MATH- TEST PAPERS: 2022

Friday, 30 December 2022

MODEL TEST PAPER -7(ICSC)

F M: 80 Time: 2.30 hrs

Section A

(Attempt all questions). 4x10= 40

1) a) Mr. D. K Vhora gets ₹6455 at the end of a year at the rate of 14% p.a. in a recurring deposit account. Find the monthly installment.

b) Solve the inequation and represent the solution set on the number line -2/3< -x/3 + 1≤2/3; x belongs to R.

c) Solve the following quadratic equation using formula (Correct to three significant figures): x² -3(x+3) = 0. (4+3+3)

2a) Prove the following identity:

Cosecx/(cosec x -1) + cosecx/(cosecx +1) = 2 sec²x.

b) A box contains 25 cards, numbered from 1 to 25. A card is drawn at random from the box. Find the probability that the number on the card is (i) even ii) Prime iii) multiple of 6.

c) i) Find the mean proportional of 8.1 and 2.5.

ii) Find the third proportion of 21/4 and 7. (3+4+3)

3a) The point p(3,4) is reflected on p' in the x-axis and o' is the image of o (origin) in the line pp' find:

i) The coordinates of p' and o'.

ii) The length of the segment pp' and oo'.

iii) The perimeter of the quadrilateral pop'o' (use graph sheet for this question)

b) If A= 3 4 B= 1 y & C= 7 0

5 x 0 1 10 5 Find x and y if 2A + B= C.

c) If A=(-4,3), B=(8,-6) in what ratio is the line joining AB, divided by the x-axis ? (3+4+3)

4). (4+3+3)

a)Find the mean by shortcut method:

Class interval frequency

05-10 10

10-15 6

15-20 4

20-25 12

25-30 8

30-35 4

35-40 2

40-45 1

45-50 3

b) For what value of k the equation (k -12)(x²+ 2x)+ 2= 0 has equal roots.

c) Find the mode and the median:

x: 10 11 12 13 14 15

f: 1 4 7 5 9 3

SECTION B

(Attempt only 4) 4x10= 40

5) a) Construct incentre of a triangle ABC, where AB= 6cm, BC= AC = 4cm.

b) Two vertices of a triangle are (3,5) and (-7,4). Find the third vertex, given that the centroid is (2,-1).

c) Is a, b,c are in continued proportion then prove that: (a+ b)/(b+ c) =a²(b - c)/c²(a - b).

6a) 90 pupils in a school have heights as below:

Height number of pupils

121-130 12

131-140 6

141-150 30

151-160 20

161-170 14

171-180 8

Using graph paper draw an ogive and estimate the median and inter quartile range.

b) If cosecx + cot x = p, then prove that cosx= (p²-1)/(p²+1). (6+4=10)

7)a) Solve the and draw the number line for the following in equation:

3≥ (x -4)/2 + x/3 ≥ 2; x belongs to R.

b) A letter is chosen from the words MONEY DATE. what is the probability that the word is chosen is i) a vowel b) a constant.

c) Find the matrix X If AX = B If

A= 4 B= -4 8

1 -1 2 (4+3+3=10)

8)a) SOLVE: 16{(a- x)/(a+ x)}³= (a+ x)/(a- x). (3+3+4=10)

b) There are two children in a family, find the probability that,

i) there is at least one girl in the family.

ii) Younger child is a boy.

c) A two digit number is 4 times the sum and three times the product of its digits. Find the numbers.

9) Shyam has a cumulative bank account and deposited ₹600 per month for a period of 4 years. If he gets ₹ 5880 as intrest at the time of maturity, find the rate of interest.

b) Solve:(using formula) 2x - 1/x = 7 (correct to 2 decimal places)

c) Use a graph paper and draw a histogram from the following data, hence estimate mode.

Class interval Frequency

00-05 10

05-10 14

10-15 28

15-20 42

20-25 50

25-30 30

30-35 14

35-40 12 3+3+4

10). 4+3+3

a) Ages of two persons A and B are in the ratio 4:3 five year hence, the ratio of their ages will change to 9:7. Find their present ages.

b) Find X and Y If the relation AB = C where A= X 3X B= 2 C= 5

Y 4Y 1 12

c) Prove: 1 - cos²x/(1+ sin x) = sinx.

Saturday, 24 December 2022

Revision PAPER (X) CBSC

28/11/23

1) If tanx = a/b, then which of the following is the value of (a sinx + b cosx)/+a sinx + b cosx).

a) (a²+ b²)/(a²- b²)

b) a/√(a²- b²)

c) b/(a²- b²)

d) √(a²+ b²)

2) Evaluate: (sinx + cosex)²+ (cosx + Secx)²= tan²x + cot²x +7.

3) If cosecx + cotx = y, find the value of cosx.

4) Eliminate k

x= a seck, y= b tank.

5) If the points (a,0),(0,b) and (1,1) are collinear, then show that 1/a + 1/b =1.

6) A number is chosen at random from the numbers 1,2,...,15. Find the probabilities that the chosen number is

a) even

b) multiple of 3

c) multiple of 4

d) multiple of 3 but not 4

e) multiple of both 3 and 4.

7) In the figure BC= 5cm,

angle B= 90°, AB= 5AE, CD= 2AE and AC= ED. Find the length of EA, CD, AB, and AC.

8) The wheel of a cart is making 5 revolution per second. If the diameter of the wheel is 84cm, find its speed in kmph. Give your answer, correct to nearest km.

9) The inner dimensions of a closed wooden box are 2m, 1.2m and 0.75m. The thickness of the wood is 2.5cm. find the cost of wood required to make the box if 1 m³ of wood cost &5400.

10) 3) A cylinder is surmounted by a cone at one end and a hemisphere at the other end, Given that common radius=3.5 cm, the height of the cylinder is 6.5 cm and the total height 12.8cm, calculate the volume of the solid correct to the nearest integer. 376 cm³

27/11/23

1) x takes 3 hours more than y to walk 30 km. If x doubles his pace, he is ahead of y by 3/2 hours. Find their speeds of walking. 10/3,5 kmph

2) In a cyclic quadrilateral ABCD, angle A= (2x+4)°, angle B= (y+3)°, angle C= (2y +10)°, angle D= (4x -5)°. Find the four angles. 70,53,110,127

3) 2x+ 3y=7; (k+1)x + (2k -1)y = 4k +1. Find the value of k for which system of equations have infinitely many solution. 5

4) Solve for x,y: a/x - b/y =0; ab²/x + a²b/y = a²+ b², where x,y≠0. a,b

5) Solve: abx²+ (b²- ac)x - BC= 0. c/b, -b/a

6) if the equation (1+ m²)x²+ 2mcx + (c²- a²)=0 has equal roots, prove that c²= a²(1+ m²).

7) The length of a hall is 5m more than its breadth . If the area of the floor of the hall is 84 m², what are the length and breadth of the hall ? 7, 12

8) Find the sum of the first 25 terms of an AP, whose nth term is given by aₙ= 7 - 3n. - 800

9) In figure AB diameter and AC is a chord of a circle such that angle BAC =30°. The tangent at C intersects AB produced in D. Prove that BC= BD.

10) Two chords AB and CD of a circle intersect each other at P outside the circle. If AB =5cm, BP= 3cm and PD =2 cm, find CD. 10cm

11) if x is an acute angle and tanx + cotx=2, find the value tan⁷x + cot⁷x.

12) If A and B are complementary angles, prove that (sinA+ sinB)²= 1+ 2 sinA cosA.

13) A bag contains 5 red balls , 8 white balls, 4 green balls and 7 black balls . If one ball is drawn at random , find the probability that it is

a) black.

b) red

c) not green

14) What is the probability that a leap year has 53 Sunday and 53 Mondays ?

15) If r₁ and r₂ be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new Sphere is (r₁³ + r₂³)¹⁾³.

25/11/23

1) The vertical and slant height of a cone are 24cm and 25cm. Calculate (i) curved surface (ii) volume of the cone.

2) The diameter of an iron sphere is 18cm. The sphere is melted and is drawn into a long wire of uniform cross section. If the length of the wire is 108m. Find its diameter.

3) Find the value of m if (x - m) is a factor of 3x³ + 2x² - 19x + 3m.

4) simplify:2tan 40/cot 50 - cosec 61/sec 29

5) Solve: 4x² - 4ax +(a² - b²) = 0

6) The horizontal distance between two towers is 140m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 60m. Find the height of the first tower.

7) Prove:

(1+cosA)/(1-cosA) = (cosecA + cotA)²

8) Find the sum of all multiples of 11 between 100 and 400. 6831

9) How many terms of the series {27+24+21+...} must be added to get the sum 132? 8 or 11

10) Of the point (x,y) be equidistant from the points (a+ b, b - a) and (a - b, a+ b), show that bx = ay.

11) Show (1+ Secx + tanx)(1- cosecx + cotx)= 2.

12) In the figure BOA is a diameter of a circle and the tangent at a point P meets BA extended at T.

If Angle PBO=30° , then find angle PTA. 90

13) In the figure O is the centre of a circle of radius 5cm,

T is a point such that OT= 13cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB. 24

10) Using step deviation method, Calculate the mean of the following frequency distribution:

Class. Frequency

50-60 9

60-70 11

70-80 10

80-90 14

90-100 8

100- 110 12

110-120 11

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

CLASS- interval. Frequency

0-10 2

10-20 8

20-30 10

30-40 5

40-50 4

50-60 3

) The marks scored by 40 pupils of a class in a test were as follows:

CLASS: 0 1 2 3 4 5

No. of pupils: 2 4 5 14 11 4

Calculate the mean mark.

MODEL TEST PAPER -6 (ICSE)

1) Mamta has a cumulative Time Deposit Account in a bank. She deposits Rs800 per month and gets Rs15198 as maturity value. If the rate of interest be 7% p.a., find the total time for which the account was held.

2) The vertical and slant height of a cone are 24cm and 25cm. Calculate (i) curved surface and (ii) volume of the cone.

3) The diameter of an iron sphere is 18cm. The sphere is melted and is drawn into a long wire of uniform cross section. If the length of the wire is 108m. Find its diameter.

4) The marks scored by 40 pupils of a class in a test were as follows:

CLASS: 0 1 2 3 4 5

No. of pupils: 2 4 5 14 11 4

Calculate the mean mark.

5) Solve: 2 ≤ 2x - 3 < 5, x belongs R and mark it on the number line.

6) Find the value of m if (x - m) is a factor of 3x³ + 2x² - 19x + 3m.

If = -3 2 B= x C= -5

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

8) What must be added to each of the number 7, 25, 19, and 35 so that the resulting numbers are in proportion ?

9) simplify:

2tan 40/cot 50 - codec 61/sec 29

10) Using the remainder theorem, find the remainder when 7x³ + 5x² - 4x - 1 is divided by (x +1).

11) Solve: 4x² - 4ax +(a² - b²) = 0

12) The horizontal distance between two towers is 140m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 60m. Find the height of the first tower.

13) Find the Equation of a straight line parallel to y-axis and passing through the point (-3,5).

14) Using step deviation method, Calculate the mean of the following frequency distribution:

Class. Frequency

50-60 9

60-70 11

70-80 10

80-90 14

90-100 8

100- 110 12

110-120 11

16) Prove:

(1+cosA)/(1-cosA) = (cosecA + cotA)²

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

CLASS- interval. Frequency

0-10 2

10-20 8

20-30 10

30-40 5

40-50 4

50-60 3

Monday, 12 December 2022

Quick Revision - IX

Week -1

EXPONENTIAL

1) Find the value of:

B) 9⁻³ x 16¹⁾⁴/6⁻² x (1/27)⁻⁴⁾³. 8

C) For what value of x, the relation 2ˣ = 3⁻ˣ is true. When x is 0

D) If (20)²ˣ = 49 then find the value of (20)⁻ˣ. ±1/7

E) If x= 5, y =3 then find the value of (x +y)ˣ⁾ʸ. 32

F) Find: (2²ⁿ - 3. 2²ⁿ ²)(3ⁿ⁻² - 2.3²ⁿ⁻²)/{3ⁿ⁻⁴(4ⁿ⁺³ - 2²ⁿ)}. 1/4

G) {(0.3)¹⁾³(1/27)¹⁾⁴(9)¹⁾⁶(0.82)²⁾³}/{(0.9)²⁾³(3)⁻¹⁾²(1/3)⁻²(243)⁻¹⁾⁴}. 3/10

H) (16)⁻³⁾⁴ x (32)¹⁾⁵ x 5⁰. 1/4

2) Solve:

A) 4ˣ = 8³. 9/2

B) 4ˣ⁺² = 2²ˣ⁺3 +2. -1

C) 4ˣ - 3.2ˣ⁺² + 2⁵ = 0. 2

D) xʸ = yˣ , x² = y³. 27/8, 9/4

E) 2ˣ⁺² + 2ˣ⁻¹ = 9. 1

F) 3²ˣ+ 9 = 10. 3ˣ. 2 or 0

G) 9. 812ˣ = 1/27ˣ⁻². 4/7

H) 4ˣ⁺² + 2²ˣ⁺³ = 96. 1

I) 3²ˣ⁻⁵ 9ˣ⁻² = 4. 5/2

J) 6²ˣ⁺⁴ = 3³ˣ. 2ˣ⁺⁸. 4

3) Simplify:

(A) xˡ/xᵐ)ˡ⁺ᵐ.(xᵐ/xⁿ)ᵐ⁺ⁿ.(xⁿ/xˡ)ⁿ⁺ˡ. 0

B) {(p² - 1/q²)ᵖ (p - 1/q)ᑫ⁻ᵖ/{(q² - 1/p²)ᑫ (q+ 1/p) ᑫ⁻ᵖ}. (p/q)ᵖ⁺ᑫ

C) [1- {1- (1- x³)⁻¹}⁻¹]⁻¹⁾³ when x = 0.1. 0.1

D) ⁵√(243)³. 27

E) ₓ√x = √(x)ˣ. 0, 4

F) {(2¹⁾³. 8²⁾³. 6⁻⁵⁾⁴. 3⁻³⁾⁴)/(9⁻¹.³√16)}⁻⁴. 2

G) (9.(4ˣ)²)/(16ˣ⁺¹ - 2ˣ⁺¹. 8ˣ). 9/14

4) Prove:

A) 1/(1+ xᑫ⁻ᵖ + xʳ⁻ᵖ) + 1/(1+ xᵖ⁻ᑫ + xᑫ⁻ʳᑫ) = 1

B) If aˣ = m, aʸ = n and a²= (mʸˣ)ᶻ then show that xyz = 1.

C) If 64ˣ =48 ʸ = 36ᶻ then prove 1/x + 1/z = 2/y.

D) If 2ᵃ = 3ᵇ = 12ᶜ then prove that ab = c(a+ 2b).

E) If pˣ = qʸ = rᶻ then prove pqr = 1.

F) If aˣ = bʸ = cᶻ and b² = ac, then prove 1/x + 1/z = 2/y.

G) If (3.6)ᵖ =(.36)ᑫ = (1)ʳ then prove 1/p = 1/q + 1/r.

H) If x¹⁾ᵃ = y¹⁾ᵇ = z¹⁾ᶜ and xyz = 1 then prove a+ b + c= 0.

I) If x² = y³ then prove (x/y)³⁾²+ (y/x)²⁾³ = x¹⁾² + y⁻¹⁾³.

J) If xʸ = yˣ then prove that (x/y)ˣ⁾ʸ = xˣ⁾ʸ ⁻¹ .

M) If x¹⁾³ + y¹⁾³+ z¹⁾³ = 0 then prove that (x + y+ z)³ = 27 xyz.

N) If x= 2¹⁾³ + 2⁻¹⁾³ then prove 2x³ = 6x +5.

O) a= 5 - 5²⁾³ - 5¹⁾³ then prove a³ - 15a² + 60a = 20

Friday, 2 December 2022

MODEL Test paper 5(x) ICSC

SECTION A (40 MARKS)

Question 1). 2x5= 10

a) If Ram bought a watch for ₹2688 which includes 12% GST and a shirt for ₹440 which includes 10% GST. Find the printed price of the watch and shirt together, without GST

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

c) If a 3a 2 5

b 4b 1 = 12 , find a and b.

d) The mean of numbers 6,y, 7, x and 14 is 8. express y in terms of x. y= 13 - x

e) Given f(x) = x/(x² -1). Find f(1/3). 2/3

Question 2. 10 marks

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

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

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

Question 3 Marks 10

a) Solve 1 < 3x -3≤ 12, x belongs to R and mark it on a number line.

b) Calculate the mean, median, and mode of the following numbers 11, 12, 10, 11, 12, 13, 14, 13, 15, 13.

c) Find the sum of the numbers between 2 to 200 divisibile by 4

Question 4 marks 10

a) If cosA= 4/5 and cos B= 24/25 ; evaluate

i) cosec² A.

ii) cot A + cot B

b) On a map drawn to a scale of 1:12500 a triangular plot of land has the following measurements: PQ = 10cm, QR= 8cm, Ang PRQ=90° . Calculate

i) The actual length of PQ in km.

ii) the area of the plot in square kilometre.

c) Find the probability of extra Sunday in February in leap year

SECTION B) 40 MARKS

Answer any four questions

Question 5. Marks 10

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

b) John has a recurring deposit account of ₹1000 per month in a bank. What will he get after 12 months if the rate be 9% p.a

c) Find the mean of the following frequency distribution:

Class Interval. Frequency

0 - 30 3

30-60 7

60-90 15

90-120 14

120-150 7

150-180 4

Question 6. Marks 10

a) Two dice are thrown. Find the probability

A) The sum of the scores is 9

B) the product of the score is 12

C) the score on second die is greater than first

D) the sum of the score is a multiple of 4

E) the sum of the score is a perfect square.

b) Prove that: sinA/(1-cotA) + cosA/(1-tanA)= sinA + cosA

Question 7. Marks 10

a) A straight line passes through the points A(-2,8) and B(10, -4). It intersect the co-ordinates Axes point E and F. P is the midpoint of the segment EF, Find:

I) the equation of the line.

ii) the coordinates of the E and F.

iii) The coordinates of the point P.

b) 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 the rows was doubled and the number of seats in each row was reduced by 15, the total number of seats increased by 400. Find

I) the number of rows in the original arrangement.

ii) the number of seats in the Auditorium after rearrangement.

Question 8. Marks 10

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

Class Interval. Frequency

0 - 20 3

20- 40 8

40- 60 10

60- 80 6

80-100 4

100-120 3

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

I) the width of the river and.

ii) the height of the tree.

Question 9. Marks 10

a) If A=. 4 -2 B= 0 2 C= -2 0

6 -3 1 -1 1 -3

Find A² - A + BC

b) 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 shot, each of which is a sphere of a diameter 5mm are dropped Into the vessel, 1/3 of the water flows out. Find the number of lead shots dropped into the vessel.

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

Thursday, 1 December 2022

XI MODEL TEST PAPER 2

Question 1. 1x10= 10

i) (i¹⁸ + (1/i)²⁵)³ is
A) 1 B) -1 C) 1 + i D) 2(1- i)

ii) If one root of 2x² - 5x + k= 0 be double the other, find the value of k

A) 2 B) 7 C) 9 D) 25/9

iii) Which term of the series 5, 8, 11, ....is 320 ?

A) 106 B) 110 B) 100 D) 98

iv) How many 9-digits numbers of different digits can be formed?

A) 3298760 B) 3456890

C) 3265920 D) 345690

v) 2³ˣ -1 is divisible by

A) 5 B) 6 C) 7 D) 8

vi) Cos 83 - cos 17 =

A) sin 50 B) sin 33 C) sin 50 sin 33 D) - sin 50 sin 33

vii) If (x+3, y -5) = (5,0) then x and y are

A) 2,5 B) -2,5 C) 2, -5 D) - 2, -5

viii) If f(x)= √x , then the value of f(125)/{f(16)+ f(1)} is

A) 0 B) 1 C) 2 D) none

ix) lim ₓ→₁ {(x²-1)/(x-1)} is

A) 1 B) -1 C) 0 D) 2

x) dy/dx of 1/√x³ is

A) x B) x³ C) 2x³ D) none

Question 2 (2x10)= 20

i) Which term of the progression 19, 91/5, 87/5,......is the first negative term?

ii) Find the coefficient of x⁴ in the expansion of (x/2 - 3/x²)¹⁰

iii) Find the value of cos (15/2)°

iv) There are 3 copies each of 4 different books. Find the number of ways of arranging them on a shelf

v) Value of (cos 10+ sin 20)/(cos 20+ sin 10)

vi) Prove by Induction: 1+5+12+22 +35+....+ n/2 (3n-1)= n²(n +1)/2

Solve: tanx + sec x =√3

vii) Find the domain and range of (4- x)/(x -4)

viii) lim ₓ→π/2 cotx/(π/2 - x).

ix) Find dy/dx with the help of 1st principal sin x

Find dy/dx of cos(sin x²)

x) The angle between the lines whose slopes are -3 and -1/2

Find the centre and radius of the circle of 2x² + 2y²= 3x - 5y +7.

Question 3 10x3 = 30

i) Solve: | 1 - i|ˣ = 2ˣ.

If x be real, Prove that the value of (2x² - 2x +4)/(x² - 4x +3) can not lie between -7 and 1.

ii) The product of first three terms of a GP is 1000. If we add 6 to its second term and 7 to its third term, the resulting three terms form an AP. Find the terms of the GP.

iii) Find the number of words which can be formed by taking two alike and two different letters from the word COMBINATION.

iv) 2 cos a = x + 1/x , Prove 2 cos 2a = x³ + 1/x³

v) Express cos 6x in terms of cos 3x

vi) Solve 3 - 2 cosx - 4 sinx - cos2x + sin2x = 0

vii) If f(x)= log {(1-x)/(1+ x)}, show that f(a) + f(b)= f{(a+ b)/(1+ ab)}.

vii) If y = 1/(a - z) show that dz/dy = (z - a)²

ix) Find the equation of the line joining the origin to the point of intersection of 4x + 3y = 8 and x+ y = 1.

Find the equation of the straight line which passes through the point (4,5) and is perpendicular to the line 3x - 2y +5= 0.

x) Find the Cartesian equation of the curve whose parametric equations is x= t, y= t².

The point diametrically opposite to the point P(1,0) on the circle x² + y² + 2x + 4y - 3= 0 is
A) (3,-4) B) (-3,4) C) (-3,-4) A) (3,4)

Question 4. 4x 5 = 20

i) Two samples of sizes 50 and 100 are given. The mean of these samples are respectively 56 and 50. Find the mean of size 150 by combining.

ii) Find the standard deviations of

Class: 0-4 4-8 8-12 12-16
F: 4 8 2 1

iii) In Two factories A and B engaged in the same industrial area, the average weekly wages (in rupee) and the standard deviations are as follows:
Factory Average S.D   workers
   A            34.5        5        476
   B            28.5       4.5     524
a) Which factory, A or B, pays out a larger amounts as weekly wages?
b) which factory, A or B, has greater variability in individual wages?

iv) Find the value of 8th decile and 75th percentile from the following:
Class.                Frequency
10-14                      3
15-19                      7
20-24                    16
25-29                    12
30-34                     9
35-39                     5

Determine the mode:
Marks       No. Of students
00-10              5
10-20              12
29-30              14
30-40              10
40-50               8
50-60.              6

Wednesday, 16 November 2022

XI. MODEL TEST PAPER -1

MODEL TEST PAPER -1

Question -1. 1x 10= 10

i) Find the GM between a³b and ab³

A) ab B) ab²C) a²b D) a²b²

ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is

A) 0 B) 0i C) o+0i D) 1

iii) If (A xB)={(3,2),(3,4),(5,2),(5,4)} find B

A) |2,4| B) [2,4] C) (2,4) D) {2,4}

iv) (A∩B) U(A - B) is

A) A B) B C) A' D) (AB)'

v) In how many ways can the letters of the word BANANA be arranged?

A) 60 B) 72 C) 144 D) 210

vi) Two dice are thrown simultaneously. The probability that the number on both the dice are same

A) 1/6 B) 1/18 C) 1/36 D) none

vii) If cosx= -1/2, what is the general value of x.

A) 30 B) 60 C) 90 D) none

viii) If the distance between the points (-3,3) and (4,y) be 5√2 units, find the value of y.

A) 3 B) 4 C) 5 D) 6 units

ix) Find the equation of straight line whose x-intercept and y-intercept are 3 and - 4 respectively.

A) x/3 + y/4 = 1 B) x/3 - y/4 = 1

C) - x/3 + y/4 = 1 D) x/3 + y/4 = -1

x) The middle term of (x/y - y/x)¹⁰ is

A) 5th B) 6th C) 7th D) 4th

Question 2. 2x10= 20

i) Find the value of sin 105°

ii) Find the nth term of the GP 12, 4, 4/3, 4/9

iii) Find the modulus of (1+ 3i)/(2- i)

iv) The mean deviation of 7,8, 4, 13, 9, 5, 16

v) Find the value of cos(-1170).

vi) Find the radius and centre of the circle x²+ y² = 36 is

vii) Find the number of arrangement of word MONDAY.

Find the number of arrangement of x²y³z⁴

viii) Find the coefficient of x⁷ in the expansion of (x² + 1/x)¹¹

ix) Find the value of sin 75 + cos 75

x) Find the area of the triangle whose vertex are (3,2,(4,-2),(-4,7) respectively.

Question 3. 3x10= 30

i) Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.

Find the equation of the hyperbola whose vertices are are (0,±3) and the foci are (0,±5)

ii) Find the square root of 5 - 12i

iii) a) lim ₓ→₁ (x²-1)/(x +1).

b) lim ₓ→₂ (x³ +1)/(x²+1+ 3x)

iv) How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7 .

Prove cos 3π/32 = 1/2 √[2+√{2+ √(2+ √2)}].

v) A die is thrown and at the same time a card is drawn from a pack of 52 playing cards. Find the probability of getting 5 and Ace of hearts.

vi) find dy/dx: y=√{(1- cos 2x)/(1+ cos 2x)).

vii) if the three consecutive vertices of a parallelogram be A(3,4-1), B(7,10,-3) and C(5,-2,7). Find the fourth vertex D.

Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.

viii) √3 sin10 + sin 20 = cos 50.

Prove cos 9 cos 27 cos 63 cos 81 = 1/16

ix) Prove that the straight lines is concurrent x+ y+5= 0, x- y+1= 0, 3x - y+ 7= 0.

x) If 2 cos a= x+ 1/x, then show that 2 cos 3a = x³ + 1/x³.

A 4 digited number is written by the digits 1,2,3 and 4 and where no digits is repeated in any number. Find the probability that the number is

A) odd

B) mutiple of 4.

Question 4. 4x5= 20

i) The midpoints of the sides of a triangle are (1,5,-1),(0,4,-2) and (2,3,4) find its vertices

Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .

ii) In how many can a committee of 6 persons be formed taken atleast 3 gentlemen and 2 ladies from 10 gentlemen and 7 ladies, where two particular ladies refuse to serve in the same committee together.

iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.

If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.

iv) In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.

Which term in the expansion of (2x² - 1/x)¹² is independent of x ? Find the value of that term.

v) The coordinates of the points A, B and C are (6,3),(-3,5) and (4,-2) respectively and that of the point P(x,y). Show that the ratio of the area of PBC and ABC is |(x+ y -2)/7|.

Find the equation of the straight line which passes through the point of intersection of the straight lines y- 2x +2= 0 and y - 3x +5= 0 and is at a distance of 7/√2 units from the origin.

Question 5. (20 marks)

i) If the standard deviations of the numbers 2, 3, 2x, 11 is 35, find the possible value of x. (2)

Find the standard deviation of

X: 10 15 18 20 25

F: 3 2 5 8 2

ii) Find the mean deviation of:

Class: 20-30 30-40 40-50 50-60

F: 3 7 12 8 (3)

iii) Find the combined standard deviations of 10 numbers when mean of x and y are 20, 30 respectively. And standard deviations of x and y are 3 and 4 respectively. (4)

iv) Two dice are thrown simultaneously. Find the probability of getting

A) a doublet

B) a multiple of 3 as the sum

C) multiple of 3 and 5

D) multiple of 3 or 5. (4)

v) Find the derivative from the first principle. √(4-x). (3)

vi) Find the equation of a circle concentric with the circle 2x²+ 2y² - 6x + 8y +1= 0. (4)

CBSE- MODEL TEST PAPER -1 (XI)

MODEL TEST PAPER -1

Question -1. 1x 10= 10

i) Find the GM between a³b and ab³

A) ab B) ab²C) a²b D) a²b²

ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is

A) 0 B) 0i C) o+0i D) 1

iii) If (A xB)={(3,2),(3,4),(5,2),(5,4)} find B

A) |2,4| B) [2,4] C) (2,4) D) {2,4}

iv) (A∩B) U(A - B) is

A) A B) B C) A' D) (AB)'

v) In how many ways can the letters of the word BANANA be arranged?

A) 60 B) 72 C) 144 D) 210

vi) Two dice are thrown simultaneously. The probability that the number on both the dice are same

A) 1/6 B) 1/18 C) 1/36 D) none

vii) If cosx= -1/2, what is the general value of x.

A) 30 B) 60 C) 90 D) none

viii) If the distance between the points (-3,3) and (4,y) be 5√2 units, find the value of y.

A) 3 B) 4 C) 5 D) 6 units

ix) Find the equation of straight line whose x-intercept and y-intercept are 3 and - 4 respectively.

A) x/3 + y/4 = 1 B) x/3 - y/4 = 1

C) - x/3 + y/4 = 1 D) x/3 + y/4 = -1

x) The middle term of (x/y - y/x)¹⁰ is

A) 5th B) 6th C) 7th D) 4th

Question 2. 2x110= 20

i) Find the value of sin 105°

ii) Find the nth term of the GP 12, 4, 4/3, 4/9

iii) Find the modulus of (1+ 3i)/(2- i)

iv) The mean deviation of 7,8, 4, 13, 9, 5, 16

v) Find the value of cos(-1170).

vi) Find the radius and centre of the circle x²+ y² = 36 is

vii) Find the number of arrangement of word MONDAY.

Find the number of arrangement of x²y³z⁴

viii) Find the coefficient of x⁷ in the expansion of (x² + 1/x)¹¹

ix) Find the value of sin 75 + cos 75

x) Find the area of the triangle whose vertex are (3,2,(4,-2),(-4,7) respectively.

Question 3. 3x10= 30

i) Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.

Find the equation of the hyperbola whose vertices are are (0,±3) and the foci are (0,±5)

ii) Find the square root of 5 - 12i

iii) a) lim ₓ→₁ (x²-1)/(x +1).

b) lim ₓ→₂ (x³ +1)/(x²+1+ 3x)

iv) How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7 .

Prove cos 3π/32 = 1/2 √[2+√{2+ √(2+ √2)}].

v) A die is thrown and at the same time a card is drawn from a pack of 52 playing cards. Find the probability of getting 5 and Ace of hearts.

vi) find dy/dx: y=√{(1- cos 2x)/(1+ cos 2x)).

vii) if the three consecutive vertices of a parallelogram be A(3,4-1), B(7,10,-3) and C(5,-2,7). Find the fourth vertex D.

Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.

viii) √3 sin10 + sin 20 = cos 50.

Prove cos 9 cos 27 cos 63 cos 81 = 1/16

ix) Prove that the straight lines is concurrent x+ y+5= 0, x- y+1= 0, 3x - y+ 7= 0.

x) If 2 cos a= x+ 1/x, then show that 2 cos 3a = x³ + 1/x³.

A 4 digited number is written by the digits 1,2,3 and 4 and where no digits is repeated in any number. Find the probability that the number is

A) odd

B) mutiple of 4.

Question 4. 4x5= 20

i) The midpoints of the sides of a triangle are (1,5,-1),(0,4,-2) and (2,3,4) find its vertices

Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .

iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.

If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.

iv) In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.

Which term in the expansion of (2x² - 1/x)¹² is independent of x ? Find the value of that term.

v) The coordinates of the points A, B and C are (6,3),(-3,5) and (4,-2) respectively and that of the point P(x,y). Show that the ratio of the area of PBC and ABC is |(x+ y -2)/7|.

Find the equation of the straight line which passes through the point of intersection of the straight lines y- 2x +2= 0 and y - 3x +5= 0 and is at a distance of 7/√2 units from the origin.

Question 5. (20 marks)

i) If the standard deviations of the numbers 2, 3, 2x, 11 is 35, find the possible value of x. (2)

Find the standard deviation of

X: 10 15 18 20 25

F: 3 2 5 8 2

ii) Find the mean deviation of:

Class: 20-30 30-40 40-50 50-60

F: 3 7 12 8 (3)

iii) Find the combined standard deviations of 10 numbers when mean of x and y are 20, 30 respectively. And standard deviations of x and y are 3 and 4 respectively. (4)

iv) Two dice are thrown simultaneously. Find the probability of getting

A) a doublet

B) a multiple of 3 as the sum

C) multiple of 3 and 5

D) multiple of 3 or 5 (4)

v) Find the derivative from the first principle. √(4-x). (3)

vi) Find the equation of a circle concentric with the circle 2x²+ 2y² - 6x + 8y +1= 0. (4)

Saturday, 12 November 2022

MODEL TEST PAPER- 1, 3,4 -XI(ISC) 22/23

MODEL TEST PAPER 4

1) Value of [i¹⁸ + (1/i)²⁵]³ is

A) 2 B) 1+ i C) 1 - i D) 2(1 - i). (1)

2) Find modulus of (1+ i)²/(3- i). (2)

3) If the roots of the equation ax² + bx + c= 0 be in the ratio 3:4, show that 12b² = 49ac. (2)

4) Simplify ) ¹⁰C₅ + ¹⁰C₄. (1)

5) What is the probability of getting 3 white balls in a draw of 3 balls from a box containing 5 white and 4 black balls? (2)

6) If n be a positive integer, show that n(n +1) is an even positive integer? (1)

7) If f(x)= (x -1)/(2x² - 7x+5)

find f '(1). (4)

8) Four persons are chosen of random from a group consisting of 3men, 2 women and 4 children. Find the probability that exactly two of them will be children. (4)

If m,n are the roots of x²+px+q = 0, then find m⁴+n⁴ in terms of p and q.

9) If 100 times the 100th term of an A. P with non-zero common difference equals to the 50 times 50th term, then find 150th term. (4)

10) Find the total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets atleast one ball. (4)

11) Find the number of ways one or more balls can be selected from 10 white, 9 green and 7 black balls. (3)

12) Find Domain and Range of

2x/(1+2x). (4)

13) Out of 64 students, the number of students taking maths is 45 and the number of students taking both maths and stats is 10. Then find the number of students taking only Statistics. (3)

14) If x=π/7, then find the value of

(tanx. tan2x + tan 2x. tan 4x+

tan 4x. tanx). (2)

Find the coordinates of the point on the y-axis which is equidistant from the point (-5,4) and (3, -2).

15) If the area of the triangle with vertices (p+1,1),(2p+1,1),(2p+2,2p) is 9 square units. Then find the value of p. (4)

16) If cos(x-y)+ cos(y-z)+ cos(x-z)= -3/2 then prove sinx+sin y+sin z= cos x+ cos y+ cos z. (3)

17) Find the co-ordinates of the foci, the eccentricity and the Equations of the directrix of the hyperbola 4x² - 9y² = 36. (4)

Find the focus, the Equation to the directrix and the length of the latus ractum of the parabola y²+22= 4x+ 4y.

18) Find the Quartiles and Quartile Deviation of the daily wages (in Rs.) of 7 persons given bellow:

12, 7, 15, 10, 19, 17, 25. (4)

19) Calculate Karl Pearson's coefficient of correation between X and Y of the following:

X: 5 7 1 3 4

Y: 2 2 4 5 6 (4)

Find Mode of

C. I: 10-20 20-30 30-40 40-50 50-60

F: 5. 12. 18 6 9

20) Express sin 10x in terms of 5x. (4)

21) a) Find the middle term of (2a - b/3)⁹

b) Expand (2/x - x/2)⁵. 2+2

22) 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). (4)

The 5th , 8th, and 11th term of a GP are P, Q and S respectively, show that Q²= PS.

23) Find the value of sin 150°. (2)

24) Solve: 2 cos x + cos 3x = 0. (4)

Sec x - cosecx = 4/3.

25)a) if f(x)= x³/2 - x²/2 + x - 16, find f(1/2).

b) if f(x)= 3x²+2 and g(x)= x+1 then find fog. 2+2

26) differentite: (x+1)(2x-3). (2)

MODEL TEST PAPER -3 (XI)

Question 1. 1x10= 10

i) For any set A, (A')' is equal to

A) A' B) A C) null set D) None

ii) If A={1,2,4}, B{2,4,5}, then (A - B) x (B - A) is

A) {(1,2),(1,5),(2,5)} B) {(1,4)}

C) = (1,4) D) none

iii) If D, G And R denotes respectively the number of degrees, grades and radians in an angle then,

A) D/100= G/90= 2R/π

B) D/90= G/100= R/π

C) D/90= G/100= 2R/π

D) D/90= G/100= R/2π

iv) If tan a= x - 1/4x, then sec a - tan a is

A) -2x, 1/2x B) -1/2x, 2x C) 2x D) 2x, 1/2x

v) The value of sin²75 - sin²15 is

A) 1/2 B)√3/2 C) 1 D) 0

vi) The complete set of values of k, for which the quadratic equation x² - Kx + k+2= 0 has equal Roots, consists of

A) 2+√12 B) 2±√12 C) 2- √12 D) - 2- √12

vii) The number of permutations of n different things taking r at a time when 3 particular things are to be included is

A) (n-3) P (r-3) B) (n-3) P r

C) nP (r-3) D) r!. (n-3) P (r-3)

viii) If 7th and 13th term of an AP is 34 and 64 respectively, then its 18th term is

A) 87 B) 88 C) 89 D) 90

ix) One card is drawn from a pack of 52 playing cards. The probability that is the card of a king or spade.

A) 1/26 B) 3/26 C) 4/13 D( 3/13

x) If the equation of a circle is Kx² + (2K -3)y² - 4x + 6y -1= 0, then the coordinates of centre are

A) (4/3,-1) B) (2/3,-1) C) (-2/3,1) D) (2/3,1)

Question 2. 2x10= 20

i) In a group of 800 people, 550 can speak Hindi and 450 can speak English. How many can speak both Hindi and English?

ii) Find the domain of (x-2)/(3- x)

iii) Prove: cos 510 cos 330+ sin 390 cos 120 = -1

iv) Show: √{(1- cos 2x)/(1+ cos 2x)} = tan x

v) Solve: sin x + cosx =√2.

vi) lim ₓ→₀ {√(1+ x + x²) -1}/x.

vii) find dy/dx : x sin x

ix) The odds in favour of an event are 3: 5. Find the probability of occurrence of this event.

x) Find the coordinates of the centre and radius of x² + y² + 6x - 8y -24= 0

xi) If A={1,2,3}, B={4}, C={5}, then verify AX(B UC)= (AXB)U(AXC)

xi) Find the 7th term in the expansion of (3x² - 1/3)¹⁰.

Find the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants

xii) Find the square root of 5 +12i.

xiii) If three points A(h,0), P(a,b) and B(0,k) lie on a line, show that: a/b + b/k = 1.

Question 3. 3x8= 24

i) If sin(A+ B)= 1 and sin(A- B)= 1/2, 0 ≤ A, B≤π/2, then find the value of tan(A+ 2B)

ii) If sinx + sin y= √3(cos y - cos x), prove sin 3x + sin 3y = 0

Show that √[2+√{2+√(2+ 2 cos 8x)}]= 2 cos x

iii) lim ₓ→₀ {(x+y)sec(x+y) - x sec x}/y.

iv) dy/dx of x²/(x sin x + cosx)².

v) Find the probability that in a random arrangement of the letters of the word SOCIAL vowels come together.

How many four different numbers, greater than 5000 can be formed with the digits 1,2,5,9,0 when repetition of digits is not allowed.

vi) How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE ?

vii) Find the equation of a line passing through the point (2,3) and parallel to the line 3x - 4y +5= 0

viii) The sum of first three terms of a GP is 13/12 and their products is -1. Find the GP.

Question 4 (Any one). 2+3= 5

i) lim ₓ→√2 (x⁴ - 4)/(x²+ 3x √2 -8)

ii) Find the equation of a circle concentric with the circle x² + y² - 6x + 12y +15= 0 and double of its area.

B) i) Using the 1st principal find sin 2x

ii) Five cards are drawn from a pack of 52 playing cards. What is the chance that these 5 will contain:

a) just one ace

b) atleast one ace?

C) i) Find two numbers whose arithmetic mean is 34 and the geometric mean is 16.

ii) Solve the equation in R:

|x + 1/3|> 8/3

__________________________________

MODEL TEST PAPER -1

Question -1. 1x 10= 10

i) The three angles of a right angled triangle are in AP. Then the angle are

A) 30,60,90 B) 35, 65, 80 C) 10, 70, 100 D) none

ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is

A) 0 B) 0i C) o+0i D) 1

iii) If m and n are the roots of x(x-3)= 4, what is the value of (m²+ n²)

A) 15 B) 16 C) 17 D) 18

iv) 2³ⁿ -1 is divisible by

A) 2 B) 3 C) 7 D) 5

v) In how many ways can the letters of the word BANANA be arranged?

A) 60 B) 72 C) 144 D) 210

vi) Two dice are thrown simultaneously. The probability that the number on both the dice are same

A) 1/6 B) 1/18 C) 1/36 D) none

vii) If cosx= -1/2, what is the general value of x.

A) 30 B) 60 C) 90 D) none

viii) If the distance between the points (-3,3) and (4,y) be 5√2 units, find the value of y.

A) 3 B) 4 C) 5 D) 6 units

ix) Find the equation of straight line whose x-intercept and y-intercept are 3 and - 4 respectively.

A) x/3 + y/4 = 1 B) x/3 - y/4 = 1

C) - x/3 + y/4 = 1 D) x/3 + y/4 = -1

x) The middle term of (x/y - y/x)¹⁰ is

A) 5th B) 6th C) 7th D) 4th

Question 2. 2x10= 20

i) Find the value of sin 105°

ii) nth terms of the series 16, 8, 4, 2.

iii) Find the modulus of (1+ 3i)/(2- i)

iv) A. M and G. M of two numbers is 6.5, 6 respectively. Find the numbers.

v) Find the value of cos(-1170).

vi) Find the radius and centre of the circle x²+ y² = 36 is

vii) Find the number of arrangement of word MONDAY.

Find the number of arrangement of x²y³z⁴

viii) Prove by induction 1+2+3+ ....n = n(n+1)/2

ix) Find the value of sin 75 + cos 75

x) Find the area of the triangle whose vertex are (3,2,(4,-2),(-4,7) respectively.

Question 3. 3x10= 30

i) How many term of the series e+(-6)+12 +(-24)+.... must be added from the first so that the sum may be -1023?

ii) Find the square root of 5 - 12i

iii) a) lim ₓ→₁ (x²-1)/(x +1).

b) lim ₓ→₂ (x³ +1)/(x²+1+ 3x)

iv) How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7 .

Prove cos 3π/32 = 1/2 √[2+√{2+ √(2+ √2)}].

v) A die is thrown and at the same time a card is drawn from a pack of 52 playing cards. Find the probability of getting 5 and Ace of hearts.

vi) find dy/dx: (x²+3x+ 1)/(x²+ 2x -1)

vii) Expand: (x - 2/x²)⁵.

Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.

viii) √3 sin10 + sin 20 = cos 50.

Prove cos 9 cos 27 cos 63 cos 81 = 1/16

ix) Prove that the straight lines is concurrent x+ y+5= 0, x- y+1= 0, 3x - y+ 7= 0.

x) If 2 cos a= x+ 1/x, then show that 2 cos 3a = x³ + 1/x³.

A 4 digited number is written by the digits 1,2,3 and 4 and where no digits is repeated in any number. Find the probability that the number is

A) odd

B) mutiple of 4.

Question 4. 4x5= 20

i) If one of the roots of the equation x² - px+ q= 0 be double the other, then prove 2p² = 9q.

Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .

iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.

If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.

iv) The value of (2x²- 2x+4)/(x²- 4x+3) doesn't lie between -7 and 1

Which term in the expansion of (2x² - 1/x)¹² is independent of x ? Find the value of that term.

v) The coordinates of the points A, B and C are (6,3),(-3,5) and (4,-2) respectively and that of the point P(x,y). Show that the ratio of the area of PBC and ABC is |(x+ y -2)/7|.

Find the equation of the straight line which passes through the point of intersection of the straight lines y- 2x +2= 0 and y - 3x +5= 0 and is at a distance of 7/√2 units from the origin.

SECTION- B

SECTION - C

1) Find the 3 yrs moving average of 2, 3, 4, 3, 4, 5, 6, 3, 5, 6, 4,3,2. (2)

2) Mean age of 20 boys and 20 girls are 30 and 40. Find the combined mean age. .(3)

3) Find correlation coefficient between x and y of following:

X: 2 3 5 6 4

Y: 1 3 5 3 4 (5)

Find the mode of:

Class: 20-30 30-40 40-50 50-60

F:. 5 12 25 7

4) Find the 52 percentile of:

Class: 1-10 11-20 21-30 31-40

F: 12. 35. 15. 3 (5)

Find median of the above question

5) Find the standard deviation of

X: 20-30 30-40 40-50 50-60

F:. 12 25 6 10 (5)

Find the combined standard deviations of 10 numbers when mean of x and y are 20, 30 respectively. And standard deviations of x and y are 3 and 4 respectively.

Friday, 4 November 2022

MODEL TEST PAPER (X) CBSC

MODEL test paper -1

SECTION -A (40 Marks)

1). 1x 10 = 10

i) Value of the quadratic polynomial p(x)= 2x²- 3x + 5 at x= -2

A) 9 B) 10 C) 11 D) 12

ii) Is 3x² - 4x +2 = 2x² - 2x +4 a quadratic equation. T/F

iii) (1- sin²x)sec²x is

A) 0 B) 1 C) sin²x D) tan²x

iv) Cosx sin(90-x)+ sinx cos(90- x) is

A) 0 B) 1 C) - sin x D) - sinx

v) If cos x= 4/5, value of tan x is

A) 1 B) 0 C) 3/2 D) none

vi) Find the diagonal of a cuboid 30cm long, 24cm wide and 18cm high.

A) 40 B) 42.42 C) 50 D) 52.52

vii) The mean of 16 numbers is 8. If 2 is added to every number, what will be the new mean ?

A) 8 B) 16 C) 10 D) 12

viii) Which term of the sequence -1, 3, 7, 11, .... is 95.

A) 30 B) 31 C) 32 D) 33

ix) If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is

A) right angled B) equal C) cyclic D) opposite.

x) A tower is 100√3m hig5. Find the angle of elevation it its top from a point 100m away from its foot.

A) 30 B) 60 C) 45 D) 90

2) 1x 10= 10

i) Write the first three terms in aₙ= n²+1.

ii) How many numbers of two digit are divisibile by 3 ?

iii) The length of a chord of a circle is 4cm. If it's distance from the centre is 1.5cm, determine the radius of the circle.

iv) Find the length of a tangent drawn to a circle with radius 5cm, from a point 13cm from the centre of the circle.

v) An unbiased die is thrown. What is the probability of getting a number between 3 and 6. (1)

vi) One root of x²- 3x - c= 0 is -2, find the value of c.

vii) If the probability of winning a game is 0.3. what is the probability of loosing it ?

viii) Probability of a sure event is __

ix) f(x)= x²+ 3x - 10, g(x)= x -2. Find out whether g(x) is a factor of f(x) or note

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

3) 2x 10= 20

i) Solve x/(x+1) + (x+1/x = 34/15, x≠0, x≠ -1.

ii) Find the sum of the deviations of the variate values 3, 4, 6, 8, 14 from their mean. (2)

iii) Find k if 2x² + Kx + 3= 0 has real roots.

iv) A two digit number is four times the sum and three times the product of its digits. Find the number.

A journey of 240 km would take half an hour less if the speed were increased by 2km per hr. Find the usual speed.

v) Determine the general term of an AP whose 7th term is -1 and 16th term 17.

vi) The 7th term of an AP is 32 and its 13th term is 62. Find AP.

vii) A solid cylinder has total surface area of 462cm². Its curved surface area is one third of its total surface area. Find the volume of the cylinder. (π= 22/7).

viii) In a ∆ABC, AD is the bisector of ang.A, meeting side BC at D . If BD= 2.5cm, AB= 5m and AC= 4.2cm, find DC.

D and E are the points on the sides AB and AC respectively of a ∆ABC such that AD= 8cm, DB= 12cm, AE= 6cm and CE= 9cm, prove that BC= 5DE/2.

ix) √{(- sinx)/(1+ sin x)}= secx - tan x.

x) An electric pole is 10m hig8. A steel wire tied to top of the pole is affixed at a point on the ground to keep the pole up right. If the wire makes an angle of 45 with the horizontal the foot of the pole, find the length of the wire. (2)

SECTION - B (40 Marks)

4) Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram. (3)

5) A ladder 15m long reaches a window is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street. (3)

6) AB and CD are two parallel chords of a circle whose diameter is AC. Prove that AB= CD. (3) OR

Equal chords of a circle subtends equal angles at the centre.

7) If x sin³a + y cos³a= sina cos a and x sin a = y cos a, prove x² + y² = 1. (3)

8) If A, B, C are the interior angles of a triangle ABC , prove that tan{(A+ C)/2}= cot(B/2). (3)

9) A man standing in the deck of a ship, which is 8m above water level. He observes the angle of elevation of the top of a hill as 60 and the angle of depression of the base of the hill as 30. Calculate the distance of the hill from the ship and the height of the hill. (3)

10) If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that 1/V = 2/S(1/a + 1/b + 1/c). (3)

11) Class interval. Frequency

0-10 7

10-20 10

20-30 15

30-40 8

40-50 10 find the mean. (4)

Class interval. Frequency

1-10 8

11-20 10

21-30 15

31-40 8

41-50 18 find the median.

12) A vessel in the shape of a cuboid contains some water. If three identical spheres are immersed in the water, the level of water is increased by 2cm. If the area of the base of the cuboid is 160cm² and its height 12cm, determine the radius of any of the spheres. (4)

13)a) If cotx = 1/√3, find the value of (1- cos²x)/(2- sin²x). (2)

b) The radius and slant height of a cone are in the ratio 4:7. If it's curved surface area is 792cm², find its radius. (π= 22/7). (2)

14) a) If P(A)= 1, then A is called

A) certain event

B) impossible event

C) possible event

D) absolute event. (1)

b) A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is

A) red

B) black or white

C) not black. (3)

15)a) 2(a²+ b²)x² +2(a+ b)x+ 1= 0.

b)Find the sum of (-5)+(-8)+(-11)+....+(-230). (1.5+ 1.5)

16)a) AB and CD are two parallel chords of a circle such that AB= 10cm and CD = 24cm. If the chords are on the opposite sides of the centre and the distance between them is 17cm, find the radius of the circle.

b) Two chords AB and CD of a circle intersect each other at P outside the circle. If AB= 5cm, BP=3 cm, and PD= 2cm, find CD.

Thursday, 3 November 2022

MODEL TEST PAPER (WBCHSE) 22/23

Full Marks --80

SET - 1

PART - A (Marks : 10)

1) Choose the correct alternatives:

i) If A={1, 2, 3, 4} and Iₐ be the identity relation on A, then

A) (1,2) belongs to Iₐ

B) (2,2) belongs to Iₐ

C) (2,1) belongs to Iₐ

D) (3,4) belongs to Iₐ

ii) The principle value of tan⁻¹(-√3) is

A) π/3 B) π/4 C) - π/ D) - π/3

iii) If A is an invertible Matrix of order 3 and |A|=5, then the value of |adj A| is equal to

A) 20 B) 21 C) 24 D) 25

iv) If f(x)= log(tan(x/2)), then the value of f'(x) is

A) sin x B) 1/(sin x/2 cos x/2)

C) - cosec x D) Cosec x

v) The value of ∫ |cos x| dx at (π, 0) is

A) 0 B) 1 C) 2 D) none

vi) The area (in square unit) bounded by the curve y =sin x, x-axis and the two ordinates x=π, x= 2π is

A) 1 B) -1 C) -2 D) 2

vii) Angle between the straight lines (x -5)/7 = (y+2)/-5 = z/1 and x/1 = y/2 = z/3 is

A) π/4 B) π/3 C) π/2 D) π

viii) the value of m for which the straight line 3x -2y+z + 3= 0= 4x - 3 + 4z +1 is parallel to the plane 2x - y + mz - 2= 0 is

A) - 2 B) 8 C) -18 D) 11

ix) If the odds against an event are 4 : 5 then the probability the occurrence of the event is

A) 5/9 B) 4/9 C) 4/5 D) 1/9

x) the variance of a binomial distribution with parameters n and p is

A) n/4 B) ≤ n/4 C) > n²/4 D) ≤ n²/4

Part - B (Marks : 70)

2) On the set Q⁺ of all positive rational numbers if the binary operation * is defined by a*b = ab/4 for all a, b belongs to Q⁺, find the identity elements in Q⁺. (2)

3) solve: tan⁻¹(cot x) + cot⁻¹(tan x)= π/4. (2)

4) prove by property

y+ z z y

z z+x x = 4xyz

y x x+ y (2)

If the matrix A= 2 5 and AB=-13 8

1 3 - 8 5 find B.

5) find: lim ₓ→₀ (mᵃˣ - nᵇˣ)/x. (2)

6) if x= t log t, y= (log t)/t, find dy/dx when t= 1. (2)

7) find: ∫(1 - 1/x²) ₑ(x + 1/x) dx. (2)

solve: dy/dx = 1 - x + y - xy

8) Show that the function f(x)= x³/3 - 6x² + 20x - 5 has neither a maximum or minimum value. (2)

9) The radius of a circular plate increases at the rate of 0.002 cm/s. How fast is the area changing when radius is 14 cm? (2)

10) Find the acute angle between the z-axis and the straight line joining the points (3,2,3) and (-3,-1,5). (2)

Prove that the equation of the plane which passes through the point (2,- 3, 5) and which is parallel to the yz- plane is x= 2.

11) Show that the probability that exactly one of the events A and B occurs is P(A)+ P(B) - 2P(AB). (2)

If X is a discrete randomal variable and ' a ' is a constant, show that, E(x - bar x)= 0.

12) Show that the relation R. on the set A={x belongs to z : 0≤ x ≤ 12}, given by R{(a,b): |a - b| is a multiple of 4 and a, b belongs to A} is an equivalence relation on A. (2)

Show that tan⁻¹(1/√3 tan(x/2)) = 1/2 cos⁻¹{(1+ 2 cos x)/(2+ cos x)}.

13) If the matrix

A= 1 x -2

2 2 4

0 0 2 and A² + 2I = 3A. Find x; here I is the unit matrix of order 3. (3)

solve bx matrix method:

2/x - 3/y + 3/z = 10, 1/x + 1/y + 1/z = 10, 3/x - 1/y + 2/z = 13.

14) Prove the determinant. (3)

1 a a²- bc

1 b b²- ca = 0

1 c c² - ab

(a²+ b²)/c c c

a (b²+ c²)/a a = 4abc

b b (c²+ a²)/b

15) If f(x)= tan⁻¹{x/(1+ 20x²)} show that f(x)= {5/(1+ 25x²)} - 4/(1+ 16x²). (3)

Let y= (sin⁻¹x)²+ (cos⁻¹x), show that, (1- x²) d²y/dx² - x dy/dx = 4

16) Evaluate ∫ dx/√(2x³/3 - x² + 1/3). (3)

Evaluate ∫ x²/(x⁴ +1) dx.

17) Solve: (1- x²) dy/dx - xy = x², given y= 2 when x = 0. (4)

Solve: (6x + 9y -7) dx= (2x + 3y -6) dy.

18) AB = 2i - 4j + 5k and BC= i - 2j - 3k in parallelogram ABCD, find a unit vector in direction parallel to the diagonal AC of the parallelogram. (4)

19) prove ∫ log(sin x) dx at (π/2, 0) = π/2 log(1/2). (4)

20) Urn A contains 1 white, 2 black and 3 red balls; Urn B contains two white, one black and one red ball; urn C contains 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are drawn. These happen to the one white and one red. what is the probability that they come from urn A ? (4)

21) the probability distribution of a random variable X is as follows:

X: 0 1 2 3 4

P(x) 0.20 0.25 0.35 0.14 0.06 Find the probability distribution of the variable Y where y= x² + 5. (5)

22) A company manufacturers two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. He earns a profit of ₹50 each on type A and ₹ 60 each on type B. formulate the above problem as a LPP to maximize the profit. (5)

Solve graphically:

Minimize Z= 3x+ 5y

Subject to the constraints x+3y≥ 3, x+ y ≥ 2, x, y≥ 0

23) A cylindrical tin can, open at the top, of a given capacity, has to be constructed. Show that the amount of the ti required will be least if the height of the can is equal to its radius. (5)

24) Show that lines r= (i+ j+ k)+ t(i - j + k) and r= (3i - k)+ s(4j - 16k) intersect and find the position vector of their point of their point of their point of intersection. (5)

Find the equation of the plane passing through the point (-1, 1,1) and (1,-1,1) and is perpendicular to the plane x+ 2y + 2z = 5.

Wednesday, 2 November 2022

MODEL TEST PAPER ISC(2022/23)

MODEL TEST PAPER -1

Full Marks- 90

SECTION A

1) (any 5) 1x5= 5

Choose the correct alternatives:

i) If A={1, 2, 3, 4} and Iₐ be the identity relation on A, then

A) (1,2) belongs to Iₐ

B) (2,2) belongs to Iₐ

C) (2,1) belongs to Iₐ

D) (3,4) belongs to Iₐ

ii) The principle value of tan⁻¹(-√3) is

A) π/3 B) π/4 C) - π/ D) - π/3

iii) If A is an invertible Matrix of order 3 and |A|=5, then the value of |adj A| is equal to

A) 20 B) 21 C) 24 D) 25

iv) If f(x)= log(tan(x/2)), then the value of f'(x) is

A) sin x B) 1/(sin x/2 cos x/2)

C) - cosec x D) Cosec x

v) The value of ∫ |cos x| dx at (π, 0) is

A) 0 B) 1 C) 2 D) none

vi) If the odds against an event are 4 : 5 then the probability the occurrence of the event is

A) 5/9 B) 4/9 C) 4/5 D) 1/9

i) solve the determinants

x² x 1 (2)

0 2 1 = 28

3 1 4

ii) Solve the equation for x:

cos(tan⁻¹x)= sin(cot⁻¹3/4). (1)

iii) using Matrix rule, solve the system of equations: 5x + 7y = -2, and 4x+ 6y = -3. (2)

iv) Evaluate: lim ₓ→₀{(sin x - x)/x³}. (1)

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

a) mutually exclusive

b) Independent. (2)

vi) Evaluate: ∫ (x sin x)/(1 + sin x) dx at (π, 0). (2)

vii) (1)

For what value of x is the given matrix 2x + 4 4

x + 5 3 a singular Matrix

viii) if y= xʸ, prove that x dy/dx = y²/(1- y log x). (2)

ix) The probability of A, B, C solving a problem 1/2, 1/3, 1/4 respectively. Find the probability that the problem will be solved. (2)

x) If f(x)= 27x³ and g(x)= ³√x, then find gof(x). (1)

3) Using properties of determinant, Prove that 1 + a 1 1

1 1+ b 1

1 1 1+ c = abc(1+ 1/a + 1/b + 1/c) (3)

4) Apply Rolle's theorem to find a point (or points) on the curve y= - 1+ cos x where the tangent is parallel to the x-axis in [0, 2π]. (3)

5) If the following function is differentiable at x= 2, then find the values of a and b.

If the following function is differentiable at x= 2, then find the values of a and b.

f(x)= x², if x ≤ 2

ax+ b, if x > 2 (3)

6) if cos⁻¹(x/a) + cos⁻¹(y/b) = k , prove that x²/a² - (2xy cos k)/ab + y²/b² = sin²k. (3)

7) if y= {x+ √(1+ x²)}ⁿ , then show that (1+ x²) d²y/dx² + x dy/dx = n²y. (3)

8) Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that at least one ball is red ? (3)

9) Evaluate ∫ (3x+1)/√(5 - 2x - x²) dx. (3)

evaluate ∫ √x/{√x + √(a - x)} dx

10) Find the equation of the tangent to the curve y= x² - 2a + 7 which is

a) a parallel to the line 2x - y +9= 0

b) Perpendicular to the line 5y - 15x= 13. (3)

11) Find the interval in which the function f given by f(x)= sin x - cos x, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing. (2)

12) solve: dy/dx + y/x = x². (2)

13) Solve: dy/dx + 1 = eˣ⁺ʸ. (2)

14) Let f: N --> N be a function defined as f(x)= 4x² + 12x + 15, show that f: N --> S is invertible (where S is the range of f). Find the inverse of f and hence find f⁻¹(31) and f⁻¹(87). (3)

15) Using matrix, solve the following equations: 5x+ 3y + z= 1z, 2x+ y+ 3z= 19, x+ 2y+ 4z= 25. (3)

If A= 1 -1 0

2 5 3

0 2 1 find A⁻¹.

16) Prove that the area of a right angled triangle of given hypotenuse is maximum, when the triangle is isosceles. (3)

Show that of all rectangle inscribed in a given fixed circle, the square has the maximum area.

17) Evaluate: ∫ x² sin⁻¹x dx. (2)

18) ∫ x dx/(x²+ 4x +3). (2)

19) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of success and show that its mean is 2/3. (3)

20) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn from 2nd. Find the probability that the ball drawn is white. (3)

SECTION - B

21) if a= 2i + j + 3k and b= 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle. (2)

22) If (8,2,0), B(4,6,-7), C(-3, 1, 2) and D(-9,-2,4) are four given points, then find the angle between AB and CD. (2)

23) Find the vector and Cartesian equation of the plane which bisects the line joining the points (3,-2,1) and (1,4,-3) at right angles. (2)

24) If a= 7i- 2j + 3k, b= i - j + 2k, c= 2i + 8j are three vectors then find a.(b x c) and (a x b). c. (4)

25) find the equation of the plane passing through the line of intersection of the planes x+ 2y+ 3z - 5= 0 and 3x - 2y - z + 1= 0 and cutting off equal intercepts on the OX and OZ axes. (4)

Find the co-ordinate of the points on the line (x -1)/2 = (y+2)/3 = (z - 3)/6 which are at a distance of 3 units from the point (1,-2,3).

26) Find the area of the region included between the parabola y= 3x²/4 and the line 3x- 2y + 12= 0. (4)

27) Find the area bounded by the line y= x, The x-axis and the ordinate8 x= 0 and x= 4. (2)

SECTION- C

28) The fixed cost of a product is ₹ 18000 and the variable cost per unit is ₹550. if the demand function is p(x)= 4000 - 150x, find the break even values. (2)

29) Given x + 4y = 4 and 3x + y= 16/3 are regression lines. Find the line of regression of x on y. (2)

30) The cost function for a commodity is C(x)= ₹(200 +20x - x²/2)

a) find the marginal cost(MC).

b) calculate the marginal cost when x= 4. (2)

31) Two regression lines represented by 2x+ 3y - 10= 0 and 4x + y - 5 = 0. find the line of Regression of y on x. (4)

Fit a straight line to the following data. Treating y as the dependent variable.

X: 1 2 3 4 5

Y: 7 6 5 4 3 Hence, estimate the value of y when x= 3.5

32) the marginal cost function of a firm is MC= 33 log x. Find the total cost function when the cost of producing one unit is ₹11. (4)

If the marginal cost of a commodity is equal to half its average cost, show that fixed cost is zero. If the cost of producing 9 units of the commodity is ₹60, find the cost function.

33) A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours in both machines and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of ₹7 and that of B at a profit of ₹4. find graphically the production level per day for maximum profit. 6