Download XII STANDARD MATHEMATICS – 10 MARKS QUESTIONS Solve by

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XII STANDARD MATHEMATICS – 10 MARKS QUESTIONS
Solve by matrix inversion method each of the following system of linear equations:
(i) x + y + z = 9, 2x + 5y + 7z = 52, 2x + y – z = 0
(ii) x – 3y – 8z + 10 = 0, 3x + y = 4, 2x + 5y + 6z = 13
Solve the following non-homogeneous system of linear equations by determinant method:
(i) x + 2y + z = 6, 3x + 3y – z = 3, 2x + y – 2z = – 3
(ii) 2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4
Examine the consistency of the following system of equations. If it is consistent then solve the
same. (i) x + y + z = 6, x + 2y + 3z = 10, x + 3y + 5z = 14
(ii) x + y + z = 8, x – y + 2z = 6, 3x + 5y – 7z = 14.
Discuss the solutions of the system of equations for all values of .
x + y + z = 2, 2x + y – 2z = 2, x + y + 4z = 2
For what values of k, the system of equations kx + y + z = 1, x + ky + z = 1, x + y + kz = 1 have
(i) no solution (ii) a unique solution and (iii) an infinite number of solutions.
Solve by matrix inversion method 2x – y + 3z = 9, x + y + z = 6, x – y + z = 2
A bag contains 3 types of coins namely Re. 1, Rs.2. and Rs.5. There are 30 coins amounting to
Rs. 100 in total. Find the number of coins in each category
Verify whether the given system of equations is consistent. If it is consistent, solve them.
2x + 5y + 7z = 52, x + y + z = 9, 2x + y – z = 0
Verify whether the given system of equations is consistent. If it is consistent, solve them
x – y + z = 5, – x + y – z = – 5, 2x – 2y + 2z = 10
Investigate for what values of ,  the simultaneous equations x + y + z = 6,
x + 2y + 3z = 10, x + 2y + z =  have (i) no solution (ii) a unique solution and
(iii) an infinite number of solutions.
For what value of  the equations x + y + 3z = 0, 4x + 3y + z = 0, 2x + y + 2z = 0 have a (i)
trivial solution, (ii) non-trivial solution.
Prove that cos (A + B) = cos A cos B – sin A sin B (ii) sin (A – B) = sin A cos B – cos A sin B.
Find the vector and Cartesian equation of the plane containing the line
x2
y2
z 1
x 1
y 1
z 1
=
=
and parallel to the line
=
=
.
2
3
3
3
1
2
Find the vector and Cartesian equation of the plane through the point (1, 3, 2) and parallel to the
x 1
y2
x2
y 1
x2
z3
lines
=
=
and
=
=
2
1
1
2
3
2
Find the vector and Cartesian equation to the plane through the point (-1, 3, 2) and perpendicular
to the planes x + 2y + 2z = 5 and 3x + y + 2z = 8.
Find the vector and Cartesian equation of the plane passing through the points A (1, -2, 3) and
x2
y 1
z 1
B(-1, 2, -1) and is parallel to the line
=
=
2
4
3
Find the vector and Cartesian equation of the plane through the points (1, 2, 3) and (2, 3, 1)
perpendicular to the plane 3x –2y + 4z – 5 = 0
x2
y2
z 1
Find the vector and Cartesian equation of the plane containing the line
=
=
2
3
2
and passing through the point (–1, 1, –1).
Altitudes of a triangle are concurrent – prove by vector method.
Prove that cos (A – B) = cos A cos B + sin A sin B (ii) sin (A + B) = sin A cos B + cos A sin B
x4
x 1
y 1
z 1
y
z 1
Show that the lines
=
=
and
=
=
intersect and hence find the
3
0
3
1
0
2
point of intersection.
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Find the vector and Cartesian equations of the plane passing through the point (2, –1, –3) and
x  2 y 1 z  3
x 1 y 1 z  2




parallel to the lines
and
.
3
2
4
2
3
2
Find the vector and Cartesian equations of the plane passing through the points (–1, 1, 1) and
(1, –1, 1) and perpendicular to the plane x + 2y + 2z = 5.
Find the vector and Cartesian equations of the plane passing through the points (2, 2, –1),
(3, 4, 2) and (7, 0, 6).
 z 1
P represents the variable complex number z. find the locus of P, if Re 
  1.
 zi
 z 1  
P represents the variable complex number z. find the locus of P, if arg 
 .
 z  3 2
q
If  and  are the roots of the equation x2 – 2px + (p2 + q2) = 0 and tan =
, show that
y p
( y  )n  ( y   )n
sin n
 q n1

sin n 
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If  and  are the roots of the equation x2 – 2x + 4 = 0, prove that n – n = i2n + 1 sin
deduce 9 – 9.
If a = cos2 + isin2; b = cos2 + isin2; c = cos2 + isin2; prove that
1
a 2b 2  c 2
(i) abc 
= 2cos( +  – )
 2cos( +  + ) (ii)
abc
abc
Solve the equation (i) x7 + x4 + x3 + 1 = 0; (ii) x4 – x3 + x2 – x + 1 = 0
(iii) x9 + x5 – x4 – 1 = 0;
3
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n
and
3
1
3 4
 and hence prove that the product of the values is 1.
Find all the values of   i

2
2


2/3
Find all the values of (– 3 – i) .
A cable of suspension bridge is in the form of a parabola whose span is 40mts. The road way is
5mts below the lowest point of the cable. If an extra support is provided across the cable 30mts
above the ground level, find the length of the support if the height of the pillars is 55mts.
Find the axis, vertex, directrix, focus, latus rectum, and length of the latus rectum of the parabola
x2 – 6x – 12y – 3 = 0.
Find the eccentricity, vertices, directrices, foci, center, length of the latus rectum of the ellipse
36x2 + 4y2 – 72x + 32y – 44 = 0.
Find the eccentricity, vertices, directrices, foci, center, length of the latus rectum of the
hyperbola 9x2 – 16y2 + 36x + 32y + 164 = 0.
A comet is moving in a parabolic orbit around the sun, which is at the focus of a parabola. When
the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an

angle of
radians with the axis of the orbit. Find (1) the equation of the comet’s orbit (2) how
2
close does the comet come nearer to the sun? (Take the orbit as open rightward).
The arch of a bridge is in the shape of a semi ellipse having a horizontal span of 40ft and 16ft
high at the center. How high is the arch, 9ft from the right or left of the center?
Find the equations of directrices, latus rectum and length of latus rectum of the hyperbola
9x2 – 4y2 – 36x + 32y + 8 = 0.
Find the equation of the hyperbola if its asymptotes are parallel to x + 2y – 12 = 0 and
x – 2y + 8 = 0, (2, 4) is the center of the hyperbola and it passes through (2, 0).
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A satellite is traveling around the earth in an elliptical orbit having the earth at a focus and of
eccentricity ½. The shortest distance that the satellite gets to the earth is 400kms. Find the
longest distance that the satellite gets from the earth.
A kho-kho player in a practice session while running realizes that the sum of the distances from
the two kho-kho poles from him is always 8mts. Find the equation of the path traced by him if
the distance between the poles is 6m.
On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height
of 4mts when it is 6mts away from the point of projection. Finally it reaches the ground 12mts
away from the starting point. Find the angle of projection.
A ladder of length 15m moves with its ends always touching the vertical wall and horizontal
floor. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of
the ladder in contact with the floor.
A water tank has the shape of an inverted circular cone with base radius 2 meters and height
4meters. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the
water level is rising when the water is 3m deep.
Two sides of a triangle have length 12m and 15m. The angle between them is increasing at a rate
of 2/min. how fast is the length of third side increasing when the angle between the sides of
fixed length is 60
Gravel is being dumped from a conveyor belt at a rate of 30ft3/min and its coarsened such that it
forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is
the height of the pile increasing when the pile is 10ft high?
Find the equation of the tangent and normal at  = /2 to the curve x = a( + sin ),
y = a(1 + cos ).
Find the equations of the tangent and normal to the curve 16x2 + 9y2 = 144 at (x1, y1)
where x1 = 2 and y1 > 0.
Find the condition for the curves ax2 + by2 = 1, a1x2 + b1y2 = 1 to intersect orthogonally.
Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the curve x = a
cos4, y = a sin4, 0    /2 is equal to a.
Show that the equation of the normal to the curve x = a cos3; y = a sin3 is
x cos - y sin = a cos2.
(a) Evaluate: lim (cos x) sin x . (b) Evaluate: lim ( x) sin x
(c) Evaluate: lim (tan x) cos x
x 0
x 0
x / 2
Find the local maximum and minimum values of f (x) = x4 – 3x3 + 3x2 – x.
A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight
river. He needs no fence along the river. What are the dimensions of the field that has the largest
area?
Find the area of the largest rectangle that can be inscribed in a semi circle of radius r.
The top and bottom margins of a poster are each 6cms and the side margins are each 4cms. If the
area of the printed material on the poster is fixed 384cms2. Find the dimensions of the poster
with the smallest area.
Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius
“a” is 8/27 (volume of the sphere).
A right circular cylinder is inscribed in a sphere of radius r. find the largest possible volume of
such a cylinder.
A piece of wire 10m long is cut into two pieces one piece is bent into a square and the other is
bent into an equilateral triangle. How the wire should be cut so that the total area enclosed is
minimum.
Trace the curve y = x3 + 1
(ii) Trace the curve y = x3
(iii) Trace the curve y2 = 2x3
 x y 
u
u 1

Using Euler’s theorem, prove that x
y
 tan x if u  sin 1 
 x y
x
y 2


63.
If w = u2 ev where u = x/y and v = y log x, find
64.
Using Euler’s theorem prove that x
 /3
65.
Evaluate

 1
/6
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 x3  y3 
u
u

y
 sin 2u if u  tan 1 
x
y
 x y 
dx
cot x
1
 x
Solve: (i) (2D2 + 5D + 2) y = e 2
(ii) Solve: (D2 + 7D + 12) y = e2x
Solve: (i) (D2 + 4D + 13) y = cos 3x
(ii) Solve: (D2 + 9) y = sin 3x
2
–2x
x
Solve: (i) (D – 13D + 12) y = e +5 e
(ii) Solve: (D2 – 6D + 9) y = x + e2x
d2y
dy
Solve
 3  2 y  2e 3 x when x = log2, y = 0 and when x = 0, y = 0
2
dx
dx
2
Solve: (D – 1) y = cos 2x – 2 sin 2x
(ii) Solve: (3D2 + 4D + 1) y = 3 e–x/3
Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at
x = –1 and 1 respectively.
The number of bacteria in a yeast culture grows at a rate, which is proportional to the number
present. If the population of a colony of yeast bacteria triples in 1 hour. Show that the number of
bacteria at the end of five hours will be 35 times of the population at initial time.
Radium disappears at a rate proportional to the amount present. If 5% of the original amount
disappears in 50 years, how much will remain at the end of 100 years.
(Take A0 as the initial amount).
The rate at which the population of a city increases at any time is proportional to the population
at that time. If there were 1, 30, 000 people in the city in 1960 and 1, 60, 000 in 1990 what
population may be anticipated in 2020.
A radioactive substance disintegrates at a rate proportional to its mass. When its mass is 10mgm,
the rate of disintegration is 0.051mgm per day. How long will it take for the mass to be reduced
from 10mgm to 5 mgm?
In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the
quantity of the substance still untransformed at the instant. At the end of one hour, 60 grams
remain and at the end of 4 hours 21 grams. How many grams of the first substance were there
initially?
Find the area between the curve y = x2 – x – 2, x-axis and the lines x = – 2 and x = 4.
Find the area between the line y = x + 1 and the curve y = x2 – 1.
Find the bounded by the curve y = x3 and the line y = x.
2
80.
w
w
and
x
y
2
 x 3  y 3
Find the length of the curve       1
a
b
Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x.
Compute the area between the curve y = sin x and y = cos x and the lines x = 0 and x = .
Find the area bounded by x-axis and an arch of the cycloid x = a (2t – sin2t), y = a (1 – cos2t).
Show that (R – {0}, ) is an abelian group. Here “” Denotes usual multiplication.
Show that the set of all non-zero complex numbers is an abelian group under the usual
multiplication of complex numbers.
Show that (Z, *) is an infinite abelian group where * is defined as a * b = a + b + 2.
Let G be the set of all rational numbers except 1 and * be defined on G by a * b = a + b – ab for
all a, b G. Show that (G, *) is an infinite group.
Prove that the set of four functions f1, f2, f3, f4 on the set of non-zero complex numbers C – {0}
1
1
defined by f1(z) = z, f2(z) = –z, f3(z) = , f4(z) = – , z C – {0} forms an abelian group with
z
z
respect to the composition of functions.
89. (i) Show that (Zn, +n) forms a group.
(ii) Show that (Zn – ([0]), 7) forms a group.
90. Show that the nth roots of unity form an abelian group of finite order with usual multiplication.
91. Show that the set G of all positive rational forms a group under the composition * defined by
ab
a*b=
for all a, b  G.
3
92. Show that the set of G of all rational numbers except – 1 forms an abelian group with respect to
the operation * given by a * b = a + b + ab for all a, b G.
93. Show that the set {[1], [3], [5], [7], [9]} forms an abelian group under multiplication modulo 11.
 a 0
 , a  R – {0} forms an abelian group under
94. Show that the set of all matrices of the form, 
 0 0
matrix multiplication.
95. Show that the set G = {2n / nZ} is an abelian group under multiplication.

kx 1e  x , x,  ,   0
96. The probability density function of a random variable x is f(x) = 
.
0,
otherwise
Find (i) k, (ii) P(X > 10)
97. If X is normally distributed with mean 6 and standard deviation 5 find (i) P(0  X  8),
(ii) P(|X – 6| < 10)
98. The mean score of 1000 students for an examination is 34 and S.D is 16. (i) How many
candidates can be expected to obtain marks between 30 and 60 assuming the normality of the
distribution and (ii) determine the limits of the marks of the central 70% of the candidates.
99. The air pressure in a randomly selected tyre put on a certain model new car is normally
distributed with mean value 31psi and standard deviation 0.2psi. (i) What is the probability that
the pressure for a randomly selected tyre (a) between 30.5 and 31.5psi (b) between 30 and 32psi.
(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5psi?
100. Obtain k,  and 2 of the normal distribution whose probability distribution function is given by
2
f(x) = ke2 x  4 x – < X < 
101. Obtain c,  and 2 of the normal distribution whose probability distribution function is given by
2
f(x) = ce  x 3 x – < X < 
102. An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls
in three draws one from the urn (i) with replacement (ii) without replacement.
103. A random variable X has the following probability mass function (1) find k
(2) Evaluate P(X < 4), P(X  5)and P(3 < X  6) (3) What is the smallest value of x for which
P(X  x) > ½
X
0
1
2
3
4
5
6
P(X = x)
k
3k
5k
7k
9k
11k
13k
104. The total life time (in years) of 5 year old dog of a certain breed is a Random variable whose
for x  0
0,

distribution function is given by F(x) =  25
. Find the probability that such a
for x  5
1  x 2 ,
88.
five year old dog will live (i) beyond 10 years (ii) less than 8 years
(iii) anywhere between 12 and 15 years.
105. The number of accidents in a year involving taxi drivers in a city follow a Poisson distribution
with mean equal to 3. Out of 1000 driver find approximately the number of drivers with (i) no
accidents in a year (ii) more than 3 accidents in a year (e – 3 = 0.0498)