28278 + Interview Questions in Maths in Class 12

1.	Statement I f(x)=2x^3-9x^2+12x-3 is increasing outside the interval (1, 2). Statement II f'(x)
Answer» Both I and II are TRUE and II is the CORRECT explanation for I Both I and II are true and II is not the correct explanation for I I is true but II is false I is false but II is true Answer :B

Discussion

2.	Prove that: int_(-a)^(a) log ((2-x)/(2+x)) dx=0
Answer» SOLUTION :`"Let " f(x) =log ((2-x)/(2+x))` `RARR ""f(-x) =log ((2+x)/(2-x))` `=-log ((2-x)/(2+x))=-f(x)` `rArr ` f(x) is an ODD FUNCTION . `rArr ""int_(-a)^(a) f(x) dx=0` `rArr int_(-1)^(1) log ((2-x)/(2+x)) dx=0` HENCE Proved

Discussion

3.	Letf : R to Rbe a continuous onto function satisfying f(x)+f(-x) = 0, forall x in R . Iff(-3) = 2 and f(5) = 4in [-5, 5], then what is the minimum number of roots of the equation f(x) = 0?
Answer» Solution :` f(x)+ f(-x) = 0` f(x) is an odd function. Since the points (-3, 2) and (5, 4) LIE on the curve, (3, -2) and (-5, -4) will also lie on the curve. For minimum number of roots, graph of the continuous function f(x) is as FOLLOWS. From the above graph off(x), it is clear that equation f (x) = 0 has at least three real roots.

Discussion

4.	If m_(1), m_(2), m_(3) and m_(4) are respectively the magnitudes of the vectors a_(1)=2hati-hatj+hatk, a_(2)=3hati-4hatj-4hatk, a_(3)=hati+hatj-hatk and a_(4)=-hati+3hatj+hatk, then the correct order of m_(1),m_(2),m_(3) and m_(4) is
Answer» `m_(3)ltm_(1)ltm_(4)ltm_(2)` `m_(3)ltm_(1)ltm_(2)ltm_(4)` `m_(3)ltm_(4)ltm_(1)ltm_(2)` `m_(3)ltm_(4)ltm_(2)ltm_(1)` Solution :Given, `m_(1)=\|a_(1)\|=sqrt(2^(2)+(-1)^(2)+(1)^(2))=sqrt(6)` `m_(2)=\|a_(2)\|=sqrt(3^(2)+(-4)^(2)+(-4)^(2))=sqrt(41)` `m_(3)=\|a_(3)\|=sqrt(1^(2)+1^(2)+(-1)^(2))=sqrt(3)` and `m_(4)=\|a_(4)\|=sqrt((-1)^(2)+(3)^(2)+(1)^(2))=sqrt(11)` `therefore m_(3)ltm_(1)ltm_(4)ltm_(2)`

Discussion

5.	An equilateral triangle is Inscribed in parabola y^(2) = 4ax whose one vertex is at origin then length of side of triangle
Answer» `6sqrt(3A)` `8sqrt(3a)` `4sqrt(a)` `2sqrt(3)a` ANSWER :B

Discussion

6.	If A = [[1,2],[4,2]] then show that \|2A\|= 4\|A\|
Answer» SOLUTION : `\|A\| = \|[1,2],[4,2]\| =2-8 = -6 2A = 2\|[1,2],[4,2]\| = \|[2,4],[8,4]\|,\|2A\|= \|[2,4],[8,4]\| = 8-32 = -24 = 4xx-6 = 4\|A\|`

Discussion

7.	State which of the following is the value of 8th term of the G.P. {2, 6, 18, 54,…}
Answer» 45 `-45` `+- 15` `+- 45` ANSWER :A

Discussion

8.	If(x ^ 3 )/((2x-1)(x + 2 )(x - 3 )) = A +(B)/(2x - 1 )+(C )/(x + 2 )+(D)/(x - 3 ) ,then A isequalto
Answer» ` (1)/(2) ` `- (1)/(50)` ` - (8)/(25) ` `(27)/(25) ` Solution : ` ( X ^ 3 )/((2x- 1 )(x + 2 ) (x- 3 ))= A+ (B)/((2x- 1 ))+(C )/((x + 2)) +(D)/((x - 3 )) ` Thequotientobtained, when `x ^ 3 `isdivided by` (2x- 1 )(x + 2 )( x - 3 ) `is` (1)/(2) ` `therefore A = (1)/(2) `

Discussion

9.	Find the coefficient of x^(2) in the expansion of (7x^(3)-2/x^(2))^(9).
Answer» SOLUTION :N/A

Discussion

10.	Let PQ:2x+y+6=0 is a chord of the curve x^(2)-4y^(2)=4. Coordinates of the point R(alpha, beta) that satisfy alpha^(2)+beta^(2)-1 le 0, such that area of triangle PQR is minimum, are given by :
Answer» `((-2)/(SQRT(5)),(1)/(sqrt(5)))` `((-2)/(sqrt(5)),(-1)/(sqrt(5)))` `((2)/(sqrt(5)),(1)/(sqrt(5)))` `((2)/(sqrt(5)),(-1)/(sqrt(5)))` Answer :B

Discussion

11.	Using properties evaluate the following definite integrals, evaluate the following: int_0^4 \|x-1\| dx
Answer» SOLUTION :`int_0^a f(X) g(x) dx = int_0^a f(a-x) g(a-x)dx` =`int_0^a f(a-x) [4- g(x)]dx` =`int_0^a 4 f(a-x) dx - int_0^a f(a-x) g(x) dx` =`4 int_f(a-x)dx - int_0^a f(x) g(x) dx` `gt 2 int_0^a f(x) g(x) dx = 4 int_0^a f(a-x)dx` = `4 int_0^a f(x) dx` THUS, `int_0^a f(x) g (x) dx = 2 int_0^a f(x) dx`

Discussion

12.	1 + ""^2C_1x + ""^3C_2 + ""^4C_3 x^3 + ….. tooo terms can be summed up if
Answer» `X LT 1` `x GT - 1` `-1 lt x lt 1` `-OO lt x lt oo` Answer :C

Discussion

13.	The point or contact of the Une 2x-y+2 = 0 with the parabola y^(2)=16x is
Answer» (2,4) (3,4) (1,4) (-2,1) ANSWER :C

Discussion

14.	Let a = I - 2j + 3k, b = 2i + 3j - k and c = lambda I + j + (2 lambda - 1) k. If c is parallel to the plane containing a,b then lambda =
Answer» 0 1 `-1` 2 Answer :A

Discussion

15.	If the four points with position vectors -2hat(i)+hat(j)+hat(k),hat(i)+hat(j)+hat(k),hat(j)-hat(k)andlamdahat(j)+hat(k) are coplanar, then lamda=
Answer» 1 2 -1 0 Answer :A

Discussion

16.	Let O be the circumcenter, H be the orthocenter, I be the incenter, and I_(1), I_(2), I_(3) be the excenters of acute-angled DeltaABC
Answer» SOLUTION :SEE the THEORY of DIFFERENT CENTRE

Discussion

17.	If the coefficients of 2nd, 3rd, 4th terms of (1+x)^n are in A.P. then n=
Answer» 4 5 7 6 Answer :C

Discussion

18.	The order and degree of the differential equation y = x(dy)/(dx) + (2)/((dy)/(dx)) is
Answer» 1, 2 1, 3 2, 1 1, 1 Answer :A

Discussion

19.	Ifthe eventsA and Bare independent if P (A ) = (2)/(3) and P (B ) = (2)/(7) thenP (A nn B)isequalto :
Answer» `(4)/(21)` `(5)/(21)` `(1)/(21)` `(3)/(21)` Answer :B

Discussion

20.	Obtain the following integrals : int sqrt(1+sinx) dx
Answer» ANSWER :`:. I=-2cos ((X)/(2))+2SIN((x)/(2))+C`

Discussion

21.	Evaluate sqrt(1 + cos 2x) dx
Answer» Answer :`SQRT(2) SIN X + C`

Discussion

22.	Integrate the following functions sqrt(x^2+4x-5)
Answer» Solution :`int SQRT(x^2+4x-5) DX- int sqrt((x+2)^2-4-5) dx` `int sqrt((x+2)^2-3^2) dx` =`(x+2)/2 sqrt((x+2)^2-3^2)-3^2/2 log\|x+2+sqrt(x^2+4x-5)\|+C` `(x+2)/2 sqrt(x^2+4x-5) - 9/2 log\|x+2+ sqrt(x^2+4x-5)\| +c`

Discussion

23.	Find the values of x and y so that the vectors2hati+3hatjandxhati+yhatj are equal.
Answer» ANSWER :X = 2 and y = 3

Discussion

24.	Find the point on the curve y^2 - x^2 + 2x - 1 =0 where the tangent is parallel to the x - axis.
Answer» Solution :Given the CURVE is `y^2 - X^2 + 2xx - 1 = 0…(1)` `rArr 2Y dy/dx - 2xx + 2 = 0` `rArr dy/dx = x- 1/y` if the tangent is parallel to `x-axis then dy/dx = 0` `x - 1/y = 0 rArr x = 1` Putting x = 1 in (1) we GET `y^2 - 1 + 2 - 1 = 0 rArr y = 0` The point is (1,0).

Discussion

25.	Find the values of each of the expression following : sin^(-1)("sin"(2pi)/3)
Answer» ANSWER :`(PI)/3`

Discussion

26.	If figure, identify the following vectors : (i) Collinear (ii) Equal (iii) Coinitial (iv) Collinear but not equal
Answer» ANSWER :(i) `VEC(a)` and `vec(d); vec(b),vec(X),vec(z);vec( c )` and `vec(y)` (ii) `vec(b)` and `vec(x);vec(a)` and `vec(d);vec( c )` and `vec(y)` (iii) `vec(a),vec(y),vec(z)` (iv) `vec(b),vec(z);vec(x),vec(z)`

Discussion

27.	If a,b,c,d are in G.P., prove that (a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2.
Answer» SOLUTION :Let a,B,C,d are in G.P. Let the COMMON ratio=r `impliesb=ar,c=ar^2,d=ar^3` LHS=`(a^2+b^2+c^2)(b^2+c^2+d^2)=(a^2+a^2r^2+a^2r^4)(a^2r^2+a^2r^4+a^2r^6)=a^4r^2(1+r^2+r^4)^2=(a^2r+a^2r^3+a^2r^5)^2(a.ar+ar.ar^2+ar^2.ar^3)^2=(ab+bc+cd)^2=R.H.S.`(PROVED)

Discussion

28.	Find int((3sinphi-2)cosphi)/(5-cos^(2)phi-4sinphi)dphi
Answer» Answer :`3 log (2-sin PHI)+(4)/(2-sin phi)+C`

Discussion

29.	A man running round a race course notes that the sum of the distances of two flag posts from him is always 10 meters and the distance between the flag posts is 8 meters. Then the area of the path he encloses (in square meters) is
Answer» ANSWER :`15 PI`

Discussion

30.	Evaluate the following integrals int secx(secx+tanx) dx
Answer» SOLUTION :`INT SECX(secx+TANX)DX = int(sec^2x+secxtanx) dx = intsec^2x dx+ int secx tanx dx = tanx+secx+c`

Discussion

31.	Find the range of x for which the following expansions are valid . (3+ (5x^(2))/(3))^(-3//7)
Answer» SOLUTION :N/A

Discussion

32.	There is an error of 002 cm is is made in measuring the radius 10 cm of a circle. Then I: Appoximate error in area is 05 sq. cm II: Approximate percentage error in area is 0*4
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II true Answer :B

Discussion

33.	ABCDEF is a regular hexagon . What is the slope of the line containing bar(FE) ?
Answer» `-1/2` `-SQRT3` `-SQRT2` `sqrt3` ANSWER :B

Discussion

34.	If tan^(4)theta +tan^(2) theta = 2, then the value of cos^(4)theta +cos^(2)theta is-
Answer» `(3)/(4)` `(3)/(2)` `(1)/(2)` 1 Answer :C

Discussion

35.	int(sinax-sinbx)/(cosax-cosbx)dx
Answer» SOLUTION :`int(sinax-sinbx)/(cosax-cosbx)dx` =`(2SIN((a-b)/2) X.cos((a+b)/2) x)/(-2sin((a+b)/2) x.sin((a-b)/2) x)dx` =`-intcot((a+b)/2)xdx` =`-2/(a+b) In abs(sin((a+b)/2)x)+C` =`1/k Inabs(sinkx)+C`

Discussion

36.	Find the area bounded by the curve x^(2) = 4y and the line x = 4y - 2.
Answer» ANSWER :`9/8`

Discussion

37.	A set contains 3n members. Let P_( n) be the probability tha S is partitioned into 3 disjointsubsetswith n members in each subset such that the three members of S are in different subsets. Then lim_(n to oo) P_(n)=
Answer» `2//7` `1//7` `1//9` `2//9` ANSWER :D

Discussion

38.	Prove that the circles x^(2) + y^(2) -8x -6y +21=0and x^(2) + y^(2) -2y -15=0have exactlytwo common tangents Also find the intersection of those tangents.
Answer» ANSWER :`(8,5)`

Discussion

39.	Ravi obtained 70 and 75 marks in first two unit tests. Find the number if minimum marks he should get in the third test to have an average of atleast 60 marks.
Answer» ANSWER :The MINIMUM MARKS =35.

Discussion

40.	Let f(x) = (x^(2) - 2x + 3)/( x^(2) - 2x - 8) , x in R - {-2, 4} The range of f is
Answer» `((-2)/(9),1]` `R - ((-2)/(9),1)` `(-INFTY, (-2)/(9)]` `R - ((-2)/(9),1]` Answer :D

Discussion

41.	A ramdom variable X can assume any positive integral value of n with a probability proportional to n with a probability proportional to (1)/(3^(n)). Find the expectation of X?
Answer» ANSWER :`3//2`

Discussion

42.	If A is of order mxxn and B is of order nxxp, then AB is a matrix of order mxxp.
Answer» ANSWER :1

Discussion

43.	If the angle between the line (x-1)/(1)=(y-2)/(k)=(z+3)/(4) and the plane x-3y+2z+5=0 is sin^(-1)((3)/(7sqrt(6))), the value of k is
Answer» 2 -2 1 -1 Answer :A

Discussion

44.	Evaluation of definite integrals by subsitiution and properties of its : int_(1)^(a)[x]f'(x)dx=......... where agt1 and [.] is a greatest integer function.
Answer» `[a]F(a)-{f(1)+f(2)+......+f[a]}` `[a]f([a])-{f(1)+f(2)+......+f(a)}` `AF(\|a\|)-{f(1)+f(2)+......+f(a)}` `af(a)-{f(1)+f(2)+......+f([a])}` ANSWER :A

Discussion

45.	P is a point on the given curve such that the normal at P to the curve meets the axist is 'x' at G. If the distance of P from the origin is same as its distance from G, then the curve is
Answer» a CIRCLE a RECTANGLE hyperbola Both (a) and (b) none of these Answer :C

Discussion

46.	If A=[(2,-1),(4,2)],B=[(4,3),(-2,1)],C=[(-2,-3),(-1,-2)], find the value of A+B+C.
Answer» ANSWER :`[(4,-1),(1,1)]`

Discussion

47.	I : If e and e' are the eccentricity of the hyperbolax^(2) //a^(2) -y^(2)//b^(2) =1 and its conjugate hyperbola the value of1//e^(2) +1//e'^(2)is 1 II : If e and e_1are the eccentricity of the hyperbolaxy =c^(2) , x^(2) -y^(2) =c^(2) " then" e^(2)+ c_1^(2)is equal to 4
Answer» only I is TRUE only II is true both I and II are ture neither I nor II true ANSWER :C

Discussion

48.	For k gt0,sum_(x=0)^inftyk^x/(x!)lim_(n toinfty)(n!)/((n-x)!)(1-k/n)^(n-x)(1/n)^x=
Answer» 0 k x 1 Answer :D

Discussion

49.	y= tan^(-1) ((ax-b)/(bx + a))" then " (dy)/(dx)\|_(x= -1)= ………
Answer» `(1)/(2)` a ab `(B)/(a)` ANSWER :A

Discussion

50.	Lt_(n rarr oo)sum_(r=0)^(n-1)((r)/(n^(2) + r^(2)))
Answer» ANSWER :`(1)/(2) LOG 2`

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Statement I f(x)=2x^3-9x^2+12x-3 is increasing outside the interval (1, 2). Statement II f'(x)

Prove that: int_(-a)^(a) log ((2-x)/(2+x)) dx=0

Letf : R to Rbe a continuous onto function satisfying f(x)+f(-x) = 0, forall x in R . Iff(-3) = 2 and f(5) = 4in [-5, 5], then what is the minimum number of roots of the equation f(x) = 0?

If m_(1), m_(2), m_(3) and m_(4) are respectively the magnitudes of the vectors a_(1)=2hati-hatj+hatk, a_(2)=3hati-4hatj-4hatk, a_(3)=hati+hatj-hatk and a_(4)=-hati+3hatj+hatk, then the correct order of m_(1),m_(2),m_(3) and m_(4) is

An equilateral triangle is Inscribed in parabola y^(2) = 4ax whose one vertex is at origin then length of side of triangle

If A = [[1,2],[4,2]] then show that |2A|= 4|A|

State which of the following is the value of 8th term of the G.P. {2, 6, 18, 54,…}

If(x ^ 3 )/((2x-1)(x + 2 )(x - 3 )) = A +(B)/(2x - 1 )+(C )/(x + 2 )+(D)/(x - 3 ) ,then A isequalto

Find the coefficient of x^(2) in the expansion of (7x^(3)-2/x^(2))^(9).

Let PQ:2x+y+6=0 is a chord of the curve x^(2)-4y^(2)=4. Coordinates of the point R(alpha, beta) that satisfy alpha^(2)+beta^(2)-1 le 0, such that area of triangle PQR is minimum, are given by :

Using properties evaluate the following definite integrals, evaluate the following: int_0^4 |x-1| dx

1 + ""^2C_1x + ""^3C_2 + ""^4C_3 x^3 + ….. tooo terms can be summed up if

The point or contact of the Une 2x-y+2 = 0 with the parabola y^(2)=16x is

Let a = I - 2j + 3k, b = 2i + 3j - k and c = lambda I + j + (2 lambda - 1) k. If c is parallel to the plane containing a,b then lambda =

If the four points with position vectors -2hat(i)+hat(j)+hat(k),hat(i)+hat(j)+hat(k),hat(j)-hat(k)andlamdahat(j)+hat(k) are coplanar, then lamda=

Let O be the circumcenter, H be the orthocenter, I be the incenter, and I_(1), I_(2), I_(3) be the excenters of acute-angled DeltaABC

If the coefficients of 2nd, 3rd, 4th terms of (1+x)^n are in A.P. then n=

The order and degree of the differential equation y = x(dy)/(dx) + (2)/((dy)/(dx)) is

Ifthe eventsA and Bare independent if P (A ) = (2)/(3) and P (B ) = (2)/(7) thenP (A nn B)isequalto :

Obtain the following integrals : int sqrt(1+sinx) dx

Evaluate sqrt(1 + cos 2x) dx

Integrate the following functions sqrt(x^2+4x-5)

Find the values of x and y so that the vectors2hati+3hatjandxhati+yhatj are equal.

Find the point on the curve y^2 - x^2 + 2x - 1 =0 where the tangent is parallel to the x - axis.

Find the values of each of the expression following : sin^(-1)("sin"(2pi)/3)

If figure, identify the following vectors : (i) Collinear (ii) Equal (iii) Coinitial (iv) Collinear but not equal

If a,b,c,d are in G.P., prove that (a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2.

Find int((3sinphi-2)cosphi)/(5-cos^(2)phi-4sinphi)dphi

A man running round a race course notes that the sum of the distances of two flag posts from him is always 10 meters and the distance between the flag posts is 8 meters. Then the area of the path he encloses (in square meters) is

Evaluate the following integrals int secx(secx+tanx) dx

Find the range of x for which the following expansions are valid . (3+ (5x^(2))/(3))^(-3//7)

There is an error of 0*02 cm is is made in measuring the radius 10 cm of a circle. Then I: Appoximate error in area is 0*5 sq. cm II: Approximate percentage error in area is 0*4

ABCDEF is a regular hexagon . What is the slope of the line containing bar(FE) ?

If tan^(4)theta +tan^(2) theta = 2, then the value of cos^(4)theta +cos^(2)theta is-

int(sinax-sinbx)/(cosax-cosbx)dx

Find the area bounded by the curve x^(2) = 4y and the line x = 4y - 2.

A set contains 3n members. Let P_( n) be the probability tha S is partitioned into 3 disjointsubsetswith n members in each subset such that the three members of S are in different subsets. Then lim_(n to oo) P_(n)=

Prove that the circles x^(2) + y^(2) -8x -6y +21=0and x^(2) + y^(2) -2y -15=0have exactlytwo common tangents Also find the intersection of those tangents.

Ravi obtained 70 and 75 marks in first two unit tests. Find the number if minimum marks he should get in the third test to have an average of atleast 60 marks.

Let f(x) = (x^(2) - 2x + 3)/( x^(2) - 2x - 8) , x in R - {-2, 4} The range of f is

A ramdom variable X can assume any positive integral value of n with a probability proportional to n with a probability proportional to (1)/(3^(n)). Find the expectation of X?

If A is of order mxxn and B is of order nxxp, then AB is a matrix of order mxxp.

If the angle between the line (x-1)/(1)=(y-2)/(k)=(z+3)/(4) and the plane x-3y+2z+5=0 is sin^(-1)((3)/(7sqrt(6))), the value of k is

Evaluation of definite integrals by subsitiution and properties of its : int_(1)^(a)[x]f'(x)dx=......... where agt1 and [.] is a greatest integer function.

P is a point on the given curve such that the normal at P to the curve meets the axist is 'x' at G. If the distance of P from the origin is same as its distance from G, then the curve is

If A=[(2,-1),(4,2)],B=[(4,3),(-2,1)],C=[(-2,-3),(-1,-2)], find the value of A+B+C.

I : If e and e' are the eccentricity of the hyperbolax^(2) //a^(2) -y^(2)//b^(2) =1 and its conjugate hyperbola the value of1//e^(2) +1//e'^(2)is 1 II : If e and e_1are the eccentricity of the hyperbolaxy =c^(2) , x^(2) -y^(2) =c^(2) " then" e^(2)+ c_1^(2)is equal to 4

For k gt0,sum_(x=0)^inftyk^x/(x!)lim_(n toinfty)(n!)/((n-x)!)(1-k/n)^(n-x)(1/n)^x=

y= tan^(-1) ((ax-b)/(bx + a))" then " (dy)/(dx)|_(x= -1)= ………

Lt_(n rarr oo)sum_(r=0)^(n-1)((r)/(n^(2) + r^(2)))

There is an error of 002 cm is is made in measuring the radius 10 cm of a circle. Then I: Appoximate error in area is 05 sq. cm II: Approximate percentage error in area is 0*4