Magnus Academy (Maths Specialist)

Wednesday, 11 November 2026

CONE

1) A cone of base radius 7cm and height 10cm. Find:

a) the volume

b) the curved surface

c) the total surface

d) the mass, if material of the cone weighs 6gm per cm³

e) the cost, at the rate of Rs 3 per kg.

2) Find the curved surface of the following cones:

a) radius of base 5cm, slant height 7cm.

b) radius of base 3cm, height 4 cm.

c) diameter of base 10cm, height 12cm

3) Find out whether the following statements is true or false:

The volume of cone is one half of the volume of the cylinder of the same radius and height.

4) Find the volume of the following cones:

a) radius of base 3cm, height 7 cm.
b) diameter of base 12cm, height 35 cm.

c) radius of base 5cm, slant height 23cm.

d) diameter of base 15cm, slant height 8.5cm

e) radius of base 13cm, height 6.3 dm

5) Find the curved surface of a cone, with radius of base 28cm and height 21cm.

6) Find the volume of a cone, with radius of base 35cm, slant height 37 cm.

7) Find the whole surface of a cone whose volume is 314.2cm³ and height 12 cm (π= 3.142).

8) What is the capacity of a cone whose base area is 154 sq.cm and height 12cm.

9) A right circular cone has radius of its base 30cm and slant height 50cm. Find its volume and total surface area.

10) Find the volume of a right circular cone, the diameter of whose base is 14cm and the area of curved surface is 550cm².

11) The height of a cone is 5m. Find the height of a cone sixteen times its volume and radius equal to its diameter.

12) How much canvas will it take to make a conical tent 11m in height and 12m in diameter at the base ?

13) The floor area of a tent in the form of a right circular cone is 3168/7 m². The canvas required for the tent is 3960/7 m². Find the air capacity of the tent.

14) Find the area to the nearest square metre of the canvas required to make a right circular conical tent 30m in perpendicular height and covering 1386 square metres of the ground. What is the cost to the nearest rupee of canvas required at the rate of Rs 4 per metres of width 1.5 metres?

15) A solid cone of base radius 6cm and height 10 cm is lowered into a cylindrical jar of radius 8cm, which contains water sufficient to submerge the cone completely. Find the rise in water level in the jar.

16) From a solid cylinder of height 12cm and base radius 5 cm a conical cavity of the same height and base is hollowed out. Find the surface area of the remaining solid.

17) A conical reservoir is 7m in diameter. Water is poured into it at the rate of 220 litres per minute. Find the height of water level in the reservoir after one hour. Give your answer to the nearest cm.

18) A cylindrical iron pillar 49 dm high and 6dm in radius is surmounted by a cone 14dm high. Find the volume of the pillar.

19) A girl fills a cylindrical bucket 32cm in height and 18cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24cm, find

a) the radius

b) the slant height of the heap.

Give your answer correct to one place of decimal.

20) A rectangular vessel is 20cm x 16cm x 11cm and is full of water. This water is poured in to a conical vessel of base radius 10cm. If the vessel is completely filled, find the height of the vessel.

21) Find the volume of a right circular conical tent whose vertical height is 8m and the area of whose base is 156m².

22) The circumference of the base of a cone is 44cm and the height is 24cm, find

a) the volume

b) the slant height

c) the curved surface area of the cone.

23) The height of a solid cylinder is 15cm and diameter 7cm. Two equal conical holes of radius 3cm and height 4cm are cut off. Find

a) the volume

b) the surface of the remaining solid.

24) The circumference of the base of a cone is 44cm and the slant height is 25cm. Find the volume and curved surface of the cone.

25) From a solid cylinder whose height is 8cm and radius is 6cm, a conical cavity of height 8cm and of base radius 6cm is hollowed out. Find the volume of the remaining solid correct to one place of decimal.

26) How many metres of cloth 1.1m wide will be required to make a conical tent, whose vertical height is 12cm and base radius is 16m ? also, find the cost of the cloth at the rate of Rs 14/ metre.

Tuesday, 1 September 2026

Arithmetic progression A to Z(XI)

ARITHMETIC PROGRESSION

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EXERCISE - A

1) Write first 4 terms in each of the sequences:

A) aₙ = (5n+2). 7, 12, 17, 22

B) aₙ = =(2n-3)/4. -1/4,1/4,3/45/4

C) aₙ = =(-1)ⁿ⁻¹ x 2ⁿ⁺¹. 4,-8,16,-32

2) find first five terms of the sequence defined by

a₁= 1, aₙ= aₙ₋₁+3 for n≥2. 1, 4, 7, 10, 13

3) find first five terms in the sequence, defined by

a₁= -1, aₙ= (aₙ₋₁)/n for n≥2. -1, -1/2,-1/6,-1/24,-1/120

EXERCISE - B

1) FIND

a) 2, 6, 10, ...... 20th term. 78

b) 1, 3, 5 ..... 100th term. 199

c) 23rd term of 7,5,3,1,.... -37

d) 20th term √2, 3√2, 5√2,... 39√2

e) nth term 8, 3, -2,-7,....... 13-5n

f) (a-3b), (a- b),(a+b) ... r th term. a+ 2br - 5b

g) 13, 8, 3,-2.... nth term. 18-5n

h) 3,5,7,9,... 8th and nth term. 17, 2n+1

i) 2, 3/2,2,.....nth & 10th term. (n+3)/2, 13/2

j) 1, (n+2)/n, (n+4)/n, (n+6)/n, .... 12th and n th term. (n+22)/n, (3n-2)/n

EXERCISE - C

1) Which term of the sequences

a) 5, 2, -1, ......is -22. 10

b) 148, 146, 144 ..... is 30. 60

c) 5, 8,11, 14,.... is 320 ? 106th

d) 64, 60, 56,...is 0 ? 17th

e) 10,13,16, ...., 43 ? 12

f) 5/6, 1, 7/6 ,.... 10/3 ? 16

g) 4,9,14,19,.... is 109? 22

h) 7,13,19,...., 205. 34

i) 84,80,76,....... Is 0 . 22

j) 2,5,8..... is 92. 31

k) -1,-5/6,-2/3,-1/2,...10/3. ? 27

l) 12+i,11+6i,10+4i,. ...is (i) purely real (ii) purely imaginary.

EXERCISE - D

1) a) Is 305 a term of the series 7, 10, 13, 16? If not find nearest term. No,101th

b) Is 90 a term of the series 4,7, 10, 13 .... ? B. No

c) Is 68 a term of A.P 7,10,13,.. no

d) Is 302 a term of 3,8,13.....? No

e) Is 301 a term of the AP 5, 11, 17, 23.....? No

EXERCISE - E

1) Find the value of k if series in AP

a) 2k-1,k+1,3-k. 0

b) 5k-1, 3k+7,8k+1. 2

c)k²+2k+2,3k²+6k+6,4k²+5k+4. -3,-2

EXERCISE - F

1) Find

a) 15th term from the end of the AP 3,5,7,9,... 201. 173

b) 19th term from the end of the AP 2,6,10,14,...86.

c) 15th term from the end of 7,10,13, ....130. 88

d) 20th term from the end of 3,8,13, ..... 253. 158

e) 12 th term from the end of 3,5,7,9,......201. 179

f) 12 th term from the end of 1,4,7,10,....,88. 55

EXERCISE - G

1) Find 10th term of the AP where 1st term is 5 & common difference is 2. 23

2) 1st term of an AP is 6, common difference is 2. find 15th term.

3) The 20th term of an A. P is 79. If 1st term is 3, find 10th term. 39

4) The 5th and 13th terms of an AP are 5 and - 3 respectively. find this AP and obtain its 16th term. 9, 8, 7, 6 and 16th term=-6

5) Find 15th term of an AP whose 10th term is - 25 & 20th term is - 55. -40

6) Find 2nd term and rth term of the A. P whose 6th term is 12 and 8th term is 22. -5, 5r - 18

7) The 6th &16th term of AP are 9√2/2 and 19√2/2. Find its 20th term. 23/√2

8) The 2nd, 31st and the last terms of an AP are 31/4,1/2 and -13/2 respectively. find 1st term and the number of terms. 8, 59

9) If the 9th term of an AP is 0, prove that its 29th term is double the 19th term.

10) how many two digit numbers are divisible by 7 ? 13

11) The 4th terms of an AP is three times the first and the 7th term exceeds twice the third term by 1. find the first term and the common difference. 3,2

12) if 7 times the 7th term of an AP is equals to 11 times its 11th term, show that its 18th term is 0.

13) If 9th term of an A. P. Is zero, prove that it's 29th term is double the 19th term.

14) If 10th times the 10th term of an A. P is equal to 15 times 15th term, show that 25th term of an A. P is zero.

15) How many Numbers of two digitare Divisible by 3.

16) The first term of an AP is 5, 11th term is 125. Find common difference.

17) The 8th term of an AP is double the 13th. Is 2nd term is double the 10th term.

18) The 4th and 8th term is 2 and 10. Find 11th term and first term.

19) A certain series in AP. Every

term of it is multiplied by 3. Is resultant series in AP ?

20) a) In 3rd term: 5th term=1: 4 Find 7th term: 12th term.

b) If 7 times the 7th term is equal to 11 times of the 11th term. Find the 18th term ?

21) The 4th term is equal to 3 times the first term and the 7th term exceeds twice the third term by 1. Find 1st term and common difference.

22) if the mth term of an AP be 1/n and its nth term be 1/m then show show that its mn th term is (1+m-n)/mn

23) If the mth term of a given AP is n and its nth term is m then show that its pth term is n+ m -p

24) If m times the mth term of an AP is equal to n times its nth term, show that m + n th term of the AP is zero.

25) If the pth, qth and rth term of an AP be a, b, c respectively. show that (q-r)a + (r-p)b +(p-q)c = 0

26) Show that the nth term from the end of an AP with the first term a, common difference d and the last term l is given by (a+l+ad-and)/a

27) If x th term of AP is y and y th term is x. Find (x+y) th term and m th term.

28) if (m+1) th term of an A. P. is twice the (n+1)th term, prove that (3m+1)th term is twice the (m+n+1)th term.

29) The 4th term of an A. P is is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.

EXERCISE - H

1) Which term of the AP

a) 19, 91/5, 92/5, ........is the negative term ? 28

b) Which term of the sequence 20, 77/4, 37/2,71/4 is the 1st negative term ? 28

c) Which term of the sequence 24,93/4,45/2,87/4 is the first negative term ? 31

EXERCISE - I

1) Find A. M between

a) 9 and 19. 14

b) 12 and -8. 2

c) -13 and -7. -10

d) 14 and 18. 16

e) (a-b) and (a+b). a

2) Fill up the gap

a) 4 , _ , _ , 10

b) _ , _ , _ , -2 , -10.

c) 34, _ , _ , _ , 48.

3) Insert 3 AM between

a) 3 and 19. 7,11,15

b) 2 and 14

c) - 24 and 0

d) 13/4 and 5/4

4) Insert 5 arithmetic means between

a) 8 and 26. 11,14,17,20,13

b)17 and 29.

c) 2 and 26

5) Insert 4 A. M between

a) 4 and 29. 9,14,19,24

b) (a - b)² and a² + b² + 8ab.

c) 4 and 19. 7,10,13,16

6) Insert 7 A. M between√2 and 17√2.

7) Find the middle term of

a) 1,6,11,......101. 11th, 51

b) 4,7,10, ....73. 12th,13th, 24

c) 5,8,11, ...., 95.

EXERCISE - J

1)a) Find three numbers in AP whose sum is 24 and product of extreme is 55 5,8,11 or 11,8,5

b) three numbers are in AP. If their sum is 27 and the product 648, find the numbers. 6,9,12

c) the sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. find these terms. 4,7,10

d) The three Numbers are in 3:7:9 if 5 is substracted from the second the resulting Number are in AP. Find the original number.

e) The sum of three terms of an A. P. is 21 and the product of the first and the third term exceeds the second term by 6, find three nos.

f) The sum of three Numbers in AP is 12 and sum of their cubes is 408. Find the numbers.

g) Divide 12 into 3 parts in AP as

their squares is 56.

2a) find the four numbers in AP whose sum is 20 and the sum of whose squares is 120. 2,4,6,8 or 8,6,4,2

b) divide 32 into 4 parts which are in AP such that the product of the extremes is to product of means as 7:15. 2,6,10,14

c) the angles of a quadrilateral are in AP whose common difference is 10°. find the angles. 75,85,95,105

d) Find the four numbers in A. P., Whose sum is 50 and in which the greatest number is 4 times the least.

e) The sum of four numbers in AP is 24. Sum of their squares is 164. Find the numbers.

f) Divide 20 into four parts in AP

and such that the product of first and fourth is to the product of second and third in the ratio 2:3.

g) Divide 32 into four parts which

are in AP. Such that the product of extremes is to the product of mean is 7:15.

EXERCISE - K

1) Find the sum of the series

a) 1,3, 5, 7... 24th term. 576

b) 2+5+8+..... 30th term

c) 6, 16/3, 14/3... 10th term. 30

d) 0.8+1.2+1.6+... to 25th terms.

e) 2+ 7 +12+ .... upto 20 terms.

f) √2,2√2,3√2...20th term. 210√2

g) 0.7,0.71,0.72...100th term. 119.5

h) (n-1)/2 + n/n + (n+1)/n + .... upto n terms.

i) 101+99+97+....+47. 2072

2) How many term

a) 26+21+16+11.....to get 11. 11

b) 18+16+14+12....to get 78. Explain the double answer. 6,13

EXERCISE - L

1) Find the series

a) 3p²+5p.

b) an²+bn.

c) 2n+ 3n²

d) 2p² + p

e) The sum of n terms of an AP is 3n²+5n. Find p th term. Find which term of the A. P is 152.

2) Find the series if

a) 4r - 3.

b) 10 - 3n

EXERCISE - M

1) The first and the last terms of an A. P having finite number of terms are respectively - 2 and 124 and the sum of the A. P is 6161. Find the number of terms.

2) The first and the last term of an A. P are respectively - 4 and 146, and the sum of A. P= 7171. Find the number of terms, common difference.

3) If the sum of first m terms of an AP is equal to the sum of the first n terms, prove that the sum of the first (m+n)th term is zero.

4) In an AP if the 12th term is -13

and sum of first four terms is 24. What is the sum of its first 10 terms.

5) If the first term of an AP is 2 and sum of the first five terms is equal to one-fourth of the sum of the next five terms. Find the sum of first 30 terms. Also find its 20th term.

6) If the sum of p terms of an AP is q and the sum of q terms is p. Show that the sum of p+q terms is -(p+q). Also find the sum of first p-q terms.

7) How many terms of an AP -6, -11/2, -5,... make the sum -25 ?

8) How many terms of the sequence 18,16,14, ... Should be taken that sum is zero.

EXERCISE - N

1)a) Find the sum of all Numbers between 100 to 300 which are Divisible by 8.

b) Find the sum of all integer from 200 to 400 not Divisible by 7.

c) Find the sum of all odd numbers of four digits which are Divisible by 9.

d) Find the sum of all integers between 50 and 500 which are Divisible by 7 ?

e) Find the sum all natural

numbers between 100 and 1000 (excluding both) which are Divisible by 5 ?

f) The first, second and the last terms of an AP are a, b, 2a. Find its sum.

MISCELLANEOUS (1)

1) The sum of the first p,q,r terms of an A.P are a,b,c respectively. Show that (aq-ar)/p + (br-bp)/q + (cp-cq)/r =0

2) The sums of n terms of three arithmetical progression S' , S" S"'. The first term of each is unity and the common difference are 1,2,3 respectively. Prove S'+S"'= 2 S".

3) The pth term of an A. P. is a and qth term is b. Prove the sum of its p+q is (p+q)/2 {a + b + (a-b)/(p-q)}

4) The sum of n terms of two arithmetic progression are in the ratio (3n+8):(7n+5). Find the ratio of their 12th terms.

5) The interior angles of a polygon in A. P. The smallest angle is 120 and the common difference is 5°. Find the number of sides of the polygon.

48) If (aⁿ + bⁿ)/(aⁿ⁻¹+ bⁿ⁻¹) is the AM between a and b then find the value of n. 1

6) There are n AM between 5 and 23. Such that the ratio of 1st mean : last mean is 1:3. Find the value of n.

7) If the AM between pth and qth terms of an AP be equal to the AM between rth and sth terms of the AP, then show that (p+q)= (r+s).

8) 7th term : 3rd term = 12 : 5 Find 13th term : 4th term.

9) If nth term of the A. P. 9,7,5, is same as the nth term of the A. P 15,12,9,.... Find n.

EXERCISE - O

1) If a,b,c are in A. P prove the following are in A. P.

a) b+c, c+a, a+b.

b) a²(b+c),b²(c+a), c²(a+b).

c)a(1/a+1/c),b(1/c+1/a),c(1/a+1/b)

d) {(b+c)²-a²},{(c+a)²-b²},{(a-b)²-c²}.

e) (a- c)² = 4(b² - ac).

f) a³ + 4b³+ c³= 3b(a²+c²).

g) a²(b+c), b²(c+a), c²(a+b) .

h) b+c -a, c+a - a, a+b - c.

i) b c - a², ca - b² , ab - c².

j) (a - c)² = 4(a - b)(b - c).

k) a² + c² +4ac = 2(a b+ b c+ c a).

l) a³+c³+6abc = 8b³.

2) If a², b²,c² are in A. P then prove that the following are in also in A. P.

3) 1/(b+c), 1/(c+a), 1/(a+b).

b) a/(b+c), b/(c+a), c/(a+b)

c) a/(b+c),b/(c+a), c/(a+b).

4) If a²(b+c), b²(c+a), c²(a+b) are in A. P prove that either a,b,c are in A. P or ab+b c+ca=0.

5) 1/a, 1/b, 1/c are in A. P prove following are in A. P.

a) (b+c)/a, (c+a)/b, (a+b)/c

b) a(b+c), b(c+a), c(a+b).

6) (b+c)/a, (c+a)/b, (a+b)/c are in A. P prove following are in A. P.

a) 1/a, 1/b, 1/c.

b) b c, c a, a b

EXERCISE -P

1) Find the sum of

a) 3.7+5.10+7.13+...... n terms.

b) 7²+8²+9²+ ...20².

c) 10²+11²+12²+......30².

d) 6³+7³+8³+.....20³.

e) 1.2²+3.5²+5.8²+.....n terms.

MIXED PROBLEMS

Sunday, 19 July 2026

QUESTION BANK- MATHS

MAXIMUM AND MINIMUM

PART-

1) Find two numbers whose sum is 24 and whose product is as large as possible. 12,12

2) Find two positive numbers x and y such that x+ y= 60 and xy³ is maximum. 15,45

3) Find two positive numbers x and y such that the sum is 35 and the product x²y⁵ is maximum. 10,25

4) Amongst all pairs of positive numbers with product 256, find whose sum is the least. 16,16

5) Find two positive numbers whose sum is 14 and the sum of whose squares is minimum . 7,7

6) Find the maximum value of ax+ by, where xy= c² and a,b,c are positive.

7) Show that all the rectangles with a given perimeter, the square has the largest area.

8) Show that of all rectangles of given area, the square has the smallest perimeter.

9) Show that of all the rectangles inscribed in a given circle, the square has the maximum area.

10) Show that the rectangle of maximum perimeter which can be inscribed of radius a is a square of side √2 a.

11) AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ ABC is maximum, when it is isosceles.

12) If the sum of the length of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between is π/3.

13) Show that the area of right angled triangle of given hypotenuse is maximum when the triangle is isosceles.

14) Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is cube.

15) An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.

16) A metal box with a square base and vertical sides is to contain 1024 cm³ of water, the material for the top and bottom costs Rs 5 per cm² and the material for the sides costs Rs 2.50 per cm². Find the least cost of the box. 8x8x16, Rs1920

17) An open box with a square base is to be made out of a given quantity of cardboard of area c¹ square units. Show that the maximum volume of the boxe is c³/6√3. cubic units.

18) The sum of the surface area of a rectangular parallelopiped with sides x, 2x and x/3 and a sphere is given to be constant. Prove that the sum of the volume is minimum, if x is equal to 3 times the radius of the sphere. also, find the minimum value of the sum of their volumes. (2/3) x³(1+ 2π/27)

19) a wire of length 36 m is to be cut into two pieces . One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces, so that the combined area of the square and the circle is minimum.

20) A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum. 2P/(π+4), P/(π+4)

21) A square piece of tin of side 24cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find this maximum volume. 1024 cm³

22) A rectangular sheet of fix perimeter with sides having their length in the ratio 8:15 is converted into an open rectangular box by folding after removing squares of equal area from all 4 corners. if the total area of the removed square is 100 square unit, the resultant box has maximum volume. Find the length of the sides of the rectangular sheet . 24

23) Show that a cylinder of a given volume which is open at the top, has minimum total surface area, provided its height is equal to the radius of its base .

24) Show that the height of the closed cylinder of given surface and maximum volume, is equal to the diameter of its base.

25) Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

26) A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300 per subscriber. The company purposes to increase the annual subscription and it is believed that every increase of Rs 1 one subscriber will discontinue the service. Find what increase will bring maximum revenue. 100

27) Find the point on the curve y²= 4x which is nearest to the point (2,1). (1,2)

28) A jet age of an enemy is flying along the curve y= x²+ 2. A soldier is placed at the point (3,2). What is the shortest distance between the soldier and the jet ? √5

Saturday, 11 July 2026

NEW BOOK- XI

Complex number

1) Simplify: 1+ i² + i⁴+ i⁶.

2) Show that: [i³⁷ - (1/i)⁴¹]³= 8i.

3) Find the conjugate of (2+ 3i)².

4) Find x and y if (3x -7)+ 5iy = 2y +3 -4(1- x)i.

5) Find the modulus of -12+ 5i.

6) Express the reciprocal of the complex number 3+ i √5 in the form of a+ ib.

7) Find the modulus of (1+ i)/(1- i) + (1- i)/(1+ i).

8) Find the solution of the equation |1- i|ˣ= 2ˣ.

9) Show that the points representing the complex numbers (3+ 3i), (-3-3i) and (-3√3+ 3√3 i) are the vertices of an equilateral triangle.

10) Express (3+ i)/(-5- 4i) in the standard form a+ ib.

11) Find the modulus of (2+ 3i)/(3+ 2i).

12) Prove that the representative points of the complex numbers 1+ 4i, 2+ 7i, 3+ 10i are collinear.

13) Find the modulus of

{(3+2i)(1+ i)(2+ 3i)}/{(3+ 4i)(4+ 5i)}.

14) If x+ iy = √{(a+ ib)/(c + id)}, show that (x²+ y²)²= (a²+ b²)/(c²+ d²).

1) 0 3) -5-12i 4) 6,4 5) 13 6) 3/14 - √5i/14 7) 0 8) 0 10) -19/41 +7i/41 11) 1 13) 13√2 /5√41

Thursday, 25 June 2026

NEW BOOK - X

Revision Algebra

1) A company declares a dividend of 14% on shares of Rs 50. If the rate of interest on investment is 10%, find the market value of the shares.

2) Mohan bought Rs 100 shares at Rs 50 above par. He invested Rs 18000.

a) What is his annual income, if dividend is 6% ?

b) He sells 50% of his shares when the price rises to Rs 200. What is his gain on this transaction ?

3) Ramji invested Rs 16000 and Rs 20000 in buying shares of two companies. these companies declared dividend of 12% and 8% re. He collects the dividends and sells out all his shares at a losss of 2% and 3% respectively on his investment. Find his total gain from these two transactions ?

4) A man bought 4000 shares each of face value Rs 10 at Rs 14 per share . At the end of the yr, the company declares dividend of 16 percent. Calculate

a) The amount of money invested by him.

b) the percentage yield on his outlat. Give answer in two significant figures..

Algebra - Test

1) Mr Desai opens a recurring deposit of Rs 2000 per month for 31 months paying simple interest of 12% p.a. Calculate the amount receives at the time of maturity. 69300

2) Which is better investment ---12% shares at 90 or 15% at 120? 12%

3) A man invests a sum of money in Rs 20 shares, paying 15% dividend, quoted at 20% premium . If his annual dividend is Rs 720, find

a) his total investment

b) the rate of interest on his investment. Rs 5760, 12.5%

4) A man bought 3200 shares of Rs 10 paying 15% per annum. He sold them when the price rose to Rs 28 and invested the proceeds in Rs fifty shares paying 11% per annum at 12% premium . Find the annual change in his income. Rs 4000

FACTOR THEOREM

1) Show that x³- 7x²+ 14x -8 is the factor of x -1, then find other factors also.

2) Using factor theorem, find out whether polynomial g(x) is a factor of f(x) or not:

a) Given f(x)= x²+ 3x -10, g(x)= x -2.

b) Given f(x)= 6x² - 5x - 6, g(x)= 2x -3.

c) Given f(x)= 2x² - 5x -12, g(x)= x - 4.

d) Given f(x)= 3x² - 2x -8, g(x)= 3x +4.

e) Given f(x)= 9x²- 4a²+ 4ay - y², g(x)= 3x +2a - y.

f) Given f(x)= x²- 3x²+ 4x -2, g(x)= x -1.

g) Given f(x)= x²(x -14)+ 37x -60, g(x)= x - 2.

h) Given f(x)= x³+ x²- 2x -30, g(x)= x -3.

3) Find the value of k when f(x)= x³ - 3x² - x + k, if g(x)= x +1 is a factor of f(x).

4) Find the values of p and q if g(x)= x +2 is the factor of f(x)= x²- px + x + q and , f(2)= 4.

5) For what value of a is x-2 a factor of f(x)= x² - ax + 6.

6) Find the values of p and q if x+3 and x-4 are factors of x³ - px²- qx + 24.

7) Factorise x³- 7x² + 14x - 8completely.

8) Find the values of p and q if g(x)= x -4 is a factor of f(x)= 4x³- px² + qx -216 and f(-1)= -225.

9) Find the values of a and b if p(x)= x +2 is a factor of q(x)= ax³- bx² + 2(x -2) and q(2)= 20.

10) Find the value of a if x+ a is a factor of f(x)= x³ + a(x² +1) - 2x + 4.

11) Find the remainder, if f(x)= x³- 2x² + x - 3 is divided by g(x)= x +2.

12) Find the remainder, if f(x)= 2x³- 3x² -4x - 5 is divided by g(x)= 2x + 1.

13) If g(x)= 2x -3 is a factor of f(x)= 2x³- 9x² + x + p, find the value of p. Hence find all the factors.

14) If g(x)= 3x - 2 is a factor of f(x)= 3x³ + x² + px + 12, find the value of p. Hence find all the factors of f(x).

15) Find the value of p if the polynomial f(x)= x³- 3x² - px + 24 is divisible by g(x)= x + 3. Hence find all the factors.

16) Find the value of q if the polynomial f(x)= 2x³ + qx² - 7x -12 is divisible by g(x)= x + 4. Hence find all the factors of f(x).

17) Find the value of q if the polynomial f(x)= x³ + 2x² - 13x + q is divisible by g(x)= x -2. Hence find all the factors f(x).

18) Find the value of p if the polynomial f(x)= x³ - px² - x +3 is divisible by g(x)= x² -1. Hence find all the factors.

19) Find the value of p and q if the polynomial f(x)= px³ - 8x -12 + qx² is divisible by g(x)= x² - 4. Hence find all the factors.

Find the value of p and q if the polynomial f(x)= px³ + qx² - 8x -12 is divisible by g(x)= x² - 4. Hence find all the factors.

20) Find the value of p and q if the polynomial f(x)= px³ + 6x² + qx + 6 is divisible by g(x)= x²+ 4x + 3. Hence find all the factors of f(x).

21) Find the value of p for which the polynomial f(x)= x³ + px² - 16x +8 has g(x)= x -2 as one of the factors. Hence find the remaining two factors.

22) Find the value of q for which the polynomial f(x)= 2x³ + qx² - x -15 has g(x)= 2x + 3 as one of the factors. Hence find all the remaining factors.

23) Factorise 6x³ - 11x² - 3x +2. Completely.

MEAN

1) Find the arithmetic mean of the numbers 5, 3, 8, 7, 6, 1. What is the new mean if 3 is added to each number?

2) The table below gives the expenditure (in rupees) on water consumption of 70 houses in a locality. Find the average expenditure (in rupees) per house.

Expenditure no of house

15-20 7

20-25 5

25-30 7

30-35 8

35-40 9

40-45 11

45-50 7

50-55 5

55-60 4

60-65 4

65-70 3

3) Find the mean by the short cut method of the following table:

X: 5 6 7 8 9 10

F: 10 12 15 11 7 5

4) Find the mean of the following frequency table by step deviation method:

Class frequency

10-15 5

15- 20 6

20-25 8

25-30 12

30-35 6

35-40 3

5) Weights of 50 eggs were recorded as given below :

Weight(in gms) no of eggs

80-84 5

85-89 10

90-94 12

95-99 12

100-104 8

105-109 2

110-114 1

Calculate the mean weight to the nearest geams.

6) Find the mean of the following data:

a) 12, 15 , 10, 9, 11, 14, 6.

b) 200, 230, 18ó, 210, 240.

7) a) if 8 is the mean of 4, 7, a, 8, 10, 12, find a.

b) if 7 is the mean of 5, 3.5, 4.5, b, 8.5, 9.5; find b.

8) Following at the height of (in inches) of 10 girls:

52, 55 , 48, 51, 47, 46, 47, 46, 50, 51

Find the mean height. If 3 each are added to each of the above height, find the new mean height.

9) Following are the runs scored by three batsman, in 6 completed innings:

A: 15, 25, 0, 32, 35, 65

B: 24, 36, 12, 14, 18, 30

C: 18, 45, 37, 3, 40, 32

Find who is the best batsman, using the above data.

10) From the following table find the mean height:

Height: 60 61 62 63 64

no of boys: 2 3 5 8 7

11) From the following table, find the mean mark:

Mark number of students

0 1

1 1

2 3

3 6

4 8

5 8

6 6

7 4

8 7

9 4

10 2

12) Find the mean weight of 70 students from the following data :

Weight (kg) no of students

50 8

52 10

54 15

56 20

58 8

60 6

62 2

64 1

13) The daily salaries of 60 workers are given below. Find the mean salary.

Salary: 10 15 20 25 30 35

Workers: 20 12 8 10 9 1.

14) If the mean of the following distribution is 7.5, find the missing frequency f of the following data :

variable frequency

5 20

6 17

7 16

8 10

9 f

10 6

11 7

12 6

15) if the mean of the following distribution is 58.2, find the missing frequency f

salary no of person

45 6

50 8

55 10

60 f

65 6

70 5

75 2

80 1

16) Find the mean for the following frequency distribution, using direct method

Marks students

0- 10 5

10-20 8

20-30 12

30-40 10

40-50 5

17) Find the mean, using direct method

Weight students

30-34 6

34-38 8

38-42 12

42-46 9

46-50 5

18) Find the mean by shortcut method :

weight student

11-15 8

16-20 12

21-25 13

26-30 16

31-35 12

36-40 9

41-45 8

46-50 2

19) Find the mean by step deviation method

weight students

10-20 6

20-30 8

30-40 12

40-50 15

59-60 10

60-70 9

20) Find the mean by step division method :

variate frequency

10-20 12

20-30 15

30-40 16

40-50 19

50-60 12

60-70 6

21) compute the mean of the following frequency table by i) direct method and ii) shortcut method

class frequency

05-10 10

19-15 6

15-20 4

20-25 12

25-30 8

30-35 4

35-40 2

40-45 1

45-50 3

22) Find the mean of the following frequency distribution:

Class frequency

00-10 6

10-20 8

20-30 12

30-40 8

40-50 6

23) Find the mean for the following distribution:

class frequency

1-10 3

11-20 5

21-30 8

31-40 6

41-50 3

24) Find the mean for the following distribution:

Class frequency

00-10 6

10-20 12

20-30 18

30-40 20

40-50 25

50-60 32

70-80 18

80-90 6

90-100 1

25) Find the mean for the following distribution:

Class frequency

01-10 8

11-20 15

21-30 16

31-40 17

41-50 20

51-60 18

61-70 11

71-80 8

81-90 2

91-100 1

26) The frequency distribution of marks obtained by 40 students as under. Calculate the arithmetic mean:

Marks student

00-08 5

08-16 3

16-24 10

24-32 16

32-40 4

40-48 2

27) find the missing frequency for the following distribution, if the mean is 12.9.

Class frequency

0-5 3

5-10 f

10-15 8

15-20 5

20-25 4

28) a box contains nails of different lengths, measured to nearest half centimetre. The frequency distribution is as follows:

Length: 2.5 3.0 3.5 4.0 4.5 5.0

F: 10 35 50 45 35 25

i) state the upper boundary of the last class.

ii) state the class size.

iii) calculate the mean.

29) The following table shows the marks of 120 students at ICSE examination in mathematics. Calculate the mean marks:

Marks no of students

30-39 1

40-49 3

50-59 11

60-69 21

70-69 43

80-89 32

90-99 9

If the marks of each student are increased by 5%. Find the new mean marks.

30) Marks obtained by 40 students in Mathematics were as follows:

23 36 54 60 65 45 40 41 21 29 35 38 68 58 47 42 75 89 95 30 34 74 78 22 84 62 46 49 37 39 66 78 25 50 91 44 37 37 48

Construct the frequency table for the above marks, against the following class intervals with 21-30......

Find mean.

Test paper -23

1) a)

b) If (-6, n+2) on reflection in origin gives (m+3, -4). Find the values of m and n. (2)

d) Given that x +2 and x-3 are factors of x²+ ax + b. Calculate the values of a and b. (2)

e) If A= 2 4

2. 1

then show that A⅖- 3A - 6I= 0 (2)

2) a) Find the equation of a line passing through the points (7, -3) and (2,-2). If this line meets x-axis at point P and y-axis at Q, find the coordinate of P and Q. (3)

3) a)

4) a) In the adjoining figure, a circle touches the side AC of a ∆ ABC at point P. BC produced at Q and BA produced at R. Show that the length of tangent BQ is equal to half the perimeter of ∆ ABC. (3)

b) The result of an examination are tabulated below:

Marks(less than) no of candidates

10 0

20 25

30 42

40 65

50 95

60 120

70 128

80 135

90 148

100 150

find median. (3

c) The time taken by a person to cover 150 km was 2.5hrs more than the time taken in return journey. if he returned at a speed of 10 kmph more than the speed of going. what was the speed per hour in each direction ? (3)

5) a)

b) A man invests Rs 8000 in buying shares of a company of face value of Rs 100 each at a premium of 10%. If he earns Rs1200 at the end of the year as dividend, find

i) the number of shares he has in the company.

ii) the dividend percent per share. (3)

6) a)

b) A shopkeeper announces a discount of 15% on his goods. if the marked price of an article is Rs 1850, how much must a customer pay for it, if GST is 9%. (3)

c) solve the graoh the solution set of -2< 2x - 6 or -2x +5≥ 13 where x belongs to R. (3)

7) a) Solve : (2x -3)/(x -1) - 4{(x -1)/(2x -3)}= 3. (3)

c) If q is the mean proportional between p and r, prove that p²- q²+ r²= q⁴(1/p²- 1/q²+ 1/r²). (3)

8) a)

b) If A=a 0 & B= 0 - b & C= 1 -1

0 2 1 0. 1 1 and BA= C², find the values of a and b. (3)

c) Show that P(3, m -5) is a point of tricsection of the line segment joining the points A(4,-2) and B(1,4). Hence find the value of m. (3)

9) a) In the given figure, two straight lines AB and CD intersect each other at point P(3,4). AB makes an angle of 45° and CD makes an angle 60° with the positive direction of x-axis. Find the equation of AB and CD. (3)

c) A line PQ is drawn parallel to base of ∆ ABC which meets sides AB and AC at points P and Q respectively. If AP= (1/3) PB , find the value of

i) ar(∆ABC)/Ar(∆APQ)

ii) ar(∆APQ)/ar(teapeziumPBCQ). (3)

10) a) In the given figure, AOB is a diameter and DC|| AB. If angle CAB= x°, find the value of the following in terms of x:

i) angle COB

ii) angle DIC

iii) angle DAC

iv) angle ADC. (3)

11) a) From a solid cylinder whose height is 16 cm and radius is 12 cm, a conice cavity of height 8cm and of a base radius 6 cm is hollowed out. Find the volume and the total surface area of the remaining solid. (3)

b) prove that : tan²A - tan²B= (sin²A - sin²B)/cos²A cos²B. (3)

c) The length of the shadow of a vertical Tower is √3 times its height. Find the angle of elevation of the Sun. (3)

TEST PAPER - 24

1)a) The lines given by ax+ 4y= 8 and 2x+ 3y +2=0 are mutually perpendicular. Find the value of a. (2)

b) AB and CD are two chords of a circle intersecting at a point P outside the circle. Find if PC= 30cm, CD= 14cm and PA= 24cm. Determine AB. (2)

d) Find the total surface area of an open pipe of length 50 cm, external diameter 20cm and internal diameter 6cm. (2)

2)a)

b) Find the equation of the lines which passes through the point (-2,3) and are equally inclined to the co-ordinate axes. (3)

c) If P= 2 6 & Q= 3 x

3 9 y 2

Find x and y such that PQ= null matrix. (3)

3) a) Using remainder theorem, factorise the expression 3x³+ 10x²+ x - 6. (3)

c) If (a²+ b²)+x²+ y²)= (ax + by)², show that a/x = b/y. (3)

4) a) If a= 4√6/(√2+ √3), find the value of

(a+ 2√2)/(a- 2√2) + (a+ 2√3)/(a- 2√3). (3)

c) Mr. Ahuja invests equal sums of money in two companies offering 8% shares of Rs 100 at a premium of Rs 25 at 6% shares of Rs 100 at a discount of Rs 25. He finds that if with his total investment he has bought equal number of shares of each company, his income would have been Rs142 less. Find his total investment . (3)

5) a)

6) a)

b) A recurring deposit account of Rs 1200 per month has a maturity value of Rs 12440.00. if the rate of interest is 8% and the interest is calculated at the end of every month, find the time of the recurring deposit account. (3)

7) a) i(plot the point A(3,5) and B(-2,4).

ii) A' is the image of A when reflected in the x-axis . Write down the coordinates of A'b and plot it.

iii) B' is the image of B when reflected in the y-axis, followed by reflection in the origin, Write down the coordinates of B'.

iv) write down the geometrical name of the figure AA'BB'.

v) Name two invariant points under the reflection in the x-axis . (3)

b) The point P(5,-4) divides the line segment AB as shown

in the figure in the ratio 2:5. Find the coordinates of points A and B. 3

8) a) The vertices of a ∆ ABC are (0,5), B(-1,-2) and C(11,7). Write down the equation of BC . Find

a) i) the equation of line through A and perpendicular to BC.

ii) the coordinates of the point P, where the perpendicular through A as obtained in (i) meets BC. (3)

9) a) Find the mean of the given data:

C. I frequency

63-70 9

70-77 13

77-84 27

84-91 38

91-98 32

98-105 16

105-112 15 (3)

10a)

b) A, B are the points (9,6) and (10,0). O is the origin, OM is a median and OP is an altitude of ∆ OAB. Find the equations of OM and OP. (3)

11) a) prove: (SinA + cosecA)² + (cosA + secA)²= 7+ tan²A + cot²A. (3)

c) If A= -1 1

a b and A²= I, find a, b.

Friday, 29 May 2026

H. W - VIII

ALGEBRAIC EXPRESSIONS

1) Find each product

a) -5x² and 7x⁵.

b) -7ab and (1/4) a²bc

c) (-24/25) x²z and (-15/16) x²zy²

d) (-8/5) x³yz² and (-3/4) xy³z²

2) Find the product:

a) -2a², 7a²b⁷, 6a⁵b⁵.

b) (-6/5) xy, (15/16) yz, (8/9) xyz

c) 5a⁴, (a²)³, (2a)²

d) (2/3) a³bc², (1/3) a²b³c, 9abc³

e) -8y²z , (4/9) xyz³, (-27/5) x³y³.

3) Find the area of a rectangle whose length and breadth (in units) are 3x²y and 4xy².

4) Find the volume of a rectangular box with given length, breadth and height (in units) respectively

a) 3x²y/4, 14y²/5, 2x⁴/7