TEST PAPER -(Multiple choice Questions- Maths)- (IX) -1

1) Choose the correct statement:

a) Reciprocal of every rational number is a rational number

b) The square roots of all positive integers are irrational numbers.

c) The product of a rational and an irrational number is an irrational number.

d) The difference of a rational number and an irrational number is an irrational number.

2) The compund intrest on ₹1000 at 10% p.a. compound annually for 2 years is

a) ₹190 b) ₹200 c) ₹210 d) ₹1210

3) If x+ 1/x= 2 then x²+ 1/x² =?

a) 4 b) 2 c) 0  d) none

4) Factorize: 12a²b +15 ab² is

a) 3a(4ab+ 5b²) b) 3b(4a²+ 5ab)

c) 3ab(4a+ 5b).  c) none

5) If x= 3, y= k is a solution of the equation 3x - 4y +7=0.0, then the value of k is

a) 16 b) -16 c) 4 d) -4

6) Sum of digits of a two digit number is 8. If the number obtained by reversing the digits is 18 more than the original number, then the original numbers is

a) 35 b) 53 c) 26 d) 62

7) Which of the following is not a quadratic equation:

a) 2x²= 3x - 5 b) (2x-1)(x -1)= 2x²-7x +2

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

d) (x+1)³= x³+2x +2

8) The value of (81/16)⁻³⁾⁴

a) 4/9 b) 9/4 c) 27/8 d) 8/27

9) If log√₃ 27= x, then the value of x is

a) 3 b) 4 c) 6 d) 9

10) Which of the following is not a criterion for congruency of triangles?

a) SAS  b) ASA c) SSA d) SSS

11) In a ∆ABC, AB= 3cm, BC= 4cm and CA= 5cm. If D and E are midpoints of AB and BC respectively, then the length of DE is

a) 1.5cm b) 2 c) 2.5 d) 3.5

12) In a ∆ ABC, if AB= 6√3cm, BC= 6cm and AC= 12cm, then angle B is

a) 120° b) 90° c) 60° d) 45°

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

a) 90 b) 95 c) 105 d) 120

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

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

15) If P and Q are any two points on a circle, then the line segment PQ is called a

a) radius of the circle 

b) diameter of the circle

c) chord of the circle

d) secant of the circle

TEST PAPER -XI (TRIGONOMETRY)(1)

1) The value of (cos10+ sin 10)/(cos10 - sin10) is

a) tan35 b) - cot 35 c) - tan35 d) tan 55

2)If tan(x/2)= t, then value of (1 - t²)/(1 + t²) is

a) cos 2x b) sec x c) cosx d) tanx

3) If 0< x <π/2, then largest angle of the triangle sides are 1, sinx and cosx is

a)2π/3 - x b) π/2 c) 2π/3 d) x

4) In triangle ABC, if a²+ b²+ c²- bc - ca - ab=0, then the value of sin²A+ sin²B + sin²C is

a) 9/4 b) 4/9 c) 3√3/2 d) 3/2

5) if tanx = -4/3, then the value of sinx.

a) 2/5  b) 4/5  or -4/5  c) 4/5 but ≠ -4/5 d) -4/5 but ≠ 4/5

6) The general solution of tan 5x= tan 3x is

a) (2n +1/π b) nπ/2 c) nπ d) none 

7) If tanx + secx = eˣ, then the value of cosx is

a) (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ) b) (eˣ - e⁻ˣ)/2 c) 2/(eˣ - e⁻ˣ) d) 2/(eˣ + e⁻ˣ)

8) The number of integral value of k for which the equation 7 cos x + 5 sin x = 2k + 1 has a the solution. is 

a) 8 b) 6 c) 7 d) 9

9) The value of (tanx + 2tan2x)/tanx is

a) less than 1 

b) greater than 5

c) cannot lie witin 1 and 5

d) either less than 1 or greater than 5

10) The trigonometrical equation sinx + cosx = 2 has

a) one solution b) two solution c) infinite number of solutions  d) no solution

11) The value of 3[sun⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2 + x)+ sin⁶(5π - x)] is equal to

a) 2 b) 1 c) 0 d) 4

12) In a triangle of ABC , if (b+ c)/11= (c + a)/12= (a+ b)/13, then the value of cosC is

a) 16/17 b) 17/36 c) 5/7 d) 5/6

13) The value of (sin 47 - sin 25 + sin 61 - sin11) is 

a) cos 7 b) sin 7  c) 2 sin7 d) 2 cos 7

14) if in a triangle ABC, A≠ B and a cosA = b cos B, then which of the following is correct ?

a) a²= b²+ c² b) c²= a²+ b²  c) b²= c²+ a² d) none

15) if m, n, p are angles such that tanm+ tan n+ tan p= tan m tan n tan p and x= cosm + i sin m, y= cos n + i sin n and z= cos p + i sin p, then the value of xyz is

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

16) if sinx + sin²x = 1, then the value of (cos¹²x + 3 cos¹⁰x+ 3 cos⁸x + cos⁶x) is

a) 1 b) 4 c) 2 d) 3 

17) If x = 2sin k/(1+ sin²k + cos k) then the value of (1+ sink- cosk)/(1+ sink) is

a) -x/2 b) x/2 c) x d) - x

18) The roots of the equation 1- cosx = sinx sin(x/2) are

a) nπ/4 b) 2nπ c) nπ d) nπ/2 where n belongs to Z

19) In triangle ABC, if b²= c²+ a² then the value of (tanA+ tanC) is

a) tanA tanC b) tanB c) c²/ab d) b²/ac

20) If x + y= (4n +1)π/4(where n is an integer) and x+ y is not an odd multiple of π/2, then the value of (sin2x + sin2y)/(cos2x + cos2y) is

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

21) If 1+ cosx= k where x is acute then the value of sin(x/2) is

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

22) The value of the expression cos 1 cos 2 cos 3 ....cos 179 is

a) 1 b) 1/√2 c) -1/√2 d) 0 

23) If ∆= a²-(b - c)², where ∆ is the area of the triangle ABC, then tanA is equal to

a)  8/17  b) 11/15  c) 15/16  d) 8/15

24) If 12cot²x - 31cosecx + 32= 0,  then the value of sinx is

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

25) The value of (cos⁶5- 15 cos⁴5sin² 5+ 15 cos²5 sin⁴5- sin⁶5) is equal to 

a) -1/√2  b) √3/2 c) 1/√2 d) 1

TEST PAPER (SET THEORY AND RELATION (OR FUNCTION)

1) For three sets A, B and C, if A∩C = B∩ C= ∅ and AU C = B UC, BC then prove that A= B.

2) For any three sets A, B and C, If A∩B= A∩ C and AUB = AUC then show that B= C.

3) Two finite sets A and B consist of m and n elements respectively. The number of subsets in A exceeds that of B by 112. Find the value of m and n.

4) A survey shows that 75% of the student of a school like Mathematics and 65% like Physics. If x% of the students like both Mathematics and Physics, find the maximum and minimum values of x.

5) In a survey of 35 students of a class it was found that 17 students like Mathematics and 10 like Mathematics but not Biology. Find the number of students who like 

a) Biology 

b) Biology but not Mathematics ,

It being given that each student takes at least one of the two subjects.

6) An enquiry into 100 candidates of failed in English of HS Examination revealed the following data:

failed in aggregate-66, failed in paper I-37, failed in paper II -59, failed in aggregate and paper I- 17, failed in aggregate and paper II -43 and failed in both papers -13.

 Find the number of candidates who failed in

a) aggregate of paper II but not paper I.

b) aggregate but not in paper I and paper II.

7) In a survey it was found that 76 men read magazine A, 30 read magazine B, 40 read magazine C and and 6 men read all the three magazines . If the total number of men who read magazines be 116, find how many men read exactly two of the three magazines.    

8) For two sets A and B, the 3 elements of A x B are (a,x),(b,y),(c,x); find B x A.    

9) If (a,b) and (b,c) are elements of A x A, find the set A and other elements of A x A.

10) Find the domain of definition of each of the following functions:

a) f(x)= 1/log₁₀(1- x)  + √(x +2).

b) f(x)=log{(3+ x)/(3- x)}.

11) If f(x+y)= f(x)+ f(y) for all real number x and y, show that f(x)=xf(1).

12) f(x + 1/x)= x²+ 1/x², find f(x).

13) If 2f(1/x)+ f(x)=3x, find f(x - 1/x).

14) If f(x)= tan(x - π/4), find f(x). f(-x).

15) A cubic function f(x) satisfies the relation f(x)+ f(1/x)= f(x). f(1/x), show that f(x)= 1+ x³ or 1- x³. Further, if f(2)= 9, show that f(4)= 65.

Multiple choice Questions- XI (Algebra) - 2

1) The complex numbers z is such that the |z|= 1, z≠ -1 and w= (z-1)/(z+1), then real part of w is

a) 1/|z +1| b) 0 c) √2/|z +1|² d) - 1/|z +1|²

2) The number of words that can be formed out of the letters of the word ARTICLE so that the vowels occupy even places is 

a) 360  b) 574  c) 300  d) 144

3) The product [32. (32)¹⁾⁶. (1/32)¹⁾³⁶.....∞] is equals to

a) 128  b) 256 c) 64  d) 512

4) The value of [⁴⁷C₄ +⁵ ᵣ₌₁∑⁵²⁻ʳC₃] is equal to 

a) ⁵²C₃ b) ⁵²C₄ c) ⁵¹C₄ d) ⁵³C₄  

5) If 1, w, w² are the cube root of unity, then the value of (x+ y)²+ (xw + yw²)+ (xw²+ yw)² is equals to

a) 3xy b) 9xy c) 6xy d) 3(x²+ y²)

6) if a,b,c are three unequal numbers such that a, b, c are in AP and (b - a),(c - b), a are in GP, then the value of a: b: c is

a)2:3:4 b) 3:5:7 c) 1:2:3 d) none

7) Each of 8 questions in a paper has an alternative. Number of ways in which a candidate can answer one or more questions is

a) 3⁸-1 b) 2⁸ c) 3⁸ d) 2⁸-1

8) The number of terms in expansion of (a+ b+ c)¹⁰ is

a) 55  b) 66 c) 33  d) 44

9) The argument of the complex number z= 1+ i tan(3π/5) is

a) -2π/5 b) 3π/5 c) 2π/5 d) -3π/5

10) If an infinite GP, first terms is equals to twice the sum of the remaining terms then its common ratio is

a)  1/4  b) -1/4 c) 1/3 d) -1/3

11) if the sum of first n terms of an AP series is n²+ 2n, then the term of the series having value 201 is.

a) 99th term  b) 100th term  c) 101th term  d) 102th term 

12) Everybody in a room shake hands everybody else. If the total number of handshakes is 66, then the number of persons in the room is

a) 11  b) 12 c) 13 d)  14

13) if r> 1, n> 2 are positive integers and the co-efficients of (r+2)th and 3rd terms in expansion of (1+ x)²ⁿ are equal, then n is equals to

a) 3r b) 3r+1 c) 2r d) 2r +1

14) How many words can be formed from the letters of the word COMMITTEE ?

a) 9! b) 9!/2! c) 9!/(2!)² d) 9!/(2!)³

15) The solution set of the inequation x+ 1/x > 2 is

a) 0 < x <∞ b) 0 ≤ x <∞ c) R -{0} d) 1 ≤ x <∞ 

16) The polar form of the complex number (i²⁵)³ is

a) cosπ + i sinπ b) cosπ - i sinπ c) cos(π/2) + i sin(π/2) d) cos(π/2) - i sin(π/2)

17) If in a distribution, n= 10, ∑x= 20, ∑x²= 200, then the value of standard deviation of the distribution is

a) 2 b) 16 c) 6 d) 4

18) The value of (²⁰C₄ + ²⁰C₃ + ²⁰C₂ - ²²C₁₈) is

a) 0 b) 1242 c) 7315 d) 960

19) How many ways are there to arrange the letters in the world GARDEN with the vowels in alphabetical orders ?

a) 240 b) 360 c) 480 d) 120

20) The complex number 1, -1, i√3 form a triangle which is

a) equilateral triangle b) isosceles triangle c) right angled triangle  d) isosceles right angled triangle

21) Two roots of the quadratic equation x²+ (i -5)x + 18+ i=0 are

a) -2+ 3i, 3+ 4i b) 2- 3i, 3+ 4i  c) -3 - 4i, 2- 3i  d)  3 - 4i, 2+ 3i 

22) How many zeros are there in (126)! ?

a) 29  b) 30 c) 31 d) 32 

23) How many numbers of 5 digits can be formed from the numbers 0, 1, 2, 3, 4 when repetition of digits is not allowed?

a) 120 b) 96  c) 144 d) 48 

24) A sample of 4 items is drawn at random from a lot of 10 items, containing 3 defectives. If x denotes the number of defective items in the sample, then P(0< x <3) is equals to

a) 3/10 b) 1/2  c) 4/5  d) 1/6

25) If a, b, c are in GP and x, y are AM of a, b and b, c respectively, then the value of (a/x + c/y) is

a) 1 b) 2 c) 0 d) 4

Multiple choice Questions - XI (Algebra) - 1

1) Two roots of the equation 2x²+ 3ix +2= 0 are

a) -i/2, 2i b) 2i, -i/2 c) 2i, i/2 d) -2i, i/2

2) Fifth term of a GP is 2; then the product of its first 9 terms is

a) 256 b) 1024 c) 512  d) none

3) The number of numbers that can be formed using the digits 1,2,3,4,5(repetition of digits is not permissible) such that the ten's digit is greater than thousand's digits, is

a) 60 b) 45 c) 30  d) none

4) Total number of 4 digit odd numbers that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7 are

a) 216  b) 375  c) 720 d) 400 

5) The number of diagonals of a polygon of 20 sides is

a) 150  b) 170 c) 125 d) 210

6) 10ⁿ + 3. 4ⁿ⁺² +5 is always divisible by (for all n ∈ N)

a) 7 b) 5 c) 17 d) 9

7) Six line segments of lengths 2, 3, 4, 5, 6, 7 units are given. The number of triangles that can be formed by these lines is

a) ⁶C₃ - 7 b) ⁶C₃ - 6 c) ⁶C₃ - 5 d) ⁶C₃ - 8

8) For all positive values of x and y the value of {(1+ x+ x²)(1+ y+ y²)/xy} is

a) > 9 b) < 9 c) ≤ 9 d) ≥ 9

9) The point of represented by the complex number (2- i) is rotated about origin through an angle of π/2 in clockwise direction. The new position of the point will be..

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

10) The value of 1³- 2³+3³-4³+.......+9³ is

a) - 425  b) 475  c) 425  d) -475

11) A point z moves in the argand plane such that |z - 3i| =2, then its locus is

a) y-axis  b) straight line  c) a circle  d) none of these

12) The number of terms of the AP 3,7,11, 15,..... to be taken so that the sum is 406, is

a) 14 b) 16 c) 20 d) 24

13) a,b,c,d are all real numbers and (a²+ b²)d² - 2(a+ c)bd + (b²+ c²)= 0, then a, b, c are in 

a) GP  b) AP  c) HP  d) none

14) The coefficient of x⁶ in the expansion of (x²+ x - 1)⁴ is

a) -2 b) -4 c) 2 d) 4 

15) How many 9 digit numbers can be formed using the digits 2,2,3,3,5,5,8,8,8 so that odd digits occupy even positions?

a) 240 b) 180 c) 120 d) 60

16) if a,b,c,d,e are in AP then the value of (a + b + 5c - 5d+e) in terms of a is

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

17) If x+ 1/x =√3, then one value of x is 

a) cos(π/3)+ i sin(π/3) b) cos(π/6)+ i sin(π/6)  c) sin(π/6)+ i cos(π/6)  d) cos(π/2)+ i sin(π/2) 

18) Two variable x and y are related by y= 8+ 2x; if the S. D of x is 3, then the SD of y will be

a) 10 b) 14 c) 11 c) 6

19) If in the binomial expansion of (1+ x)ⁿ where n is a natural number, the coefficients of 5th, 6th and 7th terms are in AP , then n is equals to

a) 7 or 13 b) 7 or 15  c) 7 or 14  d) 18 or 14

20) if z and w are two non-zero 290 complex numbers such that |zw| =1 and arg z- arg w =π/2, then (conjugate of z)w is equals to

a) - i b)  1 c) - 1 d) i

21) Solution set of the inequation is |2/(x -4)| > 1 (x≠ 4) is

a) x∈ (2,6) b) x∈( 2,4)U (4,6) c) x ∈(4,6) d) x ∈( 2,4]

22) If the fifth term of a GP is x, then the product of its first nine terms is

a) x⁵ b) x⁷ c) x⁹ d) x¹⁰

23) Number of 6-digit numbers that can be formed with the digits of the number 112233 is

a) 30 b) 60 c) 90 d) 120

24) The first term and the common difference of an AP are a and d respectively. If m-th and n-th terms of this AP are 1/n and 1/m respectively, then the value of (a -b) is

a) 1/mn b) 0 c) 1 d) (m + n)/mn

25) 3rd term from the end in the expansion of (4x/5 - 5/2x)⁸ is 

a) 4375/x⁴ b) - 4375/x⁴ c) 4325/x² d) - 4325/x²

TEST PER ON ALGEBRA(XI)

1) Find the smallest positive integer n, for which {(1- i)/(1+ i)}ⁿ is purely real. 

2) If ˣ√a = ʸ√b= ᵖ√c and a,b,c are in GP., show that x,y,p are in AP.        

3) If the roots of ax²+ bx+ c = 0 are both positive, then

A) a>0, b>0, c<0

B) a>0, b<0, c>0

C) a<0, b>0, c<0

D) a<0, b<0, c>0

4) The sum of n terms of a GP is 1/3 (2²ⁿ⁺¹ -2); find its common ratio.

5) Of the numbers 2, x, y, 12 (x>0, y>0), if three are in GP and last three in AP, find x and y.

6) If A be the sum and R the sum of the reciprocals of n terms of a GP, whose first term is 4 and last term 36, find the value of S/R.

7) Find the amplitude of (1+ sinx+ i cosx).

8) Express (1+ ix)(1+ iy)(1- ix)(1- iy) as sum of two squares.

9) If z= x + iy be a complex number, then |z²| equals to

A) z² B) conjugate of z² c) z. conjugate of z d) none

10) For what value of k, does the equation 8x²+ 2x + k=0 have its one root square of the other?

11) In an examination a candidate has to pass in each of the four subjects. The number of ways he can fail in the examination is

A) 1 b) 15 c) 16 d) 24

12) Which term in term in (x + 1/x²)⁹ is independent of x ?

a) 2nd b) 3rd c) 4th d) 5th

13) Show that the sum of the coefficient of xʳ and xʳ⁻¹ in the expansion of (1+ x)ⁿ is equal to the coefficient of xʳ in (1+ x)ⁿ⁺¹.

14) If p, q are the roots of equation ax²+ bx + c=0, evaluate 1/(ap + b) + 1/(aq + b).

15) For what value of p, the expression x²+ 4x +4 and x²+ px + 6 have a common factor?

16) For what values of m, the roots of x²- (m -1)x + m + 1/4= 0 are real and equal?

17) For what value of k, the roots of kx(1- x)= 1 will not be real?

18) In a GP, sum of n terms is 255, the last term is 128 and the common ratio is 2. Find n.

19) Find the square root of 9i.

20) The sum of n terms of a series is n/3 (4n²-1). The 5th term of the series is

a) 81 b) 84 c) 121 d) 165

21) At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 6 candidates and 4 of them are to be chosen. Find the number of ways in which a voter may vote.

22) Find the middle terms in the expansion of (1+ x)⁹?

23) The middle term of the progression 3,7,11,......147 is

a) 71 b) 73 c) 75 d) 77

24) If p,q are the roots of x²- 2x +2=0, the least integer n (>0) for which pⁿ/qⁿ =1 , is

a) 2 b) 3  c) 4  d) none

25) A polygon has 44 diagonals; the number of its sides is

a) 8 b) 7 c) 11  d) none

MISCELLANEOUS QUESTIONS -XII(2023/24)

RELATION AND MAPPING

1) For all x∈ z, the function f: z --> is defined by f(x)= 3x+4, Find the function goz --> z  such that (gof=I.         g(x)= (x -4)/3 where x∈z

2) Find the number of equivalence relation on set A={a,b,c} containing elements (b,c) and (c,b).           2

3) If f(x)= sin x, g(x)= x² and h(x)= log x, find the composite function [ho(gof)](x).    2 log(sinx)

4) Let N be the set of all natural numbers and R be the relation on N x N defined by (x,y) R(z,t) => xt(y + z)= yz(x +t). 

 Check whether R is anequivalence relation on N x N.          Yes

5) Let R be the set of real numbers and X={x ∈ R : -1 < x < 1}= Y. Is the mapping f: x --> R defined by f(x)= (2x -1){1 - |2x -1|} bijective ?     Yes

TRIGONOMETRIC FUNCTION

1) PROVE:

a) tan⁻¹{(2 sin2x)/(1+ 2 cos2x)} -1/2  sin⁻¹{(3 sin2x)/(5+ 4 cos2x)}= x.

b) tan⁻¹{(3 sin2x)/(5+ 3 cos2x)} + tan⁻¹{(1/4  tanx)} = x.

c) 2 sin⁻¹(2/√13) + 1/2 cos⁻¹(7/25)+ tan⁻¹(63/16)=π.

2) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π and x + y + z =3/2, then show that x= y= z.

3) If tan⁻¹a + tan⁻¹(1/b)= tan⁻¹3, then find the positive integral values of a and b.    1,2 or 2,7

4) If tan⁻¹[{√(1+ x²) - √(1- x²)}/{√(1+ x² + √(1- x²)}] = θ  ; then show that sin2θ  = x².

5) If (n tanθ)/cos²(α -θ) = {m tan²(α -θ}/cos²θ} then show that 2θ= α - tan⁻¹{(n - m)/(n + m)  tanα}.

6) If φ = tan⁻¹{x √3/(2k - x)} and θ = tan⁻¹{(2x - k)/k √3}, then show that one value of φ - θ is 30°

7) If sin⁻¹(x/a) + sin⁻¹(y/b) = sin⁻¹(c²/ab),  then show that b²x² + 2xy √(a²b² - c⁴) + a²y² = c⁴.

8) Solve: cos⁻¹x - sin⁻¹x = cos⁻¹(x √3).         0, ±1/2

DETERMINANT

By using properties of determinant. Prove that:

1) (x +4)        2x            2x

        2x         x +4          2x  = (5x +4)(x -4)²

        2x          2x           x+4

2) 1        1+ x           1+ x + y

     2        3+2x         1+3x +2y     =1

     3        6+3x         1+6x+3y

3) a         a²        1+ xa³ 

     b         b²        1+ xb³  

     c         c²         1+ xc³

= (1+ abcx)(a - b)(b - c)(c - a)

4) (y + z)²       x²          x² 

         y²       (z + x)²      y²  = 2xyz(x + y+ z)³

         z²            z²     (x + y)²

5)  - yz        y²+ yz        z²+ yz 

   x²+ zx         -zx           z²+ zx 

   x²+ xy      y²+ xy           -xy

= (xy + yz + zx)³

6) (y + z)²        z²           y² 

         z²        (z + x)²       x² 

         y²             x²      (x + y)²

= 2(xy+ yz + zx)³

7) ax - by - cz        ay+ bx              cx+ az

      ay + bx          by - cz- ax          bz+ cy

      cx+ az              bz+ cy          cz - ax- by

= (a²+ b²+ c²)(x² + y²+ z²)(ax + by + cz)

METRICES

1) If A=  -4       4       3  & B= 1     -1      1

               -7       1       3           1     -2     -2

                5      -3      -1           2      1      3 find AB and use this to solve the following system of equation: x - y + z=4; xx - 2y - 2z=0 ; 2x + y + 3z= 1.       3,-2,-1

2) Let X= 3       2      5

                 4       1      3

                 0       6      7 express X as sum of two matrices such that one is symmetric and other is skew-symmertric.

LIMIT, CONTINUITY AND DIFFERENTIABLE

1) lim ₓ→∞ √x{√(x +3) - √x}.                 3/2

2) lim ₓ→₀ {(x -1 + cosx)/x}¹⁾ˣ .             e⁻¹⁾²

3) lim ₓ→∞ {(x +5)/(x +1)}ˣ.                 e⁴

4) lim ₓ→π/2    (1+ cosx)³ˢᵉᶜˣ.             e³

5) lim ₓ→₀ (1+ 3x)^(x+3)/x.                 e³

6) lim ₓ→₀ {log(x²+ x+1)+ log(x²- x+1)}/(secx - cosx).       1

7) lim ₓ→₀ {xeˣ - log(x +1)}/x².            3/2

8) lim ₓ→₀ {tan(π/4  + x)}¹⁾ˣ.               e²

9) lim ₓ→₀ (sinx + cosx)¹⁾ˣ.                 e

10) lim ₓ→₀ {sin log(1+ x)}/log(sinx +1).        1

1) Prove that the function f(x)= sinπ |x| is continuous at x =0 but not differentiable at the same point.     

2) If g(x) is the universe of f(x) and f'(x)= (1+ x³)⁻¹, show that g'(x)= 1+ {g(x)}³.

3) A function f(x) is defined as follows:

                 - 2sinx,        when -π ≤ x ≤ -π/2

 f(x)=          a sinx + b, when -π/2< x < π/2

                     cosx,        when  π/2≤ x ≤ π

If f(x) is continuous in the interval -π≤ x ≤π. find the value of a and b.     -1,1

DIFFERENTIATION

1) Find dy/dx when

a) xˢᶦⁿ ʸ + yˢᶦⁿˣ =1.          

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

c) (√x)ˣ + (x)^√x.

d) y= tan⁻¹(√x - ⁴√x)/(1+ ⁴√x³).

e) xˡᵒᵍˣ + (sinx)ˣ+ 15x.

2) Find d²y/dx²

a) x= e⁻ᵗ and y = te⁻ᵗ.

b) x = a sin³t and y= a cos³t at t=π/4.

c) x= a(t - sint) and y= a(1+ cost) at t=π/2

d) x=√3(3 sint + sin3t) and y= √3(3cost + cos3t) at t=π/3.

e) if 3px²= y²(p - x⁶), show dy/dx = y³/x³ - 2y/x.

f) If y²(1- x²)= x²+1, show that (1- x⁴)(dy/dx)²= y⁴-1.

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

h) If f(x)= sin(logx) and y= f{(2x+3)/(3- 2x)} show that dy/dx= 12/(9- 4x²)  cos{log{(2x+3)/(3- 2x)}}.

i) If x= sect - cost, y= secⁿt - cosⁿt, then show that (x²+4)(dy/dx)²= n²(y²+4).

j) 

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