28278 + Interview Questions in Maths in Class 12

1.	underset(x to pi//2)lim (sqrt(1+cos^(3)x)-sqrt(1-cos^(3)x))/((pi//2-x)^(3)))=
Answer» 0 1 `pi//2` `pi//4` ANSWER :B

Discussion

2.	[a b c] [a' b' c'] =
Answer» 0 1 2 3 Answer :B

Discussion

3.	Find the maximum and minimum values, if any, of thefunctions given by g(x) = – \| x + 1\| + 3
Answer» ANSWER :MAXIMUM VALUE = 3; No MINIMUM value

Discussion

4.	2 sin^(2)x + sin^(2)2x=2,pi lt x lt pi then x equals
Answer» `SQRT(10+2sqrt(5))/(4)` `(sqrt(5)-1)/(4)` `sqrt(5+1)/(4)` `(sqrt(3)+1)/(2sqrt(2))` Answer :a,B,c,d

Discussion

5.	int_(-2)^2 \|x cos pi x\|dx is equal to :
Answer» `8/pi` `4/pi` `2/pi` `1/pi` ANSWER :A

Discussion

6.	Let a=hat(i)+hat(j)+hat(k), b=hat(i)-hat(j)+hat(k) and c=hat(i)-hat(j)-hat(k) be three vectors. A vector v in the plane of a and b whose projection on c is 1/sqrt(3), is given by
Answer» `3 hat(i)+hat(J)-3hat(K)` `3hat(i)-hat(j)+3hat(k)` `3hat(i)+hat(j)+3hat(k)` `-3hat(i)+hat(j)+3hat(k)` Solution :Here, `(v.C)/(\|c\|) = 1/(sqrt(3))` `rArr (v.c)/(sqrt(1+1+1)) = 1/(sqrt(3))` `rArr (v.c)/(sqrt(3)) = 1/(sqrt(3)) rArr v.c = 1` `rArr v.(hati-hatj -hatk) = 1 "….."(i)` we get from option (b) `v=3 hati-hatj+3hatk` ` v.c (3HATI -hatj+3hatk)` `=3+1-3=1` which satisfies Eq. 9i) `v=3hati-hatj+3hatk`

Discussion

7.	Find the value of a so that the four points with position vectors -hatj+hatk,2hati-hatj-hatk,hati+lambdahatj+hatk and 3hatj+3hatk are co-plannar.
Answer» ANSWER :`lambda=1`

Discussion

8.	Let f(x) and phi(x) are two continuous function on R satisfying phi(x)=int_(a)^(x)f(t)dt, a!=0 and another continuous function g(x) satisfying g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0, and int_(b)^(2k)g(t)dt is independent of b If m,n are even integers and p,q epsilon R, then int_(p+n alpha)^(q+n alpha)g(t)dt is equal to
Answer» <P>`int_(p)^(q)g(X)dx` `(n-m)int_(0)^(ALPHA)g(x)dx` `int_(p)^(alpha)g(x)dx+(n-m)int_(0)^(alpha)g(2X)dx` `int_(p)^(q)g(x)dx+((n-m))/2int_(0)^(2ALPHA)g(x)dx` Solution :`int_(p+m alpha)^(q+n alpha) g(t)dt=int_(p+m alpha)^(p)g(x)dx+int_(p)^(q)g(x)dx+int_(q)^(q+n alpha) g(x)dx` `=-m/2 int_(0)^(2 alpha) g(x)dx+int_(p)^(q)g(x)dx+n/2 int_(0)^(2alpha) g(x)dx` `=int_(p)^(q)g(x)dx+((n-m)/2)int_(0)^(2alpha) g(x)dx`

Discussion

9.	Consider the function defined implicity by the equation y^(2)-2ye^(sin^(-1)x)+x^(2)-1+[x]+e^(2sin ^(-1)x)=0("where [x] denotes the greatest integer function"). Line x=0 divides the region mentioned above in two parts. The ratio of area of left-hand side of line to that of right-hand side of line is
Answer» `1+PI:pi` `2-pi:pi` `1:1` `pi+2:pi` SOLUTION :`"RATIO "=((pi)/(2)+1)/((pi)/(2))=(pi+2)/(pi).`

Discussion

10.	Consider the function defined implicity by the equation y^(2)-2ye^(sin^(-1)x)+x^(2)-1+[x]+e^(2sin ^(-1)x)=0("where [x] denotes the greatest integer function"). The area of the region bounded by the curve and the line x=-1 is
Answer» `PI+1` sq. units `pi-1` sq. units `(pi)/(2)+1` sq. units `(pi)/(2)-1` sq. units Solution :`"For "-1lexlt0` `(y-e^(sin^(-1)X))^(2)=2-x^(2)` `y=e^(sin-1x)pmsqrt(2-x^(2))` `A=overset(0)underset(-1)int(e^(sin^(-1)x)+sqrt(2-x^(2)))-(e^(sin^(-1)x)-sqrt(2-x^(2)))dx` `=2overset(0)underset(-1)intsqrt(2-x^(2))dx` `=2((1)/(2)xsqrt(2-x^(2)) :\|_(-1)^(0)+(2)/(2)sin^(-1)""(x)/(sqrt(2)):\|_(-1)^(0))` `=[1+2(0-(-(pi)/(54)))]` `=(pi)/(2)+1` sq. units. `"For "0lexlt1,y=sin^(-1)xpmsqrt(1-x^(2))` `A=2overset(1)underset(0)intsqrt(1-x^(2))dx` `=2[(x)/(2)sqrt(1-x^(2)):\|_(0)^(1)+(1)/(2)sin^(-1)""(x)/(1):\|_(0)^(1)]` `=0+sin^(-1)(1)=(pi)/(2)` sq. units. `"Total area "=((pi)/(2)+1)+(pi)/(2)=pi+1.`

Discussion

11.	int_(0)^(pi//2) e^(x)((1+sin x)/(1+cos x))dx=
Answer» `E^(PI)` `e^(pi//2)` `e` 1 Answer :B

Discussion

12.	Let f:[0,1]to R be a continuous function then the maximum value ofint_(0)^(1)f(x).x^(2)dx-in_(0)^(1)x.(f(x))^(2)dx for all such function(s) is:
Answer» ANSWER :`0.06`

Discussion

13.	For any two event A, B show that P(A^(C)) P(B) - P(A^(C) nn B) = P(A nn B) - P(A) P(B)
Answer» <P> ANSWER :`P(A NN B) - P(A) P(B)`

Discussion

14.	Using the combinedmethod find the valus of the root of theequatin x^(3) - x - 1 = 0 On the interval [1,2] with an accuracy up to 0.005 .
Answer» ANSWER :`1.325`

Discussion

15.	For what values of 'k' the expression (4 - k)x^2 + 2(k + 2)x + 8k +1 will be a perfect square ?
Answer» ANSWER :0,3

Discussion

16.	A small firm manufacturers Golden chains and rings. The total number of chains and rings that it can manufacture in a day is at the most 24. The ring takes 30 minutes. The maximum time available per day is 16 hours. If the profit on the ring be Rs. 300 and one chain be Rs. 190. How many of rings and chains be produced to maximizethe profit ?
Answer» ANSWER :NUMBER of RINGS = 8 Number of CHAINS = 16

Discussion

17.	Direction ratios of line given by line(x-1)/3=(2y+6)/12=(1-z)/-7 are :
Answer» lt3,12,-7gt lt3,-6,7gt lt3,6,7gt lt3,6,-7gt Answer :C

Discussion

18.	Which of the followingmolecule has dipolemonet
Answer» `C_(6)H_(4)(OH)_(2)` (PARA) `C_(6)H_(4)Cl_(2)` (para) `C_(6)H_(5)CIBr` (para) ANSWER :A::C::D

Discussion

19.	If (0,0) is one limiting point of a coaxal system of circles whose common radical aixs is the line x + y = 1 then the other limiting point is
Answer» (1,1) (-1,-1) (1,-1) (-1,1) ANSWER :A

Discussion

20.	Nine balanced coins tossed together once. Find probability of getting at least 6 heads.
Answer» ANSWER :`(65)/(256)`

Discussion

21.	The series 1 + 1/2 + 1/3 +…..+ 1/n can be expressed by the definite integral
Answer» `int_(0) ^((pi)/(2)) tan ((theta)/(2)) (1- COS ^(n) theta) d theta` `int_(0) ^((pi)/(2)) COT ((theta)/(2)) (1- cos ^(n) theta) d theta` `int_(0) ^((pi)/(4)) cot ((theta)/(2)) (1+cos ^(n) theta) d theta` `int_(0) ^((pi)/(4)) tan ((theta)/(2)) (1+cos ^(n) theta) d theta` Answer :C

Discussion

22.	Observe the following statements Assertion (A) : int((x^(2) -1)/(x^(2)) ) e^((x^(2) +1)/(x)) " dx " = e^((x^(2)+1)/(x)) + e Reason (R) : int f^(1) (x) e^(f(x)) dx = f (x) + c Which of the following is true ?
Answer» Both A and R are TRUE and R is not the CORRECT REASON for A . Both A and R true and R is correct reason for A A is true, R is false A is false, R is ture Answer :C

Discussion

23.	Evaluate int_(0)^(100pi)sqrt(1+ cos 2 x dx)
Answer» ANSWER :`200 SQRT(2)`

Discussion

24.	Evaluate (i) int_(0)^(pi/3) (cos x)/(3+ 4 sin x)dx
Answer» ANSWER :`(1)/(4) LOG((3+2sqrt(3))/(3))`

Discussion

25.	Assume X, Y, Z, W and P are matrices of order2xx n, 3xxk, 2xxp, n xx 3 and p xx k, respectively. If n = p, then the order of the matrix 7X-5Z is:
Answer» `pxx2` `2XXN` `N xx 3` `p XXN` Answer :B

Discussion

26.	The points P (a, b+c), Q (b, c +a) and R(c, a+b) are such that PQ = QR if
Answer» a, B, C are in A.P. a, b, c are in G.P. a, b, c are in H.P. NONE of these ANSWER :A

Discussion

27.	Integrate the following functions x tan^-1x
Answer» SOLUTION :`int x tan^-1x DX` =`tan^-1x xx x^2/2 - int 1/(1+x^2) x^2/2 dx` =`x^2/2 tan^-1 x- 1/2 int (1+x^2-1)/(1+x^2) dx` `x^2/2 tan^-1 x-1/2 int (1-1/(1+x^2)) dx` =`x^2/2 tan^-1 x- 1/2(x-tan^-1 x)+C` =`x^2/2 tan^-1 -x/2 +1/2 tan^-1 x+c`

Discussion

28.	In the game of chess, the Knight can make any of the moves displayed in the diagram. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?
Answer» 8 24 38 48 Answer :D

Discussion

29.	If x^(y) = e^(x - y) prove that (dy)/(dx) = (log_(e)x)/((1 + log_(e)x)^(2)).
Answer» ANSWER :`(y(X-1))/(x(y+1))`

Discussion

30.	x, y, z in R such that x^(2)+y^(2)+z^(2)=1 and alpha=x^(2)+2y^(2)+3z^(2)
Answer» MINIMUM VALUE of `ALPHA` is 2 maximum value of `alpha` is 3 minimum value of `alpha` is 3 maximum value of `alpha` is 4 Answer :B

Discussion

31.	If A and B are independent events,show that A and B^c are independent
Answer» <P> Solution :`P(ACAPB^c)`=P(A-B) =P(A)-P`(AcapB)` =P(A)-P(A)P(B) =P(A)[1-P(B)] `THEREFORE` A. and B. are independent EVENTS.

Discussion

32.	1+ 2 (log_eq)+(2^2)/(2!) (log_ea)^2 +(2^3)/(3!) (log_e a)^3+...=
Answer» e a `a^2` `e^2` ANSWER :C

Discussion

33.	""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=
Answer» `""^((n+1))C_(R + 1)` `""^(n+2)C_r` `""^(n+2)C_(r+1)` `""^(n+1)C_r` ANSWER :C

Discussion

34.	Find (1)/(2)(A+A')and(1)/(2)(A-A), when A=[{:(0,a,b),(-a,0,c),(-b,-c,0):}]
Answer» ANSWER :`=[{:(0,a,B),(-a,0,C),(-b,-c,0):}]`

Discussion

35.	If I_(1) int sin^(-1) ((2x)/(1 +x^(2)) )dx , I_(2) = int cos^(-1) ((1-x^(2))/(1 +x^(2)) )dx , I_(3) = int tan^(-1) ((2x)/(1 - x^(2)) )dx , then I_(1) + I_(2) - I_(3)=
Answer» `2X tan^(-1) " X- log " ( 1 + x^(2)) + C ` `2 [ "XTAN"^(-1) x - log (1 + x^(2)) ] + c ` `2"xtan"^(-1) "x + log " ( 1 + x^(2)) + c ` 0 Answer :A

Discussion

36.	A right circular cone has height 9 cm and base radius is 5 cm. It is inverted and water is poured into it. If at any instant, the level of the water riese at the surface at that instant, then vessel will be full in
Answer» 25 seconds 50 seconds 75 seconds 100 seconds Answer :C

Discussion

37.	The vectors 2i - 3j + k, I - 2j + 3k, 3i + j - 2k
Answer» `-12` 14 10 15 Answer :A

Discussion

38.	Evalute the following integrals int sin^(4) xdx
Answer» ANSWER :`- (SIN^(3) " x COS x")/(4) - (3)/(8) " sin x cos x +(3)/(8)` x + C

Discussion

39.	Write the order and degree of the differential equations given by : (d^2y)/dx^2+3(dy/dx)^4+y=0
Answer» SOLUTION :ORDER = 2, DEGREE = 1

Discussion

40.	A letter is known to have come from LONDONor CLIFTON . On the envelope just two consecutive letters ON are visible . What is the probability that the letter has come from (i) LONDON (ii)CLIFTON.
Answer» ANSWER :` (12)/(17)` (II) ` (5)/(17)`

Discussion

41.	intxcos^2xdx
Answer» SOLUTION :`intxcos^2xdx=intx.(1+cos2x)/2 DX` =`1/2intxdx+1/2intx.cos2xdx` =`1/4x^2+1/2{X.(SIN2X)/2-1/4int(sin2x)/2dx}` =`1/4x^2+1/4xsin2x-1/4intsin2xdx` =`1/4x^2+1/4xsin2x+1/8cos2x+C`

Discussion

42.	Select the CORRECT order of lattice energy ?
Answer» BEO LT MGO MgS lt MgO `Mg_3N_2` lt MgO `BeF_2` lt BeO

Discussion

43.	The point on the curve y^(3)+3x^(2)=12y where the tangent is vertical is ………..
Answer» `(pm(4)/(sqrt(3)),-2)` `(pm(sqrt(11))/(3),1)` (0,0) `(pm(4)/(sqrt(3)),2)` Answer :D

Discussion

44.	If y_(1)=max\|\|z-omega\|-\|z-omega^(2)\|\|, where \|z\|=2 and y_(2)=max\|\|z-omega\|-\|z-omega^(2)\|\|, where \|z\|=(1)/(2) and omega and omega^(2) are complex cube roots of unity, then
Answer» `y_(1)=SQRT(3)`, `y_(2)=sqrt(3)` `y_(1) lt sqrt(3)`, `y_(2)=sqrt(3)` `y_(1)=sqrt(3)`, `y_(2) lt sqrt(3)` `y_(1) gt 3`, `y_(2) lt sqrt(3)` SOLUTION :`(c )` We have `\|\|z_(1)\|-\|z_(2)\|\| LE \|z_(1)-z_(2)\|` and EQUALITY holds only when `argz_(1)=argz_(2)` `implies\|\|Z-w\|-\|z-w^(2)\|\| le \|w^(2)-w\| le sqrt(3)` and equality canhold only when `\|z\|=2` and not when `\|z\|=(1)/(2)`

Discussion

45.	If a and b are randomly chosen from the set {1,2,3,4,5,6,7,8,9}, then the probability that the expression ax^(4)+bx^(3)+(a+1)x^(2)+bx+1 has positive values for all real values of x is
Answer» `(34)/(81)` `(31)/(81)` `(32)/(81)` `(10)/(27)` Solution :`(C )` The expression `AX^(4)+ax^(2)+bx^(3)+bx+x^(2)+1` `=(x^(2)+1)(ax^(2)+bx+1)` `:.x^(2)+1` is positive for all REAL `x` For `ax^(2)+bx+1` to be positive for all real `x` `a gt 0`, `b^(2)-4a lt 0` If `b=1`, a can take `9` value from `1` to `9` `b=2`, a can take `8` value from `2` to `9` `b=3`, a can take `7` value from `3` to `9` `b=4`, a can take `5` value from `5` to `9` `b=5`, a can take `3` value from `7` to `9` `b` cannot take the values `6,7,8,9`. `:.` Number of exhaustive cases `=9+8+7+5+3=32` For each of the `9` values of `a`, there are `9` corresponding values for `h`. `:.` Number of exhaustive cases `=9xx9=81` `:.` The required probability `=(32)/(81)`

Discussion

46.	If the points (0, 0), (3, sqrt(3)), ( x, y)form an equilateral triangle, then (x,y)=
Answer» `(0, 2sqrt(3)), (3,-sqrt(3))` `(1, 2sqrt(3)), (3,sqrt(3))` `(1, sqrt(3)), (3, -sqrt(3))` none ANSWER :A

Discussion

47.	If A=[{:(0,"tan"(alpha)/(2)),("tan"(alpha)/(2),0):}]and I theidentity matrix of order 2, show that I+A=(I-A)[{:(cosalpha,-sinalpha),(sinalpha,cosalpha):}].
Answer» ANSWER :`=I+A`

Discussion

48.	If the points whose position vectors are 3bar(i) -2bar(j)-bar(k), 2bar(i)+3bar(j)-4bar(k), -bar(i)+bar(j)+2bar(k), 4bar(i)+5bar(j)+lambdabar(k) are coplanar, then show that lambda = -146/17.
Answer» ANSWER :`-146//17`

Discussion

49.	A common tangent to the conic x^(2) = 6y and 2x^(2) -4y^(2)=9 is
Answer» `X-y=3//2 ` ` x+y =1` ` x+ y =9//2` ` x- y =1` ANSWER :A

Discussion

50.	Prove the following : 1 lt int_(0)^(pi//2)sqrt(sinx)dx lt sqrt(pi/2)
Answer» SOLUTION :N//A

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

underset(x to pi//2)lim (sqrt(1+cos^(3)x)-sqrt(1-cos^(3)x))/((pi//2-x)^(3)))=

[a b c] [a' b' c'] =

Find the maximum and minimum values, if any, of thefunctions given by g(x) = – | x + 1| + 3

2 sin^(2)x + sin^(2)2x=2,pi lt x lt pi then x equals

int_(-2)^2 |x cos pi x|dx is equal to :

Let a=hat(i)+hat(j)+hat(k), b=hat(i)-hat(j)+hat(k) and c=hat(i)-hat(j)-hat(k) be three vectors. A vector v in the plane of a and b whose projection on c is 1/sqrt(3), is given by

Find the value of a so that the four points with position vectors -hatj+hatk,2hati-hatj-hatk,hati+lambdahatj+hatk and 3hatj+3hatk are co-plannar.

Consider the function defined implicity by the equation y^(2)-2ye^(sin^(-1)x)+x^(2)-1+[x]+e^(2sin ^(-1)x)=0("where [x] denotes the greatest integer function"). Line x=0 divides the region mentioned above in two parts. The ratio of area of left-hand side of line to that of right-hand side of line is

Consider the function defined implicity by the equation y^(2)-2ye^(sin^(-1)x)+x^(2)-1+[x]+e^(2sin ^(-1)x)=0("where [x] denotes the greatest integer function"). The area of the region bounded by the curve and the line x=-1 is

int_(0)^(pi//2) e^(x)((1+sin x)/(1+cos x))dx=

Let f:[0,1]to R be a continuous function then the maximum value ofint_(0)^(1)f(x).x^(2)dx-in_(0)^(1)x.(f(x))^(2)dx for all such function(s) is:

For any two event A, B show that P(A^(C)) P(B) - P(A^(C) nn B) = P(A nn B) - P(A) P(B)

Using the combinedmethod find the valus of the root of theequatin x^(3) - x - 1 = 0 On the interval [1,2] with an accuracy up to 0.005 .

For what values of 'k' the expression (4 - k)x^2 + 2(k + 2)x + 8k +1 will be a perfect square ?

Direction ratios of line given by line(x-1)/3=(2y+6)/12=(1-z)/-7 are :

Which of the followingmolecule has dipolemonet

If (0,0) is one limiting point of a coaxal system of circles whose common radical aixs is the line x + y = 1 then the other limiting point is

Nine balanced coins tossed together once. Find probability of getting at least 6 heads.

The series 1 + 1/2 + 1/3 +…..+ 1/n can be expressed by the definite integral

Observe the following statements Assertion (A) : int((x^(2) -1)/(x^(2)) ) e^((x^(2) +1)/(x)) " dx " = e^((x^(2)+1)/(x)) + e Reason (R) : int f^(1) (x) e^(f(x)) dx = f (x) + c Which of the following is true ?

Evaluate int_(0)^(100pi)sqrt(1+ cos 2 x dx)

Evaluate (i) int_(0)^(pi/3) (cos x)/(3+ 4 sin x)dx

Assume X, Y, Z, W and P are matrices of order2xx n, 3xxk, 2xxp, n xx 3 and p xx k, respectively. If n = p, then the order of the matrix 7X-5Z is:

The points P (a, b+c), Q (b, c +a) and R(c, a+b) are such that PQ = QR if

Integrate the following functions x tan^-1x

In the game of chess, the Knight can make any of the moves displayed in the diagram. If a Knight is the only piece on the board, what is the greatest number of spaces from which not all 8 moves are possible?

If x^(y) = e^(x - y) prove that (dy)/(dx) = (log_(e)x)/((1 + log_(e)x)^(2)).

x, y, z in R such that x^(2)+y^(2)+z^(2)=1 and alpha=x^(2)+2y^(2)+3z^(2)

If A and B are independent events,show that A and B^c are independent

1+ 2 (log_eq)+(2^2)/(2!) (log_ea)^2 +(2^3)/(3!) (log_e a)^3+...=

""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=

Find (1)/(2)(A+A')and(1)/(2)(A-A), when A=[{:(0,a,b),(-a,0,c),(-b,-c,0):}]

If I_(1) int sin^(-1) ((2x)/(1 +x^(2)) )dx , I_(2) = int cos^(-1) ((1-x^(2))/(1 +x^(2)) )dx , I_(3) = int tan^(-1) ((2x)/(1 - x^(2)) )dx , then I_(1) + I_(2) - I_(3)=

A right circular cone has height 9 cm and base radius is 5 cm. It is inverted and water is poured into it. If at any instant, the level of the water riese at the surface at that instant, then vessel will be full in

The vectors 2i - 3j + k, I - 2j + 3k, 3i + j - 2k

Evalute the following integrals int sin^(4) xdx

Write the order and degree of the differential equations given by : (d^2y)/dx^2+3(dy/dx)^4+y=0

A letter is known to have come from LONDONor CLIFTON . On the envelope just two consecutive letters ON are visible . What is the probability that the letter has come from (i) LONDON (ii)CLIFTON.

intxcos^2xdx

Select the CORRECT order of lattice energy ?

The point on the curve y^(3)+3x^(2)=12y where the tangent is vertical is ………..

If y_(1)=max||z-omega|-|z-omega^(2)||, where |z|=2 and y_(2)=max||z-omega|-|z-omega^(2)||, where |z|=(1)/(2) and omega and omega^(2) are complex cube roots of unity, then

If a and b are randomly chosen from the set {1,2,3,4,5,6,7,8,9}, then the probability that the expression ax^(4)+bx^(3)+(a+1)x^(2)+bx+1 has positive values for all real values of x is

If the points (0, 0), (3, sqrt(3)), ( x, y)form an equilateral triangle, then (x,y)=

If A=[{:(0,"tan"(alpha)/(2)),("tan"(alpha)/(2),0):}]and I theidentity matrix of order 2, show that I+A=(I-A)[{:(cosalpha,-sinalpha),(sinalpha,cosalpha):}].

If the points whose position vectors are 3bar(i) -2bar(j)-bar(k), 2bar(i)+3bar(j)-4bar(k), -bar(i)+bar(j)+2bar(k), 4bar(i)+5bar(j)+lambdabar(k) are coplanar, then show that lambda = -146/17.

A common tangent to the conic x^(2) = 6y and 2x^(2) -4y^(2)=9 is

Prove the following : 1 lt int_(0)^(pi//2)sqrt(sinx)dx lt sqrt(pi/2)