2015
Mathematics
1. Let p(x) be any polynomial of
degree greater than or equal to one and a be any real number. If p(x) is divided by the polynomial x-a, then the remainder is equal to
a) P(-a)
b) P(a)
c) –p(a)
d) –p(-a)
2.
The sum of the roots of the quadratic equation ax2+
bx + c = 0
a) b/a
b) –b/a
c) c/a
d) –c/a
3. The 6th and 17th
terms of an AP are 21 and 54 respectively. The first term of the AP is
a) 6
b) 4
c) 3
d) 2
4. If tan(2Ѳ+30o)=cot 3 Ѳ, then Ѳ=
a) 5o
b) 10o
c) 11o
d) 9o
5. The curved surface area of
a right circular cylinder of radius r and height h is
a) πrh
b) πr(h+r)
c) 2πrh
d) 2πr(h+r)
6. Define the modulus of a
real number. 1
Ans:
7. Find the canonical
decomposition of 1764. 1
Ans:
8. Define a sequence. 1
Ans:
9. Check if the pair of
equations 2x+3y-12=0 and 6x+9y+36=0, is consistent or not. 1
Ans:
10. Write the statement of
Basic Probability Theorem. 1
Ans:
11. The length of the sides of
a triangle are 8 cm, 15 cm and 17 cm. State whether the triangle is a right
triangle or not. 1
Ans:
12. Find the area of a sector
of a circle of radius 14 cm when the sectorial angle is 45o. 1
Ans:
13. Give mathematical
definition of the probability of the occurrence of an event A. 1
Ans:
14. Prove that |-x| = |x| for
any x ϵ R. 2
Ans:
15. If a be the first term and d
be the common difference of an AP, show that the nth term is given
by
an =a+(n-1)d. 2
Ans:
16. How many terms of the AP:
1, 6, 11, 16, …….. must be taken so that
the sum is 148? 2
Ans:
17. Find the value of :
Ans:
18. A solid metallic cone is
81 cm high and the radius of its base is 6 cm. if it is melted and recast into
a solid sphere, find the radius of the sphere. 2
Ans:
19. State any three field
properties of the real numbers. 3
Ans:
20. If x 2 + px + q
and x
2 + lx + m are both divisible by x + a, show that a=
3
Ans:
21. Solve graphically:
x + y = 5
3x + 2y = 12
Ans:
22. Prove that the lengths of
the tangents drawn from an external point to a circle are equal. 3
Ans:
23. Prove that :
Ans:
24. A fair die is tossed
twice. Find the probability that the sum of the points obtained in the two
tosses is equal to 8. 3
Ans:
25. Factorise:
2b2c2 + 2c2a2 + 2a2b2
- a4 - b4 - c4
Or,
-
Ans:
26. A journey of 224 km from
Imphal to Jiribam takes 2
hours less by a car than by a passenger bus.
If the average speed of the bus is 12 km/hr less than that of the car, find the
average speed of the bus and that of the car. 4
Ans:
27. Show that the area of the
triangle whose vertices are
)
is
4
Ans:
28. State and prove Pythagoras
Theorem. 5
Ans:
Or,
Prove that the internal bisector of an angle of a triangle
divides the opposite side internally in the ratio of the other two sides.
Ans:
29. Construct a triangle
similar to a given triangle ABC with its sides equal to
of the corresponding sides of the triangle
ABC. Write the steps of construction. 2+3=5
Ans:
30. Two towers of the same
height stand on either side of a road 40 m wide. At a point on the road between
the towers the elevations of the towers are 600 and 300. Find the height of the
towers. ( Take
Ans:
Or,
A man in a boat being rowed with uniform speed away from a cliff
60 meters high observes that it takes 2 minutes for the angle of elevation of
the top to change from 600 to 450. Find the speed of the
boat.
( Take
Ans:
31. A cone is cut into
three parts by planes through the points of trisection of its altitude and
parallel to the base. Prove that the volumes of the parts are in the ratio
1:7:19. 6
Ans:
32. In an all India
examination of 100 marks, 157210 candidates completed for admission to a
particular course. The grouped frequency distribution of the marks secured by
the candidates are given below:
Marked secured : 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No. of candidates : 5606 8670 100078 25686 25700 35006 24678 16462 4270 1054
The concerned authority found that there are seats only for the
best 10% of the total number of candidates. Find the cut off marks. Will a
candidate securing 76 marks get admission? 6
Ans:
--------------------------------
Mathematics
2014
1. If |x+5| = -x, then the
value of x is:
(A) 0
(B) 5
(C)
(D)
2. The pair of linear
equations
and
has unique solution if
(A)
(B)
(C)
(D)
3. The 15th
term of an AP exceeds the 22nd term by 35; the common difference of
the AP is
(A) – 5
(B) 5
(C) -7
(D) 7
4. The circumference of a
circle is 44 cm. Its area is square centimetres is :
(A) 38
(B) 38.5
(C) 40
(D) 40.5
5. If A and B are two
independent events of a random experiments, then P(AB)=
(A) P(A)+P(B)
(B) P(A)- P(B)
(C) P(A).P(B)
(D) P(A)
P(B)
6. Write the statement of
“Euclid’s Division Lemma”. 1
7. Find the constant
remainder when
is divided by x-1 . 1
8. Define an “Arithmetic
Progression”. 1
9. The nth term
of a sequence is n2+3. Is the sequence an AP ? 1
10. The length of the sides
of a triangle are 7 cm, 24 cm and 25 cm.
Determine if the triangle is a right triangle or not. 1
11. Write a Pythogorian
relation between the trigonometric ratios sec A and tan A. 1
12. Write the formula for
the volume of a frustum of cone. 1
13. When are events of a
random experiment said to be equally likely. 1
14. If
.
2
15. Find the quadratic
equation whose roots are
2
16. Find the sum of the
first 25 terms of the AP :20,17,14,11, ……… 2
17. If
2
18. A dice is thrown once.
Find the probability of getting a number greater than 4. 2
19. Factorise:
3
20. Using the prime
factorisation method, find the HCF and LCM of 420, 504 and 924. 3
21. Solve graphically: 3
22. Prove that the tangent
at any point of a circle is perpendicular to the radius through the point of
contact. 3
23. Find the values of the
trigonometric ratios of 45o. 3
24. A chord of a circle of
radius 12 cm subtends an angle of 60o at the centre. 3
(Use
)
25. State and prove Factor
theorem. 4
26. A motor boat whose
speed is 18 km/hr in still water takes 1 hour more to go 24 km upstream than to
return downstream to the same spot. Find the speed of the stream. 4
Or,
A number consists of two digits. The product of the digits is
20. When 9 is added to the number, the digits interchange their places. Find
the number.
27. In what ratio is the
line segment joining the points (-2,3) and (3,8) divided by the y-axis. Also
find the coordinates of the point of division. 4
28. Construct a pair of
tangents to a circle from an external point. Write the steps of construction. 2+3=5
29. The angle of elevation
of a bird from the eye of a man on the bank of a pond is 30o and the
angle of depression of its reflection in the pond is 600. Find the
height of the bird above the pond if the height of the eye above water surface
is 1.5 meters. 5
Or,
A vertical tower stands on a horizontal plane and is surmounted
by a vertical flagstaff of height h. At a point on the plane, the angle of
elevation of the bottom of the flagstaff is
and that of the top of the flagstaff is
.
Prove that the height of the tower is
30. A metallic sphere of
radius 6 cm is melted and recast to from a cylinder of height 32 cm. Find the
total surface area of the cylinder. (Take
5
31. State and prove
SSS-Similarity Theorem. 6
Or,
Prove that the internal bisector of an angle of a triangle
divide the opposite side internally in the ratio of the other two sides.
32. In the following
distribution, the frequencies of two classes are missing. However the mean of
the data is given to be 50. Find the missing frequencies.
6
|
Class
|
Frequency
|
|
0-20
|
17
|
|
20-40
|
….
|
|
40-60
|
32
|
|
60-80
|
…
|
|
80-100
|
19
|
|
Total
|
120
|
-------------------
Mathematics
2013
1. If p(x) is a polynomial
of degree
1 and a is any real number, then x-a is a
factor of p(x) if and only if :
(A) p(a) =1
(B) p(a)=0
(C) p(-a) = 0
(D) p(a)=a
2. the sum of the roots of
the quadratic equation ax2+bx+c=0 is equal to
(A)
(B)
(C)
(D)
3. The number of two-digit
numbers which are divisible by 7 is:
(A) 11
(B) 12
(C) 13
(D) 14
4. If
(A) 100
(B) 110
(C) 130
(D) 150
5. Area of a sector of a
circle with radius r and angle
measured in degrees, is:
(A)
(B)
(C)
(D)
6. What is meant by a
cyclic expression? 1
7. Find the canonical
decomposition of 2013. 1
8. State the nature of the
roots of the quadratic equation
9. Find the common
difference of the A.P. whose nth term is 3n-2. 1
10. Write the statement of
Pythagoras Theorem. 1
11. Write the formula for
the volume of a frustum of cone. 1
12. How long is an arc of a
sector of a circle with radius 6 cm and angle 300? (Take
) 1
13. What is meant by a
random experiment? 1
14. Show that any square
number is of the form 4k or 4k+1. 2
15. If a be the first term
and d, the common difference of an A.P., then show that the nth term an=a+(n-1)d.
2
16. If one root of the
equation x2-px+q=0 be twice the other, show that 2p2=9p. 2
17. Prove the identity.
2
18. A solid metallic cone
is 12 cm high and radius of its base is 3 cm. it is melted and recast into a
solid sphere. Find the radius of the sphere. 2
19. State and prove
Remainder Theorem. 3
20. Factorise:
21. Solve graphically:
22. Two concentric circles
are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle
which touches the smaller circle. 3
23. Find the values of any
two trigonometric ratios of 600. 3
24. A fair coin is tossed 3
times. Find the probability that head appears exactly twice.
25. If x,y
R and xy = 0, then prove that x=0 or y =0. 4
Or,
For real numbers
where
26. In a class room, there
are a number of benches. If 4 students sit on each bench, five benches are left
vacant and if 3 students sit on each bench, 4 students are left standing. Find
the number of benches and students in the class room. 4
27. Prove that the area of
a triangle with vertices
(
:
28. Construct a triangle
similar to a given triangle ABC, with its sides equal to
of the corresponding sides of the
ABC.
Write the steps of construction. 2+3=5
29. A tower subtends an
angle of 600 at a point on the same level as the foot of the tower
and from a second point 10 meters above the first, the angle of depression of
the foot of the tower is 300. Find the height of the tower. 5
30. The radius and height
of a circular cone are 10 cm and 25 cm respectively. The area of the
cross-section of the cone by a plane parallel to its base is 154 cm2.
Find the distance of the plane form the base from the base of other cone. Find
the distance of the plane from the base of the cone. (Take
) 5
Or,
A vessel is in the form of an inverted cone of height 6 cm and
radius 4 cm. it is filled with water upto the rim. When lead shots each of
which is a sphere of radius 0.2 cm are dropped into the vessel, one-tenth of
the water flows out. Find the number of lead shots dropped into the vessel.
31. State and prove Basic
Proportionality Theorem. 6
Or,
If a perpendicular is drawn from the vertex of a right angle of
a right triangle to the hypotenuse, prove that the triangles on each side of
the perpendicular are similar to the whole triangle and to each other.
32. The expenditure for the
consumption of water per month by 100 families is given below:
Expenditure (in Rs): 30-40 40-50 50-60 60-70 70-80 80-90
90-100 100-110 110-120
No. of families : 12 18 20 15 12 11 6 4 2
---------
Mathematics
2012
1. The remainder when x3-2x+3x-1
is divided by x-2, is :
(A) 5
(B) -5
(C) 23
(D) -23
2. The sum of the first n
terms of an A.P. with first term a and common difference d, is given by:
(A)
(B)
(C)
(D)
3. The quadratic equation
x2-x+1=0 has
(A) Two unequal real roots
(B) Two equal real roots
(C) Two irrational roots
(D) No real roots
4. If
is an acute angle such that cos
(A)
(B)
(C)
(D)
5. If A and
are complementary events of each other, the
value of P(A) + P(
is:
(A) 0
(B) 1
(C)
(D) 2
6. Is there any x
R such that x2 is not positive? 1
7. Write all the possible
of the remainder when a number is divided by 3. 1
8. What is meant by the
discriminant of a quadratic equation? 1
9. For what value of k will
the roots of the equation
10. Write the statement of
the converse of Pythagoras Theorem. 1
11. How long is the radius
of a circle circumscribing a rectangle of sides 8 cm and 6 cm? 1
12. Write the expression
for the area of a sector of a circle with radius r and angle
(in degrees). 1
13. Define mutually
exclusive events associated with a random experiment. 1
14. For any x
, prove that x.0=0 2
15. If
are the roots of the quadratic equation
,
show that
. 2
16. Form the quadratic
whose roots are
.
2
17. If
2
18. A ball is drawn at
random from an urn containing 4 black and 5 red balls. Find the probability
that the ball is red. 2
19. State and prove Factor
Theorem. 3
20. By what numbers may 408
be divided so that the remainder is 23? 3
21. Solve graphically:
22. Prove that the lengths
of tangents drawn from an external point to a circle are equal. 3
23. Prove that identity:
24. A toy is in the form of
a cone mounted on a hemisphere with the same radius. The diameter of the base
of the conical portion is 6 cm and its height is 4cm. Determine the surface
area of the toy. 3
25. Factorise :
26. A manufacturing company
produced 600 cars in the third year and 700 cars in the seventh year. Assuming
that the production increases uniformly by a fixed number every year, find the
production in the 10th year. 4.
27. Prove that the
coordinates of the point R which divides the line joining
internally
in the ratio m:n are (
4
28. Draw a line segment AB
and divide it in the ration 4:3 write the steps of construction. 2+3=5
29. An aeroplane when 3000
metres high, passes vertically above another at an instant when the angles of
elevation at the observing point on the ground, are 600 and 450
respectively. How many metres higher is the one than the other? (Take
30. Two towers of the same
height stand on either side of a road 60 m wide. At a point on the road between
the towers the elevations of the towers are 600 and 300.
Find the height of the towers and position of the point. (Take
31. State and prove either
AAA –similarity Theorem or SSS-similarity Theorem. 6
32. Find the mean and
median marks from the following frequency table: 6
Marks above : 0 10 20 30 40 50 60 70 80 90 100
No. of students : 80 77 72 65 55 43 28 16 10 8 0
-----------
Mathematics
2011
1. If
(A) -3
(B) 3
(C) -4
(D) 4
2. The expression
(A)
(B)
(C)
(D)
3. The linear equations
(A)
(B)
(C)
(D)
4. For what value of k
will the equation
(A) 6
(B) 9
(C) -6
(D) -9
5. The area of a circle of
radius r, is:
(A)
(B)
(C)
(D)
6. When is an algebraic
expression said to have cyclic factors? 1
7. When is a number
?
8. Find the sum of the
following AP. 1
9. When is a line said to
be a tangent to a circle? 1
10. PA and PB are tangent segments drawn from an external point P to
a circle with centre O. If
AOB=1300, at what angle are the two tangents inclined
to each other?
11. Write the coordinates
of the mid-point of the line segment joining the points (
.
12. Prove that:
13. Express the length of
arc of a sector of a circle with radius r and angle
degrees, in terms of r and
.
14. Prove that
,
for any real number x. 2
15. Prove that in an AP
with first term a and common difference d, the nth term is given by
.2
16. Form the quadratic
equation whose roots are
2
17. Construct a pair of
tangents to a circle from an external point (traces of construction only). 2
18. In a right
2
19. Show that every odd
integer is of the form 4k+1 or 4k-1. 3
20. State and prove
Remainder Theorem. 3
21. Blur
22. Blur
23. Blur
24. A solid metallic
cylinder of height 24 cm and radius 3 cm is melted and recast into a cone of
radius 6 cm. Find the height of the cone. 3
25. State Euclid’s
algorithm for finding HCF of two given positive integers stepwise. 4
26. A man invested Rs.
36000, a part of it at 12% and the rest at 15% per annum simple interest. If he
received a total annual interest of Rs. 4890, how much did he invest at each
rate? 4
27. Two dice are thrown.
Find the probability that the sum of their points is 10. 4
28. State and prove SAS
similarity theorem. 5
Or,
Prove that the ratio of
the areas of two similar triangles is equal to the ratio of the squares of
their corresponding sides.
29. A tower subtends an
angle
at a point on the same level as the foot of
the tower and from a second point h meters above the first, the angle of
depression of the foot of the tower is
.
Find the height of the tower. 5
Or,
The angles of
depression of the top and the bottom of a 8 m tall tree form the top of a tower
are 450 and 600 respectively. Find the height of the
tower.
30. Blur
31. Construction a triangle
similar to a given triangle ABC with its sides equal to of the
corresponding sides of the
.
Write the steps of construction. 3+3=6
32. Find the mean and mode
of the following distribution.
Marks below : 10 20 30 40 50 60 70 80
No. of students : 15 35 60 84 96 127 198 250
Mathematics
2010
1. Let p(x) be a
polynomial of degree
1
and a be any real number. If p(x) is divided by x-a, then the remainder is 1
(A) –p(-a)
(B) –p(a)
(C) p(-a)
(D) p(a)
2. the value of
is
(A) 0
(B) 1
(C)
(D)
3. The length of the
shadow of a tower is 20 m when the altitude of the sun is 600. The
length of the tower in metres, is:
(A) 20
(B)
(C)
(D) 40
4. If the points (a,0),
(2,3) and (0,2) are collinear, then the value of a is:
(A) 1
(B) -1
(C) -4
(D) 4
5. The area of a sector of
a circle with radius r and sectorial angle
measured in degree is:
(A)
(B)
(C)
(D)
6. Find the quotient when
x3-1 is divided by x2+x+1. 1
7. When is a pair of
linear equations said to be a dependent pair? 1
8. For what value of k
does the pair of equations 1
2x+3y+6=0
4x+ky+12=0
9. What is meant by the
discriminant of the quadratic equation ax2+bx+c=0 ? 1
10. Define an Arithmetic
Progression. 1
11. Write the statement of
Pythagoras Theorem. 1
12. Write the coordinates
of the mid-point of the line segment joining (x1,y1) and
(x2,y2). 1
13. Find the curved surface
area of a cylinder of radius 3 cm and height 7 cm. 1
14. Define the terms (i)
sample space and (ii) event, associated with a random experiment. 1
15. If
2
16. Factorise: 2
17. Prove the formula
2
18. The nth term
of a sequence is given by an=3n+2. Show that the sequence is an AP. 2
19. The length of a tangent
to a circle from a point which is at a distance of 5 cm from the centre of the
circle is 4 cm. Find the radius of the circle. 2
20. State and prove Factor
Theorem. 3
21. Express in the form a3+
b3 +c3 -3abc and hence factorise x6 + 8x3
+ 27. 3
22. Solve graphically:
3x + y =9
2x – 3y + 16 =0
23. Prove that the tangent
at any point of a circle is perpendicular to the radius through the point of
contact. 3
24. Find the values of the
trigonometric ratios of 450. 3
25. Prove that
26. Find the least multiple
of 17 which when divided by 6, 9 and 15 leaves the same remainder 4 in each
case. 4
27. Solve by the method of
completing perfect square, the equation ax2+bx+c=0, a
0. 4
28. Prove that the coordinates
of the point R which divides the line joining P(x1, y1)
and Q(x2 , y1) internally in the ratio m:n are
. 4
29. A solid metallic cone
is 27 cm high and radius of its base is 16 cm. If it is melted and recast into
a solid sphere, find the curved surface area of the sphere. 4
30. State and prove Basic
Proportionality Theorem. 5
Or,
Prove that the internal
bisector of an angle of a triangle divides the opposite side internally in the
ratio of the other two sides.
31. Draw any line segment
and divide it internally in the ration 3:5. Write the steps of construction. 3+2=5
32. A vertical tower stands
on a horizontal plane and is surmounted by a vertical flagstaff of height h. At
a point on the plane, the angle of elevation of the bottom of the flagstaff is
Find the height of the tower. 5
Or,
The angel of elevation
of a bird from the eye of a man on the bank of a pond is 300 and the
angle of depression of its reflection in the pond is 600. Find the
height of the bird above the pond if the eye is 1.5 m above the water level.
33. Two dice are thrown and
the points on them are added together. Find which is more likely to happen that
the sum is 7 and that the sum is 8. 5
34. A bucket is in the form
of a frustum with a capacity of 45584 cm2. If the radii of the top
and bottom of the bucket are 28 cm and 21 cm respectively, find its height and
surface area. 6
35. The following is the
grouped data of the number of persons of various age groups in a hill village
in a border area of Manipur. Find the mean age and median age of the
inhabitants of the village: 6
Age group: 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No. of
Persons : 50 55 78 75 62 47 23 7 2 0
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