show that
𝐴𝑅
the elasticity of demand is equal to
. Verify this for the linear demand law p=
𝐴𝑅−𝑀𝑅
a+bx. Where p is price and x is the quantity.
[MAR-07]
5) At what point on the circle x²+y ²-2x-4y+1=0,the tangent is parallel to (i) x-axis (ii) yaxis
[JUN-07]
6) Prove that the cost function C=100+x+2x², where x is the output, the slope of AC curve
1
= (MC-AC).(MC is the marginal cost and AC is the average cost.
𝑥
[OCT-07]
7) find the equation of the tangent and normal at the point (a sec θ, b tan θ) on the hyperbola
𝑥²
𝑎²
-
𝑦²
𝑏²
=1
[MAR-08]
8) find the equation of the tangent and normal at the point (a cos θ, b sin θ) on the ellipse
+
𝑦²
𝑏²
𝑥²
𝑎²
=1
SHORT ANSWER IMPORTANT QUESTION:
(1)Find the equilibrium price and equilibrium quantity for the following demand and supply
functions, Qd = 4-0.06p and Qs= 0.6+0.11p
[SEP-07]
(2)A metal cylinder is heated and expands so that its radius increases at a rate of 0.4 cm per
minute and its height increases at a rate of 0.3 cm per minute retaining its shape.Determine the
rate of change of the surface area of the cylinder when its radius is 20 cms. and height is 40
cms.
[MAR-08]
(3) A metal cylinder when heated, expands in such a way that its radius r, increases at the rate
of 0.2 cm. per minute and its height h increases at a rate of 0.15 cm per minute. Retaining its
shape. Determine the rate of change of the volume of the cylinder when its radius is 10 cms and
its height is 25 cms.
[OCT-06]
4) The total cost C of producing x units isC = 0.00004x³ - 0.002x² + 3x + 10,000. Find the
marginal cost of 1000 units output.
5) Find the equilibrium price and equilibrium quantity for the
following demand and supply functions.
Qd = 4 0.05p and qs = 0.8 + 0.11p
[oct-06,oct-07]
6) For what values of x, is the rate of increase of x³ 5 x² +5x+8
is twice the rate of increase of x?[JUN-07]
7)Prove that for the cost function C = 100 + x + 2 x², where
1
x is the output, the slope of AC curve = (MC-AC).
𝑥
(MC is the marginal cost and AC is the average cost) [oct-07]
8) Find the equations of the tangents and normals to the following
curves
(iv) xy = 9 at x = 4.[OCT-07] (vi) x = a cos, y = b sinat =MAR-06]
9) Find the equation of the tangent and normal to the supply curve
y = x² + x + 2 when x = 6.
[OCT-06]
10) Find the equation of the tangent and normal to the demand
curve y = 36 x² when y = 11
[MAR-07]
1
11) A firm produces x tones of output at a total cost
C(x)= x³10
4x²+20x+5.
Find (i)
average cost,(ii) average variable cost, (iii) average fixed cost (iv) marginal cost,(v)
marginal average cost.
[MAR-08]
12)The demand curve for a monopolist is given by x=100-4p.(i) find the total revenue,
average revenue and marginal revenue,(ii) at what value of x, the marginal revenue is
equal to zero? [MAR-06,JUN-07]
1−2𝑥
𝐸𝑦
13) if y =2+3𝑥 ,find 𝐸𝑥 ,Obtain the values of η when x = 0 and x = 2
𝑥+7
[JUN-06]
14) For the cost function y =3x
+5,prove that the marginal cost falls continuously as the output x
𝑥+5
increases
[JUN-06]
15) For what value of x is the rate of increases of x3-5x2+5x+8 = 0 twice the rate of increase of x.
[JUN-07]
CHAPTER-IV.
S.NO
1
QUESTIONS
ONE MARK
DIFFERENTIATION-II
NO.OF QUESTIONS
4
(1X4=4)
2
3
SIX MARK
TEN MARK
TOTAL MARKS
1
2
30
(1X6=6)
(2X10=20)
LONG ANSWER IMPORTANT QUESTIONS
1) Investigate the maxima and minima of the function
2x³ + 3x² - 36x + 10.[MAR-08]
2) Find the maximum and minimum values of the function
(i) 2x³ -15x² + 24x - 15.[MAR-08],(ii)x³ - 6x² + 9x + 15.[OCT-06]
1
3)A firm produces x tonnes of output at a total costC = ( x³ -5x²+10x+5). At what level of output will
10
marginal cost and the average variable cost attain their
respective minimum?[JUN-07]
4) Find EOQ for the data given below. Also verify that carrying
costs is equal to ordering costs at EOQ.[MAR-08,JUN-06]
Item
Monthly
Ordering cost
Carrying cost
Requirements
A
9000
RS.200
RS.3.60
B
25000
RS.648
RS.10.00
C
8000
RS.100
RS.0.60
5) A manufacturer has to supply his customer with 600 units of his products per year. Shortages
are not allowed and storage cost amounts to 60 paise per unit per year. When the set up cost is
Rs. 80 find, (i) the economic order quantity. (ii) the minimum average yearly cost (iii) the
optimum number of orders per year (iv) the optimum period of supply per optimum order.[O07]
6) The annual demand for an item is 3200 units. The unit cost is
Rs.6 and inventory carrying charges 25% per annum. If the
cost of one procurement is Rs.150, determine (i) Economic
order quantity. (ii) Time between two consecutive orders
(iii) Number of orders per year (iv) minimum average yearly cost[MAR-07]
7) If u=log√(x² + y² + z²) then prove that,
8) if u=cos
x
𝜕𝑢
𝜕𝑥
+𝑦
𝜕𝑢
𝜕𝑦
𝑥+𝑦
-1
√𝑥+√𝑦
1
𝜕𝑥²
+
𝜕²𝑢
𝜕𝑦²
+
𝜕²𝑢
𝜕𝑧²
=
1
𝑥 2 +𝑦 2 +𝑧 2
,using Euler’s theorem prove that
+ 𝑐𝑜𝑡𝑢 = 0
2
𝜕²𝑢
[oct-06]
9) The demand for a commodity A is q1 =240-p12+6p2-p1p2.
Find the partial Elassticities Eq1/Ep1 and Eq1/Ep2
when p1 = 5 and p2 = 4.
[june,oct-06,mar-07]
10) The demand for a quantity A is q1 = 16 3p1 2p22 . Find(i) the partial
elasticities Eq1/Ep1 and Eq1/Ep2
(ii) the partial elasticities for p1 = 2and p2 = 1.
[oct-07,mar-08]
11) If u=tan-1
12) If f = log
𝑥²+𝑦²
𝑥+𝑦
𝑥²+𝑦²
𝑥+𝑦
then prove that x
then prove that x
𝜕𝑢
𝜕𝑥
𝜕𝑓
𝜕𝑥
+𝑦
+𝑦
𝜕𝑢
𝜕𝑦
𝜕𝑓
𝜕𝑦
1
= 𝑠𝑖𝑛2𝑢
2
=1
SHORT ANSWER IMPORTANT QUESTION:
1) Find the intervals in which the curve y = x43 x³ + 3x² + 5x + 1
is convex upward and convex downward.
[JUN-07]
2) A certain manufacturing concern has the total cost function
1
C = x² 6x + 100. Find when the total cost is minimum.
[MAR-07]
5
3) A firm produces x tons of a valuable metal per month at a
1
total cost C given by C = Rs.( x³ 5x²+ 75x + 10). Find a t
3
what level of output, the marginal cost attains its minimum.
[OCT-07]
4) A manufacturer can sell x items per week at a price of p = 600 4x rupees. Production cost
of x items works out to Rs. C where C = 40x + 2000. How much production will yield
maximum profit?
[MAR-06]
CHAPTER-V
APPLICATIONS OF
5
INTEGRATION
WEIGHTAGE
S.NO
QUESTIONS
1
ONE MARK
2
SIX MARK
3
TEN MARK
TOTAL MARKS
NO.OF QUESTIONS
3
(1X3=3)
1
(1X6=6)
2
(2X10=20)
29
LONG ANSWER IMPORTANT QUESTIONS
√𝑠𝑖𝑛³𝑥
𝛱/2
Evaluate ∫0
√𝑠𝑖𝑛3 𝑥+√𝑐𝑜𝑠³𝑥
𝑑𝑥
[mar-06]
3
𝛱/3
𝑑𝑥
2) Evaluate ∫
MAR-07,MAR-08] 3) Evaluate
𝛱/6 1+√𝑐𝑜𝑡𝑥
𝛱/3
𝑑𝑥
∫𝛱/6 1+ 𝑡𝑎𝑛𝑥MAR-06]
√
𝛱/2 𝑎𝑠𝑖𝑛𝑥+𝑏𝑐𝑜𝑠𝑥
4) Evaluate ∫
dx[MAR-07, MAR-08]
0
𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥
 Find
the area of one loop of the curve y² = x² (4-x²)
between x = 0 and x = 2.
[oct-o6]
6) Find the area of one loop of the curve y² = x² (1-x²) between
x = 0 and x = 1.
7) Find the area of the circle of radius ‘a’ using integration.
𝑥²
[mar08]
𝑦²
8) Find the area of the ellipse
+ =1
𝑎²
𝑏²
13)The demand and supply functions under pure competition are pd= 16 -x² and ps = 2x²+ 4.
Find the consumers’ surplus and producers’ surplus at the market equilibrium price.
[JUNE-06,MAR-08]
14) 10) The demand and supply law under a pure completion are given
by pd = 23 x²and ps = 2x² 4. Find the consumers’ surplus and producers’ surplus at the
market equilibrium price.
[OCT-06]
11) Under pure competition the demand and supply laws for commodity and pd = 56 x² and ps
𝑋²
= 8 + Find the consumers’ surplus and producers’ surplus at the equilibrium price.
3
[JUN-07]
12) Find the consumers’ surplus and the producers’ surplus under
market equilibrium if the demand function is pd = 20 3x x² and the supply function is ps = x
1.
[MAR-06,0CT-07]
13) The demand and supply function for a commodity are given by
Pd = 15 x and ps = 0.3x + 2. Find the consumers’ surplus and
producers’ surplus at the market equilibrium price.
[mar-07]
1
14) if the marginal revenue function is R (x) = 15 – 9x – 3x²,find the revenue function and
average revenue function
[JUN-07]
SHORT ANSWER IMPORTANT QUESTION:
9) The marginal cost function of manufacturing x units of
a commodity is 3x² - 2x + 8. If there is no fixed cost find the
total cost and average cost functions.
[OCT-06]
10) If the marginal revenue for a commodity is
MR = 9 - 6x² + 2x, find the total revenue and demand
function.
[MAR-07]
11) The elasticity of demand with respect to price p for a
𝑥−5
commodity is
, x > 5 when the demand is ‘x’. Find the
𝑥
demand function if the price is 2 when demand is 7. Also find
the revenue function.
[JUNE-06]
12) The elasticity of demand with respect to price for a
commodity is a constant and is equal to 2. Find the demand
function and hence the total revenue function, given that when
the price is 1, the demand is 4.
[MAR-06]
CHAPTER-VI
6 DIFFERENTIAL EQUATIONS 
S.NO
1
2
3
QUESTIONS
NO.OF QUESTIONS
ONE MARK
4
(1X4=4)
SIX MARK
2
(2X6=12)
TEN MARK
1
(1X10=10)
TOTAL MARKS
26
SHORT ANSWER IMPORTANT QUESTION:
𝑑𝑦
1
1) SOLVE:𝑑𝑥 + 𝑦𝑐𝑜𝑠𝑥 = 2 𝑠𝑖𝑛2𝑥
[MAR-06]
2) SOLVE:𝑐𝑜𝑠𝑥 𝑑𝑥 + 𝑦𝑠𝑖𝑛𝑥 = 1
3) Solve:(D2+10D+25)y = 5𝑒 𝑥
[JUN-06]
[MAR-06]
𝑑𝑦
𝑑²𝑦
𝑑𝑦
4) SOLVE: 𝑑𝑥² + 4 𝑑𝑥 + 4𝑦 = 5 + 𝑒 −𝑥
[JUN-06]
𝑑𝑦
5) SOLVE:(1-x2)𝑑𝑥 − 𝑥𝑦 = 1
𝑑𝑦
𝑦
[oct-06]
𝑦²
6) SOLVE:2 𝑑𝑥 = 𝑥 − 𝑥²
𝑑𝑦
𝑥
𝑑𝑦
2𝑥𝑦
[[oct-06]
𝑦
7) SOLVE:𝑑𝑥 = 𝑦 + 𝑥
[MAR-07]
1
8) SOLVE:𝑑𝑥 + 1+𝑥² = (1+𝑥 2 )² given that y = 0 when x = 1
[MAR-07]
𝑑𝑦
9) SOLVE:𝑑𝑥 + 𝑦𝑐𝑜𝑡𝑥 = 𝑠𝑖𝑛2𝑥
[JUN-07]
1
𝑥
2
10) SOLVE:(4𝐷2 − 8𝐷 + 3)𝑦 = 𝑒
[JUN-07]
11) SOLVE:(𝑥 + 𝑦)𝑑𝑦 + (𝑥 − 𝑦)𝑑𝑥 = 0
[oct-07]
𝑑𝑦
12) SOLVE:𝑑𝑥 + 𝑦𝑐𝑜𝑡𝑥 = 4𝑥𝑐𝑜𝑠𝑒𝑐𝑥 if y = 0 when x = Π/2
[MAR-08]
13) The slope of a curve at any point is the reciprocal of twice the ordinate of the point. The curve also passes
through the point (4, 3). Find the equation of the curve
LONG ANSWER IMPORTANT QUESTIONS
𝑑𝑃
2𝑃³−𝑋³
1) The net profit p and quantity x satisfy the differential equation𝑑𝑥 = 3𝑥𝑃² .Find the relationship between the
net profit and demand given that p = 20 when x = 10.
𝑑𝑝
𝑑²𝑝
2) Suppose that Qd = 305p +2ssss 𝑑𝑥 + 𝑑𝑥 and Qs = 6 + 3p. Find the equilibrium price for market clearance.
[JUN-06,MAR,JUN-07,MAR-08]
3) Solve:(𝐷2 − 13𝐷 + 12)𝑦 = 𝑒 −2𝑥 + 5𝑒 𝑥
4) Solve:(𝐷2 − 14𝐷 + 49)𝑦 = 3 + 𝑒 7𝑥
7 INTERPOLATION AND
FITTING A STRAIGHT LINE
S.NO
QUESTIONS
NO.OF QUESTIONS
1
2
3
ONE MARK
2
(1X2=2)
SIX MARK
2
(2X6=12)
TEN MARK
1
(1X10=10)
TOTAL MARKS
24
SHORT ANSWER IMPORTANT QUESTION:
1) Find the missing term from the following data.
x
F(x)
1
100
2
-------
3
126
4
157
2) Find y when x = 0.2 given that
x:
0
1
2
3
4
y:
176
185
194
202
212
3) If y75 = 2459, y80 = 2018, y85 = 1180 and
Y90 = 402 find y82.
4) Using Lagrange’s formula find the value of y when x = 42 from the following table
x:
40
50
60
70
y:
31
73
124
159
5) Using Lagrange’s formula find y(11) from the following table
x:
6
7
10
12
y:
13
14
15
17
6) From the following data, find f(3)
x:
1
2
3
4
5
f(x) :
2
5
14
32
7) Find the missing term from the following data.
x:
0
5
10
15
20
25
y:
7
11
14
-24
32
8) From the following data find the area of a circle of diameter 96 by using Gregory-Newton’s formula
Diameter x : 80
85
90
95
100
Area y : 5026
5674
6362
7088
7854
9) From the following data find y(25) by using Lagrange’s formula
x:
20
30
40
50
y:
512
439
346
243
10) 13) If f(0) = 5, f(1) = 6, f(3) = 50, f(4) = 105, find f(2) by using
Lagrange’s formula
11) Apply Lagrange’s formula to find y when x = 5 given that
x:
1
2
3
4
7
y:
2
4
8
16
128
12) Fit a straight line for the following data.
x:
0
1
2
3
4
y:
1
1
3
4
6
LONG ANSWER IMPORTANT QUESTIONS
13) Fit a straight line to the following data:
x: 4
8
12
y: 7
9
13
14) Fit a straight line to the following data.
x : 100
200
300
y : 90.2
92.3
94.2
16
17
20
21
24
25
400
96.3
500
98.2
600
100.3
15) Fit a straight line y = ax + b to the following data by the method of least squares.
X
0
1
3
6
8
y
1
3
2
5
4
16) From the data, find the number of students whose height is between 80cm. and 90cm.
Height in cms x : 40-60 60-80
80-100
100-120
120-140
No. of students y : 250 120
100
70
50
8PROBABILITY
DISTRIBUTIONS
S.NO
1
2
3
QUESTIONS
ONE MARK
SIX MARK
TEN MARK
TOTAL MARKS
NO.OF QUESTIONS
4
(1X4=4)
1
(1X6=6)
2
(2X10=20)
30
SHORT ANSWER IMPORTANT QUESTION:
1)