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1. Mirzapur Cadet College, Tangail Q Higher Mathematics
1. If S = {x : x ∈ Ñ and x (x −1) = x2 − x} then Find S′ = Ñ\S. What can you say about S? 4
Or, For any finite sets A and B. Prove that, n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
2. Answer any two of the following : 3×2=6
(a) If x − 2 is a factor of x4 − 5x3 + 7x2 − a, then show that a = 4
(b) Resolve into factor : a6 + 18a3 + 125
1 1 1 3
(c) If a3 + b3 + c3 = abc , then show that, bc + ca + ab = 0 or a = b = c.
3. Show that S = ô, where, S = {n : n ∈ ô and 5n − 2n is divisible by 3} 4
3 3 3 n2 (n + 1)2
3
Or, Use the method of mathematical induction to show that for all n ∈ ô; 1 + 2 + 3 +........ + n =
4
3 3 3
4. If x a + y b + z c = 0 and a2 = bc then show that ax3 + by3 + cz3 = 3xyza. 5
logk (1 + x) 1+ 5
Or, If logk x = 2, then show that x = 2 .
5. Find the domain of the function stated by given F(x) and determine whether the function is one-one or not. F(x) = (x − 1)2 . 5
x2 y2
Or, Sketch the graph of the relation S, S = {(x, y) : + = 1}.
4 25
|x|
6. Solve : + x2 = 2. 4
x
x(x − 4)
Or, Solve and show the solution set on the number line : < 0.
(x − 5)
x + y x − y 10
7. Solve : x − y + x + y = 3 ; x2 = 3 + y2 4
Or, Determine the simultaneous solutions of the inequalities : 2x − 3y − 1 > 0 and 2x + 3y − 7 < 0.
8. Impose a condition on x under which the infinite series 4
1 1 1
+ + + ........... (up to infinity) will have a sum and find that sum.
(x + 1) (x + 1)2 (x + 1)3
9. State and prove that of Apollonius theorem. 6
Or, Prove that the circum-centre, the centroid and orthocenter of any triangle are collinear.
10. The medians AD and BE of the triangle ABC intersect at G. The line drawn through G parallel to DE meets AC in F. Prove
that AC = 6EF. 4
Or, The chords AC and BD of the semicircle described on AB as diameter intersect in P. Prove that AB2 = AC.AP + BD.DP.
11. To construct a triangle having given the base, the vertical angle and the difference of the other two sides. (Essential for
describe and construction) 5
Or, To draw a circle which touches a given circle at a given point and passes through a point outside the circle. (Essential for
describe and construction).
12. With the help of vectors prove that the straight line segment. Joining the middle points of two sides of a triangle is parallel to
and half the third side. 4
Or, Prove by vector method that the diagonals of a parallelogram bisect each other.
13. A spherical ball of circumference 44 cm. exactly fits into a cubical box. Find the volume of the box unoccupied. 4
Or, A rectangular block of copper of length 21 cm., breadth 12cm. and height 11 cm. is melted and form into a uniform solid
wire of radius 0.7 cm. Find length of wire.
14. Answer any three of the following questions : 4×3=12
(a) What is the height of a hill which subtends angle 7 at a point 540 km. away from the hill.
(b) If acosA − bsinA = c, show that asinA + bcosA = ± a2 + b2 − c2.
(c) If 2sinA = 2−cosA find the value of sinA.
sin B + cos B
(d) If tanB = 3/4. And sinB is negative, then find the value of,
sec B + tan B
(e) Solve the equation: cosθ + sinθ = 2.
15. Find arithmetic mean of the following data: 5
Number 25-34 35-44 45-54 55-64 65-74 75-84 85-94
Students 5 10 15 20 30 16 4
Or, Find the standard deviation from the following frequency distribution table.
Class 5-14 15-24 25-34 35-44 45-54 55-64 65-74 75-84
Frequency 10 20 30 40 50 60 70 80
2 D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ

2. Mymensingh Girls' Cadet College, Mymensingh Q Higher Mathematics

1. If S = {x : x ∈ Ñ and x2 + 1 = 0}, then find S′ = Ñ\S, What can you say about S? 4
Or, For any finite sets A, B, C, prove that
n (A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C).
2. Answer any two of the following : 3×2=6
(i) If x – 2 is a factor of x4 – 5x3 + 7x2 – a, then show that a = 4.
(ii) Resolved into factor : a6 + 18a3 + 125
1 2 4 8 16
(iii) Simplify : + + + – .
1 + x 1 + x2 1 + x4 1 + x8 1 – x16
n n
3. Show that S = ô where S = {n : n ∈ ô and 5 – 3 is divisible by 2} 4
Or, Use the method of mathematical Induction to show that, for all n ∈ ô.
1 1 1 1 n
+ + + .............. + = .
1.2 2.3 3.4 n(n + 1) n + 1
2 –2 a –1 b–a
(a – b ) (a – b )
4. Simplify : 2 5
(b – a–2)b (b + a–1)a–b
logk(1 + x)
Or, If = 2, then prove that 2x = 1 + 5.
logk x
5. Find the domain of the function stated by given F(x) = x – 1. 4
Or, Sketch the graph of the relation and determine from the graph whether the relation is a function S = {(x, y) : y = 2x2}.
6. Solve : 32x–2 – 5.3x–2 – 66 = 0 4
(2x – 3) (x – 2)2
Or, Solve and show that the solution set on the number line: > 0.
x+1
2 2
7. Solve : x + y = 61; xy = – 30. 4
Or, Draw the graph of the solution sets of each pair of the inequalities : 3x – 3y > 5 and x + 3y ≤ 9.
1 1 1
8. Impose a condition on x under which the infinite series : + + + ............ (up to infinity) will have a
x + 1 (1 + x)2 (1 + x )3
sum. Find that sum. 4
9. State and prove the Appollonius theorem. 6
Or, Prove that in an acute angle triangle the perpendiculars drawn from the vertices to opposite side bisect the angle of the
Pedal triangle.
10. E is the middle point of the median AD of the triangle ABC and produced BE meets AC at F. Prove that AC = 3AF and BF = 4EF. 4
Or, Prove that if from a point P outside a cricle a tangent PT and a secant PBA are drawn to the circle, then prove that
PT 2 = PA.PB
11. To draw a cricle which touches a given circle at a given point and passes through a point out side the circle. (The sign and
description of construction are necessary) 5
Or, To draw a triangle having given the base, the vertical angle and the sum of the other Two angles. (The sign and
description of construction are necessary.)
12. With the help of vectors, prove that the straight line segment, joining the middle points of two sides of a triangle is parallel
to and half the third side. 4
Or, If a, b, c, d are the position vectors respectively of the points A, B, C, D then show that ABCD will be a parallelogram
if and only if b – a = c – d.
13. A hostel building is to be constructed for 70 students such that each student required 4.25 square meters of floor and 13.6
cubic meters of space. If the hostel room is 34 meters long, what will be its breadth and height? 4
Or, Three spherical balls of glass of redii 6, 8 and r cm are melted and formed into 9 cm a single solid sphere. Find the
value of r.
14. Answer any three of the following questions :− 4 × 3 = 12
(a) Define redian angle and prove that radian angle is a constant angle?
(b) The angles of a triangle are in A. P and the largest angle is double the least, Express the angles in radians.
(c) If secA + sinA = m, tanA – sinA = n, then prove that m2 – n2 = 4 mn.
π 5π 8π 9π
(d) Evaluate: sin2 + sin2 + sin2 + sin2 .
7 14 7 14
(e) Solve: 2sin2A + 3cosA = 0 where, 0° ≤ A ≤ 360°.
15. The frequency distribution table of the marks in mathematics obtained by 50 students of class IX is given below. Find the
frequency polygon of the given data : 5
Marks obtained 31-40 41-50 51-60 61-70 71-80 81-90 91-100 Total
No. Of students 6 8 10 12 5 7 2 50
Or, Find the standard deviation from the following frequency distribution table:
x 0 1 2 3 4 5 6 7
f 5 10 15 18 25 19 11 6
D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ 3

3. Comilla Cadet College, Comilla Q Higher Mathematics

1. A, B, C considered as subsets of universal set U, show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 4
Or, If S = {x : x ∈ Ñ and x(x − 1) = x2 − x} then find S′ = Ñ\S
2. Answer any two of the following questions : 3×2=6
(a) Resolve into factors : b2c2 (b2 − c2) + c2a2 (c2 − a2) + a2b2 (a2 − b2)
(b) If (a + b + c)(ab + bc + ca) = abc, then show that (a + b + c)3 = a3 + b3 + c3
x2
(c) Resolve 2 into partial fraction.
x − p2
3. Using the method of mathematical induction show that S = ô, where S = {n : n ∈ ô and 22n−1 + 1 is divisible by 3} 4
n2(n + 1)2
Or, Using the method of mathematical induction show that 13 + 23 + 33 + .......... + n3 =
4
1 1 2
4. If xyz ≠ 0, aa = bb = cc and b2 = ac then show that + = 5
x z y
Or, Show that log a b log b c log c a = 8
1
5. Find the dom and range of the function F(x) = and show that the function is one-one or not 4
3x − 2
Or, Draw the graph of the relation S = {(x, y) : y = x2 − 4x + 7} and determine from the graph whether S is function or not.
6. Solve : 32x−2 − 5.3x−2 − 66 = 0 4
3x + 4 x + 2
Or, Solve : and show the solution set on the number line <
5x + 3 2x + 3
7. Solve 2x2 + 3xy + y2 = 20, 5x2 + 4y2 = 41 4
Or, Draw the graph of the solution set of the following pair of inequalities: x + 2y – 4 > 0 and 2x – y – 3 > 0
Ω Ω
8. Express 1.231 as a rational fraction. 4
9. The internal bisector any angle of a triangle divides opposite sides or internally in the ratio of the other two sides. 6
Or, The circumcentre the centroid and the orthocentre of any triangle are collinear.
10. The medians of a ∆ABC meet at G. Prove that AB2 + BC2 + CA2 = 3(GA2 + GB2 + GC2). 4
Or, The perpendiculars AD, BE, CF from the vertiees to the opposite sides of the triangle ABC meet at O. Prove that
AO.OD = BO.OE = CO.OF
11. To construct a triangle having given the base, the vertical angle and the difference of the other two sides. (Sign and
description of the construction are required) 5
Or, Draw three circles of different radii such that they touch each other. (Sign and description of the construction are required)
12. Prove by vector method that the straight line segment joining the middle points of two sides of a triangle is parallel to and
half the third side. 4
Or, If the diagonal of quadrilateral bisect each other. Prove that it is a parallelogram.
13. An iron sphere of diameter 4 cm. is flattened into a circular iron sheet of thickness 2/3cm. What is the radius of the sheet? 4
Or, A spherical ball of circumference 44cm. exactly fits into a cubical box. Find the volume of the box unoccupied.
14. Answer any three of the following questions : 4 × 3 = 12
(a) Prove that the radian is a constant angle.
12
(b) If cosA = and A is positive acute angle then find tan A + C cosec A
13
(c) If cosθ − sinθ = c then prove that asinθ + bcosθ = ± a2 + b2 − c2
π 3π 5π 7π
(d) Simplify cos2 + cos2 + cos2 + cos2
8 8 8 8
(e) Solve : cosecθ. cotθ = 2 3 where 0° < θ < 90°
15. Find the arithmetic mean by short cut method from the frequency distribution table: 5
Class 15-19 20-24 25-29 30-34 35-39 40-44 45-49
Frequency 3 13 21 15 5 4 2
Or, Find the mean deviation from the following frequency distribution table.
x 5 10 15 20 25 30 35 40
f 8 10 12 15 20 18 13 8
4 D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ

4. Feni Girls' Cadet College, Feni Q Higher Mathematics

1. If S = {x : x ∈ Ñ and x (x −1) = x2 − x} then Find S′ = Ñ\S. What can you say about S ? 4
Or, For any finite sets A and B. Prove that, n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
2. Answer any two of the following questions : 3×2=6
(a) If x − 2 is a factor of x4 − 5x3 + 7x2 − a, then show that a = 4
(b) Resolve into factors : (a + b + c) (ab + bc + ca) − abc
1 2 4 8 16
(c) Simplify : 1 + x + 1 + x2 + 1 + x4 + 1 + x8 + 16
x −1
3. Use the Method of Mathematical Induction to show that, 4
1 1 1 1 n
for all, n ∈ ô : 1.3 + 3.5 + 5.7 + ..... +
(2n − 1) (2n + 1) = 2n + 1
Or, Show that S = ô, Where S = {n : n ∈ ô and 5n − 2n is divisible by 3}.
3 2
−
4. If a2 + 2 = 32 + 3 and a ≥ 0 then show that 3a3 + 9a = 8.
3 5
x − x2 − 1
Or, Show that logk = 2logk (x − x2 − 1 )
x + x2 − 1
1
5. Find the domain of the functin and determine whether the function is one-one or not. F(x) = x − 2 . 4
2 2
x y
Or, Sketch the graph of the relation and determine from the graph whether the relation is a function : S = {x, y) : 4 + 25 = 1}.
6. Solve : 11x − 6 = 4x + 5 − x − 1 . 4
x−3 x−2
Or, Solve and show the solution set on the number line : x − 4 > x − 1 .
7. Solve : x2 + y2 = 25, xy = 12 4
Or, Draw the graph of the following inequalities : y ≤ 2x
8. Impose a condition on x under which the infinite series 4
1 1 1
x + 1 + (x + 1)2 + (x + 1)3 + ......... (up to infinite) will have a sum and find that sum.
9. Prove that The sum of the squares on any two sides of a triangle is equal to twice the square on half the third side together
with twice the square on the median which bisects the third side. 6
Or, The rectangle contained by the diagonal of a quadrilateral inscribed in a circle is equal to the sum of the two rectangles
contained by its opposite sides.
10. The medians AD and BE of the triangle ABC intersect at G. The line drawn through G parallel to DE meets AC in F. Prove
that AC = 6EF. 4
Or, The tangent drawn at the point P of a circle of drawn C intersect the other two parallel tangents at Q and R, prove that
PQ. PR = CP 2
11. To construct a triangle having given the base, the vertical angle and the difference of the other two sides. 5
[N.B : description and signs of the figure are essential]
Or, Draw a circle touching a given straight line at any point and also touching a given circle at a given point.
[N.B : description and signs of the figure are essential]
12. If D, E, F are the middle points respectively of the sides BC, CA, AB of the triangle ABC, prove that AD + BE + CF = 0. 4
Or, Prove with the help of vectors that the straight line drawn from the middle point of one side of a triangle and parallel to
another passes through the middle point of the third side.
13. The height of a right circular cone is 24 cm. and its volume is 1232 cubic cm. What is its slant height? 4
Or, Find the surface and volume of a sphere of radius 6 cm.
14. Answer any three of the following questions : 4×3=12
(a) Prove that: 'Radian is a constant angle'.
(b) Prove that: tan2A + cot2A + 2 = sec2A. cosec2A.
(c) If sin A + cos A = 1, show that; sin A − cos A = ± 1.
(d) Solve the equation: 6sin2A − 11 sinA + 4 = 0 for 0° ≤ A ≤ 90°.
A+B A+B C C
(e) In any ∆ABC, show that: sin 2 + tan 2 = cos 2 (1 + cosec 2 ).
15. Find the arithmetic mean of the obtained marks of a student using shortcut method from the following frequency distribution
table: 5
Obtained Marks 25-34 35-44 45-54 55-64 65-74 75-84 85-94
Number 0f students 5 10 15 20 30 16 4
Or, Find the standard deviation from the following frequency distribution table :
Class Interval 5-14 15-24 25-34 35-44 45-54 55-64 65-74 75-84
Frequency 10 20 30 40 50 60 70 80
D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ 5

5. Pabna Cadet College, Pabna Q Higher Mathematics

1. If A = {a, b}, B = {2, 3} and C = {3, 4} then show that A × (B ∪ C) = (A × B) ∪ (A × C) 4
Or, For any subset A, B of the universal set U prove that (A ∩ B)′ = A′ ∪ B′.
2. Answer any two of the following questions : 3×2=6
(a) If x = b + c − a, y = c + a − b and z = a + b − c, then show that x3 + y3 + z3 − 3xyz = 4(a3 + b3 + c3 − 3abc).
x2 − 9x − 6
(b) Resolve into partial factors :
x(x − 2)(x + 3)
(c) Resolve into factor (a − b)3 + (b − c)3 + (c − a)3
3. Show that S = ô where S = {n : n ∈ ô and 5n − 2n is divisible by 3} 4
2 2 2 2 n(n + 1)(2n + 1)
Or, Use the method of Mathematical induction to show that 1 + 2 + 3 +... ... ... + n =
6
1 1
4. If x = (a + b)3 + (a − b)3 and a2 − b2 = c2 then show that x3 − 3cx − 2a = 0 5
a−1 b−1 c−1
Or, If xy = p, xy = q and xy = r then show that (b − c) logk p + (c − a) logk q + (a − b) logk r = 0
1
5. Find the domain and Range of the function F(x) = and determine whether the function is one-one or not. 4
x−2
2 2
Or, Sketch the graph of the relation S = {(x, y) : x + y = 9 and y ≥ 0} and determine from the graph whether the relation is
a function.
2x x−1
6. Solve : 6 +5 = 13 4
x−1 2x
3x + 4 x + 2
Or, Solve : < and show the solution set on the number line.
5x + 3 2x + 3
x+y x−y 5 2
7. Solve : + = , x + y2 = 90 4
x−y x+y 2
Or, Solve : 8yx − y2x = 16, 2x = y2
1 1 1
8. Impose a condition on x under which the infinite series + + + ...... ∞ will have a sum and find that sum. 4
x + 1 (x + 1)2 (x + 1)3
9. Prove that a straight line drawn parallel to one side of a triangle cuts the other two sides or those sides produced proportionally. 6
Or, Prove that the rectangle contained by the diagonal of a quadrilateral inscribed in a circle is euqal to the sum of the two
rectangel contained by its opposite sides.
10. If P is any point on the base BC of the isosceles triangle ABC then prove that AB2 − AP 2 = BP.PC 4
Or, The chords AC and BD of the semicircle discribed on AB as diameter intersect in P. Prove that AB2 = AC. AP + BD. BP.
11. To constant a triangle having the base, the vertical angle and the difference of the other two sides. 5
Or, To draw a circle which touches a given circle at a given point and passes through a point outside the circle.
12. If a, b, c are the position vectors of A, B, C respectively and if the point C divides AB in the ratio m t n internally, then
mb + na
c= 4
m+n
Or, If the diagonals of a quadrilateral bisect each other, Prove that it is a parallelogram.
13. A solid sphere of diameter 8 cm. is melted and fromed into a uniform hollow sphere of outer diameter is 10 cm. Find the
thickness of the second sphere. 4
Or, A spherical ball of circumference 44 cm. exactly fits into a cubical box. Find the volume of the box unoccupied.
14. Answer any three of the following questions :− 3 × 4 = 12
(a) Prove that Radian is a constant angle.
tanθ + secθ sin θ + 1
(b) Prove that =
tanθ − secθ + 1 cos θ
(c) If acos θ − bsinθ = c, show that asin θ + bcos θ = ± a2 + b2 − c2
π 5π 8π 9π
(d) Evaluate sin2 + sin2 + sin2 + sin2
7 14 7 14
(e) Solve : 4(cos2θ + sinθ) = 5, where 0° < θ < 360°
15. Using short cut method find the arithmetic mean from the following frequency distribution table: 5
xi 1 2 3 4 5 6 7 8 9 10
fi 5 20 30 40 50 35 21 12 10 8
Or, Find the mean deviation from the following frequency distribution table.
x 5 10 15 20 25 30 35 40 45 50
f 8 12 15 10 20 18 13 8 3 2
6 D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ

6. Joypurhat Girls' Cadet College, Joypurhat Q Higher Mathematics

1. If S = {x : x ∈ Ñ and x(x −1) = x2 − x}, determine S′ = Ñ\S. What can you say about S? 4
Or, Show that A × (B ∪ C) = (A × B) ∪ (A × C).
2. Answer any two of the following questions : 3×2=6
(a) If (x − 2) is a factor of the polynomial x4 − 5x3 + 7x2 − a, show that a = 4
(b) If x = b + c − a, y = c + a − b, z = a + b − c show that x3 + y3 + z3 −3xyz = 4 (a3 + b3 + c3 − 3abc)
a2 + bc b2 + ca c2 + ab
(c) Simplify : + +
(a −b)(a − c) (b − c)(b − a) (c − b)(c − a)
3. Use the method of mathematical induction to show that, for all n ∈ ô,
n2(n + 1)2
13 + 23 + 33 + ... ... ... + n3 = 4
4
2n−1
Or, Show that, S = ô, where S = {n : n ∈ ô and (2 + 1) is divisible by 3}
x y z 2 1 1 2
4. If a = b = c and b = ac, show that + = 5
x z y
x − x2 − 1
Or, Show that logk = 2logk(x − x2 − 1 )
x + x2 − 1
5. If A = {−2, −1, 0, 1, 2} and S = {(x, y) : x ∈ A, y ∈ A and y = x2}. Express S in roster method and also find the domain and
range of S. 4
Or, Sketch the graph of the relation S and determine whether the function is one-one or not.
x−1 3x + 2
6. Solve : +2 =3 4
3x + 2 x−1
|x|
Or, Solve : + x2 = 2
x
7. Solve : 18yx − y2x = 81, 3x = y2 4
x−2 y+1 z+4
Or, Solve : = = , 2x + 3y − 4z = 13
3 4 5
8. Find the sum of the (infinite) geometric series : 1 + 0.1 + 0.01 + ... ... ... 4
9. State and prove Apollonius Theorem. 6
Or, Prove that if two chords of a circle cut at a point within it, the rectangles contained by the segment of one is equal to
that contained by the segment of the other.
10. If AM and DN are the heights of the similar triangles ABC and DEF, prove that AM t DN = AB t DE. 4
Or, The chords AC and BD of the semicircle described on AB as diameter intersect in P. Prove that AB2 = AC. AP + BD. BP
11. Construct a triangle having given the base, vertical angle and sum of the other two sides. (Sign of the figure and description
are necessary) 5
Or, Construct a circle which touches a given straight line at a given point in it and passes through another given point
outside that line. (Sign of the figure and description are necessary)
12. Prove by Vector method that the diagonals of a parallelogram bisect each other. 4
Or, If a and b are non zero, then a = mb is true only if a is parallel to b.
13. The area of the curved surface of a right circular cylinder 100 sq. cm and its volume is 150 cu. cm. Find the radius of the
base and the height. 4
Or, A conical tent has a height of 7.50 metres. How much canvass will be required if it is desired to enclose a land of area
2000 sq. metres.
14. Answer any three of the following questions :− 4 × 3 = 12
(a) Define radian angle. Prove that it is a constant angle.
tan θ + sec θ − 1 sin θ + 1
(b) Show that =
tan θ − sec θ + 1 cos θ
(c) If sin A + sin A = 1, prove that cos2 A + cos4 A = 1
2

(d) Solve : 6sin2θ − 11 sin θ + 4 = 0; θ < 90°

π 3π 5π 7π
(e) Evaluate : sin2 + sin2 + sin2 + sin2
4 4 4 4
15. Draw the histogram and frequency polygon for the following data : 5
Height (cm) 146-155 156-165 166-175 176-185 186-195
No. of persons 5 35 25 15 20
Or, Find the standard deviation for the following data :
xi 1-50 51-100 101-150 151-200 201-250 251-300 301-350
fi 7 0 9 13 5 4 2
D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ 7

7. Rangpur Cadet College, Rangpur Q Higher Mathematics

1. For any subsets A, B of the universal set U prove that (A ∪ B)′ = A′ ∩ B′. 4
Or, Show that the set S = {3n : n = 0 or n ∈ ô}
2. Answer any two of the following questions : 3×2=6
(a) Use the Remainder Theorem to find the remainder when P(x) = 2x3 − 5x2 + 7x − 8 is divided by x − 2.
1 1 1 3
(b) If 3 + 3 + 3 = then show that bc + ca + ab = 0 or a = b = c.
a b c abc
x3 + 2x2 + 1
(c) Resolve into partial fractions : 2
x + 2x − 3
3. Show that S = ô, where S = {n : n ∈ ô and 22n−1 + 1 is divisible by 3} 4
1
Or, Show that by the Method of Mathematical Induction 12 + 22 + 32 + ... ... ... + n2 = n(n + 1) (2n + 1)
6
a
()⎧ 1 a −b ⎫ a+b
2 2

4. Simplify : ⎨⎩ xa a−b ⎬⎭ 5
Or, If a3−x. b5x = a5+x. b3x then show that xlogk (b/a) = logk a.
5. F(x) = (x − 1)2; find the domain of the function stated by given F(x) and determine whether the function is one-one or not. 4
Or, Sketch the graph of the relation S = {(x, y) : x2 + y2 = 9 and x ≥ 0}
6. Solve : 41+x + 41−x = 10 4
x−4 x−6
Or, >
x−2 x−3
7. Solve : y = 4, y2 = 2x
x
4
Or, x + y + z = a + b + c
ax + by + cz = a2 + b2 + c2
x y z
+ + =3
a b c
1 1 1
8. Impose a condition on x under which the infinite series + + + ........... (up to infinity) will have a sum
x +1 (x +1)2 (x + 1)3
and find that sum. 4
9. State and prove the "Theorem of Apollonius". 6
Or, State and prove the "Ptolemy's Theorem".
10. The medians of a ∆ABC meet at G. Prove that AB2 + BC2 + CA2 = 3(GA2 + GB2 + GC2). 4
Or, A straight line drawn through the vertex A of the parallelogram ABCD meets the side BC at M and the side DC at N.
prove that BM.DN is a constant.
11. Construct a circle which touches a given straight line at a given point in it and passes through another given point outside
that line. 5
Or, Draw a circle touching a given straight line, at a given point in it and also touching another circle.
12. Prove with the help of vectors that the straight line joining the middle points of the diagonals of a trapezium is parallel to
half the difference of the parallel sides. 4
Or, Prove with the help of vectors that the straight lines joining the middle points of the adjacent sides of a quadrilateral
form a parallelogram.
13. The outer diametre of a hollow sphere is 13 cm and the thickness of the iron is 2 cm. A solid sphere is formed with the iron
used in the hollow sphere, what will be its radius. 4
Or, If the volume of a right circular cone is V, the area of its curved surface is S, radius of the base is r, height is h and
π.r2
semi-vertical angle is α, then show that S = square unit.
sinα
14. Answer any three of the following questions :− 4 × 3 = 12
(a) Prove that "Radian is a constant angle".
(b) If acosA − bsinB = c, show that asinA + bcosB = ± a2 + b2 + c2
(c) 3 sinθ + cosθ = 2, when 0 < θ < 2π
π 13π 16π 47π
(d) Find the value of cos2 + cos2 + cos2 + cos2
15 30 15 30
5 sin θ + cos(−θ) 51
(e) If tanθ = and cos θ is negative, show that = .
12 sec(−θ) + tan θ 26
15. The frequency distribution table of the marks obtained in mathematics by students of class are given below. Find the mean
deviation of the marks obtained. 5
Obtained marks 25-34 35-44 45-54 55-64 65-74 75-84 85-94
Number of students 5 10 15 20 30 16 4
Or, The marks obtained in Physics by 60 students of class X are given below. Find the standard deviation of the marks
obtained.
Numbers 51-60 61-70 71-80 81-90 91-100
Students 10 15 20 10 5
8 D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ

8. Jhenidah Cadet College, Jhenidah Q Higher Mathematics

1. If A = {a, b}, B = {1, 2} and C = {2, 3}, then show that, A × (B ∪ C) = (A × B) ∪ (A × C). 4
Or, For any finite set A and B prove that, n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
2. Answer any two of the following : 3×2=6
(a) If (a + b + c) (ab + bc + ca) = abc, then show that (a + b + c)3 = a3 + b3 + c3.
(b) Resolve x3 + y3 + z3 – 3xyz into factor.
x2 – 9x – 6
(c) Resolve into partial fraction : .
x(x – 2) (x + 3)
3. If m, n, k ∈ ô and m < n, then show that m + k < n + k. 4
2 2 2 n(n + 1) (2n + 1)
2
Or, By the method of mathematical induction show that, for all n ∈ ô, 1 + 2 + 3 + .............. + n = .
6
4. If ax3 + by3 + cz3 = 3axyz and a2 = bc, then show that, x3 a + y3 b + z3 c = 0. 5
Or, xya–1 = p,xyb–1 = q,xyc–1 = r, then show that (b – c) logk p + (c – a) logkq + (a – b) logkr = 0.
5. Find the domain of the function F(x) = (x – 1); and determine whether the function is one-one or not. 4
Or, Sketch the graph of the relation S = {(x, y) : x2 + y2 + 2x + 4y + 1 = 0} and determine from the graph whether the
relation is a function or not.
2x x–1
6. Solve: 6 +5 = 13. 4
x–1 2x
3x–5 2x–6
5 .b
Or, Solve: = a2x–6 (a > 0, b > 0, 5b ≠ a)
5x+1
7. Solve: x2 + xy + y2 = 3, x2 – xy + y2 = 7. 4
Or, Draw the graph of the solution sets of the inequalities x – 3y – 6 < 0 and 3x + y + 2 < 0.
1 1 1
8. Impose a condition on x under which the infinite series, + + + ......... will have an infinite sum and
x + 1 (x + 1)2 (x + 1)3
find that sum. 4
9. Prove that, the sum of the squares on any two sides of a triangle is equal to twice the square on half the third side together
with twice the square on the median which bisect the third side. 6
Or, Prove that, in an acute-angled triangle the perpendiculars drawn from the vertices to opposite side bisect the angle of
the pedal triangle.
10. If P is any point on the base BC of the isosceles triangle ABC, then prove that AB2 – AP 2 = BP.PC. 4
Or, BE and CF are respectively perpendicular on AC and AB of the triangle ABC. Prove that, ∆ABC t ∆AEF = AB2 t AE2.
11. To construct a triangle having given the base, the vertical angle and the difference of the other two sides. (Sign of
construction and description should be given) 5
Or, To construct a circle which touches a given circle at a given point and passes through another given point outside the
circle. (Sign of construction and description should be given)
12. By the method of vector, prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. 4
Or, Prove with the help of vectors, that the straight line joining the middle points of the non-parallel sides of a trapezium is
to half the sum of the parallel sides and parallel to that sides.
13. The area of surface of a rectangular solid is 198 square metre and its dimensions are in the ratio 3 t 2 t 1, find the volume
and length of diagonal of the solid. 4
Or, The area of the curved surface of a right circular cylinder is 100 sq. cm and its volume is 150 cubic cm. Find the radius
of the base and the height.
14. Answer any three of the following questions :− 4 × 3 = 12
(a) Give the definition of radian. Prove that radian is a constant angle.
(b) A man running along a circular track at the speed of 5 km/hour covers an arc in 36 seconds which subtends an angle
56° at the centre. Find the diameter of the circle.
(c) If acosθ – bsinθ = c, show that asinθ + bcosθ = ± a2 + b2 – c2.
π 13π 16π 47π
(d) Evaluate: cos2 + cos2 + cos2 + cos2 .
15 30 15 30
A+B A+B C C
(e) For any triangle ABC, prove that, sin + tan = cos (1 + cosec )
2 2 2 2
15. Using short-cut method find the arithmetic mean from the following frequency distribution table: 5
Class interval 25-34 35-44 45-54 55-64 65-74 75-84 85-94
Frequency 5 10 15 20 30 16 4
Or, Find the mean deviation from the following frequency distribution table:
Class 5-14 15-24 25-34 35-44 45-54 55-64 65-74 75-84
Interval Frequency 10 20 30 40 50 60 70 80
D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ 9

9. Faujdarhat Cadet College, Chittagong Q Higher Mathematics

1. For any subset A, B of universal set U, Show that (A ∪ B)′ = A′ ∩ B′ 4
Or, Of 30 students in a class, 19 have taken up Economics, 17 Geography, 11 Civics, 12 Economics and Geography 7
Economics and Civics, 5 Geography and Civics and 2 have taken up all the these subjects. How many students have taken
up none of these three subjects?
2. Answer any two of the following questions : 3×2=6
(a) Resolve into factor, 2a3 − 3a2 + 3a − 1
x2 − zy y2 − zx z2 − xy
(b) If = = ≠ 0. Then show that (a + b + c) (x + y + z) = ax + by + cz
a b c
a + 2a2 + 1
3
(c) Resolve into partial factor 2
a + 2a − 3
3. Show that S = ô. Where S = {n : n ∈ ô and 22n−1 + 1 is divisible by 3} 4
2 3 n−1 a(rn − 1)
Or, Using method of Mathematical induction. Show that, a + ar + ar + ar + .......... + ar = ; r ≠ 1 and n ∈ ô
r−1
1 1
4. If x = (a + b)3 + (a − b)3 and a2 − b2 = c3 then prove that x3 − 3cx − 2a = 0 5
ab logk(ab) bc logk(bc) ca logk(ca) a b c
Or, If = = then show that, a = b = c
a+b b+c c+a
1
5. Find the domain and range of the function such that F : Ñ → Ñ, F(x) = 4
x−3
Or, Sketch the graph of the relation S and determine from the graph where the relation is function or not:
x2 y2
S = {(x, y) : + = 1}
9 4
|x|
6. Solve : + x2 = 2 4
x
2x + 4 x + 3
Or, Solve and show the solution set in number line : <
x−3 x−1
x 2x x 2
7. Solve : 18y − y = 81, 3 = y 4
Or, Draw the graph of the solution sets of each pair of the following inequalities : x − 3y − 6 < 0, 3x + y + 2 < 0
1 2 4 8
8. Find the sum of the given series up to infinity if the sum exists − 2 + 3 − 4 + ..................... ∞ 4
5 5 5 5
9. Prove that, the ratio of the areas of two similar triangles is equal to the ratio of the squares on corresponding sides. 6
Or, Prove that, in a acute angle triangle the perpendiculars drawn from the vertices to opposite side bisect the angle of the
pedal triangle.
10. The medians of a triangle ABC meet at G. Prove that (AB2 + BC2 + CA2) = 3(GA2 + GB2 + GC2) 4
Or, ABC is a triangle in which ∠C is a right angle. If CD is drawn perpendicular to the hypotenuse. Prove that CD2 = AD. BD
11. Construct a triangle having given the altitude, the median bisecting the base and an angle adjacent to the base. (The sign of
the construction and description are imperative.) 5
Or, Draw a circle touching a given circle at a given point in it and also touching another circle. (The sign of the construction
and description are imperative.)
12. Prove with the help of vectors that the straight line drawn from the middle point of one side of a triangle and parallel to
another passes through the middle point of third side. 4
Or, If a, b, c are the position vectors of A, B, C respectively and if the point C divides AB in the ratio m t n internally, then
mb + na
show that, c =
m+n
13. A ball of circumference 44 cm. exactly fits into a cubical box. Find the volume of the box unoccupied. 4
Or, A rectangular block of copper of length 21 cm, breadth 12 cm and height 11 cm. is melted and formed into a uniform
solid wire of radius 7 cm. Find the length of the wire.
14. Answer any three of the followings : 4 × 3 = 12
(a) What is the height of a hill which subtends an angle 7′ at a point 540 km away from the hill.
tanθ + secθ − 1 sinθ + 1
(b) Prove that, =
tan θ − secθ + 1 cosθ
(c) If 2sinA = 2 − cosA, what is the value of sinA
(d) Solve : 2sin2 θ + 3cos θ = 0. where 0° ≤ θ ≤ 360°
A+B A+B C C
(e) In any triangle ABC, show that, sin + tan = cos (1 + cosec )
2 2 2 2
15. From the following table of frequency distribution, Find the arithmetic mean: 5
xi 1 2 3 4 5 6 7 8 9 10
fi 5 20 30 40 50 35 21 12 10 4
Or, Marks obtained by 60 students in maths are given below. Find the standard deviation :
Class 51-60 61-70 71-80 81-90 91-100
No of students 10 15 20 10 5
10 D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ

10. Sylhet Cadet College, Sylhet Q Higher Mathematics

1. For any finite sets A & B, show that n(A ∪ B) = n(A) + n(B) − n(A ∩ B). 4
Or, If A & B ⊃ U, then prove that, (A ∪ B)′ = A′ ∩ B′
2. Answer any two : 3×2=6
(a) Resolve into factors, a3 − a2 − 10a − 8
2a2 − bc 2b2 − ca 2c2 − ab
(b) Simplify, + +
(a − b)(a − c) (b − c)(b − a) (c − a)(c − b)
x3
(c) Resolve into partial fraction : 2
x − a2
−3 −3 −3 −1
(d) If a + b + c = 3(abc) ; then show that, ab + bc + ca = 0 & a = b = c.
3. If S = {n t n ∈ ô & (5n − 2n) is divisible by 3}, then show that S = ô. 4
Or, By Mathematical Induction Formula show that, (where, n ∈ ô)
a(rn − 1)
a + ar + ar2 + ar3 + ......... + arn−1 = ,r≠1
r−1
3 3 3
4. If x a + y b + z c = 0; prove that ax3 + by3 + cz3 = 3axyz. 5
1 1 1
Or, Show that, + + =1
loga abc logb abc logc abc
1
5. Determine the domain & range of F(x) = . Is F one-one function or not? Why? 4
(x − 2)
2 2
Or, Draw the graph of S = {(x, y) : x + y = 9 & x ≤ 0} & Test weather S is function or not?
2x x−1
6. Solve, 6 +5 = 13 4
x−1 2x
2x 3 (x−1) 2
Or, Solve, a − (a + a)a + a = 0 (a > 0, a ≠ 1)
7. Solve, x2 − 2xy + 8y2 = 8, 3xy − 2y2 = 4 4
Or, Solve, ax.a(y+1) = a7, a2y . a(3x+5) = a20
Ω Ω
8. Express into rational fraction: 2.305 4
9. State & prove that the Ptolemy's theorem. 6
Or, Prove that the sum of the squares on any two sides of a triangle is equal to twice the square on half the third side
together with twice the square on the median which bisects the third side.
10. If E is the middle point of the median AD of the triangle ABC & produced BE meets AC at F. Prove that AC = 3AF & BF = 4EF.4
Or, The chords AC & BD of the semicircle described on AB as diameter intersect in P. Prove that, AB2 = AC. AP + BD. DP
11. Construct a triangle having given the altitude, the medians bisecting the base and an angle adjacent to the base. 5
Or, Construct a circle which passes through two given points & whose radius is equal to a given line segment.
12. If a, b, c, are the position vectors of A, B & C respectively & C divides AB in the ratio m t n internally. Prove that
mb + na
c= . 4
m+n
Or, Prove that with the help of Vectors that the st. line joining the mid. Point of the diagonals of a trapezium is parallel to &
half the difference of the parallel sides.
13. If the curved surface area & volume of cylinder are 100cm2 & 150cm3. Circular surface area = ? 4
Or, The length, breadth and the height of a rectangular parallelepiped in the ratio 21 t 16 t 12 and its diagonal is 87 centimeters.
Determine the area of the whole surface of the solid.
14. Answer any three. 4 × 3 = 12
x2 − 1
(i) If secθ + tanθ = x, then show that sinθ = 2
x +1
2 2 1
(ii) If 7 sin θ + 3cos θ = 4; then tan θ = ± .
3
π 5π 8π 9π
(iii) Evaluate, sin2 + sin2 + sin2 + sin2
7 14 7 14
(iv) Solve : 4(cos2θ + sinθ) = 5; where 0° ≤ θ ≤ 360°.
(v) The distance of the sun from the earth is 14.9 × 107 Km & the diameter of the sun subtends an angle 32′ at the centre
of the earth; what is the diameter of the sun?
15. Determine the mean in short-cut method. 5
Daily income 91-100 81-90 71-80 61-70 51-60 41-50 31-40
No of people 10 15 08 12 20 13 07
Or, Determine the mean deviation.
Class interval 51-55 56-60 61-65 66-70 71-75
Frequency 05 10 20 15 10
D”PZi MwYZ Q ¸i“Z¡cY© ¯‹zjmg‡ni cÖvK-wbe©vPwb cix¶vi cÖkœ 11

11. Barisal Cadet College, Barisal Q Higher Mathematics

1. Prove that the set S = {1, 4, 9, 16, 25 .........} of squares of natural numbers is an infinite set. 4
Or, If A= {3, 4, 5} and B = {4, 5, 6} then show that P(A) ∩ P(B) = P(A ∩ B)
2. Answer any two of the following : 3×2=6
(a) Resolve into factors : x6 + 18x3 + 125
(b) If (a + b + c) (ab + bc + ca) = abc, then prove that (a + b + c)3 = a3 + b3 + c3.
x3
(c) Resolve into partial fraction : 2 .
x –9
1 1 1 1 n
3. Prove by mathematical induction that + + + ........ + = . 4
1.3 3.5 5.7 (2n – 1) (2n + 1) 2n + 1
Or, If S = {n : n ∈ ô and 22n–1 + 1 is divisible by 3} then show that, S = ô.
4. If ax = by = cz and b2 = ac then show that x–1 + z–1 = 2y–1. 5

Or, If a3–x b5x = a5+x b3x, then show that x logk⎛ ⎞= logka.
b
⎝a ⎠
5. Find the domain of the function, F(x) = (x – 1)2 and verify the function is one-one or not? 4
Or, Sketch the graph of the relation S = {(x, y) : x2 + y2 = 9 and y ≥ 0} and determine from the graph whether the graph is a
function or not.
x–1 3x + 2
6. Solve : +2 = 3. 4
3x + 2 x–1
x+2 x–3
Or, Solve and show the solution set on the number line: > .
x+1 x–4
7. Solve: 8y x – y2x = 16, 2x = y2. 4
Or, Determine the simultaneous solutions of the inequalities 2x – 3y – 1 ≥ 0 and 2x + 3y – 7 ≤ 0.
1 1 1
8. Under what condition upon x the infinite series + + + ............. (upto infinity) will have an infinite sum and
x + 1 (x + 1)2 (x + 1)3
find that sum. 4
9. The sum of the squares on any two sides of a triangle is equal to twice the square on half the third side together with twice
the square on the median which bisects the third side. 6
Or, The rectangle contained by the diagonal of a quadrilateral inscribed in a circle is equal to the sum of the two rectangles
continued by its opposite sides. Prove it.
10. In the isosceles triangle ABC, BC DE and BC is the base. DE intersects AB and AC at the points D and E respectively.
Prove that BE2 – CE2 = BC. DE. 4
Or, Let ∆ABC be a right angled triangle and ∠C = 90°. If CD ⊥ AB then prove that CD2 = BD. AD.
11. Divide a given line segment internally and externally in a given ratio. (sign and description for construction must be
presented) 5
Or, Draw a cricle which touches a given circle at a given point and passes through a point outside the circle. (sign and
description for construction must be presented)
12. Prove with the help of vectors that the diagonals of a parallelogram bisect each other. 4
Or, Prove with the help of vectors that the straight line joining the middle points of non-parallel sides of a trapezium is to
and half the sum of parallel sides.
13. The area of the surface of a rectangular solid is 198 sq. metres and its dimensions are in the ratio 3 : 2 : 1. Find the volume
and the lenght of diagonal. 4
Or, If the height of a right circular cone is 8 cm, and the radius of its base is 6 cm. find the area of the whole surface and the
volume.
14. Answer any three of the following questions :− 4 × 3 = 12
(a) The angle of a triangle are in A.P. and the largest angle is double the least. Express the angles in radians.
1 + cosθ
(b) Prove that, = cosecθ + cotθ.
1 – cosθ
(c) If sinθ + cosθ = 1, then show that sinθ – cosθ = ± 1.
5 sinθ + cos (–θ) 51
(d) If tanθ = and cosθ is negative, show that, = .
12 sec(–θ) + tanθ 26
2
(e) Solve : 2sin θ = 3cosθ; (0° < θ < 360°)
15. Draw the histogram for the following data: 5
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. Of students 5 10 25 35 15 20
Or, Find the standard deviation of the following data:
6, 8, 10, 15, 12, 20, 8.5, 5.5, 9, 10.