28278 + Interview Questions in Maths in Class 12

1.	Consider the function f(x)=cos^(-1)([2^(x)])+sin^(-1)([2^(x)]-1), then (where [.] represents the greatest integer part function)
Answer» the DOMAIN of `f(X)` is `x in(-oo, 0]` the range of `f(x)` is SINGLETON `f(x)` is an even function `f(x) is an odd function Answer :B

Discussion

2.	Which of the following is/are incorrect?
Answer» let `F:RTOR` such that `f(x)=2X+[(x(x^(2)-1))/(4(x^(4)-x^(2)+1))+1/8]+[x]+sinxcosx` then (where [.] denotes the greatest integer function) `f` is one-one onto let `f:RtoR` such that `f(x)=(x^(3)+2x^(2))/(sinx+2)` then `f` is one-one onto let `f:Rto[1,oo)` such that `f(x)=2^(\|x-1\|^(sqrt(2))` then `f(x)` is one-one ito let `f:R-{0}toR` such that `f(x)=\|x\|ln\|x\|` then`f(x)` is one-one onto Solution :(A) `(x-1/x)/(4(x^(2)+1/(x^(2))-1))=(x-1/x)/(4((x-1//x)^(2)+1))` sop, `-1/8 le (x-1/x)/(4((x-1/x)^(2)+1))le 1/8` `:. [(x(x^(2)-1))/(4(x^(4)-x^(2)+1))+1/8]=0` So, `f(x)=2x+[x]+1/2sin2x` injective but not onto (B) `f^(')(x)=-x^(3) cos x +x^(3)(3sinx+2cosx)+4x(sinx+2)` MANY one onto

Discussion

3.	Find (dy)/(dx) of the functions x^(y) + y^(x)=1
Answer» ANSWER :`(-(y^(X) LOG y + x^(y-1) .y))/(x^(y).log x+ y^(x-1)x)`

Discussion

4.	Find vec abs x, if for a unit vector veca , (vecx-veca).(vecx+veca)=8
Answer» 3 9 2 sqrt 3 Answer :A

Discussion

5.	If A=[{:(1,0,-1),(2,1,3),(0,1,1):}], then verify that A^(2)+A=A(A+I) where I is 3xx3 unit matrix.
Answer» ANSWER :`=A(A+I)`

Discussion

6.	Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that the die shows a number greater than 4' given that there is at least one tail.
Answer» ANSWER :`(2)/(9)`

Discussion

7.	Solve for x the equation sin px+sin(p+1)x+sin(P+2)x+sin(P+3)x sin(p+4)x=0
Answer» Answer :`x=2mpipm COS^(-1)(-1pmsqrt(5))/(4),(m INI)`

Discussion

8.	The sum of all possible numbers greater than 10,000 formed by using {1, 3, 5, 7, 9} is
Answer» 6666600 932460 6660000 1968 Answer :A

Discussion

9.	The mod -amplitude form of sintheta-i cos theta is
Answer» `CISTHETA` `CIS(pi/2+theta/2)` `cis(pi/2+theta)` NONE ANSWER :D

Discussion

10.	x(dy)/(dx) + y - x + xy cot x = 0 (x ne 0)
Answer» ANSWER :`y = (1)/(X) - COT x + (C)/(x sin x)`

Discussion

11.	int_(0)^(1)(xe^(x))/((x+1)^(2))dx=
Answer» ANSWER :`(e-2)/(2)`

Discussion

12.	Integrate the following functions : xcos3x
Answer» ANSWER :`(xsin3x)/(3)+(COS3X)/(9)+C`

Discussion

13.	If cos x +cosy+cosalpha=0, sinx+siny+sinalpha=0 then cot((x+y)/2)=
Answer» `sinalpha` `COS alpha` `TAN alpha` `COT alpha` ANSWER :D

Discussion

14.	Find the roots of the following quadratic equations i) 6sqrt5 x^(2) – 9x -3sqrt5 = 0 ii) x^(2) - x - 12 = 0 iii) 2x^(2) - 6x + 7 = 0 iv) 4x^(2) - 4x+17 = 3x^(2) -10x-17 v) x^(2) + 6x + 34 = 0 vi) 3x^(2) + 2x - 5 = 0
Answer» Answer :i) 5,4 ii) -3, 4 III) `-3 +- 5i` iv) `(2)/(sqrt3), 4sqrt3` v) `-3+-5i` VI) `(-5)/(3)`, 1

Discussion

15.	Find an anti derivative (or integral) of the following functions by the method of inspection. sin 2x
Answer» ANSWER :`-(1)/(2) COS 2x+c`

Discussion

16.	Find the area of the triangle whose vertices are (-2, -3), (3, 2) and (-1, -8) using determinant method.
Answer» ANSWER :15 SQ. UNITS

Discussion

17.	Find the coefficient of x^9 and x^10 in(2x^2 - 1/x)^20
Answer» ANSWER :`0, ""^20C_10. 2^10`

Discussion

18.	The order of the differential equation y=C_(1)e^(c_(2)+x)+C_(3)e^(c_(4)+x is
Answer» 1)3 2)1 3)4 4)2 Answer :B

Discussion

19.	S is defined in Z by (x,y ) in S hArr \|x-y\| le 1. S is ........
Answer» REFLEXIVE and TRANSITIVE but not SYMMETRIC. Reflexive and symmetric but not transitive. symmetric and transitive but not reflexive. an EQUIVALENCE relation Solution :N/A

Discussion

20.	If the value of (a + b + c)= 0 then determinant \|{:(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b):}\| is equal to ,
Answer» 0 1 2 3 Answer :A

Discussion

21.	int _(0) ^(tn2) x e ^(-x) dx
Answer» ANSWER :`1/2 LN ((E)/(2))`

Discussion

22.	A line meetscoordinate axes in A and B. A circle is circumscribed about the triangle OAB . If m and n are distances of tangent to circle at origin from the point A and B respectively then diameter of the circle is
Answer» `(m+n)/(2)` `(3(m+n))/(4)` `m+n` `2(m+n)` ANSWER :C

Discussion

23.	The sum of all the rational roots of the equation 6x^(6) - 25x^(5) + 31x^(4) - 31x^(2) + 25x - 6 =0 is
Answer» 3 3.5 `(25)/(6)` 2.5 Answer :D

Discussion

24.	If N denotes the set of all positive integers and if : NtoN is defined by f(n)= the sum of positive divisors of n, then f(2^(k).3), where k is a positive integer, is
Answer» `2^(k+1)-1` `2(2^(k+1)-1)` `3(2^(k+1)-1)` `4(2^(k+1)-1)` ANSWER :D

Discussion

25.	Evalute the following integrals int (tan^(2) x + 1) dx
Answer» ANSWER :TAN X + C

Discussion

26.	Statement-1: In a triangleABC, maximum value of sinA sinB sinC is (3sqrt(3))/(8) Statement-2: In a triangleABC , sin^(2)A + sin^(2)B + sin^(2)C= 2-cos^(2) C cos(A-B). Statement-3: In a triangleABC, sin^(2)A+sin^(2)B + sin^(2)C le 9//4
Answer» TTT TFT TFF FTF ANSWER :A

Discussion

27.	Let R = {(a,a^3) \| ais a prime number less than 10}. Find R^(-1).
Answer» Solution :R = {(a,a^3)\| a is a prime NUMBER LESS than 10. `R^(-1)` = {(8,2),(27,3),(125,5),(343,7)

Discussion

28.	Simply (sin75^(@)-sin15^(@))/(cos75^(@)+cos15^(@)) Prove that (a)(sin3A+sinA)SinA+(Cos3A-cosA)cosA=0 (b)cos20^(@)cos50^(@)cos60^(@)=(1)/(16) (c) (sin8thetacostheta-sin6 thetacos 3 theta)/(cos 2 theta cos theta-sin 3 thetasin4 theta)=tan 2 theta
Answer» ANSWER :`(1)/SQRT(3)`

Discussion

29.	The points A, B and C with position vectors veca, vecb and vec c respectively lie on a circle centered at origin O. Let G and E be the centroid of DeltaABC and DeltaACD respectively where D is mid point of AB. Q.If[vec(AB) vec(AC) vec(AB) xxvec(AC)]=lambda[vec(AE) vec(AG) vec(AE) xx vec(AG)], then the value of lambda is :
Answer» -18 18 -324 324 Answer :D

Discussion

30.	Let A and B are two matricesof same order i.e. where A=[(1,-3,2),(2,K,5),(4,2,1)], B=[(2,1,3),(4,2,4),(3,3,5)] {:("Column - I", "Column - II" ),("(A) If A, B and C be" 2xx2 "matrices",(P)AB=BA),( "with entries the set of real",),("numbers. Define * as follows:",), (A*B=(1)/(2)(AB+BA)", then",(Q) A(B+C)=AB+AC),("(B)If A, B and C "2xx2" matrices",),( "with entries from the set of",),( "real numbers. Define * as follows:",(R)A*A=A^(2)),(AB=(1)/(2)(AB'+A'B)", then",),("(C) If A, B and C" 2xx2 " matrices",(S)A*I=A),("with entries from the set of real",) ,("numbers. Define as follows:",),(AB=(1)/(2)(AB-BA)", then",(T)AI=O):}
Answer» <P> Answer :(A) - P, Q, R, S; (B) - Q; (C) - Q, T

Discussion

31.	Equation of a line which is tangent to both the curves y=x^(2)+1 and y=-x^(2) is
Answer» `y=sqrt(2)x-(1)/(2)` `y=sqrt(2)x+(1)/(2)` `y=-sqrt(2)x+(1)/(2)` `y=-sqrt(2)x-(1)/(2)` Answer :B::C

Discussion

32.	{:(" "Lt),(n rarr oo):} [(1)/(1-n^(2))+(2)/(1-n^(2))+...+(n)/(1-n^(2))]=
Answer» `1/2` `- 1/2` 2 -2 Answer :B

Discussion

33.	If a+pi/2lt2tan^(-1)x+3cot^(-1)xltb then 'a'and 'b' are respectively.
Answer» 1)`0 and 2pi` 2)`0 and pi` 3)`(-pi)/2 and 2pi` 4)`pi/2 and 2pi` ANSWER :D

Discussion

34.	Integration of some particular functions : int(x+2)/(x^(2)+4x-7)dx=....+c
Answer» `(1)/(2)(X^(2)+4x-7)^(2)` `(1)/(2)LOG sqrt(\|x^(2)+4x-7)\|` `log sqrt(\|x^(2)+4x-7\|)` `log\|x^(2)+4x-7\|` ANSWER :C

Discussion

35.	Find the equation of the circle which passes through the origin and intersects each of the following circles orthogonally. x^2+y^2-4x-6y-3=0 x^2+y^2-8y+12=0
Answer» ANSWER :`2(x^2+y^2)-7x+2y=0`

Discussion

36.	If y=x^x-2^(sin x)" find " (dy)/(dx).
Answer» ANSWER :`x^(x)(1+log x)-2^(SIN x)cos x log 2`

Discussion

37.	Find k if the following pairs of circles are orthogonal x^2+y^2-6x-8y+12=0 x^2+y^2-4x+6y+k=0
Answer» ANSWER :`x^2+y^2+6x-3y=0`

Discussion

38.	Verify that the function y = c_(1) e^(ax) cos bx + c_(2)e^(ax) sin bx, where c_(1),c_(2) are arbitrary constants is a solution of the differential equation (d^(2)y)/(dx^(2)) - 2a(dy)/(dx) + (a^(2) + b^(2))y = 0
Answer» ANSWER :` = e^(ax)[0 XX SIN BX + 0 cos bx] = e^(ax) xx 0 = 0 = R.H.S`

Discussion

39.	The real part of (1-costheta + I sin theta)^(-1) is
Answer» `(1)/(2)` `(1)/(1+costheta)` `tan""(theta)/(2)` `COT""(theta)/(2)` Answer :A

Discussion

40.	(x)/((x+1)(x+2))
Answer» Solution :` int (x)/((x+1)(x+2)) DX =I (" say ")` ` " LET " (x)/((x+1)(x+2)) =(A)/((x+1))+(B)/((x+2))` `RARR (x)/((x+1)(x+2))=(A(x+2)+B(x+1))/((x+1)(x+2))` ` rArrx=A (x+2)+B(x+1)` `rArr x=-1 " then "-1= A(-1+2)+0` `A=-1` `rArr x=-2 " then "-2=0 +B (-1)` `B=2` `:. I = int (x)/((x+1)(x+2))dx ` `= int ((-1))/(x+1)dx+ int (2)/(x+2)dx` `=- LOG (x+1) +2 log (x+2)+c` ` =- log (x+1)+log (x+2)^(2)+c` ltb rgt `=log \|(x+2)^(2)/(x+1)\|+c`

Discussion

41.	If the set A contains 5 elements and the set B also contains 5 elements, then find the number of bijective functions from A to B.
Answer» ANSWER :51=120

Discussion

42.	The points A, B and C with position vectors veca, vecb and vec c respectively lie on a circle centered at origin O. Let G and E be the centroid of DeltaABC and DeltaACD respectively where D is mid point of AB. Q.If OE and CD are mutually perpendicular, then which of the following will be necessarily true ?
Answer» `\|vecb-veca\|=\|VEC C-veca\|` `\|vecb-veca\|=\|vecb-vec c\|` `\|vec c-veca\|=\|vec c-vecb\|` `\|vecb-veca\|=\|vec c-veca\|=\|vecb-vec c\|` ANSWER :A

Discussion

43.	The points A, B and C with position vectors veca, vecb and vec c respectively lie on a circle centered at origin O. Let G and E be the centroid of DeltaABC and DeltaACD respectively where D is mid point of AB. Q.If GE and CD are mutually perpendicular, then orthocenter of DeltaABC must lie on :
Answer» MEDIAN through A median through C angle BISECTOR through A angle bisector through B Answer :B

Discussion

44.	If sec theta is the eccentricity of a hyper bola then the eccentricity of the conjugate hyperbola is
Answer» `TAN THETA` `cot theta` `COS theta` `cosec theta` Answer :D

Discussion

45.	int(x^((1)/(2)))/(sqrt(1-x^(3)))dx=(2)/(3)go f(x)+c then....
Answer» `f(X)= SQRT(x)G(x)=sin^(-1)x` `f(x)=x^((3)/(2)),g(x)= sin x` `f(x)=x^((2)/(3)),g(x)= sin^(-1) x` `f(x)=x^((3)/(2)),g(x)= sin^(-1) x` Answer :D

Discussion

46.	How many R_(2)SiCl_(2) unit are required to the formation of single chain silicone having 10-Si-O-Si linkages.
Answer» SOLUTION :`9toR_(1)SiCl_(2)("CHAIN initiating units")` `2toR_(3)SICL("chain terminating units")`

Discussion

47.	Let ABC be a triangle inscribed in a circle and letl_(a)=(m_(a))/(M_(a)), l_(b)=(m_(b))/(M_(b)), l_(c )=(m_(c ))/(M_(c )) where m_(a), m_(b), m_(c ) are the lengthsof the angle bisectors of angles A, B and C respectively , internal to the triangle and M_(a), M_(b) and M_(c ) are the lengths of these internalangle bisectors extended until they meet the circumcircle. Q. The minimum value of the expression (l_(a))/(sin^(2)A)+(l_(b))/(sin^(2)B)+(l_(c ))/(sin^(2)C) is :
Answer» 2 3 4 none of these ANSWER :B

Discussion

48.	Let ABC be a triangle inscribed in a circle and letl_(a)=(m_(a))/(M_(a)), l_(b)=(m_(b))/(M_(b)), l_(c )=(m_(c ))/(M_(c )) where m_(a), m_(b), m_(c ) are the lengthsof the angle bisectors of angles A, B and C respectively , internal to the triangle and M_(a), M_(b) and M_(c ) are the lengths of these internalangle bisectors extended until they meet the circumcircle. Q. l_(a) equals :
Answer» `(sinA)/(SIN(B+(A)/(2)))` `(sinBsinC)/(sin^(2)((B+C)/(2)))` `(sinBsinC)/(sin^(2)(B+(A)/(2)))` `(sinB+sinC)/(sin^(2)(B+(A)/(2)))` Answer :C

Discussion

49.	Let ABC be a triangle inscribed in a circle and letl_(a)=(m_(a))/(M_(a)), l_(b)=(m_(b))/(M_(b)), l_(c )=(m_(c ))/(M_(c )) where m_(a), m_(b), m_(c ) are the lengthsof the angle bisectors of angles A, B and C respectively , internal to the triangle and M_(a), M_(b) and M_(c ) are the lengths of these internalangle bisectors extended until they meet the circumcircle. Q.The maximum value of the product(l_(a)l_(b)l_(c))xxcos^(2)((B-C)/(2)) xx cos^(2)(C-A)/(2)) xx cos^(2)((A-B)/(2)) is equalto :
Answer» `(1)/(8)` `(1)/(64)` `(27)/(64)` `(27)/(32)` ANSWER :C

Discussion

50.	Consider a triangle ABC in xy-plane with D, E and F as the middle points of the sides, BC, CA and AB respectively. If the co-ordinates of the point D,E,F are ((3)/(2),(3)/(2)),((7)/(2),0)and (0,(-1)/(2)) then: STATEMENT-1: Circumcentre of triangle ABC lies inside the triangle. STATEMENT-2: Orthocentre, centroid, circumcentre and incentre of triangle DEF are collinear but of triangle ABC are not-collinear. STATEMENT-3: Distance between centrioid and ortocente of DeltaABC is (5sqrt2)/(3).
Answer» F F T T T F F T F T F T ANSWER :1

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Consider the function f(x)=cos^(-1)([2^(x)])+sin^(-1)([2^(x)]-1), then (where [.] represents the greatest integer part function)

Which of the following is/are incorrect?

Find (dy)/(dx) of the functions x^(y) + y^(x)=1

Find vec abs x, if for a unit vector veca , (vecx-veca).(vecx+veca)=8

If A=[{:(1,0,-1),(2,1,3),(0,1,1):}], then verify that A^(2)+A=A(A+I) where I is 3xx3 unit matrix.

Consider the experiment of tossing a coin. If the coin shows head, toss it﻿ again but if it shows tail, then throw a die. Find the﻿ conditional probability of the event that the die shows a number greater than 4' given that there is at least﻿﻿ one tail.﻿

Solve for x the equation sin px+sin(p+1)x+sin(P+2)x+sin(P+3)x sin(p+4)x=0

The sum of all possible numbers greater than 10,000 formed by using {1, 3, 5, 7, 9} is

The mod -amplitude form of sintheta-i cos theta is

x(dy)/(dx) + y - x + xy cot x = 0 (x ne 0)

int_(0)^(1)(xe^(x))/((x+1)^(2))dx=

Integrate the following functions : xcos3x

If cos x +cosy+cosalpha=0, sinx+siny+sinalpha=0 then cot((x+y)/2)=

Find the roots of the following quadratic equations i) 6sqrt5 x^(2) – 9x -3sqrt5 = 0 ii) x^(2) - x - 12 = 0 iii) 2x^(2) - 6x + 7 = 0 iv) 4x^(2) - 4x+17 = 3x^(2) -10x-17 v) x^(2) + 6x + 34 = 0 vi) 3x^(2) + 2x - 5 = 0

Find an anti derivative (or integral) of the following functions by the method of inspection. sin 2x

Find the area of the triangle whose vertices are (-2, -3), (3, 2) and (-1, -8) using determinant method.

Find the coefficient of x^9 and x^10 in(2x^2 - 1/x)^20

The order of the differential equation y=C_(1)e^(c_(2)+x)+C_(3)e^(c_(4)+x is

S is defined in Z by (x,y ) in S hArr |x-y| le 1. S is ........

If the value of (a + b + c)= 0 then determinant |{:(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b):}| is equal to ,

int _(0) ^(tn2) x e ^(-x) dx

A line meetscoordinate axes in A and B. A circle is circumscribed about the triangle OAB . If m and n are distances of tangent to circle at origin from the point A and B respectively then diameter of the circle is

The sum of all the rational roots of the equation 6x^(6) - 25x^(5) + 31x^(4) - 31x^(2) + 25x - 6 =0 is

If N denotes the set of all positive integers and if : NtoN is defined by f(n)= the sum of positive divisors of n, then f(2^(k).3), where k is a positive integer, is

Evalute the following integrals int (tan^(2) x + 1) dx

Statement-1: In a triangleABC, maximum value of sinA sinB sinC is (3sqrt(3))/(8) Statement-2: In a triangleABC , sin^(2)A + sin^(2)B + sin^(2)C= 2-cos^(2) C cos(A-B). Statement-3: In a triangleABC, sin^(2)A+sin^(2)B + sin^(2)C le 9//4

Let R = {(a,a^3) | ais a prime number less than 10}. Find R^(-1).

Simply (sin75^(@)-sin15^(@))/(cos75^(@)+cos15^(@)) Prove that (a)(sin3A+sinA)SinA+(Cos3A-cosA)cosA=0 (b)cos20^(@)cos50^(@)cos60^(@)=(1)/(16) (c) (sin8thetacostheta-sin6 thetacos 3 theta)/(cos 2 theta cos theta-sin 3 thetasin4 theta)=tan 2 theta

Equation of a line which is tangent to both the curves y=x^(2)+1 and y=-x^(2) is

{:(" "Lt),(n rarr oo):} [(1)/(1-n^(2))+(2)/(1-n^(2))+...+(n)/(1-n^(2))]=

If a+pi/2lt2tan^(-1)x+3cot^(-1)xltb then 'a'and 'b' are respectively.

Integration of some particular functions : int(x+2)/(x^(2)+4x-7)dx=....+c

Find the equation of the circle which passes through the origin and intersects each of the following circles orthogonally. x^2+y^2-4x-6y-3=0 x^2+y^2-8y+12=0

If y=x^x-2^(sin x)" find " (dy)/(dx).

Find k if the following pairs of circles are orthogonal x^2+y^2-6x-8y+12=0 x^2+y^2-4x+6y+k=0

Verify that the function y = c_(1) e^(ax) cos bx + c_(2)e^(ax) sin bx, where c_(1),c_(2) are arbitrary constants is a solution of the differential equation (d^(2)y)/(dx^(2)) - 2a(dy)/(dx) + (a^(2) + b^(2))y = 0

The real part of (1-costheta + I sin theta)^(-1) is

(x)/((x+1)(x+2))

If the set A contains 5 elements and the set B also contains 5 elements, then find the number of bijective functions from A to B.

The points A, B and C with position vectors veca, vecb and vec c respectively lie on a circle centered at origin O. Let G and E be the centroid of DeltaABC and DeltaACD respectively where D is mid point of AB. Q.If GE and CD are mutually perpendicular, then orthocenter of DeltaABC must lie on :

If sec theta is the eccentricity of a hyper bola then the eccentricity of the conjugate hyperbola is

int(x^((1)/(2)))/(sqrt(1-x^(3)))dx=(2)/(3)go f(x)+c then....

How many R_(2)SiCl_(2) unit are required to the formation of single chain silicone having 10-Si-O-Si linkages.

Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that the die shows a number greater than 4' given that there is at least one tail.