MATH- TEST PAPERS: February 2024

Monday, 26 February 2024

MATH- TEST - VIII

ALGEBRAIC IDENTITIES

1) The square root of a²+ 1/a²+ 2 is

a) a+ 1/a b) a- 1/a c) a²+ 1/a² d) a²- 1/a²

2) The square root of a+ 1/a - 2 is

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

3) The value of (a+ b)²/{(b -c)(c - a)} + (b + c)²/{(a - b)(c - a) + (c + a)²/{(a - b)(b - c)} is

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

4) The square root of the expression (1/abc) (a²+ b²+ c²) +2(1/a + 1/b+ 1/c) is

a) (a+ b+ c)/abc b) √a + √b + √c c) √(bc/a) + √(ca/b) + √(ab/c) d) √(a/bc)+ √(b/ca) + √c/ab)

5) The square root of x²/9 + 9/4x² - x/3 - 3/2x + 5/4 is

a) 2x/3 + 3/2x - 1/2 b) x/3 + 3/2x +1 c) 3/x + 2/3x - 1/2 d) x/3 + 3/2x - 1/2

6) The square root of the expression is (xy + xz - yz)² - 4xyz(x - y) is

a) xy + yz - 2xyz b) x + y - 2xyz c) xy + z - y d) xy + yz - xz

7) The square root of a²/4 + 1/a² - 1/a + a/2 - 3/4 is

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

8) The expression (4a + 5b+ 5c)² - (5a + 4b+ 4c)² + 9a² is a perfect square of the expression

a) √3(b + c) b) 3(a+ b + c) c) 3(b+ c) d) 3(-b + c - a)

9) The expression (3a + 2b+ 3c)² - (2a + 3b+ 2c)² + 5b² is a perfect square of the expression √(a + b+ c) b) √((a + b) c) √5(a +c) d) √5(a - b+ c)

10) If a/b + b/a =2, then (a/b)¹⁰ - (b/a)¹⁰ is equal to

a) (2¹⁰-1)/2¹⁰ b) 2 c) 0 d) (2²⁰+1)/2¹⁰

11) If ab c = 6 and a + b + 6 = 6, then 1/ac + 1/ab + 1/bc =

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

12) √{(a + b+ c)²+ (a + b- c)²+ 2(c² -b²- a²- 2ab) is equal to

a) 2c b) 2a c) 2b d) a + b+ c

13) If a/b + b/a = -1, then a³- b³=

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

14) If a+ b=8 and ab= 12, then a³+ b³=

a) 244 b) 224 c) 144 d) 284

15) If (a + 1/a +2)²=4, then a²+ 1/a²=

a) 12 b) 13 c) 14 d) -14

16) If x+ 1/x =7, then x³- 1/x³=

a) 9√5 b) 144√5 c) 135√5 d) √5

17) {(a - b)³ - (a + b)³}/2 + a(a²+ 3b²)=

a) a³- b³ b) (a + b)³ c) a³+ b³ d) (a - b)³

18) If x+ 1/x =5, then x²+ 1/x²=

a) 25 b) 10 c) 23 d) 27

19) If x+ 1/x =2, then x³ + 1/x³=

a) 64 b) 14 c) 8 d) 2

20) If x+ 1/x =4, then x⁴ + 1/x⁴=

a) 196 b) 194 c) 192 d) 190

21) If x+ 1/x =3, then x⁶ 1/x⁶=

a) 927 b) 414 c) 364 d) 322

22) If x² + 1/x² =102, then x- 1/x=

a) 8 b) 10 c) 12 d) 13

23) If x³+ 1/x³ =110, then x + 1/x =

a) 5 b) 10 c) 15 d) none

24) If x³- 1/x³ =14, then x- 1/x=

a) 5 b) 4 c) 3 d) 2

25) If a+ b+ c= 9 and ab+ bc+ ca= 23, then a²+ b²+ c²=

a) 35 b) 58 c) 127 d) none

26) (a - b)³+ (b - c)³+ (c - a)³=

a) (a+ b+ c)(a²+ b²+ c²- ab - bc - ca)

b) (a - b)(b - c)(c - a)

c) 3(a - b)(b - c)(c - a) d) none

27) a+ b= 3 and ab = 2, then a³+ b³=

a) 6 b) 4 c) 9 d) 12

28) If a- b =-8 auab = -12, then a³- b³=

a) -244 b) -240 c) -224 d) -260

29) if the volume of a cuboid is 3x²- 27, then its possible dimensions are

a) 3, x, -27x b) 3, x -3, x+3 c) 3, x², 27x d) 3,3,3

30) 75 x 75 +2 x 75 x 25+25 x 25 is equal to

a) 10000 b) 6250 c) 7500 d) 3750

31) (x - y)(x+ y)(x²+ y²)(x⁴+ y⁴) is equal to

a) x¹⁶- y¹⁶ b) x⁸- y⁸ c) x⁸+ y⁸ d) x¹⁶+ y¹⁶

32) If x⁴+ 1/x⁴ =623, then x + 1/x=

a) 27 b) 25 c) 3√3 d) -3√3

33) If x- 1/x = 15/4, then x + 1/x =

a) 4 b) 17/4 c) 13/4 d) 1/4

34) If 3x+ 2/x = 7, then 9x² - 4/x² =

a) 25 b) 35 c) 49 d) 30

35) If a²+ b²+ c²- ab - bc - ca = 0, then

a) a+ b = c b) b + c = a c) c + a= b d) a= b= c

36) If a+ b + c = 0, then a²/bc + b²/ca + c²/ab is

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

37) If a¹⁾³ + b¹⁾³ + c¹⁾³= 0, then

a) a+ b+ c= 0 b) (a+ b + c)³= 27abc c) a+ b + c = 3abc d) a³+ b³+ c³= 0

38) If a+ b + c = 9, then ab+ bc + ca =23, then a³+ b³+ c³- 3abc=

a) 108 b) 207 c) 669 d) 729

39) {(a² - b²)³+ (b²- c²)³+ (c²- a²)³}/{(a - b)+ (b - c)+(c - a)}=

a) 3(a + b) (b +c)(c +a) b) 3(a - b)(b - c)(c - a)} c) (a - b) (b - c)(c - a) d) (a + b) (b +c)(c+ a)

40) The product (a + b)(a - b)(a²- ab+ b²)(a²+ ab+ b²) =

a) a⁶+ b⁶ b) a⁶- b⁶ c) a³- b³ d) a³+ b³

41) The product (x²-1)(x⁴+ x²+1) is equal to

a) x⁸-1 b) x⁸+1 c) x⁶-1 d) x⁶+1

42) If a/b + b/a = 1, then a³+ b³=

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

43) If 49a²- b = (7a + 1/2)(7a - 1/2), then the value of b is

a) 0 b) 1/4 c) 1/√2 d) 1/2

44) One of the factors of (5x +1)² -(5x -1)² is

a) 5 + x b) 5- x c) 5x -1 d) 20x

45) If 9x² - b =(3x + 1/2)(3x - 1/2), then the value of b is

a) 0 b) 1/√2 c) 1/4 d) 1/2

46) The Coefficient of x in (x +3)³ is

a) 1 b) 9 c) 18 d) 27

47) The value of 249²- 248² is

a) 1 b) 477 c) 487 d) 497

48) Which of the following is a factor of (x + y)³-(x³+ y³)?

a) x²+ 2xy + y² b) x² - xy + y² c) xy² d) 3xy

49) If x/y + y/x = -1 (x,y ≠ 0), the value of x³- y³ is

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

50) If x + y=2 and xy = 1, then x⁴+ y⁴=

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

51) If x² + y²+ xy =1 and x + y = 2, then xy=

a) -3 b) 3 c) -3/2 d) 0

52) If a, b, c are natural numbers such that a²+ b²+ c²= 29 and ab + bc + ca = 26, and a+ b + c=

a) 9 b) 6 c) 7 d) 10

53) If 2x + y/3= 12 and xy = 30, then 8x³+ y³/27=

a) 1008 b) 168 c) 106 d) none

ASSERTION- REASON

Each of the following examples contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.

a) Statement-1 and statement-2 are true; Statement-2 is a correct explanation for statement-1 .

b) Statement -1 and Statement-2 are true; Statement -2 is not a correct explanation for statement-1.

c) Statement -1 is true, statement -2 is false.

d) Statement -1 is False, Statement -2 is true.

1) Statement -1(A): √{(a+ b + c)+ (a - b + c)+2(b²- a²- c²- 2ac)}= 2b

Statement-2 (R): (x + y+ z)²= x²+ y²+ z²+ 2(xy + yz + zx). a

2) Statement -1(A): a³+ b³+ 3ab -1= (a+ b -1)(a²+ b²+ a+ b - ab +1)

Statement-2 (R): a³+ b³+ c³- 3abc= (a+ b + c)(a²+ b²+ c²+ ab + bc + ca). c

3) Statement -1(A): (a - b)³+(b - c)³+(c - a)³= 3(a - b)(b - c)(c - a)

Statement-2 (R): If a+ b + c = 0, then a³+ b³+ c³= 3abc. a

4) Statement -1(A): a²+ b²+ c²- ab - bc - ca = 0 if and only if a= b = c.

Statement-2 (R): (a+ b + c)²= a²+ b²+ c²+ 2ab + 2bc + 2ca. b

5) Statement -1(A): a+ b + c = 6 and 1/a + 1/b + 1/c = 3/2, then a/b + a/c + b/a + b/c + c/a + c/b = 6

Statement-2 (R): (a + b + c)²= a²+ b²+ c²+ 2(ab + bc + ca). b

6) Statement -1(A): if a+ b + c = 0, then a³+ b³+ c³= 3abc

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).

7) Statement -1(A): (a+ b + c)² = a²+ b²+ c²-2(ab+ bc + ca)

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).

8) Statement -1(A): a³ + 3ax/8 + cpx³/64 - 1/8 = (a + x/4 - 1/2)(a²+ x²/16 + 1/4 - ax/4 + x/8 + a/2)

Statement-2 (R): a³+ b³+ c³ + 3abc = (a + b + c)(a²+ b²+ c²+ ab + bc + ca).

9) Statement -1(A): If a+ b + c =0, ab + bc+ ca = 11, then a²+ b² + c²= 14

Statement-2 (R): (a+ b+ c)³ = a²+ b²+ c²+ 2(ab + bc + ca).

10) Statement -1(A): {(x²- y²)³+(y²- z²)³+(z³- x²)³}/{(x - y)³+(y - z)³+ (z - x)³}= (x + y)(y+ z)(z + x).

Statement-2 (R): If a + b + c= 0, then a³+ b³+ c³= 3abc.

11) Statement -1(A): (1/abc) (a²+ b² + c²)+ 2(1/a + 1/b+ 1/c) is √(a/bc) + √(b/ca) + √(c/ab).

Statement-2 (R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c²-ab - bc - ca).

ALGEBRAIC EXPRESSIONS

1) (a - b)³+ (b - c)³+ (c - a)³ is equal to

a) 2a³+ 2b³+ 2c³

b) (a - b) (b - c)(c - a)

c) 0

d) 3(a - b)(b - c)(c - a)

2) If x + y=12 and xy = 27, then x³+ y³=

a) 765 b) 756 c) 657 d) 675

3) If x+ y= -4 then x³+ y²- 12xy +64=

a) -64 b) 128 c) 0 d) none

4) If x = 2y+6, then x³- 8y³- 36xy=

a) 216 b) -216 c) 36 d) -36

5) (a+ b+ c){(c - b)²+ (b - c)²+ (c - a)²}=

a) a³+ b³+ c³- 3abc b) a³+ b³+ c³ c) 2(a³+ b³+ c³- 3abc) d) 3abc

6) If a³+ b³= 5 and a+ b=1, then ab=

a) -4/3 b) 4/3 c) -3/4 d) 3/4

7) If a³+ (b - a)³ - b³ = k(a - b), then k=

a) ab b) 3ab c) -3ab d) 3

8) If a+ b+ c= 0, then a²/bc + b²/ca + c²/ab=

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

9) The factor of x³ - x²y - xy²+ y³, are

a) (x+y)(x²- xy+ y²)

b) (x+y)(x²+ xy+ y²)

c) (x+y)²(x- y)

d) (x-y)²(x + y)

10) The factor of x³ - 1 +y³ + 3xy, are

a) (x-1+y)(x²+1+ y²+ x + y - xy)

b) (x+1+y)(x²+1+ y²+ 1- x - y - xy)

c) (x-1+y)(x²-1- y²+ x + y - xy)

d) 3(x-1+y)(x²-1+ y²)

11) The factor of 8a³+ b³- 6ab +1 are

a) (2a+ b-1)(4a²+ b²+1- 3ab- 2a)

b) (2a- b+1)(4a²+ b²+1- 4ab- 2a+ b)

c) (2a+ b+1)(4a²+ b²+1- 2ab- b -2a)

d) (2a+ b-1)(4a²+1- 2ab- b- 4a)

12) (x + y)³ -(x - y)³ can be Factorized as

a) 2y(3x²+ y²) b) 2x(3x²+ y²) c) 2y(3y²+ x²) d) 2x(x²+ 3y²)

13) The expression (a - b)³ + (b - c)³+ (c - a)³ can be Factorized as

a) (a- b)(b - c)(c - a)

b) 3(a- b)(b - c)(c - a)

c) -3(a- b)(b - c)(c - a)

d) (a+ b+c)(a²+b² + c²- ab - bc - ca)

14) The value of {(2.3)³ - 0.027}/{(2.3)²+ 0.69+ 0.09}, is

a) 2 b) 3 c) 2.327 d) 2.273

15) The value of {(0.013)³ +(0.007)³}/{(0.013)² - 0.013 x 0.007+ (0.007)²} is

a) 0.006 b) 0.02 c) 0.0091 d) 0.00185

16) The factors of a² - 1 - 2x - x², are

a) (a - x +1)(a - x -1)

b) (a + x +1)(a - x +1)

c) (a + x +1)(a - x -1) d) none

17) The factors of x⁴+ x²+ 25, are

a) (x²+ 3x +5)(x²- 3x +5)

b) (x²+ 3x +5)(x²+ 3x -5)

c) (x²+ x +5)(x²- x +5) d) none

18) The factors of x²+ 4y²+ 4y - 4xy - 2x - 8, are

a) (x - 2y -4)(x - 2y +2)

b) (x - 2y +2)(x - 4y -4)

c) (x + 2y -4)(x + 2y +2) d) none

19) The factors of x³- 7x + 6, are

a) x(x -6)(x -1)

b) (x² -6)(x -1)

c) (x +1)(x +2)(x -3)

d) (x +3)(x -2)(x -1)

20) The expression x⁴+ 4 can be Factorized as

a) (x²+ 2x +2)(x²- 2x +2)

b) (x²+ 2x +2)(x²+ 2x -2)

c) (x²- 2x -2)(x²- 2x +2)

d) (x²+2)(x²- 2)

21) If 3x = a+ b + c, then the value of (x - a)³+ (x - b)³+(x - c)³ -3(x - a)(x - b)(x - c), is

a) a+ b + c b) (a - b)(b - c)(c - a) c) 0 d) none

22) If (x + y)³ - (x - y)³ - 6y(x²- y²)= ky³, then k=

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

23) If x³- 3x²+ 3x +7= (x +1)(ax²+ bx + c), then a+ b + c=

a) 4 b) 12 c) -10 d) 3

24) If x/y + y/x = -1 (x,y ≠ 0), then the value of x³- y³ is

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

25) Which of the following is a factor of (x + y)³ - (x + y³)?

a) x²+ y²+ 2xy

b) x²+ y²- xy

c) xy² d) 3xy

Assertion- Reason based

Each of the following examples contains STATEMENT-1(Assertion ) and STATEMENT-2( (Reason) and has following four choices (a), (b), (c) and (d ), only one of which is the correct choice.

a) Statement-1 and Statement -2 are True; statement-2 is a correct explanation for statement-1

b) Statement-1 and statement-2 are True ; Statement -2 is not a correct explanation for Statement-1.

c) Statement -1 is True , Statement -2 is False .

d) Statement -1 is False , Statement -2 is True .

1) Statement -1 (A): The value 1000³ - 900³ - 100³ is 270000000

Statement -2 (R): If a+ b + c= 0, then a³+ b³+ c³= 3abc.

2) Statement -1 (A): The value of (0.093³+ 0.007³)/(0093² - 0.093 x 0.007 + 0.007²) is 0.1.

Statement -2(R): a³+ b³= (a+ b)(a²- ab + b²).

3) Statement -1(A): a³(b - c)³+ b³(c - a)³+ c³(a - b)³= 3(a - b)(b - c)(c - a)

Statement -2(R): if a+ b + c = 0, then a³+ b³+ c³= 3abc.

4) Statement -1(A): (a+ b + c){(a - b)²+ (b - c)²+ (c - a)²}= 2(a³+ b³+ c³ - 3abc)

Statement -2(R) If a+ b + c = 0 then (a+ b)³+ (b + c)³+ (c + a)³= - 3abc.

5) Statement -1(A): The product of (x²+ 4y²+ z²+ 2xy + xyz - 2yz) and (-z + x - 2y) is x³- 8y³- z³ - 6xyz

Statement -2(R): a³+ b³+ c³ - 3abc = (a + b + c)(a²+ b²+ c² - ab - bc - ca).

Statement -1(A): a²+ b²+ c²- ab - bc - ca = 0 if and only if a= b= c.

Statement -2(R): a³+ b³+ c³ - 3abc = (a+ b + c)(a²+ b²+ c²- ab - bc - ca).

6) statement-1(A): (a - b)³+ (b - c)³+ (c - a)³= 3(a - b)(b - c)(c - a)

Statement -2:(R): if a+ b + c = 0, then a³+ b³+ c³= 3abc.

7) Statement-1(A): if 3x= a+ b + c, then (x - a)³+ (x - b)³+ (x - c)³= 3(x - a)(x - b)(x - c)

Statement -2(R): if a+ b + c= 0, then a³+ b³+ c³= 3abc.

8) Statement -1(A): if a+ b + c = 5 and ab + bc+ ca= 10, then a³+ b³+ c³ - 3abc = 25

Statement-2(R): a³+ b³+ c³ - 3abc = (a+ b + c){(a+ b + c)² -3(ab + bc+ ca)}.

9) Statement -1(A): If a,b,c are all non-zero such that a+ b + c = 0, then a²/bc + b²/ca + c²/ab = 3

Statement-2 (R): If a+ b + c = 9 and a²+ b² + c²= 35, then ab + bc+ ca= 23.

10) Statement -1(A): The value of (0.027³+ 0.023³)/(0.027²- 0.027 x 0.023 + 0.023²) is 0.05

Statement-2 (R): a³- b³= (a- b)(a²- ab + b²).

DIRECT & INVERSE PROPORTION

1) 15 man can dig a 20m long trench in one day. How many should be employed for digging 140 metres long tranch of the same type in one day ?

2) The shadow of Qutub Minar, which is 72m high is 80m at a particular time on a day. Find the height of an electric pole that cast a shadow of 10m under similar condition.

3) if 721 men construct a bridge in 48 days then in how many days 1442 men can do this work ?

4) Raman takes 20 minutes to reach his school at an average speed of 6 km/h. if he is required to reach school in 24 minutes, what should be his speed ?

5) In a military camp there is a food for 30 days for 50 soldiers. Assuming that average meal of every soldier is same. If 25 more soldiers join them, how many days this food will last ?

6) Mahesh goes 5 km with a speed of 10 km/hr. if he doubles his speed, find the time taken to cover the same distance.

7) Amar weaves 35 seats of chairs in 7 days. huow many days will he take to weave seat of chairs?

8) If a minute hand makes an angle of 30° in 5 minutes. Find the angle covered between 7:10 pm to 7:30 pm.

9) A train is running at a peed of 18 km/he. if it crosses a pole in 35 seconds , find the length of the train.

10) A vegetable vendor has Rs2000 to buy potatoes available at the rate of Rs8 per kg. If the price of potatoes increases by 25%, find how much potatoes he can purchase at the same amount.

11) If 14 kg of pulses cost Rs 441, what is the cost of 22 kg of pulses?

12) A car takes 2 hours to reach a destination by travelling at 60 km/h. How long will it take while traveling at 80 km/h ?

TEST PAPER

SECTION - A (8 x1 = 8)

1) CP of 360 books= SP of 270 books. Find profit %.

2) Find the single discount equivalent to two successive discounts of 20% and 10%.

3) If 14 kg of pulses cost Rs 441, what is the cost of 22 kg is a pulses ?

4) If x and y vary inversely and x= 40, find y when constant of variation=8

5) Complete (a+ b)² - (a - b)²= ?

a) 2ab b) 4ab c) (a+ b)² d) (a - b)²

6) Simplify x²- y²=? , when x =7 and y= 5.

a) 22 b) 23 c) 24 d) 25

7) Two cylinder have same base radius r. if their height are 5m and 15cm. What is the ratio of their volume ?

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

8) Lateral surface area of a cube is 100m², find the length of each edge.

a) 5m b) 6m c) 8m d) 10m

SECTION - B

(Any 10) (10x3=30)

9) P sells an articles to Q at 10% profit. Q sells it to R at 25% profit. if R pays Rs250 for it. What did P pay for it.

10) Ahmed buys a plot of land for Rs96000. He sales 2/5 of it loss of 6%. At what gain percent should he sale the remaining part of the plot to gain 10% on the whole ?

11) Whats should be the price marked on a washing machine whose cost is Rs14409, if the retailer wants to get a profit of 10% giving a discount of 12% ?

12) A factory requires 42 machines to produce a given number of articles in 56 days. How many machines would be required to produce the same articles in 48 days ?

13) A vegetable vendor has Rs2000 to buy potatoes available at the rate of Rs8 per kg. if the price of potatoes increases by 25%, find how much potatoes he can purchase at the same amount.

14) a) Factorise: a) 7x²+ 35x +42. b) x²/98 - y²/50 c) 9p⁴- 24p²q² + 16q²- 256r² d) x²y²- 6xyz + 9z²

15) If the side of a cube is increased by 12% by how much percent does it volume increase ?

16) Three solid cube of side 1cm, 6cm and 8cm are melted to form a new cube , find the surface area of the new cube thus formed .

17) 8) Numbers 1 to 10 are written on 10 separate slips and kept in a box and mixed up well. One slip is chosen from the box without looking into it. Find the probability of

a) getting a number 5

b) getting a number less than 8

c) getting a number greater than 7

d) getting a two digit number.

18) A letter is chosen from the word EQUATION. Find the probability that the letter is a constant

19) A bag contains 8 white balls. 5 green balls and 7 balls. They are mixed thoroughly and one ball is drawn random. Find the probability of getting

a) red ball

b) a green ball

c) a yellow ball

d) a white ball rolling

20) Read the table carefully and answer the following questions:

Marks. Students

10-20 7

20-30 12

30-40 19

40-50 11

50-60 21

60-70 10

70-80 8

80-90 02

90-100 10

If the passing marks in the test is 30, then

a) How many students have failed in the examination?

b) If A⁺ is awarded to students with 90 marks, how many students have achieved A⁺ marks?

c) How many students have passed the examination?

d) If student getting 60 or more is declared 1st division then find the number of students who have been put in 1st division.

21) The data shows India's total population (in millions) from 1951 to 2011. Represent the given data by bar graph.

Years. Population

1951 360

1961 432

1971 540

1981 684

1991 852

2001 1020

2011 1210

Test- Factorization

1) Factorize: - x²+ 5x - 6

2) The value of (348)²- (347)² is

a) (1)² b) 685 c) 695 d) 705

3) The product of x/(1 -3y)(3y+ x/2)(x²/4 + 9y²).

4) If 49x²- y = (7x + 1/2)(7x - 1/2), then the value of y is

a) 0 b) 4 c) 1/√2 d) 1/2

5) If the area of a rectangle is 4x²+ 4x - 3, then find its possible dimensions.

6) Factorize: 1/2 - x²/50.

7) If x+ 1/x = 8, then the value of x²+ 1/x is

a) 62 b) 64 c) 66 d) 60

8) If 9x²- 36x + k is a perfect square then the value of k is

a) 25 b) 5 c) 36 d) 81

9) Factorize: a³- 2√2 b³.

10) 162x⁴- 50.

Saturday, 24 February 2024

CBSCE- 9 - TEST PAPER

TEST PAPER- 1

1) Attempt all (1 x3= 3)

a) If cosecθ = 13/12, then the value of tanθ is

i) 12/5 ii) 5/12 iii) 5/13 iv) 5/12

b) If tanA = x/y, then cosA is

i) x/√(x²+ y²) ii) y/√(x²+ y²) iii) (x²- y²)/√(x²+ y²) iv) (x²- y²)/(x²+ y²)

c) secθ=

i) hypotenuse/base ii) perpendicular/base iii) base/hypotenuse/base iv) none

2) Answer any 2 (2 x2=4)

a) Given A is an acute angle and cosecA=√2, find the value of (2sin²A + 3 cot²A)/(tan²A - cos²A).

b) If cot θ= 15/8, then find the value of (2sin²θ+ 3 cosθ)/(5 sinθ - 3 cos²θ).

c) If sinθ = 12/13, then find tan²θ - sec²θ.

3) If tanθ= p/q, find the value of (p sinθ - q cosθ)/(p sinθ + q cosθ).

If sinx = √3/2 then value of {(2+ 2 sinx)(1- tanx)}/{(1+cotx)(2- 2 cosecx)}.

ICSE - IX - TEST PAPER (2023/24)

TEST PAPER (EXPANSION/FACTORIZATION)

1) Evaluate: (6x²- x +8)(x²-3).

2) If x - 1/x =4, find a) x²+ 1/x² b) x⁴ + 1/x⁴.

3) Evaluate: (83)² - (17)² with the formula.

4) Factorise:

a) x³- 3x² + x -3.

b) 63x²y² -7

c) 1- 6x + 9x².

d) 7x²- 19x - 6

5) ab - a - b +1=?

a) (1- a)(1- b) b) (1- a)(b- 1) c) (a- 1)(b - 1) d) (a- 1)(1- b)

6) 3+ 23x - 8x²= ?

a) (1- 8x)(3+ x)

b) (1+ 8x)(3 - x)

c) (1- 8x)(3- x) d) none

7) 7x²- 19x - 6=?

a) (x -3)(7x +2)

b) (x + 3)(7x - 2)

c) (x -3)(7x -2)

d) (7x -3)(x +2)

8) 12x²+ 60x +75=?

a) (2x +5)(6x +5)

b) (3x + 5)²

c) 3(2x +5)² d) none

9) 10p²+ 11p +3=?

a) (2p +3)(5p + 1)

b) (5p + 3)(2p +1)

c) (5p -3)(2p -1) d) none

10) 8x³- 2x =?

a) (4x -1)(2x -1)x

b) (2x² + 1)(2x - 1)

c) 2x(2x -1)(2x +1) d) none

Fill in the blanks

11) x²- 18x +81= (___)

12) 4- 36x²= (___)(___)(___)

13) x²- 14x +13= (__)(___)

14) 9z²- x² - 4y²+ 4xy = (___)(___)

15) abc - ab - c +1= (__)(__)

Test Paper-2

Section - A (10 x 1=10)

1) In the given figure,

angle BPC=19°, arc AB= arc BC= arc CD. Then, measure of angle APD is

a) 38° b) 59° c) 57° d) 76°

2) The given figures show two congruent circles with centre O and O'.

Arc AXB subtends an angle of 75° at the centre and arc A'YB' subtends an angle of 25° at the centre O'. Then , the ratio of arc AXB to A'YB' is

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

3) Greatest chord of a circle is called its

a) radius b) diameter c) chord d) secant

4) In the given figure,

if OA = 5cm, AB= 8cm and OD is perpendicular to AB, then CD is equals to

a) 2 cm b) 3 cm c) 4cm d) 5 cm

5) If a straight line APQB is drawn to cut two concentric circles,

then

a) AP> BQ b) AP= BQ c) AP< BQ d) AQ> PB

6) The radius of a circle is 5cm and the length of one chord is 8cm. The distance of the chord from the centre.

a) 4cm b) 3cm c) 5cm d) 6cm

7) In a quadrilateral ABCD, AB|| DC and AD= BC= 5.5cm, and one of the angles is 80°, then the other angles are

a) 90,90,100 b) 120,80,80 c) 80,100,100 d) 110,85,85

8) Which of the following is not true for a parallelogram ?

a) opposite sides are equal

b) opposite angles are always bisected by the diagonals

d) diagonals bisect each other.

9) ABCD is a rhombus

in which angle BCD= 100, then (x + y) equals to

a) 40 b) 60 c) 80 d) 70

10) Given a trapezium PQRS such that PQ= 12cm, RS= 5cm, PQ|| SR, PS= QR= 8cm, If Ang R = 130°, then angle P is

a) 130 b) 50 c) 150 d) 120

Section - B

(13x4= 52)

11) The given figure

show the circumcircle of an equilateral triangle. If the radius of the circumcircle is 20cm, find the length of each side of the equilateral triangle.

12) In a circle of radius 7.5cm, AB and BC are two equal chords,

each of length 9cm. Find the length of the chord AC.

13) Two circles of radii 17cm and 10cm intersect at two points and the length of the common chord is 16cm. Find the distance between their centres.

14) In the given figure,

the straight lines l, m and n are parallel to each other and G is the midpoint of CD. Calculate

a) BG, if AD= 7cm

b) CF, if GE= 2.5cm

c) AB, if AC= 9cm.

d) ED, if FE= 4cm.

15) ABCD is a quadrilateral P,Q,R, S are the midpoints of AB, BC, CD and DA respectively.

If AC= 6cm, BD= 8.6cm, calculate PQ, QR, SR and PS.

16) ABCD and AEFG are two parallelogram.

If Angle C= 58, determine angle F.

17) In the given figure, ABCD is a parallelogram,

AX and CY are respectively the bisectors of opposite angle A and C. If Angle DCB= 80, find the measure of angle DAX.

18) In the given figure ABCD is a parallelogram. E is the midpoint of CD and through D a line is drawn parallel to EB to meet CB produced at G and it cuts AB at F.

Show that

a) AD= (1/2) GC

b) DG= 2EB

Prove that the line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and equal to half their difference.

19) PQRS is a parallelogram. PO and QO are respectively the angle bisectors of angle P and Q. Line LOM is drawn parallel to PQ.

Prove that

a) PL= QM

b) LO= OM.

20) The base of a right angled triangle is 24cm and its hypotenuse is 25cm. Find the area of the triangle.

21) The area of a triangle is 216cm² and its sides are in the ratio 3:4:5. Find the perimeter of the triangle.

22) If the length and breadth of a rectangular room are each increased by 1m, then the area of floor is increased by 21m². If the length is increased by 1m and breadth is decreased by 1m, then the area is decreased by 7m². Find the perimeter of the floor.

23) A rectangular lawn 60m by 40m has two roads, each 5m wide, running in the middle of it, and parallel to length and the other parallel to breadth. Find the cost of gravelling them at Rs3.60 per m²

24) Find the area of a trapezium ABCD in which AB|| DC, AB= 77cm, BC= 25cm, CD= 60cm and DA= 26cm

TEST PAPER - 1

(Full marks- 80)

1) Attempt all the questions (1 x 10= 10)

a) After rationalising the denominator of 7/(3√3 - 2√2), we get the denominator as

i) 13 ii) 19 iii) 5 iv) 35

b) The compound interest on 1000 at 10% per compounded annually for 2 years is

i) Rs190 ii) Rs200 iii) Rs210 iv) Rs1210

c) Factorization of 63x²- 112y² is

i) 63(x - 2y)(x +2y) ii) 7(3x + 2y)(3x -2y)

d) If x = 3, y = k is a solution of the equation 3x - 4y=-7 then the value of k is

i) 16 ii) - 16 iii) 4 iv) - 4

e) The roots of the quadratic equation x²- 3x -4=0 are

i) -4,1 ii) 4, -1 iii) 4,1 iv) -4, -1

f) The value of (log8 - log2)/log32 is

a) 2/5 b) 1/4 c) -2/5 d) 1/3

g) Three angles of a quadrilateral are 75° ,90° and 75°. The fourth angle is

i) 90° ii) 95° iii) 105° iv) 120°

h) Two parallelograms are on equal bases and between the same parallels. The ratio of their areas is

i) 1:2 ii) 1:1 iii) 2:1 iv) 3:1

i) if d is the diameter of a circle, then its area is

i) πd² ii) πd²/2 iii) πd²/4 iv) 2πd²

j) The distance between the points (0,5) and (-5,0) is

i) 5 units ii) 5√2 units iii) 2√5 units iv) 10 units

2) Any 10 (2 x 10=20)

a) Find the amount of the compound interest on Rs8000 at 5% per annum for 2 years.

b) Find the co-efficient of x² and x in the product of (x -5)(x +3)(x+7).

c) If x - y= 8 and xy= 5, find x²+ y².

d) Solve for x and y: 3x + 4y=10; 2x - 2y=2.

e) The sum of two number is 43. If the larger is doubled and the smaller is tripled, the difference is 36, Find the two numbers.

f) Solve: 2x²- 5x = 0.

g) Evaluate : [(64)²⁾³ (2)⁻² + 8⁰]⁻¹⁾²

h) Evaluate: 2 log 5 + log 8 - (1/2) log 4.

i) in ∆ABC, AB= AC and D is point on AB such that AD = DC= BC. Show that angle BAC = 36.

j) If the sides of a triangle are in the ratio 3:4:5, prove that it is right angled triangle.

k) A chord of length 16cm is drawn in a circle of a diameter 20cm. calculate its distance from the centre of the circle.

3) Any 10 (2 x 10= 20)

a) If the area of an equilateral triangle is 81√3 cm², find its perimeter.

b) Find the area of a quadrilateral whose diagonals are of length 18 cm and 12 cm, and they interesect each other at right angles.

c) Find the radius and circumference of a circle whose area is 144π cm².

d) The volume of a cube is 729 cm³. Find its surface area and the length of a diagonal.

e) if sin x = 5/13 and x is acute angle, find the value of tan x + 1/cos x.

f) Evaluate: 2√2 cos45 cos 60 + 2√3 sin 30 tan 60 - cos 0.

g) Evaluate: cos²26+ cos64 sin 26 + tan36/cot54.

h) Find points on the x-axis which are at a distance of 5 units from the point (5,-4).

i) The mean of 5 numbers is 20. If one number is excluded, mean of the remaining numbers becomes 23. Find the excluded number.

j) Explain the meaning of class mark.

k) In a right angled triangle, if hypotenuse is 20cm and the ratio of the other two sides is 4:3, find the sides.

l) In ∆ ABC, AB= AC, Angle A= (5x + 20)°and each of the base angle is 2/5 th of Angle A. Find the measure of Angle A.

4) The compound interest on a sum of money for 2 years is Rs410 and the simple interest on the same sum for the same and at the same rate is R400. Find the sum and the rate of interest

On what sum will the difference between the simple and compound interest for 3 years at 10% p.a. is Rs232.50? (4)

5) Constrct a histogram for the following data:

Weekly earnings (in Rs) No of workers

150-165 8

165-180 14

180-195 22

195-210 12

210-225 15

225-240 6 (4)

6) a) Constuct a rectangle ABCD with AB= 5 cm and AD= 3 cm. (2)

b) Draw the graphs : 2x + y + 3= 0. (2)

7) If AD, BE, CF are medians of ∆ABC, prove that 3(AB²+ BC²+ CA²)= 4(AD²+ BE²+ CF²).

Prove that each angle of a rectangle is 90°.

a) If the diagonals of a rhombus are equal, prove that it is a square.

b) If the angle of a quadrilateral are equal, prove that it is a rectangle. (4)

7) Two Cubes, each with 12 cm edge, are joined end to end. Find the surface area of the resulting cuboid. (4)

8) If log {(x+ y)/2)}= (1/2) (logx + logy), show that x²+ y²= 6xy. (4)

9) a) If an angle of a parallelogram is two-thirds of its adjacent angle, find the angle of the parallelogram. (2)

b) If 2ˣ = 3ʸ = 6⁻ᶻ, show that 1/x + 1/y + 1/z = 0. (2)

10) Factorize: x⁴+ 9x²y²+ 81y⁴

Factorize: x²+ x⁵. (2)

Saturday, 17 February 2024

TEST PAPER- XII (2023/24)

TEST PAPER -5 (2025)

Time: 3 hrs: Max. Marks: 80

GENERAL INSTRUCTIONS

1. This question papers contains 5 section A, B, C, D and E. Each Section is compulsory. However, there are internal choices in some questions .

2. Section A has 18 MCQ's and 02 Assertion -Reason based questions of 1 mark each .

3. Section B has 5 very Short Answer (VSA) type questions of 2 marks each.

4. Section C has 6 Short Answer(SA) type questions of 3 marks each.

5. Section D has 4 Long Answer (LA) type questions of 5 marks each.

6. Section E has 3 source based/case/passage based/integrated units of assessment (4 marks each) with sub-parts.

• SECTION - A

(Multiple Choice Questions) Each question carries 1 mark

1) The direction angles of line l are

a) α, β, γ

b) -α, -β, -γ

c) π -α, π-β, π-γ d) none

2) ∫ log tanx dx at (π/2,0) is equals to

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

3) If tan⁻¹(x²+ y²)= a, then dy/dx is equals to

a) x/y b) -x/y c) y/x d) -y/x

4) If sin⁻¹x = y, then

a) 0≤ y ≤ x

b) -π/2 ≤ y ≤ π/2

c) 0< y < π

d) -π/2 < y < π/2

5) ∫ eᵅˣ {af(x)+ f'(x)} dx is equals to

a) eᵅˣ f(x)+ C

b) aeᵅˣ f(x)+ C

c) eᵅˣ f'(x)+ C

d) eˣ f(ax)+ C

6) The degree of the differential equation (d²y/dx²)² + (dy/dx)²= x sin(dy/dx) is

a) 1 b) 2 c) not defined d) 3

7) If A= a b

c d then determinants of A is written as

a) a b b) a c c) a c d) a b

c d d b b d d c

8) The matrix 2x +y 4x= 7 7y - 13

5x - 7 4x y x + 6 then the value of x + y is

a) 1 b) 2 c) 4 d) 5

9) The area bounded by y= - x²+ 2x +3 and y=0 is

a) 32 sq units

b) 32/3 sq units

c) 1/32 sq units

d)1/3 sq units

10) The function f(x)= x³- 6x²+ 12x - 18 is

a) strictly increasing function

b) increasing function

c) decreasing functions

d) strictly decreasing function

11) if the relation R defined on the set A={1, 2, 3,4, 5, 6} is R={(a,b); b= a+1}, then R is

a) reflexive

b) reflexible and symmetric

c) not reflexible

d) reflexive but not run transitive

12) For Matrix A= 1 -2

3 5 , (A') A is equals to

a)10 13 b) 10 13 c) 13 29 d) 1 10

13 29 29 13 10 13 1 10

13) The projection of the vector 7i + j - 4k on 2i + 6j + 3k is

a) 7/8 b) 8/7 c) 1/7 d) 1/8

14) If A is a matrix of order 3 x 3 such that |A|= 5, then |A (adj A)| is equal to

a) 25 b) 125 c) 5 d) 1/125

15) Solve (2y -1)dx - (2x +3) dy =0

a) (2x +3)(2y -1)= C

b) (2y -1)(2x +3)= C

c) (y -1)(2x +3)= C

d) (2y -1)(x +3)= C

16) If y= (1+ x¹⁾⁶)(1+ x¹⁾³)(1- x¹⁾⁶), then dy/dx at x=1 is equal to

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

17) ∫ sin²(x/2) dx is equals to

a) (x - sinx)/2+ C

b) (x + sinx)/2+ C

c) (sinx -x)/2+ C

d) (x/2 + sinx) + C

18) If a= i + j + 2k and b = 3i + 2j - k, then the value of (a+ 3b). (2a - b) is

a) 15 b) 5 c) -15 d) 10

Assertion -Reason Based Questions

In the following questions , a statement of Assertion (A) is followed by 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 A.

c) A is true but R is false.

d) A is false but R is true.

19) If R is the relation in the set A ={1, 2, 3, 4, 5 } given by R={(a,b): |a - b| is even }.

Assertion (A): R is an equivalence relation.

Reason (R): All elements of {1,3,5} are related to all elements of {2,4}.

20) Assertion (A): If A= 2 3 -1

1 4 2

and B= 2 3

4 5

2 1 then AB and BA both are defined.

Reason (R): For the two matrices A and B, the product AB is defined , if number of columns in A is equal to the number of rows in B.

SECTION B

This Section compromises of very short answer type questions (VSA) of 2 marks each)

21) Find the cartesian and vector equation for the line passing through the points A(-1,1,2) and B(2,4,5).

The x-cordinate of a point on the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its x-cordinate.

22) Find the domain of the function f(x)= cos⁻¹x + sin⁻¹2x.

23) If y= x¹⁾ˣ, find dy/dx

If x = a(θ + sinθ), y= a(1- cosθ), find dy/dx.

24) Evaluate ∫ (x⅖-1)/(x²+4) dx.

25) Find the values of λ and μ, for which (2i + 6j + 27k) x (i + λj + μk)= 0.

SECTION C

(This Section compromises of short answer type questions (SA) of 3 marks each)

26) Show that for a≥ 1,

f(x)= √3 sinx - cosx - 2ax + b is decreasing in R.

27) Find the particular solution of the differential equation

eˣ tany dx + (2- eˣ) sec²y dy = 0, given that y=π/4 when x=0.

Find the particular solution of the differential equation

(3xy + y²) dx + (x²+ xy) dy =0, for x= 1, y= 1.

28) Evaluate ∫ (1+ x²)/(1+ x⁴) dx.

Evaluate ∫ x(logx)² dx.

29) If A={1, 2,3,.....9} and R is the relation in Ax A defined by (a,b) R(c,d), if a + d = b + c for (a,b), (c,d) in A x A.

Prove that R is an equivalence relation. Also, obtain the equivalence class [(2,5)].

If f: X --> Y is a function. Define a relation R on X given by R ={(a,b): f(a)= f(b)}. Show that R is an equivalence relation on X.

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

r= i + j + λ(2i - j + k) and 2i + j - k + μ(3i - 5j + 2k).

31) Find the area of the region bounded by the line y= 3x +2, the x-axis and the ordinates x =-1 and x =1.

SECTION D

(This Section compromises of long answer type questions (LA) of 5 marks each)

32) For any two vectors a and b, show that (1+ |a|²) (1+ |b|²)= {(1- a.b)}²+ |a+ b + (ax b)|².

33) Solve the following system of equations by metrix method, where x ≠ 0, y≠ 0 and z≠ 0.

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

Determine the product of

-4 4 4 & 1 -1 1

-7 1 3 1 -2 -2

5 -3 -1 2 1 3 and then use to solve the system equations x - y + z = 4, x - 2y - 2z = 9 and 2x + y + 3z =1.

34) Solve the following LPP maximize

Z= 12x + 16y subject to constraints,

x + y ≤ 1200,

y≤ x/2,

x ≤ 3y + 600, x, y ≥ 0

Solve the LPP maximize Z= 40x + 50y

Subject to constraints

3x + y ≤9, x + 2y ≤ 8, x, y ≥ 0

35) Evaluate ∫ (x +1)/{x(1+ xeˣ)²} dx.

SECTION E

(This Section compromises of 3 case study/passage based questions of 4 marks each)

36) Consider the given equation dy/dx + Py = Q.

The above equation is known as linear differential equation, Here, IF = ₑ∫P dx and solution is given by y. IF = ∫ (Q. IF) dx + C. Now, consider the given equation

(1+ sinx) dy/dx + y cosx + x =0.

On the basis of above information, answer the following questions .

a) Find the value of P and Q.

b) Find IF.

c) Find the general solution of the given equation.

If y(1+ sinx)= -x²/2 + C and y(0)=1, then find y and y(π/2).

37) A random variable X has the following probability distribution

x: 0 1 2 3 4 5 6 7 8

P(x): a 3a 5a 7a 9a 11a 13a 15a 17a

On the basis of above information, answer the following questions.

a) Find the value of a.

b) Find P(X= 4).

c) Find P(X>5) and P(0≤ X ≤ 2).

Find (1≤ x ≤4) and P(3<x ≤6).

38) An electronic assembly consists of two subsystem say A and B as shown below.

From previous testing procedures , the following probabilities are assumed to be known P(A fails)= 0.2, P(B fails alone)= 0.15, P(A and B fail)= 0.15.

On the basis of above information, answer the following questions .

a) Find the probability P(B fails) and probability P(A fails alone).

b) Find the probability P(whole system fail) and the probability of P(A fails/B has failed ).

1c 2c 3b 4b 5a 6c 7a 8d 9b 10b 11c 12a 13b 14b 15b 16b 17a 18c 19c 20a

21) (x +1)/1= (y -1)/1= (z -2)/1; r= - i + j + 2k + λ(i + j+ k) or -1

22) (-1/2,1/2)

23) x¹⁾ˣ{(1- logx)/x²} or tan(θ/2)

24) x - (5/2) tan⁻¹(x/2)+ C

25) λ = 3 and μ = 27/2

27) tany = 2- eˣ or |y²+ 2xy|= 3/x²

28) (1/√2) tan⁻¹{(x²-1)/√(2x)}+ C or (x²/2) (logx)² - (x²/2) logx + x²/4+ C

29) {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)} 30) 10/√59

31) 13/3 sq units

33) x=1/2, y = 1/3 and z = 1/5 or

8 0 0

0 8 0

0 0 8 x=3, y=-2, z= -1

34) max= 16000 at (800,400) or min= 230 at (2,3)

35) log(xeˣ) - log(1+ xeˣ) + 1/(1+ xeˣ) + C

36) P= cosx/(1+ sinx) , Q= -x/(1+ sinx)

1+ sinx

y(1+ sinx)= -x²/2 + C or (8- π²)/16

37) 1/81

1/9

5/9, 1/9 or 8/27, 11/27

38) 0.30 and 0.05

0.35 and 0.5

μ λ μ ⁻¹

TEST PAPER -4

θ μ λ μ

TEST PAPER -4

SECTION : A (80 Marks)

Question 1) (10x2= 20 Marks)

i) If A= 2 3

4 5 , find Inverse of A.

ii) Show that the function f: R-> R, given by f(x) = | x | is neither one one or onto.

iii) Show sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹ x = π/2

iv) If y= tan⁻¹(secx+ tanx), find d²y/dx²

v) ∫ x eˣ dx

vi) lim ₓ→₀ (log cos x)/sin²x

vii) Prove without expanding:

a - b 1 a a 1 b

b - c 1 b = b 1 c

c - a 1 c c 1 a

viii) If x > 1/2, show that the function f(x)= x(4x²-3) is strictly increasing.

ix) Solve 2ˣ⁻ʸ dx + 2ʸ ⁻ˣ dy = 0

x) A and B are two Independent events with P(A)= 2/5 and P(B)= 1/3, Evaluate P(AUB).

Question 2). (4)

Prove: 1+a²-b² 2ab - 2b

2ab 1 - a²+b² 2a

2b - 2a 1 - a² - b²

= (1+a²+b²)³.

Question 3). (4)

If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2 then Prove that, x² + y² + z² = r².

Question 4). (4)

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement. Find the probability that none is red.

Question 5). (4)

If y= (tan⁻¹x)², show that (1+x²) d²y/dx² + 2x(1+x²) dy/dx - 2= 0.

Question 6). (4)

Evaluate ∫ 2⁴ˣ sin 3x dx.

Evaluate:

∫ (cosx + x sinx)/{x(x+ cosx) dx

Question 7) (4)

Find the equations of the tangent to the curve y= x² - 2x +7 which is:

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

b) Perpendicular to 5y-15x= 13

Show that the maximum value of 2x + 1/2x is less than its minimum value.

Question 8). (4)

Solve by matrix inversion method

x+2y+z= 7; x + 3z= 11; 2x - 3y=1

Show that

a + b+ 2c a b

c b+c + 2a b

c a. c+a+2b = 2(a+b+c)³.

Question 9). (6)

Given x+y= 3, find the maximum and minimum values of 9/x + 36/y

A closed right circular cylinder is has a volume of 2156cm³. What will be the radius of the base so that total surface area is minimum.

Question 10). (4)

Show that the function f in A= R - {2/3} defined as f(x)=(4x-3)/(6x-4) is one-one and onto .

Question 11). 2+2=4

Solve:

a) dy/dx + y secx = tan x.

b) tan x dy/dx= 1+y² where x= π/2 and y= 1.

Question 12). 3+3= 6

a) Evaluate ∫ |sin x| dx at (π/2,-π/2).

b) Prove ∫ {log(1+x)}/(1+x²) dx at (1,0) = π/8 . Log 2.

Question 13). (3x2= 6)

a) If x= sint and y= cos pt, p is constant, then find the value of (1-x²) d²y/dx² - x dy/dx.

b) If m² = p² cos² t + q²sin²t, then show that m+ d²m/dt² = p²q²/m²

Question 14). 3+3=6

A) It is known that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. Assuming that males and females are in equal proportion, find the probability of the person to be male.

B) Rajiv and Robin play 12 games of chess. Rajiv wins 6 games, Robin wins 4 games and and 2 games end in a draw. They agree to play 3 more games. Calculate the probability that out of these 3 games, two games end in a draw.

Evaluate:

A) ∫ x² sin⁻¹x dx

B) ∫x² eᵃˣ dx at (a,0)

SECTION C. (20 Marks)

Question 15). 2+2+2

A) Given demand function x= 50- 0.5 P and cost function C=50+40x, find price for break-even price.

B) 4x+y-10= 0, 2x + 5y -14= 0 are two regression lines. Find the correlation coefficient between variables x and y.

C) The total cost C(x) of a firm is C(x)= 0.0005x³ - 0.7x² - 30x + 3000 where x is the output. Determine:

a) average cost (AC)

b) Marginal cost (MC)

Question 16). (4)

The two lines of Regression for a distribution (x,y) are 3x+2y= 7 and x+4y= 9. Find the regression coefficient X on Y and Y on X.

Treating x as an independent variable. Find the line of best fit for the following date:

X: 15 12 11 14 13

Y: 25 28 24 22 30

Hence, predict the value of y when x= 10.

Question 17) (4)

The marginal cost function of manufacturing x units of a commodity is 6+10x - 6x². The total cost of producing one unit of the commodity is ₹ 12. Find the total and average cost functions.

If c= 2x{(x+4)/(x+1)} + 6 is the total cost of production of x units of a commodity, show that marginal cost falls continuously as a x increases.

Question 18). (6)

A small firm manufacturers gold rings and chains. The combined number of rings and chains manufactured per day is almost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹300 and on a chain is ₹190, how many of each should be manufactured daily so as to maximize the profit?

TEST PAPER - 1

Time : 3 hours. Full Marks: 100

Group A:

b) Fill in the gaps: The value of the determinant

3 1975 1978

4 1982 1986 is ______

5 1995 2000

c) State whether the following statement is true or false:

" The product of two non-zero matrices be a non-zero matrix.

d) If y= logₑlogₑx, x > 1 which one is true

i) x dy/dx = 1

ii) (xlogₑx) dy/dx = 1

iii) (logₑx) dy/dx= 1

iv) (logₑx) dy/dx = x

e) If x = a(θ - sinθ) and y= a(1+ cosθ) then which one of the following is the value of dy/dx.

i) - cot(θ/2) ii) - cotθ iii) -tan(θ/2) iv) cot(θ/2)

f) If y= cos²x, then which one of the following is the value of d²y/dx² ?

i) - 2cos2x ii) 2 cos2x iii) - 2sin2x iv) cos2x

g) ∫ tan²dx = ______

h) ²₀∫ dx/(x²+4) =π/16. T/F

i) ∫ sin⁵x dx at (π/2, -π/2)= _____

j) The gradient of the tangent at the point (8,-4) to the parabola y²= 8(x - 6) is -1. T/F

2) a) Examine whether AB= BA for the two matrices A= 1 5 & B= 0 0

1 3 0 1. (2)

b) If y= (Secx)ᵗᵃⁿˣ , then find dy/dx. (2)

3) If x³+ y³= 2xy, then find the value of dy/dx at the point (1,1). (2)

4) Evaluate: ∫ 2sinx/(5+ 3 cosx) dx

Evaluate: ∫ xe²ˣ dx

5) What are the order and degree of the following differential equation?(d²y/dx²)² + 5(dy/dx)³+ 2y =0.

) An integer is chosen at random from 100 integers 1,2,3,.....,100. What is the probability that the selected integer is divisible by 5 or 7? (4)

) ᵗᵗ⁻¹ˣʸ³⁺²ˣ⁻¹ˣˣ⁻ˣˣ³₂

Wednesday, 14 February 2024

LAST REVISION NOTES - CLASSXII (2023/24)

DETERMINANTS

Using properties of determinants

1) x² y² z²

x³ y³ z³

xyz yzx zxy

= xyz(x - y)(y - z)(z - x)(xy + yz+ zx)

2) y+ z x+ y x

z+ x y+ z y = (x³+ y³+ z³- 3xyz)

x + y z + x z

3) a b- c c- b

a- c b c - a

a - b b - a c

= (a+ b - c)(b + c - a)(c + a - b)

4) (b+ c)² a² a²

b² (c + a)² b² = 2abc(a+ b+ c)³

c² c² (a+ b)²

5) y+ z z y

z z+ x x = 4xyz

y x x + y

6) b²c² bc b + c

c²a² ca c + a = 0

a²b² ab a + b

7) 1 1 1

a² b² c²

a³ b³ c³

= (a - b)(b - c)(c - a)(ab + bc+ ca)

8) - a² ab ac

ba - b² bc = 4a²b²c²

ac bc - c²

9) 1 + a 1 1

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

1 1 1+ c

10) a b c

a² b² c²

bc ca ab

= (a- b)(b - c)(c - a)(ab + bc+ ca)

11) a b c

a- b b - c c - a = a³+ b³+ c³ - 3abc

b + c c + a a+ b

12) a- b - c 2a 2a

2b b - c - a 2b = (a + b + c)³

2c 2c c - a - b

13) a²+1 ab ac

ba b²+1 bc = a²+ b²+ c²+ 1

ca cb c²+1

14) 1 374 1893

1 372 1892 =1

1 371 1891

15) b+ c c + a a+ b 2a 2b 2c

q+ r r+ p p+ q = 2p 2q 2r

y+ z z+ x x+ y 2x 2y 2z

16) 1 a a² - bc

1 b b²- ac = 0

1 c c² - ab

17) 42 6 1

28 4 7 = 0

14 2 3

18) c - a a- b b - c

a - b b - c c - a = 0

b - c c - a a - b

19) 219 198 181

240 225 198 = 0

265 240 219

20) a b - c c - b

a- c b c - a

a - b b - a c

= (a+ b - c)(b + c - a)(c + a - b)

21) b²+ c² a² a² a² bc ac + c²

b² c²+ a² b² = a²+ab b² ac

c² c² a²+ b² ab b²+bc c²

22) 1 a bc 1 1 1

1 b ca = a b c

1 c ab a² b² c²

23) b 1 a

c -a 1 = 1+ a²+ b²+ c²

1 -b - c

24) Using Cramer's rule solve:

a) x + y=2; 2x - z = 1; 2y - 3z = 1. 3/4,5/4,1/2

b) x - y=1 ; x + z =-6; x +y -2z =3. -2,-3,-4

c) 3x +4y+ z=5 ; x -3y + 2z =-8; -4x +2y -9z =2. 7,-3,-4

25) Solve:

a) 3 - x -1 1

-1 5 - x -1. = 0. 2,3,6

1 -1 3 - x

b) x + a b c

c x+ b a = 0

a b x+ c 0, -(a+ b+ c)

c) 2 - x 3 3

3 4- x 5= 0

3 5 4 - x 0, -1, 11

d) x +2 1 -3

1 x - 3 x -2 = 0

-3 -2 1 2,12

e) x² x 1

0 2 1 = 28

3 1 4 2, -17/7

f) If x, y and z are all different and

x x² 1+ x³

y y² 1+ y³ = 0

z z² 1+ z³ Then show that xyz = -1

INVERSE TRIGONOMETRIC FUNCTION

Prove:

1) a) 4 tan⁻¹(1/5) - tan⁻¹(1/70)+ tan⁻¹(1/99)= π/4

b) sin⁻¹{x/√(1+ x²)} + cos⁻¹{(x+1)/√(x²+ 2x+2)}= tan⁻¹(x²+ x+1)

c) cot(π/4 - 2 cot⁻¹3)= 7.

d) sin⁻¹(√3/2)+ tan⁻¹(1/√3)= 2π/3.

e) sin⁻¹(1/√17)+ cos⁻¹(9/√85)= tan⁻¹(1/2).

f) tan⁻¹(1/3) + tan⁻¹(1/5) + tan⁻¹(1/7)+ tan⁻¹(1/8)= π/4.

g) tan⁻¹(1/2 tan2A) + tan⁻¹(cotA) +tan⁻¹(cot³A)=0.

h) 2(tan⁻¹1 + tan⁻¹(1/2) +tan⁻¹(1/3)=π.

i) tan⁻¹x + cot⁻¹(x +1) = tan⁻¹(x²+ x +1).

j) tan⁻¹(1/4) + tan⁻¹(2/9) =(1/2) cos⁻¹(3/5).

k) (1/2) tan⁻¹x= cos⁻¹√[{1+ √(1+ x²)}/2√(1+ x²)].

l) sin⁻¹(4/5) + cos⁻¹(2/√5) = cot⁻¹(2/11).

m) sec²(tan⁻¹2) + cosec²(cot⁻¹3) = 15.

n) sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16)=π.

o) tan⁻¹[{√(1+ x²)+ √(1- x²)}/{√(1+ x²) - √(1- x²)} = π/4 + (1/2) cos⁻¹x².

p) cos⁻¹(4/5) + cot⁻¹(5/3) = tan⁻¹(27/11).

q) tan(2 tan⁻¹a) = 2 tan(tan⁻¹a + tan⁻¹a³).

r) cot⁻¹{(PQ+1)/(p - q)} + cot⁻¹{(QR +1)/(q - r)} + cot⁻¹{(rp +1)/(r - p)}= 0.

s) sin[sin⁻¹(1/2) + cos⁻¹(3/5)] = (3+4√3)/10.

t) cos[tan⁻¹(15/8) - sin⁻¹(7/25)] = 297/425.

2) Solve:

a) tan⁻¹(2+ x) + tan⁻¹(2- x) = tan⁻¹(2/3). ±3

b) sin⁻¹(5/x) + sin⁻¹(12/x) =π/2. ±13

c) sin⁻¹6x + sin⁻¹(6√3 x)=π/2. ±1/12

d) sin⁻¹{2a/(1+ a²)} + sin⁻¹{2b/(1+ b²)}= 2 tan⁻¹x. (a+ b)/(1- ab)

e) sin[2 cos⁻¹cot(2tan⁻¹x)]= 0. x²= 3± 2√2

f) cos(sin⁻¹x)= 1/9. ±4√5/9

g) tan⁻¹2x + tan⁻¹3x= π/4. 1/6

h) tan⁻¹{1/(2x+ 1)} + tan⁻¹{1/(4x+1)} = tan⁻¹(2/x²). 0, -2/3,3

i) tan⁻¹(1+ x) + cot⁻¹(x -1) = sin⁻¹(4/5)+ cot⁻¹(3/4).

j) tan⁻¹(x -1) + tan⁻¹x + tan⁻¹(x +1)= tan⁻¹3x.

k) sin⁻¹x + sin⁻¹2x= π/3. 1/6

3) Find the value of:

a) sin cot⁻¹cos(tan⁻¹x). √{(1+ x²)/(2+ x²)}

b) cos[2 cos⁻¹x + sin⁻¹x ] at x= 1/5. -2√6/5

c) tan{2tan⁻¹(1/5) - π/4}. -7/17

d) tan (1/2)[cos⁻¹(√5/3)] (1/2) (3- √5)

e) cos[cos⁻¹(-√3/2) + π/6]. -1

f) sin[π/3 - sin⁻¹(-1/2)]. 1

4) If tan⁻¹a + tan⁻¹b+ tan⁻¹c =π, then show a+ b + c = abc

5) If tan⁻¹x + tan⁻¹y + tan⁻¹z=π/2, then show xy+ yz + zx =1.

6) If cos⁻¹(x/2) + cos⁻¹(y/3) = K, show that 9x²- 12xy cos K + 4y²= 36 sin²K

DIFFERENTIATION

Find dy/dx of following:

1) x = a sin³t and y= a cos³t. - cot t

2) y= eˣlog tan2x. eˣ(4/sin4x + log tan2x)

3) y= (cos x)ᶜᵒˢˣ. -(cosx)ᶜᵒˢˣ sinx (1+ log(cosx).

4) y=√{(1- cosx)/(1+ cosx)}. Cosecx

5) ₑsinx². 2x cosx² ₑsinx²

6) x= a(cos t + t sin t), y= a(sin t - t cos t) at t=π/4. 1

7) logₑ{x + √(x²+ k²)}. 1/√(x²+ k²)

8) tan⁻¹ [{√(1+ x²) -1}/x]. 1/2(1+ x²)

9) (x+1)(x -2)/√x. 3√x/2 - 1/2√x + 1/√x³

10) y= 2t/(1+ t²) and x= (1- t²)/(1+ t²). (t²-1)/2t

Prove:

1) If y= (sin⁻¹x)² then show (1- x²) d²y/dx² - x dy/dx = 2.

2) If ₑmcos⁻¹ x, show that (1- x²) d²y/dx² - x dy/dx = m²y.

3) If xʸ = eˣ⁻ʸ then show dy/dx = logₑx/(1+logₑx)².

4) If xʸ yˣ = 5, then show dy/dx = [(logy + y/x)/(logx + x/y)

5) If y= sin⁻¹x/√(1- x²), then show that (1- x²) dy/dx - xy =1.

6) If xᵖ yᑫ = (x + y)ᵖ⁺ᑫ, then show dy/dx = y/x.

7) If y= xʸ, then show x dy/dx = y²/(1- y log x).

8) If √(1- x⁴)+ √(1- y⁴)= K(x²- y²), then show y√(1- x⁴) dy/dx = x √(1- y⁴).

9) If sin(xy)+ cos(xy)= 1 and tan(xy)≠ 1, then show that dy/dx = -y/x.

10) If y= {x + √(x²- 1)}ᵐ, then show (x²-1)(dy/dx)²= m² y².

11) If y= sin⁻¹x/√(1- x²), show that (1- x²) dy/dx - xy =1.

12) If ₑaˣ, show that d²y/dx² - 2a dy/dx + (a²+ b²) y= 0.

13) If √(1- x²) + √(1- y²)= (x - y) then show that √{(1- y²)/(1- x²)}

MAXIMUM AND MINIMUM

1) A box to be constructed from square metal sheet of side 60 cm, by cutting out identical squares from the four corners and turning up the sides. Find the length of the side of square to be cut out so that the box has maximum volume. 16000cm³

2) A rectangle is given whose area is constant. Prove that the sum of the length of its sides at least when it is a square.

3) Find the volume of the largest cone that can be inscribed in a sphere of radius R.

4) Assuming that the stiffness of a beam of a rectangular cross section varies as the breadth and as the cube of depth, what must be the breadth of stiffness beam that can be cut from a log of diameter a.

5) How should a wire 20cm long be divided into two parts. If one part is to be bent into a circle p, the other part is to be bent into a square and the two plane figures are to have areas the sum of which is minimum .

6) Prove that the right circular cone of maximum volume which can be inscribed in a sphere of radius a has a height of 4a/3.

7) An open tank with a square base of side 'x' metres and vertical height 'h' metres is to be constructed so as contain 'c' cubic metres of a water. Show that the expenses on lining the inside of the tank with lead would be least if h= x/2.

8) A right -angled triangle ABC with constant area S is given. Prove that the hypotenuse of the triangle is least when the triangle is isosceless.

9) The sum of three positive numbers is 26. The second number is thrice as large as the first. If the sum of the square of these numbers is least, find the numbers. 4,12,10

10) The length of the perimeter of a section of a circle is 20cm. Give an expression for the area of the sector in terms of r(the radius of the circle) and hence, find the minimum area of the sector p.

11) ABC is a right angled triangle of given area S. Find the sides of the triangle for which the area of circumscribed circle is least.

12) Prove that f(x)= log x do not have maximum or minimum .

13) Show that the height of a closed cylinder of given volume and minimum surface area is equal to its diameter.

14) An open box with a square base is to be made out of quantity of cardboard whose axis is c² unit, show that the maximum volume of the box is c³/6√3 units.

FUNCTION/ CONTINUITY/ TANGENT - NORMAL

1) Find the value of k, for which

f(x)= {√(1+ kx) - √(1- Kx)}/x , if -1≤ x < 0

(2x+1)/(x -1), if 0≤ x < 1

is continuous at x= 0. -1

2) Find the value of the constant k so that the function f, defined below, is continuous at x= 0, where f(x)= {(1- cos4x)/8x²}, if x≠ 0

k , if x= 0. 1

3) Find the value of p and q, for which

f(x)= (1- sin²x)/3 cos²x, if x < π/2

p , if x=π/2

q(1- sinx)/(π- 2x)², if x >π/2

is continuous at x=π/2. p=1/2 and q=4

4) f(x)= (1- cos4x)/2, when x< 0

a, when x= 0

√x/√{(16+ √x) -4}, when x > 0

and f is continuous at x= 0, find the value of a. 8

5) Find the value of k if the function defined by

f(x)= 2x -1, x < 2

k, x=2 is continuous x=2

x+1, x > 2. 3

6) Let 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. √(x -4)

7) Show that the function f(x)= |x -1|, x belongs to R, is continuous at x= 1.

8) Find the equations of the normal to the curve y= x³+ 2x +6 which are parallel to the line x + 14y +4= 0. x+ 14y =254 and x + 14y +86=0

9) State the reason why the relation R= {(a, b): a ≤ b²} on the set R of real numbers is not reflexive.

10) Let f: [0, ∞] --> R be a function defined by f(x)= 9x² + 6x - 5. Prove that f is not invertible . Modify only the codomain of f to make f invertible and then find its inverse. (-1+√(6+ y))/3

11) Let R be a relation defined on the set of natural numbers N as follows:

R={(x,y): x ∈ N, y ∈ N and 2x + y =24}

Find the domain and range of the relation R. Also, find if R is an equivalence relation or not. {1,2,3,4,....11}; {2,4,6,8,10,12,....22}; no

12) The equation of the tangent at (2,3) on the curve y²= ax³+ b is y=4 x - 5. Find the values of a and b. 2, -7

13) Show that the relation R defined by (a,b) R (c,d) => a + d= b + c on the set N x N is an equivalence relation.

14) If the function f: R --> R be given by

f(x)= x²+ 2 and g: R--> R be given by g(x)= x/(x -1), x≠ 1, find f o g and g o f and hence find f o g(2) and g o f (- 3). 6, 11/10

15) Let Z be the set of all integers and R be the relation on Z defined as R= {(a,b): a, b∈ Z and (a - b) is divisible by 5}. Prove that R is an equivalence relation.

16) Show that the function f: R --> R defined by f(x)= x/(x²+1) is neither one-one nor onto.

REGRESSION

1) Out of the following two regression lines, find the line of regression of x on y.

x + 4y = 3 and 9x + 3y = 16. 9x + 3y =16

2) if 4x - 5y + 33 =0 and 20x - 9y - 107=0 are two lines of regression. Find:

a) the mean value of x and y. 13, 17

b) the regression coefficients bᵧₓ and bₓᵧ. 4/5, 9/20

c) the correlation coefficient between X and Y. 0.6

d) the standard deviations of y, if the variance of x is 9. 4/5

e) the value of y for x= 3. 9

f) the value of x for y=2. 25/4

3) the equation of two lines of regression are 4x + 3y + 7 = 0 and 3x + 4y + 8 = 0.

a) find the mean value of x and y. -4/7,-11/7

b) find the regression coefficients bᵧₓ and bₓᵧ. -3/4,-3/4

c) find the correlation coefficient between X and y. -3/4

4) the regression lines are represented by 4x + 10y =9 and 6x + 3y =4. Find the regression line of x on y. 4x + 10y = 9

5) Find the equations of the lines of regression for the data:

X: 1 2 3 4 5

Y: 7 6 5 4 3

And hence find an estimate of the variable y for x= 3.5 from the appropriate line of regression. x+ y= 8, 4.5

6) you are given the following data:

X Y

A. M 36 85

S. D 11 8

Correlation copetion between X and Y is 0.66. find

a) the two dignition co-efficients . 0.9075

b) the two regression equations. y= 0.48x+6; x= 0.9075y - 41.41

c) the most likely value of y when x=10. 72.52

INTEGRATION

1) ∫ 2cos2x/(1+ sin2x) dx at (π/4,0). Log2

2) ∫ sinx cosx/(cos²x + 3 cosx +2) dx at (π/2,0). Log(9/8)

3) ∫ (x +3)/√(x²+ 4x+5) dx. √(x²+ 4x+5)+ log|√{1+ (x+2)²} + (x +2)|

4) Show that the area included between the x-axis and the curve a²y = x²(x + a) is a²/12.

5) ¹₀∫ x tan⁻¹x dx. π/4 -1/2

6) ∫e⁻²ˣ sinx dx. -2e⁻²ˣ/5 sinx - (1/5) e⁻²ˣ cosx + c

7) ∫ sin2x log(tan x) dx at (π/2,0). 0

8) ∫ cosx/(sinx + √sinx) dx. 2log|√sinx +1|+ c

9) ∫ √secx/(√secx + √cosecx) dx at (π/2,0). π/4

10) ∫ log sinx dx at (π/2,0). -(π/2) log2

11) ∫ eˣ[(1+ sinx)/(1+ cosx)] dx. eˣ(cosecx - cotx)+ c

12) ⁵₋₅∫ |x +2| dx. 29

13) ∫ (x sinx cosx)/(cos⁴x + sin⁴x) dx at (π/2,0). π²/16

14) ᵇₐ∫ (logx)/x dx. (1/2) logab log(b/a)

15) ∫ (x -1){(x -3)(x -2)²} dx. 2log(x -3) - 2log(x -2)+ 1/(x -2)+ c

16) ¹⁾²₀∫ (sin⁻¹x)/(1- x²)³⁾²dx. π/6√3 - log(2/√3)

17) ∫ (log(logx))/x dx. Logx log(logx)- logx + c

18) ∫ dx/√(1+ sinx). √2 log[log(π/4+ x/2) - cot(π/4+ x/2)]+ c

19) ∫ √tanx/sin2x dx. √tanx + c

20) ∫ dx/(1+ tanx) at (π/2,0). π/4

21) Prove ∫ (x cosx)/(1+ cosx) dx at (2π,0)= 2π².

22) ∫ (1+ tan²x)/√(1- tan²x) dx. sin⁻¹(tanx)+ c

23) ∫ x²(ₑx³) cos(2ₑx³) dx. (1/6) sin(2ₑx³)+ c

24) ∫ x²sin⁻¹x dx. (-1/9) √(1- x²)³

25) Prove ∫ (3 sinx + 4 cosx)/(sinx + cosx) dx at (π/2,0)= 7π/4.

26) Solve the differential equation: x(x²- x²y²) dy + y(y²+ x²y²) dx =0. Log(x/y)

27) ∫ tan⁻¹√{(1- x)/(1+ x)} dx. (1/2) (x cos⁻¹x - √(1- x²))+ c

28) x dy/dx - y =√(x²+ y²). y+ √(x²+ y²)= cx²

29) eʸ(1+ x²) dy - (x/y) dx = 0. eʸ(y -1) - log√(1+ x²)= c

30) cos²x dy/dx + y = tanx. y eᵗᵃⁿˣ = eᵗᵃⁿˣ(tanx -1)+ k

31) (x²- yx²) dy + (y²+ xy²) dx = 0. (1/x + 1/y) - log(xy)= c

32) dy/dx = e ˣ⁻ʸ + x² e⁻ʸ. eʸ = eˣ + x³/3 + c

33) dy/dx - eʸ⁺ˣ = eˣ⁻ʸ. tan⁻¹eʸ= eˣ + c

34) (1- x²) dy/dx - xy = x given y=2, when x=0. y√(1- x²)+ √(1- x²)= 3

35) x(x - y)dy + y² dx = 0. y/x = logy + c

36) (1+ y²) dx/dy = tan⁻¹(y)- x. ₑtan⁻¹y. x = ₑtan⁻¹y.(tan⁻¹y-1)+ c

37) ydx -(x + 2y²) dy = 0. x= 2y²+ cylinder

38) 2 dy/dx = y/x + y²/x². Log{(y -x)/y}= (logx)/2 + c

39) Find the area of the region lying in the first quadrant bounded by the parabola y²= 4x, the x-axis and the ordinate x= 4. 32/2 sq.unit

40) sinx dy/dx = cos²x sinx tan(x/2). y cot(x/2) = x/2 + (sin2x)/4+ c

41) Find the area of the figure bounded by the graph of function. y= x² and y= 2x - x². 1/3 sq.unit

42) Calculate the area bounded by the curve y= x(2- x) and the lines x= 0, y= 0, x=2. 4/3

43) tanx dy/dx + 2y = secx. (Sin²x) y+ cosx = c

44) dy/dx = {(1+ cos²x)sin²x}/{(1+ sin²y) cos²y}. y- x = tanx + cot y + c

45) ∫(sin²x + cos²x + (x³+ 2x)/√x)dx. x + (2/7)√x⁷+ (4/3)√x³+ c

46) ∫ x(logx)² dx. (x²/2) (logx)²- (x²/2) logx + x⁴/4+ c

47) ∫ x e²ˣ cosx dx. e²ˣ[(x/√5) cos(x - tan⁻¹(1/2)]

48) ∫ eˣ(x logx +1)/x dx. eˣ logx+ c

49) ∫ dx/(sinx + cosx). Log(cosecx - cotx)+ (1/4) log{(1+ cosx).(1- cosx)}

50) ∫(logx)² dx. x(logx)²- 2x logx + 2x+ c

51) ∫ dx/(1- sinx). tanx + secx + c

52) ∫ dx/{x√(x⁶-1)}. (Sec⁻¹x³)/3 + c

53) ∫ dx/(1+ 2sinx + cosx) at (π/2,0). (1/2) log 3

54) ∫ dx/√(1- x²) at (1,-1/2). 2π/3

55) ∫ √(1+ sin(x/2)) dx. 4(sin(x/4) - cos(x/4))+ c

56) ∫ (10x⁹+ 10ˣ log 10)/(10ˣ + x¹⁰) dx. Log(10ˣ + x¹⁰)+ c

57) ∫ {cot(logx)}/x dx. Log(sin(logx))+ c

58) ∫(logx)√x dx. √x³[(2/3) (logx)² -(8/9) (logx)+ 16/27]+ c

59) ∫ (x + sinx)/(1+ cosx) dx. x tan(x/2)+ c

60) ∫sin(logx) dx. (1/2) [sinx (logx)- cos(logx)+ c]

61) ∫e⁻ˣ sinx dx. (1/13) e²ˣ(2 sin3x - 3 cos3x)+ c

62) ∫ eˣ⁾²{(2- sinx)/(1- cosx)} dx. -2 eˣ⁾² cot(x/2)+ c

63) ∫ x⁴dx/(x+1)(x²+1). - log(x+1)+ (1/2) log(x²+1)+ c

64) (x - y -2)dx - (2x - 2y -3)dy= 0. (2y- x +4)+ log(x - y -1)+ c

65) dy/dx + 2xy/(1+ x²) = 1/(1+ x²)². y=( tan⁻¹x)/(1+ x²)+ c

66) 3eˣ tany dx + (1- eˣ) sec²y dy =0. 3 log(eˣ -1)= log(tany)+ c

67) dy/dx = y/x + tan(y/x). Log sin(y/x)= logx + c

68) dy/dx = (4x + y+1)². (1/2) tan⁻¹{(4x + y+1)/2}= x + c

69) (1+ y²) dx = (tan⁻¹y - x)dy. x. ₑtan⁻¹y = (tan⁻¹y-1) ₑtan⁻¹y+ c

70) dy/dx = x(2 logx +1). y= x² logx - 4 log2

71) dy/dx + y/x = x² given y=1, when x= 1. xy= x⁴/4+ 3/4

72) (x +2y²) dy/dx = y, y> 0. x= 2y²+ cylinder

73) y - x dy/dx = a(y²+ dy/dx). Logy - log(1- at)= log(x +a)+ c

74) dy/dx = x²(x -2) given y= 2, when x= 0. y= x⁴/4 - 2x³/3

75) dy/dx = sin⁻¹x. y= x sin⁻¹x + √(1- x²)+ c

76) (x +1) dy/dx = e³ˣ(x +1)². y/(x+1) = e³ˣ/3+ c

77) dy/dx = (1+ x)y²/{x²(y -1)}. log y + 1/y = log x - 1/x + c

78) Find area bounded by the curve y²= 4x and the line y=3 and y-axis. 9/2 sq.units

79) calculate the area enclosed between the axes and the curve (y - 2)²= 8x. 1/3

80) calculate the area under the curve y= 2√x included between the lines x=0 and x=1. 4/3

81) Indicate the region bounded by the curve y= x log x and y= 2x - 2x² and obtain the area enclosed by them. 7/12

82) ∫ √tanx dx. (1/√2) tan⁻¹{tanx -1)/(√2 tanx) + 1/2√2 log|(tanx -√2 tanx +1)/(tanx + √2 tanx +1|+ c

Linear programming

1) Solve the following inequations simultaneously:

3x - 2y <4, x+ 3y> 3 and x + y ≤ 5. A(-1/3,10/9), B() C(11/5,14/5)

2) Solve the following linear equations simultaneously:

2x + y -3≤ 0, 2x+ y -6> p.

3) A shopkeeper deals in two items-wall hangings and artificial plants. He has Rs 15000 to invest and a space to store atmost 80 pieces. A wall hanging cost him Rs 300 and an artificial plant Rs150. He can sell a wall hanging and a profit of Rs50 and an artificial plant at a profit of Rs18. Assuming that he can sale all the items that he buys , formulate a linear programming problem in order to Maximize his profit. Z= 50x + 18y, subject to constraints: 2x + y≤ 100, x+ y≤ 80 x≥ 0, y≥ 0.

4) A brick manufacture has to depots A and B, with stocks of 30000 and 20000 bricks respectively. He receives orders from three builders P, Q and R for 15000, 20000 and 15000 bricks respectively. The cost (in Rs) of transportation 1000 bricks to the builders from the depots given below :

Transportation cost per 1000 bricks (inRs)

P Q R

A 40 20 20

B 20 60 40

The manufacturer wishes to find how to fulfill the order so that transportation cost is minimum.

Formulate the LPP.

5) A hosewife wishes to mix together two kinds of food A and B in such a way that the mixture contains atleast 10 units of vitamins A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg of food A and B are as below:

Vitamin A vitamin B vitamin C

Food A 1 2 3

Food B 2 2 1

1 kg of food A costs Rs6 and 1 kg of food B cost Rs10. formulate the above problem as a linear programming problem, to find the least cost of the mixture which will produce the diet. Rs52

6) Determine the maximum and minimum values of the following functions and the values of x and y where they occur.

a) f(x,y)= 3x + 5y, vertices at (4,8),(2,4),(1,1),(5,2). Max 52 at(4,8), min 8 at (1,1)

b)bf(x,y)= x + 4y, vertices at (0,7),(0,0),(6,2),(5,4). Max 528 at(0,7), min 0 at (0,0)

7) Find the minimum and maximum value of the following functions and the values of x and y where the occur:

a) Q = x + 3y, subject to x,y ≥0, 5x +2y≤20, 2y ≥x. Max 30 at(0,10)), min 0 at (0,0)

b) f(x,y) = 10x + 12y, subject to x,y ≥0, 2x +5y≥22, 4x + 3yy ≥28, 2x + 2y ≤ 17. Max 97 at(5/2,6), min 562/7 at (37/7,16/7)

8) solve graphically the linear programming problem to minimise the cost c= 3x + 2y subject to the following constraints : 5x + y ≥10, x +y≥ 6, x+ 4y ≥12, x≥0, y≥ 0. Min 13 at (1,5)

9) Maximize Z= 3x + 4y, if possible, subject to constraints: x - y ≤ -1, - x +y≤0, x, y ≥0. No solution set

10) A man has Rs1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost Rs180 and Rs120 respectively. He has storage capacity of 10 bags only. He earns a profit of Rs11 and Rs9 per bag of rice and wheat respectively. Formulate an LPP to maximize the profit and solve it . 5 kg rice bags 5 wheat bags, max profit Rs100.

11) A manufacturer makes two types of tea-cups, say, A and B. Three machines are needed for their manufacturing and the time (in minutes) required for each cup on the machines is given below:

Machines

Cup I II III

A 12 18 6

B 6 0 9

Each machine is available for a maximum of 6 hours per day. If the profit on each cup A is 75 paise and on each cup B is, 50 paise , show that 15 tea-cups of type A and 30 of type B should be manufactured in a day to get maximum profit.

12) A factory owner purchases two types of machines. A and B, for his factory. The requirements and limitations for the machines are as follows:

Machine Area o.b.m lab.f.e.m d.o(in units)

A 1000 sq.m 12men 60

B 1200 sq.m 8men 40

He has an area of 9000 sq.m available and 72 skilled men who operate the machines. How many machines of each type should he buy to maximize the daily output ? 4 type A, 3 type B or 6 type of A, no machine of type B

13) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A. And 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. it cost Rs5 per kg to purchase food I and Rs7 per kg to purchase food II. Determine the minimum cost of such a mixture. 2kg food I, 4kg food II. Min cost Rs38

14) A factory manufacturers are two types of screw, A and B each types requiring the use of two machines-- an automatic and a hand-operated. It take 4 minutes on the automatic and 6 minutes on the hand-operated machine to minutes a package of screw A. While it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a package of screw B. Each machine is available for at most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs7 and of screws B at a profit of Rs10. Assuming that he can sell all the screws he can manufacture, how many package types of each type should the factory owner produce in a day in order to maximize his profit ?

determine the maximum profit. Max profit Rs410 at (30,20)

Application of Derivatives in Commerce and Economics

1) A Television manufacturer finds that the total cost for the production and marketing of x number of television sets is:

C(x)= 300x²+ 4200x + 13500.

Each product is sold for Rs8400. Determine the break even points. 5 or 9

2) C(x)= 5x + 350 and R(x)= 50x - x², are respectively the total cost and the total revenue functions for a company that produces and sales x units for a particular product. Find

a) the breakeven values. 35 or 10

b) the values of x that produce a profit. 10< x < 35

c) the values of x that result in a loss. x< 10 or x >35

3) The demand function for a manufacturer , product is P= 180 - x)/4 where x is the number of units and P is the price for unit. At what value of x will there be maximum revenue ? What is the maximum revenue ? 90, Rs2025

4) The cost function C(x) of a firm is given by C(x)= 3x² - 6x + 5. Find

a) the average cost. 2.5

b) the marginal cost when x= 2. 6

5) The total cost C(x) of a firm is given by C(x)= 0.005x³ - 0.2 x²- 30x + 2000, where x is the output. Determine

a) the average cost . 0.005x²- 0.2x - 30+ 2000/x

b) the marginal average cost. 0.01x 0.2 - 2000/x²

c) the marginal cost. 0.015x²- 0.4x - 30

6) The average cost function AC for a commodities is given by AC= x + 5 + 36/x in terms of output x. Find the total cost C and the marginal MC as the function of x. Also, find the outputs for which AC increases. C= x²+ 5x +36, 2x +5, x> 6

7) The demand function for a commodity is given by P= ae⁻ˣ⁾³⁰⁰, where P is the price per unit. Given that price is Rs7 per unit when 600 units of products are produced. Find the total average revenue and marginal function. Also find the price units when the marginal revenue is zero. 7xe⁽⁶⁰⁰⁻ˣ⁾/³⁰⁰,7(300-x)/300 e⁽⁶⁰⁰⁻ˣ⁾/³⁰⁰,7e

8) The demand function of a monopolist is given by P= 1500- 2x- x². Find the marginal revenue for any level of output. Also , find marginal revenue(MR), when x=10. 1500-4x -3x², 1160

9) Find the relationship between the slopes of marginal revenue curve and the average revenue curve , for the demand function x= (b - P)/a where x denotes the numbers of units sold at the price P per unit. Slope of MR curve is twice the slope of AR curve

10) In a factory, it is found that the number of units (x) produced in a day depends upon the number of workers(n) and is obtained by the relation, x= 5n/√(n +5). The demand function of the product is P= 2/x + x.

Determine the marginal revenue, n= 20. 40

11) A company is selling a certain products. The demand function of the product is linear. The company can sell 2000 units when the price is Rs8 per unit and when the price is Rs4 per unit, it can sell 3000 units. Find

a) the demand function. 4000 - 250x

b) the total revenue function. 4000x - 250x²

12) suppose the cost to produce some commodity is a linear function of output. Find cost as a function of output, if cost are Rs 4900 for 250 unit and Rs 5000 for 350 units . C= 10x +1500

13) A manufacturer can sell x items of commodity of price Rs(330 - x) each . Find the revenue function. If the cost of producing x items is Rs(x² + 10x + 12), determine the profit function . R(x)= 330x - x², P(x)= 320x - 12 - 2x²

14) C(x)= 5x + 350 and R(x)= 50x - x² are respectively the total cost and total revenue functions for a company that produces and sells x units of a particular product. Find

a) the break-even values. 10,35

b) the value of x that produces profit . 10< x < 35

c) the values of x that result in loss. x>35 or x < 10

15) The total cost C(x), associated with the production and making of x units of an item is given by C(x)= 0.005x³ - 0.02x² + 30x + 5000; find

a) the average cost function. Ac= 0.005x²- 0.02x + 30+ 5000/x

b) the average cost of output of 10 units. 530.3

c) the marginal cost function. 0.015x²- 0.04x +30

d) the marginal cost when 3 units are produced. Rs30.015

16) If the total cost function C= 2x²- 3x +8, find the average cost function and marginal cost function, and marginal cost when 10 units are produced. 2x + 8/x -3, 4x - 3, 37

17) Verify the cost functions,

C(x)= ax{(x + b)/(x + c)} + d, a, b , c, d > 0

that the marginal and marginal cost curves fall continuously with increasing output.

18) Suppose a manufacturer can sell x items per week at a price P= 20 - 0.01x rupees each when it cost, y= 5x - 2000 rupees to produce x item. Determine the number of items he should produce per week for maximum profit. 750 items

19) If C= 2x{(x+4)/(x +1)} + 6 is the total cost of production of x units of a certain product, show that the marginal cost falls continuously as the output x increases.

20) For the demand function P= a/(x + b) - c, where ab > 0, show that the marginal revenue decreases with increase of x.

21) Find the relationship between the slopes of marginal revenue curve and the average revenue curve , for the demand function, P= a - bx.

PROBABILITY (mixed as B. D)

1) An unbiased die is thrown 3 times. If getting 3 or 5 is considered a success. Find the probability of atleast two success. 7/27

2) There are 10 persons who are to be seated around a circular table. Find the probability that 2 particular person will always sit together. 2/9

3) A bag contains 20 balls marked from 1 to 20. One ball is drawn at random from the bag. What is the probability that the ball is drawn is marked with a number which is multiple of 5 or 7 ? 3/10

4) A card is drawn at random from a pack of 52 playing cards. What is the probability that the card drawn is neither is neither a spade nor a queen. 9/13

5) What is the probability that a leap year has 53 Sundays. 2/7

6) The probability that a boy will not pass MBA exams is 3/5 and that a girl will not pass is 4/5. Calculate the probability that at least one of them passes exam. 13/25

7) Four dices are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of atmost 2 successes. 11/16

8) The probability of A, B and C solving a problem are 1/3, 2/7 and 3/8 respectively. If all the three try and solve the problem simultaneously, find the probability that only one of them will solve it. 25/56

9) A bag contains 5 black and 6 red balls. Another bag contains 8 black and 5 red balls . A ball is then drawn from the first bag and put in the second. A ball is then drawn from the second. Find the probability that the ball drawn is black. 45/154

10) A bag has four red and 5 black balls, a second bag is 3 red and 7 black balls. 1 ball is drawn from the first bag and 2 from the second. Find the probability that two balls are black and one is red. 7/15

11) A and B throw two dices each. If A gets a sum of 9 on his two dice, then find the probability of B getting higher sum. 1/54

12) A card is drawn at random from a pack of 50 playing cards. What is the probability that the card drawn is neither a spade nor a queen ? 9/13

13) A speaks truth in 55% cases and B speaks truth in 75% cases . Determine the probability of cases in which they are likely to contradict each other in stating the same fact. 47.5%

14) A coin is tossed 5 times. What is the probability getting at least three heads ? 1/2

15) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is multiple of 3 or 7 ? 2/5

16) In a certain city , the probability of not reading the morning newspaper by the residents is 1/2 and that not reading the evening newspaper is 2/5. The probability of reading both the newspapers is 1/5. Find the probability that a resident reads either the morning of evening or both the papers. 9/10

17) A problem in mathematics is given to four students A, B , C and D. Their chances of solving the problem respectively are 1/2, 1/3, 1/4 and 1/5. What is the probability that the problem is solved ? 4/5

18) A pair of dice is thrown. If the two numbers appearing on them are different, find the probability that the sum of number is 6. 5/36

19) A and B play a game in the which A's chance of winning the game are 3/5. In a series of 6 games find the probability that A will win at least 4 games. 1701/3125

20) In an examination 30% of the students failed in mathematics, 20% failed in chemistry and 10% failed in both. A student is selected at random find the probability that:

a) the student has failed either in mathematics or in chemistry. 2/5

b) the students has failed in mathematics, it is known that he has failed in chemistry. 1/2

21) A candidate is selected for interview of management trainees for 3 companies. For the first company, there are 12 candidates, for the second there are 15 candidates and for the third, there are 10 candidates. Find the probability that he is selected in atleast one of the companies. 23/100

22) One number is choosen at random from the number 1 to 21, find the probability that may be a prime number. 8/21

23) The probability that a contractor will get a plumbing contract is 2/3 and an electric contract is 4/9. If the probability of getting at least one contract is 4/5. Find the probability that he will get both the contracts. 19/45

24) A pair of dice is thrown 4 times, if getting a even number is considered a success. Find the probability of 3 success. 5/16

25) In a given race the odds in favour of four horses A, B, C and D are 1:3, 1:4, 1:5 and 1:6 respectively. Assume that a dead heat is impossible, find the chances that one of them win the race. 319/420

26) In a single throw of 2 dice, what is the probability of getting a total of at least 10 ? 1/6

27) Assume that on an average one telephon out of 10 is busy. 6 telephone numbers are randomly selected and called. Find the probability that 4 of them will be busy. 243/200000

28) On an average aut of 12 games of the chess played by A and B , A wins 6, B wins 4 and 2 games end in a tie. A. and B plays a tournament of 3 games, calculate the probability that A and B wins alternate game, no game is tied up. 5/36

29) A bag 20 tickets with marked numbers 1 to 20. One ticket is drawn at random. Find the probability that it will be a multiple of 2 or 5. 0.36

30) In a lecture class 52% students cannot read what is written on board, 46% can not hear and 32% can neither read nor hear ? What percent of students can read or hear on board ? 66%

PROBABILITY/B/P

1) A company has two plants which manufacture scooters. Plant I manufactures 80% of the scooters while Plant II manufactures 20% of the scooters. At plant I, 85 out of 100 scooters are rated as being of standard quality, while at plant II only 65 out of 100 scooters are rated as being of standard quality. If a scooter is of standard quality, what is the probability that it came from plant I. 0.84

2) A firm produces steel pipes in 3 plants A, B and C with the daily production of 500, 1000 and 2000 units respectively. It is known that fractions of defective output produced by 3 plants are respectively 0.005, 0.008, and 010. A pipe is selected at random from a days total production and found to be defective. What is probability that it came from first plant . 5/61

3) if a bulb factory machine A, B and C manufacturers 60% , 30% and 10% respectively. 1%, 2% and 3% of the bulbs produced by A, B and C are defective. A bulb is drawn at random from the total production and found to be defective. Find the probability that bulb has been produced by machine A . 2/5

4) Bag I contains 5 green and 3 red balls, another bag II contains 4 green 6 red balls . A red balls has been drawn from one of the bag. Find the probability that it was drawn from bag I . 5/13

5) Bag I contains 2 white and 3 red balls and bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag. II . 25/32

6) A coin is tossed four times. If X is the number of heads observed. Find the probability distribution of X .

7) Find the mean, variance and s.d of the number of heads in three tosses of a coin. 3.2, 3/4, √3/2

8) If the sum of mean and the variance of a binomial distribution for 5 trials is 1.8, find the distribution. (4/5 + 1/5)⁵

9) A card from a pack of 52 cards is lost. From the remaining cards of pack, two cards are drawn are to be hearts. Find the probability of missing card to be a heart. 11/50

REGRESSION EQUATIONS