CLASS - VIII (FINAL REVISION)

23/9/25

RATIO 

1) Write the following ratios in lowest terms:

a) 4/3: 6

b) 7/1: 11/2

c) 9 months and 3 years 

d) 15kg to 210g

2) Divide 720 into a given ratio 3:5.

3) A profit of Rs 2500 is to be divided between three persons in the ratio 9:6:10. How much does each person get ?

4) Are the ratios 1:3 and 3:2 equivalent?

LAST TIME REVISION - XI

TEST- 7/10/25

1) Simplify: 1+ i²+ i⁴+ i⁶.    

2) Write in the form of a + ib where √-1= i

a) √-144 + √441.         

b) √-27 x √12 - √-125 x √-5.     

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 the complex number -12 + 5i.     

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

 25/9/25

TRIGONOMETRIC FUNCTIONS 

1) If cotA= 3/4, Find the value of 3 cosA + 5 sinA, where A lies in the first quadrant.   

2) if cos120°= -1/2, find the value of sin120° and tan 120°.    

3) prove that sec(-1680°). sin 330== -1.

4) If A, B, C, D are the angles of the cyclic quadrilateral, show that cosA + cosB + cosC + cosD= 0.    

5) If tan25°= a, prove that (tan155° - tan115°)/(1+ tan155° tan115°)= (1- a²)/2a.

6) If A, B, C be the angles of a triangle, show that 

{Sin(B+ C)+ sin(C+ A)+ sin(A + B)}/{sin(π+ A)+ sin(3π+B)+ sin(5π+ C)}= -1

7) prove that  cosx/(1- sinx) + (1- sinx)/cosx = 2 secx.

8) If secx =√2 and 3π/2< x < 2π, find the value of 

(1+ tanx + cosecx)/(1+ cotx - cosecx).   

23/9/25

COMPLEX NUMBERS 

1) Simplify:

a) i³⁸.          

b) i¹⁵.       

c) i⁻⁶.        

d) 1/i.        

e) (5i) × 7.       

f) (3i)(4i).     

g) 21/14i.        

h) 5/i³.           

i) √-9 + √-16.      

j) (21/4) √-48 - 5 √-27.      

k) √-18 . √-2.           

l) 20/√-5.           

2) Write the complex numbers that represent the following points in the plane.

a) (3,4).     

b) (0,3).     

c) (-1/3,-1/5).     

Also, represent their conjugates.

3) Find the real numbers x and y if (x - iy)(3+ 5i) is the conjugate of -6 - 24i.   

4) Find the modulus of 

a) 5 - 12i³.         

b) (1+ i)/(1- i)  - (1- i)/(1+ i).        

c) (2+ 3i)/(3+ 2i).           

d) (2+ 5i)(3+ 4i).          

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

5) Find the solution of the equation 

|1- i)|ˣ= 2ˣ.        

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

7) Express in the standard form a+ ib

a) 1/(3- 8i).            

b) (3+ i)/(+5- 4i).     

c) (1+ i)/(1- i).        

d) {(1- i)/(1+ i)}²      

e) {(1- i)/(1+ i)}³.     

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

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

REVISION PAPER CLASS- VIII

CUBE ROOTS 

1) Find the cube root of 

a) 6859 b) 17576 c) 1728

2) Find the smallest number by which the number must be multiplied so that the the number is a perfect cube

a) 120393 b) 3087

3) Find the smallest number by which the number must be divided so that the the number is a perfect cube

a) 33275 b) 29160

4) By what smallest number should 55125 be multiplied so that the product becomes a perfect cube? Also find the cube root of the product.

5) Divide 259875 by the smallest number so that the quotient is a perfect cube. Also find the cube root of the quotient.

6) Find the cube root of (-512).

7) Evaluate: a) ³√(27 x 64) 

b) ³√(125(-64)).

c) ³√(216/729)

d) ³√(-125/343).

8) Find the cube root of 

a) 0.216

b) 9.261

9) The volume of a cube is 5832 cm³. Find the length of its side.

10) Three numbers are in the ratio 1:2:3. The sum of their cubes is 7776. Find the numbers.

CLASS- IX MATHS(FULL REVISION)

21/9/25

1) Evaluate:

a) (sin27°/cos63°)² - (cos63°/sin27°)².

b) cos40°/sin50° - (1/2)(cos35°/sin55°).

c) tan18°/cot72° - sec7°/cosec83°.

d) (cos²20 + cos²70)/(sin²59 + sin²31°).

e) cos80°/sin10° + cos59° cosec31°.

f) sin63° cos27° + cos63° sin27°

g) sin48° sec42 + cos48° cosec42°.

h) sec37°/cosec53° + sin42°/cos48°.

i) tan5° tan10° tan15° tan75° tan80° tan85.

j) (sin10° sin20° sin30°)/(cos80° cos70° cos60°).

k) cos80°/sin10° + cos59°/sin31°.

l) sin²35 + sin²55.

m) sec50° sin40° + cos40° cosec50°.

n) cosec²74° - tan²16.

o) sec²12° - cot²78.

p) cot54/tan36° + tan20°/cot70° - 2.

20/9/25

1) The Simple interest on a certain sum of money for 3 years at 5% p.a is Rs1200. Find the amount due and the compound interest on this sum of money at the same rate after 3 years, intrest is reckoned annually.      Rs9261 

2) A sum of Rs9600 is invested for 3 years at 10% p.a for compound interest:

a) What is the sum due at the end of the first year ?  Rs 10560

b) What is the sum due at the end of the second year ?    11616

c) Find the compound interest earned in 2 years.    Rs2016

d) Find the difference between the answer in (b) and  (a) and find the interest on this sum for 1 year.       Rs105.60

e) Hence , write down the compound interest for the third year .    Rs1161.60

3) The compound interest on a certain sum of money at 5% per annum for 2 years is Rs246. Calculate the simple interest on the sum for 3 years at 6% p.a.    Rs432

4) What sum of money will amount to Rs3630 in two years at 10%p.a. compound interest ?   Rs3600

5) On a certain sum of money, the difference between the compound interest for a year, payable half-yearly, and the simple interest for a yeris Rs180. Find the sum lent out, if the rate of interest in both the cases is 10% p.a.     Rs72000

6) A man borrows Rs5000 at 12% compound interest p.a., intrest payable every 6 months . He pays back Rs1800 at the end of every 6 months . Calculate the third payment he had 18 months in order to clear the entire loan .     Rs2024.60

7) Calculate the component for the second year on Rs8000 invested for 3 years at 10% p.a.     Rs880

8) A man invests Rs5000 for 3 years at a certain rate of interest compound annually. At the end of 1 year it amounts Rs5600. Calculate :

a) The rate of interest per annum.    12%p.a

b) The interest accrued in the second year.    Rs672

c) The amount at the end of the third year.     Rs7924.64

9) A man invests Rs46875 at 4% per annum compound interest for 3 years. Calculate :

a) The interest for the first year.    Rs1875

b) The amount standing to his credit at the end of the second year.    Rs50700

c) The interest for the 3rd year.   Rs2028

10) A person invests Rs5600 at 14% p.a compound interest for 2 years. Calculate :

a) The interest for the first year.    Rs784

b) The amount at the end of the first year.    Rs6384

c) The interest for the second year, correct to nearest Rs.    Rs894

11) The compound interest , calculated yearly, on a certain sum of money for the second year is Rs880 and for the third year it is Rs968. Calculate the rate of interest and the sum of the money.       10%, Rs8000

12) A certain sum money amounts to Rs5292 in two years and to Rs556.60 in three years, intrest being compounded annually. Find the rate percent.     5%

13) At what rate percent, per annum compound interest, would Rs80000 amount to Rs88200 in two years; interest being compounded yearly ?     5%

14) A sum of money is lent out at compound interest for 2 years at 20% p.a. compounded being reckoned yearly. If the same sum of money was lent out at compound interest at the same rate per annum, CI being reckoned half-yearly, it would have fetched Rs482 more by the way of intrest. Calculate the sum of money lent out.   Rs20000

REVISION - X- I

REFLECTION 

1) A Triangle ABC is such that the coordinates A, B and C are (2,0),(1,1) and (0,2) respectively. Write down the coordinates of the triangle obtained by reflecting ∆ ABC in the line y=0.  Also reflect (2,0) in the line x=0.

2) Draw the unit square, whose vertices are (2,2),( 4,2),(4,4) and (2,4). Reflect the square in the y-axis and then reflect the image in the origin. What single transformation would give the same final result ?

3) A man leaving point A must take water from a river and deliver it to a man at point B. Use reflection to find the shortest path.

PAPER - 6

1) Ashok invested Rs 12500 in shares of a company paying 8% per annum. If he bought Rs 20 shares for Rs 25, find his annual dividend.

2) The mid-point of AB is P(-2,4). The coordinates of the point A and B are (a,0) and (0,b) respectively. Find a and b.

3) If A= 4    3 & B= x & C= 6

             -5    0         -2         y with the relation AB= C then 5 find the value of x and y.

4) Calculate the median and mode of the following set of numbers:

 9, 0, 2, 8, 5, 3, 5, 4 ,1, 5, 2, 7.

5) Solve the following inequation and represent the solution set on the number line.

 30 - 4(2x - 1)>  - 8. x belongs to positive integers.

6) Solve : y - √(3y -6)= 2.

7) ay triangle whose area is 12cm²,  is transferred under enlightenment about a point in space. If the area of the image is 108cm²,  find the dilation factor of the enlargement.

8) Point P(a,b) is reflected in x-axis to (5,-2).

a) Write down the values of a and b.

b) P" is the image of P when reflected in the y-axis. Write down the coordinates of P''.

c) Name the single transformation that maps P to P".

9) If A= 1    2 & B= 2   1 & C= 1   3

             -2    3          3   2          3   1 

Find C(B - A).

10) In the figure, AB || CD  and O is the centre of the circle.

If angle ADC=24°, find angle AEB.

11) If a,b,c are in continued proportion, show that:

(a²+ b²)/b(a+ c)  = b(a+ c)/(b²+ c²).

12) Mr. Gupta invested Rs 8000 in 8%(Rs 100) shares, selling at Rs80. After a he sold these shares at Rs 75 each and invested the proceed in Rs 100 shares selling at Rs 90 with a dividend of 12%. Calculate 

a) his income from the first investment.

b) his income from the second investment.

c)  the increased percentage return on his original investment.

13) If -5 is a root of the quadratic equation x²+ kx - 130= 0, find k. Hence, find the other root.

14) An open cylindrical vessel of internal diameter 49cm and height 64 cm stands on a horizontal platform. Inside this is placed a solid metallic right circular cone whose base has a diameter of 21/2cm and whose height is 12cm. Calculate the volume of water required to fill the tamk. Take π to be 22/7.

15) The perimeter of a rectangular plot is 180m and its area is 1800m². If the length is x m,  Express the breadth in terms of x. Hence , form an equation in x.  Solve the equation and find the length and the breadth of the rectangle.

16) Prove : (1+ tan²x)/(1+ cot²x)= sin²x/cos²x.

17) The I.Q of 50 pupils was recorded as follows :

I. Q scores  no of pupils 

80-90.            6 

90- 100          9

100-110        16 

110-120        13 

120-130        4 

130-140         2

Draw a histogram for the above data and estimate the mode.

18) in the given figure, find 

a) the co-ordinates of the points A, B and C.

b) the slope of BC .

c) the equation of the line AP (|| BC).

d) the coordinate of the point X and Y where line AP meets the x-axis and y-axis respectively.

e) the ratio in which point A devidas the line segment XY.

19) Factorise , by factor theorem, the expression 2x³+ 13x²+ 17x -12.

20) in the given figure,

if angle BAD= 65°, angle ABD=70° and angle BDC=45°, calculate

a) angle BCD, ADB

b) Show that AC is a diameter.

21)  The angle of elevation of a cloud from a point 50m above a lake is 30° and the measure of the angle of its depression of its reflection in the lake is 60°.  Find the height of the cloud.

22) A solid cylinder of radius 14cm and height 21cm is melted down and recast into spheres of radius 3.5cm each. Calculate the number of spheres that can be made. (π= 22/7).

PAPER- 5

1) Solve: 21x²- 8x -4=0.

2) Find the coordinates of the image of (5,-4) after reflection in

a) x= 0

b) y= 2.

3) List the solution set of the following and inequation and graph the solution set:

(1/2) + 8x > 5x - 3/2, x belongs to Z.

4) Calculate the ratio in which the line joining A(6,5) and B(4,3) is divided by the line y= 2.

5) In the figure,

BC is parallel to DW.

Area of triangle ABC= 25 cm², area of trapezium BCED= 24 cm² and DE = 21cm.

Calculate the length of BC.

6) Calculate the mean, median and mode of the following numbers :

13,11,15,13,14,15,13,17,12,16.

7) Given A= 1    1

                     8    3 evaluate A² - 3A.

8) In the figure,

I is the incentre of the circle. AI produced to meet the circle in D. Calculate 

a) angle DCB b) angle IBC c) angle BID  d) angle BIC

Given angle BAC= 50° and angle ABC= 64°.

9) Show that: √{(1- cosA)/(1+ cosA)}= sinA/(1+ cosA)

10) In the figure,

AB it is a common tangent to two circles intersection at C and D. Write down the measure of (angle ACB + angle ADB).

11) The surface area of a solid metallic sphere is 1256 cm½. It is melted and recast into right circular cones of radius 2.5 cm and height 8cm. Calculate 

a) the radius of the solid sphere.

b) the number of cones recast (π= 3.14).

12) A dividend of 9% was declared on Rs 100 shares selling at a certain price. If the rate of return is 15/2%, calculate 

a) the market value of the share.

b) the amount to be invested to obtain an annual dividend of Rs 630.

13)  in the figure AB and CD are the lines 2x - y +6=0 and x - 2y = 4 respectively.

a) write down the coordinates of A, B, C and D.

b) prove that the triangles OAB and ODC are similar .

c) Is figure ABCD cyclic ?

14) The hotel bill for a number of people for overnight stay is Rs 4800. If there were 4 more, the bill each person had to pay would have reduced by Rs 200. Find the number of people staying overnight.

15) ABCD is a rhombus. The coordinates of A and C are (5,8) and (-1,2) respectively. Write down the equation of BD.

16) The following cable shows the distribution of the heights of a group of factory workers:

Ht(cm)     no of workers 

140-145      6

145-159     12

150-155     18

155-160     20

160-165     13

165-170      8

170-175      6

a) Determine the cumulative frequencies 

b) draw the cumulative frequency curve on a graph

c) From your graph, write down the median height in cm.

PAPER -4

1) A Colour TV is marked for sale for Rs16500 which includes GST at 10%. Calculate the tax in rupees.

2) Find the remainder when 2x³- 3x²+ 7x -8 is divided by x -2.

3) Given a/b = c/d, prove that: (3a - 5b)/(3a + 5b)= (3c - 5d)/(3c + 5d).

4) Two numbers are the ratio of 7: 11. If 15 is added to each number, the ratio becomes 5 : 7. Find the numbers .

5) Find the value of x, which satisfies the inequation: 

-2≤ 1/2  - 2x/3 ≤ 11/6, x belongs to N.

Graph the solution on the number line.

6) Priti deposited Rs 1500 per month in a bank for 8 months under the recurring deposit scheme. What will be the maturity value of her deposits , if the rate of interest is 12% per annum and interest is calculated at the end of every month.

7) solve for x and give your answer correct 2 decimal places:

3x²- 5x = 1.

8) The catalogue price of washing machine is Rs 16000. The shopkeeper gives a discount of 5% on the listed price. He gives a further off season discount of 12% on the balance. But GST at 5% is charged on the remaining amount. Find :

a) The GST paid by the customer.

b) The final price he has to pay for the washing machine.

9) If 3 tan²A - 1=0, then show that cos3A= 4 cos³A - 3 cosA.

10) A plot of land has an area of 400000 m². it is represented on the map by an area of 40 cm². Find:

a) the scale factor of the map.

b) what distance on the map would a distance of 2.4km.

11) Use graph paper for this question.

The point A(4,7) was reflected in the origin to get the image A'.

a) write down the coordinate of A'.

b) If M is the foot of the perpendicular from A to the x-axis. find the coordinates of M.

c) If N is the foot of the perpendicular from A' to the x-axis, find the coordinates of N.

d) name the figure AMA'N.

e) find the area of the figure AMA'N.

12) Prove: sinx(1+ tan x)+ cos x(1+ cot x)= cosecx + secx.

13) A(14,7), B(6,-3) and C(8,1) are the vertices of a triangle ABC . P is the midpoint of AB, and Q is the midpoint of AC. Write down the coordinates of P and Q. Show that BC= 2PQ.

14) A, B and T are 3 points on a circle.

The tangent at T meets BA produced at P. Given that angle ATB= 32 and that the angle APT= 78, calculate the angle subtended by BT at the centre of the circle.

15) If A= 4   3 & B= x & C= 6

               -5   0         -2          y with the relation AB= C. Find x and y.

16) A ma invests Rs7500 on buying shares of face value of Rs 100 each at a premium of 50% in a company. If he earns Rs 550 at the end of the year as dividend, find 

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

b) what is dividend percentage per share ?

17) write down the equation of the line whose gradient is 4/3 and which passes through P,  where P divides the line segment joining A(-2,-3) and B(5,4), in the ratio 2:5.

18) A vertical Tower is 40m high . A man standing at some distance from the tower knows that cosines of the angle of elevation of the top of the Tower is 30°. How far is he standing from the foot of the tower?

19) An exhibition tent is in the form of cylinder surmaunted by a cone. The height of the tent above the ground is 67m and the height of the cylindrical part is 40m. If the diameter of the base is 144m, find the quantity of canvas required to make the tent. Allow 10% extra for folds and for stitching. Give your answer to the nearest m².

20) Using the data given below , construct the cumulative frequency table and draw the ogive. From the ogive determine the median.

Mark    no of students 

00-10      3

10-20      8

20-30      12

30-40      14

40-50      10

50-60       6

60-70       5

70-80       2

21) In the given figure,

find TP if AT= 20cm and AB= 15cm.

22) Factorise the expression with the help of the factor theorem f(x)= 6x³- 7x²- 7x + 6. Hence, find the values of x when f(x)= 0.

PAPER- 3

1) The price of a TV set inclusive GST of 9% is 40221. Find the marked price.

2) If x: y= 4:3, find (5x +8y): (6x - 7y).

3) Using the reminder theorem, find the remainder when y³- 7y¹+ 15y - 19 is divided by y- 3.

4) State and draw the locus of a point eqidistance from two parallel lines.

5) The given figure, the medians QS and RT of a ∆ PQR meet at G. prove that:

a) ∆ TGS~ ∆ RGQ

b) QG= 2 GS from (a) above.

6) Solve the following inequation and graph the solution on the number line:

2x -5≤ 5x +4 < 11, x belongs to R.

7) The marks of 20 students in a test were as follows : 5, 6, 8, 9, 10, 11, 11, 12, 13,13, 14, 14, 15,15, 16,16 18, 19 20. Calculate:

a) the mean 

b) the median 

c) the mode

8) If the matrix 

A= 1 -4 & B= -3   2 & C= 4   0

      4  1           4   0          0  -3   find 

a) A² b) BC  c) A²+ BC .

8) The point A(3,4) is reflected to A' in the x-axis, and O' is the image of O(the origin) when reflected in the AA'. Using graph paper, give 

a) the coordinates of A' and O'.

b) the lengths of the segments AA' and OO'.

c) the perimeter of the quadrilateral AOA'O'.

d) the geometrical name of the figure AOA'O'.

9) Prove the following identity:

1/(sinA + cosA)  + 1/(sinA - cosA)= 2sinA/(2 sin²A -1).

10) In the given figure, AB is the diameter of a circle with centre O. Angle BCD is 130°. Find 

a) angle DBA 

b) angle BAD.

11) Find the equation of a line passing through the point (-4,6) and having the x-intercept of 8 units.

12) A man wants to buy 72 shares available at Rs 150 (per value of Rs 100).

a) How much should he invest ?

b) if the dividend is 7.5%, what will be his annual income ?

c) if he wants to increase his annual income by Rs 300, how many extra shares should be buy ?

13) The following table gives the weekly wages of workers in a factory:

 weekly wages (Rs).  No. of workers 

150-150                        5

155-160                       20 

160-165                       10 

165-170                       10 

170- 175                       9

175-180                        6

180-185                       12

185- 190                       8  Calculate 

a) the mean 

b) the model class 

c) the numbers workers getting weekly wages, below Rs 180.

d) the number of workers getting Rs 165 or more, but less than Rs 185 as weekly wages.

14) A hollow sphere of internal and external diameters 8 cm and 16 cm respectively , is melted into a cone of base diameter 16 cm.  Find the height of the cone.

15) The shadow of a vertical tower AD on level ground is increased by 30m, when the altitude of the sun changes from 45° to 30° as shown in the given figure.

 Find the height of the tower and give your answer correct to 1/10 of a metre.

16) The marks obtained by 240 students in a mathematics test is given below:

Marks   No. if students 

00-10       10 

10-20       18 

20-30       32 

30-40       44 

40-50       52

50-60       26

60-70       22

70-80       12

80-90       16

90-100      8

Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive and using ogive, estimate:

a) the median

b) the lower quartile 

c) the number of student who obtained more than 75% in the test :

d) the number students who did not passing inthe test if the pass percentage was 40.

17) P(2,4), Q(3,3) and R(7,5) are the vertices of a ∆ PQR. Find 

a) the coordinates of the centroid G of ∆ PQR.

b) the equation of a line, through G and parallel to PQ.

18) An aeroplane travelled a distance of 800 km at an average speed of x kmph. On the return journey, the speed was increased by 40 kmph. Write down an expression for the time taken for:

a) the onward journey .

b) the return journey .

If the return journey took 40 minutes less then the onward journey, write down an equation in x and find its value.

Paper - 2

1) The point P(a,b) is reflected in the x-axis to obtain the point Q(3,-4). Find a and b.  (1)

2) If A= a  3a & B= 2 & C= 5 

              b  4b         1          12 find a and b when the relation AB= C.     (1)

3) The mean of the number 6, y, 7, x and 14 is 8. Express y terms of x.    (1)

4) Solve using the quadratic formula, x²- 5x -2=0. Give your answer correct to 3 significant figures.        (2)

5) If (8a + 5b)/(8c + 5d)= (8a - 5b)/(8c - 5d), prove that a/b = c/d.     (1)

6) Find the value of k, if x - k is a factor of x³- kx²+ x + 4.       (1)

7) Solve 1< 3x -3≤ 11, x ∈ R and mark it on a number line.     (1)

8) Calculate the mean, median and mode of the following numbers : 12, 11, 10, 11, 12, 13, 14, 13, 15, 13.    (2)

9) In the diagram,

chords AB and CD of the circle are produced to meet at O. Given that CD= 4cm, DO= 12cm and BO= 6cm, calculate AB .    (2)

10) If cosA= 4/5 and cosB= 24/25; evaluate 

a) cosec²A

b) cotA + cotB.      (2)

11) on a map drawn to a scale 1:125000, a triangular plot of land has the following measurements :

PQ=10cm, QR= 8cm, angle PRQ= 90°. Calculate 

a) the actual length of PQ in km.

b) the area of the plot in square kilometres.    (2)

12) The work done by (2x -3) men in (3x +1) days and work done by (3x +1) men in (x +8) days are in the ratio of 11:15. Find the value of x.    (2)

13) Find the mean of the following frequency distribution:

Class interval    frequency 

00-30                   3

30-60                   7 

60-90                  15

90-120                14 

120-150               7 

150-180               4         (3)

14) A man invests Rs 30800 in buying shares of nominal value Rs 56 at 10% premium . The dividend on the shares is 18% per annum. Calculate 

a) The number of shares he buys.

b) The dividend he receives annually.

c) The rate of interest he gets on his money.       (3)

15) prove that: sinx/(1- cotx) + cosx/(1- tanx)= sinx + cosx.    (2)

16) A straight line passes through the points A(-2,8) and B(10,-4). It intersects the coordinate axes at points E and F. P if the midpoint of the segment EF. 

Find 

a) the equation of the line.

b) the coordinate of E and F.

c) the coordinates of the point P.     (3)

17) In an auditorium, seats were arranged in rows and columns . The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 15, the total number of seats increased by 400.  Find 

a) The number of rows in the original arrangement.

b) the number of seats in the auditorium after rearrangement.    (3)

18) Draw a histogram and hence estimate the mode for the following frequency distribution:

Class     frequency 

00-20        3 

20-40        8 

40-60       10 

60-80        6

80-100      4

100-120    3         (3

19) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 40m away from the bank, he finds the angle of elevation to be 30°. Calculate:

a) the width of the river and

b) the height of the tree.    (3)

20) Find a and b, if

a= 3  -2 & B= 2a & C= 4 & D= 2

    -1   4           1            5          b with the relation AB + 4C = 3D.  (2)

21) A vessel is in the form of an inverted cone. Its height is 15cm and the diameter of its top which is open, is 5cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of diameter 5mm are dropped into the vessel, 1/3 of the water flows out. Find the number of lead shots dropped into the vessel.    (3)

22) In the given circle

with diameter AB, find the value of x.   (2)

23) Find the value of k for which the lines kx - 7y + 5=0 and 6x - 2y +9=0 are perpendicular to each other.     (3)

Paper -1

1) Find the rate of GST levied on a car that was sold at a price 3 times its marked price.  (1)

2) When 7x²- 3x + 8 is divided by (x -4), find the remainder (using remainder theorem).  (1)

3) Calculate the length of the tangent drawn to a circle of diameter 8cm from a point 5cm away from the centre of the circle.   (1)

4) If x²,4 and 9 are in continued proportion , find the value of x.   (1)

5) If x ∈Z, find the solution set for the inequation 5< 2x -3≤ 14 and graph the solution on a number line.    (1)

6) Find p and q if g(x)= x +2 is a factor of f(x)= x³- px + x + q and f(2)= 4.    (2)

7)       1      -2         0

If X=  -3      4 & Y= 1

a) Find the matrix Z such X + Z is a zero matrix.

b) Find the matrix M such that X + M = X.

c) Find XY.         (3)

8) a) If 7 is the mean of 5, 3, 0.5, 4.5, b, 8.5, 9.5, find b

b) If each observation is decreased in value by 1 unit, what would the new mean be ?   (2)

9) In the figure below,

AB is a chord of the circle with centre O and BT is tangent to the circle at B, if angle OAB= 32°, Find the value of x and y.    (2)

10) Construct a regular pentagon of side 3cm. Draw the lines of symmetry.   (2)

11) The volume of a cylinder 14cm long is equal to that of a cube having an edge 11cm. Calculate the radius of the cylinder.    (3)

12) A piece of butter 3cm by 5cm by 12cm is placed on a hemispherical bowl of radius 3.25cm. Will the butter overflow when it melts completely.    (3)

13) A company with 10000 shares of Rs 50 each declares an annual dividend of 5%.

a) What is the total amount of dividend paid by the company ?

b) What would be the annual income of a man who has 72 shares in the company?

c) if he receives only 4% on his investment, find the price he paid for each share.   (3)

14)a) State the equation of the mirror line, if point A(5,0) on reflection is mapped as A'(-5, 0).

b)  State the equation of the the mirror line, if point B(4,-3) on reflection is mapped as B'(4,3).

c) Point C(-3,5) on reflection in y=2 is mapped as C'. Find the coordinates of C.   (3)

15) Tanya standing on a vertical cliff in a jungle observes two rest-horses in a line with her on opposite sides deep in the Jungle below. If their angles of depression are 30° and 45° and the distance between them is 200mp, find the height of the cliff.   (3)

16) Find the equation of a line that passes through (1,3) and is parallel to the line y= -3x +2.   (2)

17) In the given figure,

calculate 

a) angle APB

b) angle AOB.     (2)

18) The midpoint of the line joining A(2,p) and B(q,4) is (3,5). Find the numerical values of p and q.     (2)

19) From the following table, find:

a) average wage of a worker. Give your answer, to the nearest paise 

b) Modal class.

Wages in Rs   No of workers 

Less than 10     15 

Less than 20     35 

Less than 30     60

Less than 40     80

Less than 50     96

Less than 60    127

Less than 70    190

Less than 80    200        (3)

20) Examine the ogive given below

which shows the marks obtained out of 100 by a set of students in an examination and answer the following questions:

a) How many students are there in the set ?

b) How many students obtained 40% marks ?

c) How many students obtained 90% and above ?

d) What is the median marks?        (4)

21) Show that: √{(1+ cosx)/(1- cosx)}= cosecx + cot x.       (2)

REVISION - XI

BINOMIAL THEOREM 

1) Expand  (2/x - x/2)⁵, x≠ 0.

2) Using binomial theorem , write the value of (a+ b)ⁿ + (a - b)ⁿ and hence find the value of (√3+ √2)⁶ - (√3 - √2)⁶.    

3) Find the 9th term in the expansion of (3x - 1/2x)⁸, x≠ 0.   

4) Find the term independent of x in (2x²- 1/x)¹².     

5) Find the middle terms in the expansion of (3- x³/6)⁷.    

6) Use binomial theorem to evaluate (10.1)⁵.      

7) Examine whether or not there is any term containing x⁹ in the expansion of (2x¹ - 1/x)²⁰.      

8) In the binomial expansion of (a - b)ⁿ, n≥ 5, The sum of 5th and 6th terms is zero, then a/b equals 

a) (n -5)/6 b)  (n -4)/5 c)  5/(n -4) d) 6/(n -5) 

      

9) If the coefficient of rth and (r +4)th terms are equal in the expansion of (1+ x)³⁰, then the value of r will be 

a) 7 b) 8 c) 9 d) 10

10) If the coefficient of x² and x³ in the expansion of (3+ ax)⁹ be same, then the value of a is.

a) 3/7 b) 7/3 c) 7/9 d) 9/7

11) Using binomial theorem, the value of (0.999)³ correct to 3 decimal places is

a) 0.999 b) 0.998 c) 0.997 d) 0.995

  

QUADRATIC EQUATION 

1) 5ˣ⁺¹ + 5²⁻ˣ = 5³ +1.      

2) √{x/(1- x)} + √{(1- x)/x} = 13/6.    

3) (x +1)(x +2)(x +3)(x +4)= 120.    

4) Prove that both the roots of the equation x²- x -3=0 are irrational.

5) For what values of m will the equation x¹- 2mx + 7m -11= 0 have 

a) equal roots.   

b) reciprocal roots ?    

6) if the roots of 2x²- 5x + k =0 be double the other, find the value of k.  

7) If α, β be the roots of the equation x¹- x -1=0, determine the value of

a) α²+ β².   

b) α³+ β³.    

8) If the roots of the equation ax²+ bx + c=0 be in the ratio 3:4, show that 12b²= 49ac.

9) if x is real, prove that the quadratic expression

a) (x -2)(x +3)+ 7 is always 

b) 4x - 3x²- 2 is always negative.

10) What is the minimum value of x²- 4x +3=0.   

11) For what real values of a, will the expression x²- ax +1 - 2a², for the real x,  be always positive ?     

12) If x be real, prove that the value of (2x⅖- 2x +4)/(x²- 4x +3) cannot lie between -7 and 1.

13) if the roots of the equation qx²+ 2px + 2q=0 are real and unequal, prove that the roots of the equation (p + q)x²+ 2qx + (p - q)= 0 are imaginary.

14) If α, β be the roots of x²- px + q=0, find the value of α⁵β⁷+ α⁷β⁵ in terms of p and q.    

15) If the difference between the roots of the equation x¹+ ax +1=0 is less than √5, then the set of possible value q of a is

a) (3,∞) b) (-∞,-3) c) (-3,3) d) (-3,∞).   

16) let α, β be the roots of the equation x²- px + r=0 and α/2, 2β be the roots of the equation x²- qx + r=0, then the value of r is

a) (2/9) (p - q)(2q - p)

b) (2/9) (q- p)(2p - q)

c) (2/9) (q - 2p)(2q - p)

d) (2/9) (2p - q)(2q - p).         

17) α, β are the roots of ax²+ 2bx + c=0 and α + β, β + δ are the roots of Ax²+ 2Bx + C=0, then what is (b¹- ac)/(b²- ac) equal to ?

a) (b/B)² b) (a/A)² c) (a²b²/A²B²) d) (ad/AB)  e) none  

18) If α, β are the roots of the equation x²- 2x -1=0, then what is the value of α²β⁻² + α⁻²β² ?

a) -2 b) 0 c) 30 d) 34    

19) If the roots of the equation x²+ px + q=0 are thn30° and tan15°, then value of 2+ q - p is 

a) 1 b) 2 c) 3 d) 0     

20) If the roots of the quadratic equation x²- 2kx + k²- 5 =0 are less than 5, then k lies in the interval 

a) (5,6) b) (6,∞) c) (- ∞,4) d) [4,5].   

21) If α and β are the roots of ax²+ bx + c =0 and if px²+ qx + r=0 has roots (1- α)/α and (1- β)/β then r= 

a) a+ 2b b) a+ b + c c) ab+ bc+ ca d) abc.     

22) The equation x²- 6x + a=0 and x²- cx + 6 =0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. then the common root is

a) 1 b) 4 c) 3 d) 2     

23) If α, β are the roots of the equation λ(x²- x) + x + 5=0 and λ₁ and λ₂ are two values of λ obtained from α/β + β/α = 4/5, then λ₁/λ₂² + λ₂/λ₁² equals 

a) 4192 b) 4144 c) 4096 d) 4048      

24) If α, βbe the roots of x² - a(x -1)+ b =0, then value of 

1/(α² - aα)  + 1/(β² - aβ) + 2/(a+ b) is 

a) 4/(a+ b) b) 1/(a+ b) c) 0 d) -1          

REVISION - XII

1) The area(in square unit) of the region bounded by the curve x²= 4y, the line x= 2 and x-axis is -

a) 1 b) 2/3 c) 4/3 d) 8/3

2) Let(a sec θ, b tanθ) and Q(a secα, b tanα) where θ + α=π/2, be two on the hyperbola x²/a²- y²/b²= 1. If (h,k) be the point of intersection of the normals at P and Q, then the value of k is 

a) (a²+ b²)/a

b) - (a²+ b²)/a

c) (a²+ b²)/b

d) -(a²+ b²)/b

3) The equation of the tangent to the curve (1+ x²)y = 2- x where it crosses the x-axis is 

a) x+ 5y = 2 b) x- 5y = 2 c) 5x - y = 2 d) 5x + y = 2 

4) The area (in square unit) bounded by the parabola y²= 4ax and x²= 4ay is

a) 64a²/3 b) 32a²/3 c) 16a²/3 d) 7a²/3

5) Equations of the tangent and the normal drawn at the point (6,0) on the ellipse x²/36 + y²/9 = 1 respectively are

a) x= 6, y=0 

b) x+ y= 6, y - x +6=0

c) x= 0, y=3 

d) x= - 6, y=0

6) The area (in square unit) of the figure by the curve y = cosx and y = sinx and the ordinates x= 0, x=π/4 is

a) √2 +1 b) √2 - 1  c) 1/√2 d) (√2 -1)/√2

7) The straight line x + y = a will be a tangent to the ellipse x²/9+ y²/16= 1 if the value of a is

a) 8 b) ±10 c) ±5 d) ±6

8) The equation of the tangent to the parabola y²= 8x which is perpendicular to the line x - 3y +8= 0 is

a) 3x + y + 2 = 0 

b) 3x - y - 1 = 0

c) 9x - 3y + 2 = 0

d) 9x + 3y + 2 = 0

9) The area (in square unit) bounded by the Parabola y²= 8x and its latus rectum is 

a) 16/3  b) 25/3 c) 16√2/3 d) 32/3

10) If the curve y²= 4x and xy= k cut orthogonally, then the value of k² will be 

a) 16 b) 32 c) 36 d) 8

11) The area (in square unit) bounded by the curve -3y²= x -9 and the lines x= 0, y= 0 and y= 1 is

a) 8/3 b) 3/8 c) 83

12) If the slope of the normal to the curve x³= 8a²y at P is (-2/3), then the coordinates of P are

a) (2a,a) b) (a,a) c) (2a, -a) d) none

13) If a> 2b > 0, then the positive value of m for which the line y= mx - b √(1+ m²) is a common tangent to the circles x²+ y²= b² and (x - a)²+ y²= b² is 

a) 2b/√(a²- 4b²) 

b) √(a²- 4b²)/2b 

c) 2b/(a- 2b) 

d) b/(a- 2b) 

14) The area in square unit of the region bounded by the lines y= |x -1| and y= 3- |x| is 

a) 6 b) 2 c) 4 d) 3

15) The minimum value of f(x)= x²+ 250/x is 

a) 55 b) 25 c) 50 d) 75

16) If f(x)=kx³- 9x²+ 9x +3 is increasing function then 

a) k< 3 b) k ≤ 3 c) k> 3 d) k is indeterminate

17) If f(x)= 1/(4x²+ 2x +1), then its maximum value is 

a) 2/3 b) 4/3 c) 3/4 d) 1

18) If f(x)= 1/(x +1)  - log(1+ x), x> 0, then f(x) is 

a) a decreasing function 

b) an increasing function 

c) neither increasing nor decreasing 

d) increasing when x> 1.

19) Let α, β be the roots of x²+ (3- λ)x - λ=0, then the value of λ for which  α²+ β² is minimum, is

a) 0 b) 1 c) 3 d) 2

20) The function f(x)=2x³- 3x² -12x +4 has

a) no maxima and minima 

b) one maximum and one minimum 

c) two maxima 

d) two minima

21) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a, is

a) 3a/2 b) √2 a/3 c) 2a/√3 d) a/√3

22) Maximum value of (logx)/x in [0, ∞) is 

a) (log2)/2 b) 0 c) 1/e d) e

23) Let the function f: R--R be defined by f(x)=2x + cosx; then f(x)

a) has maximum value at x=0

b) has minimum value at x=π

c) is a decreasing function 

d) is an increasing function 

24) The maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is

a) a- b b) a+ b c) √(a²+ b²) d) √(a²-  b²)

25) The velocity v of a particle moving along a straight line is given by a+ bv²= x², where x is the distance of the particle from the origin. Then the acceleration of the particle is 

a) x/b b) bx  c) x/a  d) b/x

26) If the slope of the tangent at (x,y) to a curve passing through the point (2,1) is (x²+ y²)/2xy , then the equation of the curve is

a) 2(x²- y²)= 3x

b) 2(x²- y²)= 3y

c) x(x²- y²)= 6

d) 2px(x² + y²)= 6

27) A particle moves uniform acceleration f along a straight line. If v be it's velocity at time t and s be the distance described during the interval, then 

a) s= 2vt - ft²

b) s= vt - ft²/2

c) s= vt/2 - ft²

d) s= vt - ft²/2

28) A particle moving in a straight line traverses a distance x in time t, if t= x²/2 + x, then the retardation of the particle is 

a) equal to its velocity 

b) constant 

c) is equal to the cube of its velocity 

d) equal to the square of its velocity 

29) If y= 3x²+ 2 and if x changes from 10 to 10.1, then the approximate change in y will be 

a) 8 b) 6 c) 5 d) 4

30) The rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to 

a) 1/r² b) r² c) r d) 1/r

Paper - 4

C- TEST PAPER- (1)

Section A

(Multiple Choice Questions) Each question carries 1 mark

1) If A= [aᵢⱼ] is a square metrix of order 2 such that 

aᵢⱼ= {1,  when i≠ j

         0, when i= j then A² is 

a) 1  0 b) 1  1 c) 1  1 c) 1  0

     1  0     0  0     1   0     0. 1

2) if A and B are invertible square matrices of the same order, then which of the following is not correct ?

a) adjacent A= |A|. A⁻¹

b) det(A⁻¹)= [det(A)]⁻¹

c) (AB)⁻¹= B⁻¹A⁻¹

d) (A+ B)⁻¹= B⁻¹ + A⁻¹

3) if the area of the triangle with vertices (-3,0),(30) and (0,k) is 9 square. units, then the value/s of k will be 

a) 9 b) ±3 c) -9 d) 6

4) If f(x)= {kx/|x|,   if x< 0

                    3,         if x≥ 0 is continuous at x= 0,  then the value of k is 

a) -3 b) 0 c) 3  d) any real number

5) The lines r= i+ j - k + λ(2i + 3j - 6k) and r= 2i - j + k + μ(6i + 9j - 18k), where λ and μ are scalars) are

a) coincident b) skew c) intersecting  d) parallel 

6) The degree of the differential equation 

[1+ (dy/dx)²]³⁾² = d²y/dx² is 

a) 4 b) 3/2 c) 2 d) not defined 

7) The corner points of the bounded feasible region determined by a system of linear constraints are (0,3),(1,1) and (3,0). Let Z= px + qy, where p,q> 0. The condition on p and q, so that the minimum of Z occurs at (3,0) and (1,1) is 

a) p= 2q b) p= q/2 c) p= 3q d) p= q

8) ABCD is a rhombus whose diagonals intersect at E. Then EA+ EB + EC + ED equals to 

a) 0 b) AD c) 2BD d) 2AD

9) For any integer n, the value of 

∫ₑsin²x cos³(2n +1)x dx at (π,0) is 

a) -1 b) 0 c) 1 d) 2

10) The value of |A|, if

       0      2x -1     √x

A= 1- 2x    0       2√x

     -√x    -2√x      0

Where x ∈ R⁺, is 

a) (2x +1)² b) 0 c) (2x +1)⅔ d) none 

11) The feasible region corresponding to the linear constraints of a Linear programming problem is given below. 

Which of the following is not a constraint to the given linear programming problem.

a) x + y ≥ 2

b) x + 2y ≤ 10

c) x - y ≥ 1

d) x - y ≤ 1

12) If a= 4i + 6j and b= 3j + 4k, then the vector form of the component of a along b is

a) 18(3i + 4k)/5

b) 18(3j + 4k)/25

c) 18(3j + 4k)/5

d) 18(2i + 4k)/25

13) Given that A a square metrix of order 3 and |A|= -2, then |adjacent(2A)| is equal to 

a) -2⁶ b) 4 c) -2⁸ d) 2⁸

14) A problem in Mathematics is given to 3 students whose chances of solving it are 1/2, 1/3, 1/4, respectively. If the events of their solving the problem are independent, then the probability that the problem will be solved, is 

a) 1/4 b) 1/3  c) 1/2 d) 3/4

15) The general solution of the differential equation ydx - x dy = 0; (given x, y> 0), is of the form

a) xy= c b) x = cy² c) y= cx d) y= cx²

(Where c is an arbitrary positive constant of integration)

16) The value of λ for which two vectors 2i - j + 2k and 3i + λj + k are perpendicular is,

a) 2 b) 4 c) 6 d) 8

17)  The set of all points, where the functions f(x)= x + |x| is differentiable, is

a) (0,∞) b) (-∞,0) c) (-∞,0) U (0,∞) d) (-∞,∞)

18) If the direction cosines of a line are < 1/c, 1/c, 1/c > , then 

a) a< c < 1 b) c> 2 c) c=±√2  d) c=±√3

Assertion - Reason Based Questions 

In the following questions , a statement of Assertion(A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices .

   a) Both A and R are true and R is the correct explanation of A.

   b) Both A and R are true but R is not the correct explanation of A

   c) A is true but R is false

   d) A is false but R is true.

19) Let f(x) be a polynomial function of degree 6 such that 

d/dx [f(x)]= (x -1)³(x -3)², then 

Assertion (A): f(x) has a minimum at x= 1.

Reason (R): When d/dx [f(x)]< 0.

∀ x  ∈ (a - h, a) and d/dx [f(x)]> 0,

∀ x ∈ (a, a+ h), where h is an infinitesimally small positive quantity, then f(x) has a minimum at x= a, provided f(x) is continuous at x= a.

20) Assertion (A): The relation 

f: {1,2,3,4} --> {x, y, z, p} defined by 

f={(1,x),(2,y),(3,z)} is a bijective function.

Reason (R) The function 

f: {1,2,3} ---> {x,y,z,p} such that 

f: {(1,x),(2, y),(3, z)} is one-one.

SECTION - B

(This section comprises of very short Answer type questions (VSA) of 2 marks each)

21) Find the value of sin⁻¹[cos(33π/5)]

OR

Find the domain of sin⁻¹(x²-4).

22) Find the interval in which the functions 

f: R---> R defined by f(x)= xeˣ,  is increasing.

23) If f(x)= 1/(4x²+ 2x +1), x ∈ R, then find the maximum value of f(x).

OR

Find the maximum profit that a company can make, if the profit function is given by P(x)= 72 + 42x - x², where x is the number of units and P is the profit in rupees.

24) Evaluate ¹₋₁ ∫ logₑ {(2- x)/(2+ x)} dx.

25) Check whether the function f: R--->R defined by f(x)= x³+ x, has any critical point/s are not ? If yes, then find the point/s.

SECTION - C

(This section comprises of short Answer type questions (SA) of 3 marks each)

26) Evaluate ∫ (2x²+3)/{x²(x²+9)} dx, x≠ 0.

27) The random variable X has a probability distribution P(X) of the following form,  where 'k' is some real number.

             k,    if X= 0

             2k,  if X = 1

P(X)=.  3k,  if X= 2

              0,  otherwise 

a) Determine the value of k.

b) Find P(X< 2).

c) Find P(X > 2).

28) Evaluate ∫ √{x/(1- x³)} dx, x ∈(0,1),

OR

Evaluate ∫ logₑ(1+ tanx) dx at (π/4,0).

29) Solve the differential equation 

yeˣ/ʸ dx = (xeˣ/ʸ + y²) dy, (y≠ 0).

OR

Solve the differential equation 

(cos²x)dy/dx + y = tanx; (0≤ x<π/2).

30) Solve the following Linear Programming Problem graphically

Minimize Z= x + 2y

Subject to the constraints , x +2≥ 100, 2x - y≤ 0, 2x + y ≤ 200, x, y ≥ 0.

OR

Solve the following Linear Programming Problem graphically 

Maximum Z= - x + 2y.

subject to constraints , x≥3, x+ y ≥ 5, x + 2y ≥ 6, y≥ 0.

31) If (a + bx)eʸ/ˣ = x, then prove that 

x d²y/dx²= {a/(a + bx)}².

SECTION - D

(This section section comprises of long answer type questions (LA) of 5 marks each)

32) Make a rough sketch of the region

{(x,y): 0≤ y ≤ x½+1,

0≤ y ≤ x +1, 0≤ x ≤ 2) and find the area of the region , using the method of integration.

33) Let N be the set of all natural numbers and R be a relation on N x N defined by (a, b) R(c,d) <=> ad= bc for all 

(a, b),(c,d) ∈ N x N. Show that R is an equivalence relation on N x N.  Also, find the equivalence class of (2,6) i.e., [(2,6)].

OR

Show that the function 

f: R---> {x ∈ R : -1< x < 1} defined by

f(x)= x/{1+ |x|} , x ∈ R is one-one and onto function.

34) Using the matrix method, solve the following system of linear equations .

2/x + 3/y + 10/z = 4, 

4/x - 6/y + 5/z = 1,

6/x + 9/y - 20/z = 2.

35) Find the coordinates of the image of the point (1,6,3) with respect to the line.

r= (j+ 2k) +  λ(i + 2j + 3k), where λ is a scalar. Also, find the distance of the image from the y-axis.

OR

An aeroplane is flying along the line 

r= λ(i - j + k), where λ is a a scalar and another aeroplane is flying along the line 

r= i - j + μ(-2j + k), where μ is a scalar . At what points on the lines should they reach, so that the distance between them is the shortest ? Find the shortest possibly distance between them.

SECTION - E

(This section comprises of 3 case - study/passage -based questions of 4 marks each 

36) Read the following passage and answer the questions given below.

      In an office 3 employees James, Sofia and Oliver process incoming copies of a certain form. James processes 50% of the forms. Sophia processes 20% and Oliver the remaining 30% of the forms. James has an error rate of 0.06, Sophia has an error rate of 0.04 and Oliver has an error rate of 0.03.

Based on the above information, answer the following questions .

a) Find the probability that Sophia processed the form and committed an error.

b) Find the total probability of committing an error in processing the form .

c) The manager of the company wants to do a quality check. During inspection, he selects a form at random from the days output of process from. If the form selected at random has an error; then find the probability that the form is not processed by James.

OR

Let E be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that James, Sophia and and Oliver processed the form.

Find the value of ³ᵢ₌₁∑ P(Eᵢ/E).

37) Read the following passage and answer the questions given below.

Teams A, B, C went for playing a tug of war game. Team A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area.

Team A pulls with force F₁ = 6i + 0j kN,

Team B pulls with force F₂ = -4i + 4j kN,

Team C pulls with force F₃ = -3i - 3j kN,

a) What is the magnitude of the force of Team A ?

b) Which team will win the game ?

c) Find the magnitude of the resultant force exerted by the teams.

OR

In what direction is the ring getting pulled?

38) Read the following passage and answer the questions given below.

      The relation between the height of the plant (y in cm) with respect to its exposure to the sunlight is governed by the following equation y= 4x - x²/2, where x is the number of days exposed to the sunlight, for x≤ 3.

i) Find the rate of growth of the plant with respect to the member of days exposed to the sunlight.

ii)  Does the rate of growth of the plant increase or decrease in the first three days ? What will be the height of the plant after 2 days ?

PAPER - III

1) What is the domain of the f(x)= (x²-1)/(x -4) ?

2) For what values of x will the given matrix 

-x     x      2

2      x     -x

x     -2    -x be a singular?

3) Evaluate ∫ sinx cos²x(sin²x + cosx) dx at (π/2,-π/2).

5) Evaluate sin(2 tan⁻¹ (1/3))

6) If a     b

        c    -a is a square root of the 2 x 2 identity matrix, then what is the relation between a,b and c ?

8) Find the solution of the equation of determinant

cosx    sinx       cosx

-sinx    cosx      sinx = 0

-cosx  -sinx      cosx

10) Evaluate: ∫ ₂2ˣ 2ˣ dx

12) Let f(x)= x²/(x²+1) for x ≥ 0. Then find f⁻¹(x).

13) Find the probability of drawing a diamond card in each of the two consecutive draws from a well shuffled pack of cards, if the card drawn is not replaced after the first draw.

14) Solve the equation 

sin[2cos⁻¹(cot(2tan⁻¹x)]= 0.

Or

Show that 

2 tan⁻¹[tan(x/2) tan(π/4 - y/2)]= tan⁻¹{sinx cosx/(siny + cosx)}

15) Find the differential equation of all straight lines which are at fixed distance 'p' from the origin.

16) 

17) Evaluate:∫ √(2x²-1)/(1- x²) dx

Or

∫ dx/((x¹⁾² + x¹⁾³).

18) If c= a(sink - k cost) and y= a(cost + k sink) find dy/dx at k=π/4.

20) Find the inverse of A

A= 2   1     3

     4   -1     0

   -7     2     1

21)        1,          if x ≤ 3

If f(x)= ax+ b,   if 3< x < 5

              7,         if 5 ≤ x

Determine the value of a and b so that f(x) is continuous.

Or

Determine the value of a, b, c for which function 

         (sin(a+1)+ sinx)/x, x< 0

f(x)=    c ,                          x=0

         (√(x + bx²) - √x)/b√x³, x> 0

may be continuous at x=0

4) If a, b, c are three mutually perpendicular unit vectors then what is the value of|a+ b + c| ?

7) a and b are unit vectors. If a+ b is a unit vector then what is the angle between a and b?

9) Equation of the plane passing through (2,3,-1) and is perpendicular to the vector 3i - 4j + 7k.

11) If a,b,C are vectors such that a.b= a.c, ax b = ax c, a≠ 0, then show that b= c

19) A line with direction ratios < 2,7, -5> is drawn to intersect the lines

(x -5)/3 = (y-7)/-1 = (z+2)/1 and (x +3)/-3 = (y-3)/2  = (z -6)/4 

Find the coordinates of the points of intersection.

PAPER - II

1) If A= 3    1

              7    5 find x and y so that A²+ xI₂ = yA.

2) Evaluate: tan[2 tan⁻¹(1/5) - π/4].

3) State the reason why the relation 

R={(a,b): a≤ b²}

on the set R of real numbers is not reflexive.

4) ∫ (xeˣ)/(x +1)² dx.

5) Deepak rolls two dice and gets a sum more than 9. What is the probability that the number on the first die is even?

6) Y= tan⁻¹{5x/(1- 6x²)}, -1/√6< x <1/√6, then show that dy/dx = 2/(1+ 4x²) + 3/(1+ 9x²).

7) Solve: (y + xy) dx + y(1- y²) dy = 0.

8) Let f: [0,∞)--> R be a function defined by f(x)= 9x²+ 6x -5. Show that f is not invertible. Modify only the codamain of f to make f invertible and then find its inverse.

9) Using properties of determinants, Prove that 

b²+ c²   ab        ac

  ba      c²+ a²   bc= 4a²b²c²

  ca       cb      a²+ b².    

10) Show that: tan[sin⁻¹(1/√17) + cos⁻¹(9/√85)]= 1/2.

11) Discuss the continuity of the function 

f(x}=  2x -1,   x< 1/2

          3 -6c,   x≥ 1/2

at x = 1/2

12) Find differentiation: 

y= tan⁻¹[{√(1+ x²) -1}/x].

13) ∫ (tanx + tan²x)/(1+ tan²x) dx.

14) ³₁∫ (3x² +1) dx.

15) Show that the equation of the normal at any point t on the curve x= 3 cos t - cos³t and y= 3 sin t - sin³ t is 4 (y cos³t - x sin³t)= 3 sin 4t.

16) The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

17) Solve: (tan⁻¹y - x) dy = (1+ y²) dx.

18) Bag I had two and three black balls, Bag II has four red and one black ball and Bag III has three white and two black balls. A bag is selected at random and a ball is drawn at random. What is the probability of drawing a red ball?

19) Using matrix method, solve 

x - 2y - 2z -5=0; -x + 3y +4=0; -2x + z - 4=0.

20) Find inverse of 

-1    1     2 

1     2     3 

3     1     1

21) Show that the right circular cone of least curved surface area and given volume has an altitude equal to √2 times the radius of the base.

22) The sum of the surface areas of a cuboid with sides x, 2x and x/3 and a sphere is given to be constant. Show that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.

23) Evaluate: ∫ x²/{(x² +4)(x² +9)} dx.

24) Three persons A, B and C apply for a job of manager in a private company. Chances of their selection (A,B and C) are in the ratio 1:2:4. The probability of A, B and C can introduce changes to improve profits of the company are 0,8,0.5,0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.

25) The vectors from origin to the points A and B are a= 3i - 6j +2k and b= 2i + j - 2k respectively. Find the area of the triangle OAB.

26) Find the Cartesian equation of the line passing through the points(3,-2,-5) and (1,-4,-7).

27) Show that (a x b)²= a.a   a. b

                                         a.b   b.b

28) If the vectors ai+ j + k, i + bj + k and i+ j+ ck are coplanar (a,b,c≠1), then show that 1/(1- a)   + 1/(1- b)  + 1/(1- c)= 1.

29) Find the shortest distance between the lines whose vector equations are 

r= (1- t)i + (t - 2)j +(3- 2t)k, and 

r= (s +1)i + (2s -1)j - (2s +1)k.

30) Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes x +2y + 3z -7=0 and 2x - 3y + 4z = 0.

31) Sketch the graphs of y= x(4- x) and find the area bounded by the curve, x-axis and the lines x=0 and x= 5.

32) The marginal cost of production of a commodity is 30+ 2x. It is known that fixed costs are Rs 120. Find 

a) Find the total cost of producing 100 units.

b) Find the cost of increasing output from 100 to 200 units.

33) You are given the following two lines of regression. Find the regression of Y on X and X on Y and satisfy your answer.

34) The cost function for a commodity is C(x)= 200+ 20x - x²/2 (in Rs)

a) Find the marginal cost MC.

b) Calculate the marginal cost when x= 4 and interpret it.

35) Fit a straight line to the following data, treating y as the dependent variable:

x: 14  12  13  14  12

y: 22  23  22  24  24

Hence, predict the value of y when x= 16.

36) You are given the following data:

                                 x         y 

Arithmetic mean.   36      85

Standard deviation 11      8

37) Given the total cost function for x units of a commodity as C(x)= x³/3  + 3x² - 7x + 16. Find 

a) the marginal cost 

b) the average cost 

c) show that the marginal cost is given by {x MC - C(x)}/x²

38) Given the price of a commodity is fixed at Rs 55 and its cost function is C(x)= 30x + 250.

a) Determine the break even point.

b) What is the profit when 12 items are sold?

39) An aeroplane can carry a maximum of 200 passes. A profit of Rs 1000 is made in each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast four times as many passengers prefer to travel by economy class then by executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit earned?

Paper - 1

1) Without expanding at any stage, find the value of the determinant:

      2     x      y+ z

∆= 2     y      z+ x

      2     z      x+ y         (2)

2) Solve: sin⁻¹cos (sin⁻¹x)=π/3.     (2)

3) Find the value of k if

M= 1   2

       2   3 and M² - kM - I₂ =0.     (2)

4) Evaluate : ∫ sin³⁾²x/(sin³⁾²x + cos³⁾²x) dx at (π/2,0).      (2)

5) Find dy/dx, if x= at² and y= 2at.       (2)

6) Find the differential equation of the family of curves y= Aeˣ + Be⁻ˣ, where A and B are arbitrary constants.     (2)

7) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 - 6x - 2x².       (2)

8) A family has two children. What is the probability that both children are boys given that atleast one of them is a boy?    (2)

9) Given that the events A and B are such that P(A)= 1/2 , P(A U B)= 3/5 and P(B)= k. Find k if

a) A and B are mutually exclusive 

b) A and B are independent.        (2)

10) Ley R⁺ be the set of all positive real numbers and f: R⁺ ---[4,∞): f(x)= x² + 4. Show that inverse of f exists and find f⁻¹.     (2)

11) Using properties of determinants Prove 

x    x²    1+ px³

y    y²    1+ py³ 

z    z²    1+ pz³ = (1+ pxyz)(x - y)(y - z)(z - x), where p is any scalar.    (3)

12) Prove that tan⁻¹(1/2) = π/4 - (1/2) cos⁻¹(4/5).     (3)

13) Show that the function f(x)= |x -1|, x ∈ R, is continuous at x= 1 but not differentiable.     (2)

14) If y= ₑa cos⁻¹x, where -1≤ x ≤ 1 then show that: (1- x²)y₂ - xy₁ - a²y = 0.   (3)

15) ∫ (6x +7)/√{(x -5)(x -4)} dx.    (3)

16) Evaluate : ³₁∫ (x² + x)dx

17) Find the equation of the normal to the curve y= x³+ 2x + 6 which are parallel to the line x + 14y + 4=0.      (2)

18) A circle disc of radius 3cm is being heated. Due to expansion, its radius is increasing at the rate of 0.05 cm/s. Find the rate at which its area is increasing when the radius the is 3.2 cm.      (2)

19) Solve the following differential equation: x dy/dx + 2y = x² logx.     (2)

20) Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values 'x' has the following form, where 'k' is some unknown constant.

P(X= x) = 0.1      if x= 0

                 kx       if x= 1 or 2

            k(5- x),    ifx= 3 or 4

                 0,        otherwise 

a) Find the value of k.

b) What is the probability that you study 

      i) atleast two hours?

      ii) exactly two hours?

      iii) atmost 2 hours?          (3)

21) If the matrix 

A= 3   -2    3 & B= -1    -5     -1

      2    1   -1          -8    -6      9

      4   -3    2         -10    1      7

With the relation AB, hence solve the system of equations 

3x - 2y + 3z = 8; 2x + y - z = 1; 4x - 3y + 2z = 4.     (3)

22) Find inverse of 

1    3      -2

-3   0      -1

2    1       0       (3)

23) Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius R is 4r/3.     (3)

24) An open topped box is to be made by removing equal squares from each corner of a 3m by 8m rectangle sheet of aluminium and folding up the sides. Find the volume of the largest such box.      (3)

25) ∫ (3x +5)/(x³- x² - x +1) dx.     (3)

26) A, B, C throw a die one after the other in the same order till one of them gets a 6 and wins the game. Find their respective probability of winning if A starts the game.    (3)

27) Find the cost of increasing from 100 to 200 units if the marginal cost in Rs per unit is given by the function MC = 0.003 x²- 0.01x + 2.5.     (1)

28) Given that the observation are (9,-4),(10,-3),(11,-1), (13,1),(14,3), (15,5),(16,8), find the two lines of regression. Estimate the value of y when x= 13.5.   (2)

29) Find the regression coefficients bᵧₓ and bₓᵧ and the two lines of regression for the following data.

X: 2   6   4   7   5

Y: 8   8   5   6   2

Also compute the correlation coefficient.    (2)

30) If the demand function is given by x= (600- p)/8, where the price is Rs p per unit and the manufacturer produces x units per week at the total cost of Rs x²+ 78x + 2500, find the value of x for which the profit is maximum.     (3)

31) The fixed cost of a new product is Rs 35000 and the variable cost per unit is Rs 500. If the demand function p= 5000 - 100x, find the break even value/s.    (3)

32) A toy company manufacturers two types of dolls, A and B. Market test and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demands for the dolls of type B is atmost half of that for dolls of type A. Further, the production level of type A can exceed three timee the production of dolls of other type by atmost 600 units. If the company makes a profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit ?     (3)