14949 + Interview Questions in Maths in Class 11 Page 137 InterviewSolution

6801.	Show that f(x) = [x] (x in R) is continuous at only those real numbers that are not integers.
Answer» ANSWER :a

Discussion

6802.	Simplify (x + sqrt(1+x^(2)))^(3) -(x-sqrt(1+x^(2)))^(3)
Answer» ANSWER :`SQRT(1+x^(2))[2+ 8X^(2)]`

Discussion

6803.	Consider the function f:(-oo,oo)to\|(-oo,oo) defined by f(x)=(x^(2)-a)/(x^(2)+a),agt0 . Whichof the followingis not true ?
Answer» MAXIMUM value of f is not attained EVEN thoughf is bounded. f(x) is increasing on `(0,oo)` and has minimum at x=0 f(x) is DECREASING on `(-oo,0)` and has minimum at x=0. f(x) is increasing on `(-oo,oo)` and has NEITHER a local maximum nor a local minimum at x=0 Answer :D

Discussion

6804.	(6x + 7)/(sqrt((x -4)(x - 5)))
Answer» Answer :`6sqrt(X^2 - 9X + 20) + 34 [LOG(x - 9/2) + SQRT(x^2 - 9x + 20)] + c`.

Discussion

6805.	A circle centred at the vertex of parabolax^(2)= 4y intersects it at theends of its latus-rectum . Equation of this cirlce is
Answer» `x^(2) + y^(2) = 5` `x^(2) + y^(2) = 4` `x^(2) + y^(2) = 1` `x^(2) + y^(2) = 2` SOLUTION :N/A

Discussion

6806.	Using the value of cos 315 ^(@) , find the value of sin 157 ""(1)/(2) ^(@and cos 157 (1)/(2) ""^(@).
Answer» ANSWER :`=1/2 SQRT (2 + SQRT2)`

Discussion

6807.	Number of points with integral coordinates which lie inside the triangle formed by (0,21), (21,0) and (0,0) is
Answer» `sum19` `sum20` `sum21` `sum22` ANSWER :A

Discussion

6808.	Evaluate the following limits in Exercises lim_(xto3)x+3
Answer» ANSWER :6

Discussion

6809.	2 tan h^(-1) ((1)/(2)) =
Answer» 0 `log_(E) 2` `log_(e)3` `log_(e)4` Answer :C

Discussion

6810.	If tan((alpha)/(2)) and tan (( beta)/(2)) are the roots of the equation 8x^(2)-26x+15=0 then cos (alpha+ beta)=
Answer» `(627)/(725)` `(-625)/(725)` `-1` 1 Answer :B

Discussion

6811.	Let f(x) be a polynomical function in x satisfyingthe conditionf(x) f(1//x)=f(x) +f(x) and f(3) = 28x ne 0 Find the coordinatesof (f(-2), f(0))
Answer» (-2, 1, 9) (-7, 1, 9) (-5, 1, 9) (0, 0, 0) Answer :B

Discussion

6812.	Given:(i)1-cot^2 theta =cosec^2 theta(ii)Is cos2 A +cos2 B=0then sin^2 A +sin^2 B =2(iii)tan2theta=(2tantheta)/(1-tan^2theta)(iv)sinx is periodic function with period pi/2Find which pair is true:
Answer» (i) and (II) are TRUE (i) and (IV)are true (ii) and (III) are true (iii) and (iv) are true Answer :C

Discussion

6813.	If a,b,c are real numbers such that (3a+2b)/(c+d) +( 3)/( 2) = 0then the equation ax^(3) + bx^(2) + cx + d = 0has
Answer» at least ONE ROOT in [-2,0] at least one root in [0,2] at least TWO roots in [-2,2] No root in [-2,2] Answer :B

Discussion

6814.	If a sin^(3)x+b cos^(3)x= sin x cos x and a sin x= b cos x then a^(2)+b^(2)=
Answer» `1//2` 2 1 `1//3` ANSWER :C

Discussion

6815.	Which value of theta listed below leads to 2^(sin theta)gt1 and 3 cos (theta) lt 1?
Answer» `70^(@)` `140^(@)` `210^(@)` `280^(@)` Answer :B

Discussion

6816.	A committee of 7 members has to be formed from 9 boys and 4 girls . In how many ways can this be done when the committee consists of exactly 3 girls.
Answer» ANSWER :`504`

Discussion

6817.	Consider the expansion of ((4x)/5-5/(2x))^9 Find the 7th term in the expansion
Answer» SOLUTION :`10500/x^3]`

Discussion

6818.	cos h^(-1)(2) =
Answer» `log_(E)(2+sqrt(3))` `log_(e)(2+sqrt(5))` `log_(e)(2-sqrt(5))` `log_(e)(2+sqrt(2))` ANSWER :A

Discussion

6819.	0leale3,0leble3 and the equation x^(2)+4+3cos(ax+b)=2x has atleast one solution then the value of a+b
Answer» `PI//2` `pi//4` `pi//3` `pi` ANSWER :D

Discussion

6820.	cos ^(2) ((A)/(2))+ cos ^(2) ((B)/(2))+cos ^(2)(( C )/(2))=
Answer» ` 1- ( r )/( R ) ` ` 1- ( R )/( r ) ` ` 2+ (r ) /( 2 R) ` ` 1 - ( r ) /(2R ) ` ANSWER :C

Discussion

6821.	Identify the quantifier and write the negation of the statement ''There exists a number which is equal to its square.''
Answer» Answer : ''There EXISTS''. The NEGATION is There does not EXIST a number which is equal to its square.

Discussion

6822.	4sqrt(x)-2
Answer» ANSWER :`(2)/(SQRT(X))`

Discussion

6823.	Find the component statements of thecompound statement "All integers are positive or negative" ?
Answer» ANSWER :All INTEGERS are POSITIVE; all integers are NEGATIVE (FALSE).

Discussion

6824.	Simiplify :(1+ i)^(-1)
Answer» ANSWER :`(1-i)/(2)`

Discussion

6825.	Consiser the system of equations x cos^(3)y+3xcosysin^(2)y=14to(1) and xsin^(3)y+3xcos^(2)y=13to(2) The value of x is/are
Answer» `+-5sqrt(5)` `+-SQRT(5)` `+-1/(sqrt(5))` `+-1/(5sqrt(5))` ANSWER :A

Discussion

6826.	If f(x)=2x^(2)+3x-5,x=3,deltax=0.1 then deltaf=A (2) If f(x)=x^(2)+4x,x=2,deltax=0.1 then deltaf=B (3) If f(x)=x^(2)+3x,x=3,deltax=0.1 then deltaf=C The ascending order of A, B C is
Answer» A, B, C B, C, A C, A, B A, C, B Answer :B

Discussion

6827.	Find the points at which the functions f(x) = x^(3) - 6x^(2) + 9x + 15
Answer» ANSWER :15

Discussion

6828.	If the area enclosed by the curves f(x)=cos^(-1)(cosx) and g(x)=sin^(-1)(cosx)" in "x in [(9pi)/4, (15pi)/4] is api^(2)//b (where a and b are coprime), then the value of (a-b) is
Answer» ANSWER :1

Discussion

6829.	For an G.P. a = 16 and fifth term is 81 then common ratio ........
Answer» ANSWER :`3/2`

Discussion

6830.	Find the middle term (terms ) in the expansion of (3x - (x^(3))/6)^(9)
Answer» ANSWER :`= (189)/8 xx x^(7)`

Discussion

6831.	If the mean and variance of the observations x_(1), x_(2), x_(3),…,x_(n) are barx and sigma^(2) respectively and a be a nonzero real number, then show that the mean and variance of ax_(1),ax_(2),ax_(3),…,ax_(n)" are "abarx and a^(2) sigma^(2) respectively.
Answer» Solution :Let `barx` be the mean of `x_(1),x_(2),x_(3),x_(n)` and a be nonzero real number. Then, `barx=(1)/(n)(x_(1)+x_(2)+x_(3)+ +x_(n)).` Let `y_(i)=ax_(i)" for each i=1, 2, 3,n. Then",` `bary=(1)/(n)(y_(1)+y_(2)+y_(3)+...+y_(n))` `=(1)/(n)(ax_(1)+ax_(2)+ax_(3)+ +ax_(n))=acdot(1)/(n)(x_(1)+x_(2)+x_(3)+...+x_(n))=abarx.` Thus, `bary=a barx.` Now, the variance of new observations is given by `"variance "(y)=sigma_(1)^(2)` `=(1)/(n)cdotoverset(n)underset(i=1)Sigma(y_(i)-bary)^(2)` `=(1)/(n)CDOT overset(n)underset(i=1)Sigma(ax_(i)-abarx)^(2)" "[because y_(i)=ax_(i)" for each i and "bary=abarx]` `=a^(2)cdot(1)/(n)cdotoverset(n)underset(i=1)Sigma(x_(i)-barx)^(2)` `=a^(2)cdot{"variance"(x)}=a^(2)sigma^(2).` `THEREFORE" new variance"=a^(2)sigma^(2).` `"REMARK "sigma_(1)=SQRT(a^(2)sigma^(2))=\|a\|cdot sigma.`

Discussion

6832.	Using the following data. Find out the trend using Quarterly moving average and plot them on graph
Answer» ANSWER :TREND VALUES are `43,38,51,63,53,75,56,38,52,12,47,88,50,25`

Discussion

6833.	This section contains 2 questions. Each question contains STATEMENT-I (Assertion) and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. Statement - 1 : The maximum and minimum values f(x) = (1)/( 3 sin x + 4 cos x-2) does not exist Statement - 2 : The given faction is an unbounded function.
Answer» STATEMENT -1 is True, Statement-2 is True, Statement -2 is a correct EXPLANATION for Statement-3 Statement-1 is True, Statement-2 is True, Statement-2 NOT a correct explanation for Statement-1. Statement-1 is True, Statement-2 is FALSE. Statement-1 is False, Statement-2 is True. Answer :A

Discussion

6834.	For 0 lt theta lt 2 pi sin^(-1)(sin theta)gtcos^(-1)(sin theta) is true when
Answer» `((PI)/4,pi)` `(pi,(3pi)/2)` `((pi)/4,(3pi)/4)` `((3pi)/4,2pi)` Answer :C

Discussion

6835.	Find the equation of line which makes a triangle witharea (50)/( sqrt(3) ) unit with axis and perpendicular from origin makes an angle (pi)/(6) with positive side of X-axis.
Answer» ANSWER :`SQRT(3) X + y=10`

Discussion

6836.	An A.P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the series.
Answer» ANSWER :3,7,11,15 ………..

Discussion

6837.	Evaluate the following limits in Exercises lim_(xto0)(cosx)/(pi-x)
Answer» ANSWER :`(1)/(PI)`

Discussion

6838.	If f(x)=log_(e)(1-x)andg(x)=[x] then find: (i) (f+g)(x) (ii) (fg)(x) (iii) ((f)/(g))(x) (iv) ((g)/(f))(x). Also find (f+g)(-1),(fg)(0),((f)/(g))(-1),((g)/(f))((1)/(2)).
Answer» Solution :Clearly, `log_(e)(1-x)` is defined only when `1-xgt0,i.e.,xlt1`. `:."dom "(F)=(-oo,1)`. ALSO, dom (g)=R. `:."dom "(f)nn"dom "(g)=(-oo,1)nnR=(-oo,1)` (i) `(f+g)"(-oo,1)toR` is GIVEN by `(f+g)(x)=f(x)+g(x)=log_(e)(1-x)+[x]`. (ii) `(fg):(-oo,1)toR` is given by `(fg)(x)=f(x)xxg(x)={log_(e)(1-x)}xx[x]`. (iii) `{x:g(x)=0}={x:[x]=0}={0,1)`. `:."dom "((f)/(g))="dom "(f)nn"dom "(g)-{x:g(x)=0}` `=(-oo,1)nnR-[0,1)=(-oo,0)`. `:.(f)/(g):(-oo,0)toR` is given by `((f)/(g))(x)=(f(x))/(g(x))=(log_(e)(1-x))/([x])`. (iv) `{x:f(x)=0}={x:log_(e)(1-x)=0}={0}`. `:."dom "((g)/(f))="dom "(g)nn"dom "(f)-{x:f(x)=0}`. `=Rnn(-oo,1)-{0}=(-oo,0}uu(0,1)`. `:.(g)/(f):(-oo,0)uu(0,1)toR` `((g)/(f))(x)=(g(x))/(f(x))=([x])/(log_(e)(1-x))`. Now, we have: `(f+g)(-1)=f(-1)+g(-1)=[-1]+log_(e)(1+1)=(log_(e)2)-1`. `(fg)(0)=f(0)xxg(0)=log_(e)(1-0)xx[0]=(log_(e)1xx0)=(0xx0)=0`. `((f)/(g))(-1)=(f(-1))/(g(-1))=([-1])/(log_(e)(1+1))=(-1)/(log_(e)2)`. `((g)/(f))((1)/(2))=g((1)/(2))/(f((1)/(2)))=([(1)/(2)])/(log_(e)(1-(1)/(2)))=([05.])/(log_(e)((1)/(2)))=0`.

Discussion

6839.	The sum of first two terms of a G.P. is (9)/(2). Its 6^(th) term is 8 times its 3^(rd) term. Find the sequences.
Answer» ANSWER :`(3)/(2), 3, 6, 12`

Discussion

6840.	If A and G are the arithmetic and geometric means respectively of two numbers then prove that the numbers are (A+sqrt(A^(2)-G^(2)))and(A-sqrt(A^(2)-G^(2)))
Answer» SOLUTION :N/a

Discussion

6841.	Expand (1-x^2)^4
Answer» ANSWER :`1 - 4x^2 + 6x^4 - 4x^6 + x^8`

Discussion

6842.	B,C are two points on 3x+4y-15=0 such that OB+OC=10 then maximum possible area of triangle OBC is
Answer» 12 10 15 13 Answer :A

Discussion

6843.	If p^(th), q^(th), r^(th) terms of a geometric progression are the positive numbers a,b,c respectively, then the angle between the vectors (log a^(2)) I + (log b^(2)) j + (log c^(2))k and (q - r) I + (r - p) j + (p - q) k is
Answer» `PI/4` `pi/2` `pi/6` `pi` ANSWER :B

Discussion

6844.	Let f(x) = (sqrt(x - 2 sqrt(x - 1)))/(sqrt(x - 1) - 1)x, then
Answer» `f'(10) = 1` `f'(3//2) =-1` DOMAIN of `f(x) ` is `x GE 1` RANGE of `f(x) ` is `(-2, -1] uu (2, oo)` Answer :A::B::D

Discussion

6845.	If the parabola y^(2) = 4ax passes through the point (3,2), then the length of its latus rectum is ……
Answer» `(2)/(3)` `(4)/(3)` `(1)/(3)` 4 Answer :B

Discussion

6846.	(sin 3 thea - sin theta sin ^(2) (2 theta ))/( sin theta + sin 2 theta cos theta ) = cos x implies x =
Answer» `4 theta ` `2 theta ` `theta ` `3 theta ` Answer :B

Discussion

6847.	Let cup be the set of all boys and girls in a school, G be the set of all girls in the school, B be the set of all boys in the school and S be the set of all students in the school who take swimming. Some but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible inter relationship among sets cup, G, B and S.
Answer» ANSWER :`(##KPK_AIO_MAT_XI_C01_E12_026_A01##)`

Discussion

6848.	Find the equation of each of the following parabolas : (i) Directrix x = 0, focus at (6, 0) (ii) Vertex at (0,4), focus at (0,2) (iii) Focus at (-1, -2), directrix x - 2y + 3 = 0.
Answer» Answer :`4X^(2) + y^(2) + 4XY + 4x + 32y + 16 = 0.`

Discussion

6849.	Mean and standard deviation of 100 items are 50 and 4, respectively . Find the sum of all the item and the sum of the squares of the items.
Answer» 250000 251600 49910 499100 Answer :B

Discussion

6850.	Find the values of thetabetween 0^(@) and 360^(@) which satisfy the equation 3 cos 2 theta - sin theta = 2.
Answer» ANSWER :`330^(@)`

Discussion

Explore topic-wise InterviewSolutions in .

Show that f(x) = [x] (x in R) is continuous at only those real numbers that are not integers.

Simplify (x + sqrt(1+x^(2)))^(3) -(x-sqrt(1+x^(2)))^(3)

Consider the function f:(-oo,oo)to|(-oo,oo) defined by f(x)=(x^(2)-a)/(x^(2)+a),agt0 . Whichof the followingis not true ?

(6x + 7)/(sqrt((x -4)(x - 5)))

A circle centred at the vertex of parabolax^(2)= 4y intersects it at theends of its latus-rectum . Equation of this cirlce is

Using the value of cos 315 ^(@) , find the value of sin 157 ""(1)/(2) ^(@and cos 157 (1)/(2) ""^(@).

Number of points with integral coordinates which lie inside the triangle formed by (0,21), (21,0) and (0,0) is

Evaluate the following limits in Exercises lim_(xto3)x+3

2 tan h^(-1) ((1)/(2)) =

If tan((alpha)/(2)) and tan (( beta)/(2)) are the roots of the equation 8x^(2)-26x+15=0 then cos (alpha+ beta)=

Let f(x) be a polynomical function in x satisfyingthe conditionf(x) f(1//x)=f(x) +f(x) and f(3) = 28x ne 0 Find the coordinatesof (f(-2), f(0))

Given:(i)1-cot^2 theta =cosec^2 theta(ii)Is cos2 A +cos2 B=0then sin^2 A +sin^2 B =2(iii)tan2theta=(2tantheta)/(1-tan^2theta)(iv)sinx is periodic function with period pi/2Find which pair is true:

If a,b,c are real numbers such that (3a+2b)/(c+d) +( 3)/( 2) = 0then the equation ax^(3) + bx^(2) + cx + d = 0has

If a sin^(3)x+b cos^(3)x= sin x cos x and a sin x= b cos x then a^(2)+b^(2)=

Which value of theta listed below leads to 2^(sin theta)gt1 and 3 cos (theta) lt 1?

A committee of 7 members has to be formed from 9 boys and 4 girls . In how many ways can this be done when the committee consists of exactly 3 girls.

Consider the expansion of ((4x)/5-5/(2x))^9 Find the 7th term in the expansion

cos h^(-1)(2) =

0leale3,0leble3 and the equation x^(2)+4+3cos(ax+b)=2x has atleast one solution then the value of a+b

cos ^(2) ((A)/(2))+ cos ^(2) ((B)/(2))+cos ^(2)(( C )/(2))=

Identify the quantifier and write the negation of the statement ''There exists a number which is equal to its square.''

4sqrt(x)-2

Find the component statements of thecompound statement "All integers are positive or negative" ?

Simiplify :(1+ i)^(-1)

Consiser the system of equations x cos^(3)y+3xcosysin^(2)y=14to(1) and xsin^(3)y+3xcos^(2)y=13to(2) The value of x is/are

If f(x)=2x^(2)+3x-5,x=3,deltax=0.1 then deltaf=A (2) If f(x)=x^(2)+4x,x=2,deltax=0.1 then deltaf=B (3) If f(x)=x^(2)+3x,x=3,deltax=0.1 then deltaf=C The ascending order of A, B C is

Find the points at which the functions f(x) = x^(3) - 6x^(2) + 9x + 15

If the area enclosed by the curves f(x)=cos^(-1)(cosx) and g(x)=sin^(-1)(cosx)" in "x in [(9pi)/4, (15pi)/4] is api^(2)//b (where a and b are coprime), then the value of (a-b) is

For an G.P. a = 16 and fifth term is 81 then common ratio ........

Find the middle term (terms ) in the expansion of (3x - (x^(3))/6)^(9)

If the mean and variance of the observations x_(1), x_(2), x_(3),…,x_(n) are barx and sigma^(2) respectively and a be a nonzero real number, then show that the mean and variance of ax_(1),ax_(2),ax_(3),…,ax_(n)" are "abarx and a^(2) sigma^(2) respectively.

Using the following data. Find out the trend using Quarterly moving average and plot them on graph

For 0 lt theta lt 2 pi sin^(-1)(sin theta)gtcos^(-1)(sin theta) is true when

Find the equation of line which makes a triangle witharea (50)/( sqrt(3) ) unit with axis and perpendicular from origin makes an angle (pi)/(6) with positive side of X-axis.

An A.P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the series.

Evaluate the following limits in Exercises lim_(xto0)(cosx)/(pi-x)

If f(x)=log_(e)(1-x)andg(x)=[x] then find: (i) (f+g)(x) (ii) (fg)(x) (iii) ((f)/(g))(x) (iv) ((g)/(f))(x). Also find (f+g)(-1),(fg)(0),((f)/(g))(-1),((g)/(f))((1)/(2)).

The sum of first two terms of a G.P. is (9)/(2). Its 6^(th) term is 8 times its 3^(rd) term. Find the sequences.

If A and G are the arithmetic and geometric means respectively of two numbers then prove that the numbers are (A+sqrt(A^(2)-G^(2)))and(A-sqrt(A^(2)-G^(2)))

Expand (1-x^2)^4

B,C are two points on 3x+4y-15=0 such that OB+OC=10 then maximum possible area of triangle OBC is

If p^(th), q^(th), r^(th) terms of a geometric progression are the positive numbers a,b,c respectively, then the angle between the vectors (log a^(2)) I + (log b^(2)) j + (log c^(2))k and (q - r) I + (r - p) j + (p - q) k is

Let f(x) = (sqrt(x - 2 sqrt(x - 1)))/(sqrt(x - 1) - 1)x, then

If the parabola y^(2) = 4ax passes through the point (3,2), then the length of its latus rectum is ……

(sin 3 thea - sin theta sin ^(2) (2 theta ))/( sin theta + sin 2 theta cos theta ) = cos x implies x =

Find the equation of each of the following parabolas : (i) Directrix x = 0, focus at (6, 0) (ii) Vertex at (0,4), focus at (0,2) (iii) Focus at (-1, -2), directrix x - 2y + 3 = 0.

Mean and standard deviation of 100 items are 50 and 4, respectively . Find the sum of all the item and the sum of the squares of the items.

Find the values of thetabetween 0^(@) and 360^(@) which satisfy the equation 3 cos 2 theta - sin theta = 2.