28278 + Interview Questions in Maths in Class 12

1.	Each side of a square has length 4 units and its center is at (3,4). If one of the diagonals is parallel to the line y=x, then anser the following questions. ,brgt The radius of the circle inscribed in the triangle formed by any two vertices of the square and the center is
Answer» `2(sqrt(2)-1)` `2(sqrt(2)+1)` `sqrt(2)(sqrt(2)-1)` none of these Solution :It is given that one of the diagonals of the square is parallel to the line `y=x` Also, the length of the DIAGONAL of the square is `4 sqrt(2)` Hence, the equation of one of the diagonals is `(x-3)/(1//sqrt(2))=(y-4)/(1//sqrt(2))=r = +-2 sqrt(2)` Hence, `x-3= y-4= +- 2` or `x=5,1` and `y=6,2` Hence, two of the VERTICES are `(1,2)` and `(5,6)` The other diagonal is parallel to the line `y= -x` , so that its equation is `(x-3)/(-1//sqrt(2))=(y-4)/(1 //sqrt(2))=r = +- 2 sqrt(2)` Hence, the two vertices on this diagonal are (1,6) and (5,2) `AB =4, AC= 4 sqrt(2)` `:. AE =2 sqrt(2)` In first figure, `EF+FA = AE` or `r +sqrt(2) r = 2 sqrt(2)` or `r= (2sqrt(2))/( sqrt(2)+1)=2sqrt(2)(sqrt(2)-1)` In second figure, `EG +FG =EF` or `sqrt(2)r +r=2` or `r= (2)/(sqrt(2)+1)=2(sqrt(2)-1)`

Discussion

2.	Cinnamic acid (Ph-CH=CH-CO OH) is nitrated preferablyat __________ and it is __________ then nitration of benzene.
Answer» m-position,faster o & p-positions, faster m-position, slower o & p-position, slower Solution :groups with dual EFFECT are `o//p` DIRECTING DEACTIVATING

Discussion

3.	Draw the rough sketch of y= log x. Using integration find the area bounded by the curve,x-axis and x=2
Answer» SOLUTION :N/A

Discussion

4.	Let ' be the binary operation on N given by a'b= L.c.m. of a and b. Is ** commutative ?
Answer» SOLUTION :`aB`=L.c.m. of a and b = L.c.m of b and a =`ba` for all `a,b in N` `THEREFORE **` is COMMUTATIVE

Discussion

5.	If x is positive, the first negative term in the expansion of (1 + x)^(27/5 is
Answer» `5^(TH)` TERM `8^(th)` term `6^(th)` term `7^(th)` term Answer :D

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6.	f(x) = (1//2) (x^(2) - 1)
Answer» G(X) = LOG x L = 1 none of these Answer :A

Discussion

7.	Show that the function f(x) = log cos xis : (i) Strictly decreasing in ]0,pi/2[ (ii) Strictly increasing in ]pi/2, pi [
Answer»

Discussion

8.	Area of the triangleformed by the lines x+y =2 and angel bisectors of the pairof st linesx^(2)+2y=1 is
Answer» 2sq units 4 SQ units 6 sq units 8 sq units Answer :A

Discussion

9.	if statements p and r are false and q is true, then trueth value of ~pimplies(q ^^ r)vv ris
Answer» <P>T F Either T or F Neither T nor F Solution :Truth value of `~p IMPLIES(Q ^^ r) VV r` is false.

Discussion

10.	If a is a non - zerointerger andb is a postivenumbersuch ab^(2) = log_(10)b,the mediamof the set {0,1,a,b,(1)/(b)} is
Answer» 1 all values of DATA are equal to standard deviation B `(1)/(b)` ANSWER :C

Discussion

11.	From 1^(st) 101 natural numbers, 4 numbers are selected at random. Find the probability that the selected numbers are in A.P. with greatest possible common difference.
Answer» ANSWER :`(2)/(.^(101)C_(4))`

Discussion

12.	Let vec(u),vec(v),vec(w) be such that vec(u)+vec(v)+vec(w)=0. If \|vec(u)\|=3,\|vec(v)\|=4,\|vec(w)\|=5, then vec(u).vec(v)+vec(v).vec(w)+vec(w).vec(u)=
Answer» 47 25 -25 -47 Answer :C

Discussion

13.	If f (x) is a function such that f''(x)+f(x)=0 and g(x)=[f(x)]^(2)+[f'(x)]^(2)and g(3)=3 then g(8)=
Answer» 0 3 5 8 Answer :D

Discussion

14.	If x and y are positive intergers such that the greastest common factor of x^(2) y^(2) and xy^(3) is 45, then which of the following could y equal?
Answer» 45 15 9 3

Discussion

15.	int_0^4(dx)/(sqrt(x^2+9)
Answer» SOLUTION :`int_0^4(DX)/(SQRT(x^2+9)) =` `int_0^4(dx)/(sqrt(x^2+3^2))` `=[In(x+sqrt(x^2+3^2)]_0^4` `=In9-In3=In3`

Discussion

16.	The set of all values of 'a' for which the expression (ax^(2)+2x-3)/(2x-3x^(2)+a) assumes all real values for real vales of x, is
Answer» [2, 3] `R-(2, 3)` `PHI` `[1, 5]` ANSWER :C

Discussion

17.	Find all the values of following . (1 + i)^(2//3)
Answer» ANSWER :`2^(1/3)CIS(4k+1)pi/6,k=0,1,2`

Discussion

18.	If veca, vecb, vecc are three non-zero vector such that each one of then is perpendicular to the sum of the other two vectors, then the value of \|veca+vecb+vecc\|^(2) is :
Answer» `(5)/(SQRT(2))` `(2)/(sqrt(5))` `5sqrt(2)` `sqrt(5)` ANSWER :C

Discussion

19.	Let f: R rarrR be defined by f(x)=2x+6 which is bijective mapping then f^(-1)(x) is given by
Answer» `(X)/(2)-3` `2x+6` `x-3` `6x+2` ANSWER :A

Discussion

20.	Using differentials, find the approximate value of each of the up to 3 places of decimal. (3.968)^((3)/(2))
Answer» ANSWER :7.904

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21.	Evaluate P(A cup B), if 2P(A) = P(B) =(5)/(13) and P(A\|B)=(2)/(5).
Answer» ANSWER :`(11)/(26)`

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22.	Using differentials, find the approximate value of each of the up to 3 places of decimal. (15)^((1)/(4))
Answer» ANSWER :`=1.96875`

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23.	{:("Column A","q is an integer greater than 1. Let stand for the smallest positive integer factor of q that is greater than 1.", "Column B"),(,,):}
Answer» If COLUMN A is larger If column B is larger If the COLUMNS are equal If there is not enough INFORMATION to decide Answer :C

Discussion

24.	The sides of a triangle are the straight lines x +y = 1 , 7y = x and y + x = 0. Then which of the following is an interior point of the triangle?
Answer» circumcentre centroid incentre orthocentre Answer :A

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25.	A merchant plans to sell two types of personal computers - a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.
Answer» ANSWER :The maximum VALUE of Z = 4500x + 5000y is 11,50,000 at B(200, 50).

Discussion

26.	Evaluate (ii) int_(0)^(pi/2)(x)/(sin x + cos x) dx
Answer» Answer :`(PI)/(2sqrt(2))log (SQRT(2)+1)`

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27.	Find the differential equation that will represented family of all circles havingcentres on the x axisand the radiusis unity
Answer» ANSWER :1

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28.	Evaluate the integrals in exercise. overset(2)underset(1)((1)/(x)-(1)/(2x^(2)))e^(2x)+dx
Answer» ANSWER :`E^(2)((e^(2)-2))/(4)`

Discussion

29.	underset(x to 0)(lim) (5^(x) + 4^(x) - 2^(x) -1)/(5x) is equal to
Answer» `log10//5` `1` `0` logx. Answer :A

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30.	If A and B are two events, then P(A uu B)=P(A nn B) if and only if
Answer» P(A)+P(B)=1 P(A)=P(B) P(A)+P(B) `gt` 1 P(A)+P(B) `LT` 1 Answer :B

Discussion

31.	The remainder obtained when the polynominal 1+x+x^(3)+x^(9)+x^(27)+x^(81)+x^(243) is divided by x-1 is
Answer» 3 5 7 11 Answer :C

Discussion

32.	For each of the differential equations given in find a particular solution satisfying the given condition : (1+x^(2))(dy)/(dx)+2xy=(1)/(1+x^(2)), y=0 when x=1
Answer» Answer :`y (1 + X^(2)) = TAN^(-1)x - (pi)/(4)`

Discussion

33.	If the set A contains 5 elements and the set B contains 6 elements , then the number of one -one and onto mapping from A to B is ....
Answer» 720 120 0 None of these Solution :N/A

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34.	The maximum and minimum distance of a satellite moving around the earth is an elliptic orbit having the earth at a focus are 82000 km and 24000 km. Find the distance of earth from other focus.
Answer» ANSWER :58000 KM

Discussion

35.	Select the Correct Option If A is an invertible matrix of order 2, then det(A^-1) is equal to
Answer» 0 detA 1 1/detA Answer :D

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36.	If A=[{:("cos"(2pi)/(3),-"sin"(2pi)/(3)),("sin"(2pi)/(3),"cos"(2pi)/(3)):}] then , A^(3)= …….
Answer» `[{:(0,1),(1,0):}]` `[{:(1,0),(0,1):}]` `[{:(1,1),(0,0):}]` `[{:(0,0),(1,1):}]` ANSWER :B

Discussion

37.	Find the minimum integral value of k for which the equation e^(x)=kx^(2) has exactly three real distinct solutions.
Answer» Solution :Given equation is `e^(x)=kx^(2)`. `RARR k=e^(x)/x^(2)` Now let `f(x)=e^(x)/x^(2)` `THEREFORE f^(')(x) = ((x-2)e^(x))/(x^(3))` `f^(')(x)=0 therefore x=2`, which is the point of MINIMA. Also `f(2) =e^(2)/4` `underset(x to infty)"lim"e^(x)/x^(2)=infty` (using L' Hospital rule twice) `underset(x to infty) e^(x)/x^(2)=0` `underset(x to 0)"lim"e^(x)/x^(2)=infty` Further `f(x) gt 0, AA x in R`. From this INFORMATION, the graph of `f(x)` is as shown in the following figure. Thus, three real distinct solutions for `k lt e^(2)/4, k in I`. So, `k_("min") =2`.

Discussion

38.	If f(x) = \|x\|^(3), show that f''(x) exists for all real x and find it
Answer» ANSWER :`-6X, X LT 0`

Discussion

39.	Express the following differential equations in the form f(x)dx+g(y)dy = 0 (i) (dy)/(dx) = (2y)/(x) (ii) x+y(dy)/(dx) = 0 (iii) (dy)/(dx) = e^(x-y) + x^(2).e^(-y) (iv) (dy)/(dx) + x^(2) = x^(2)e^(3y)
Answer» ANSWER :(i) `(dx)/(x) - (dy)/(2) = 0` (ii) x dx + y dy = 0 (III) `(x^(2) + E^(x))dx - e^(y) dy = 0` (iv) `x^(2)dx - ((1)/(e^(3Y) - 1))dy = 0`

Discussion

40.	AFruitsbasketcontains4 organes, 5applesand6 mangoes. Thenumberofwaysapersom makeselectionoffruitsfromamongthefruits in tebasketis
Answer» 210 209 36 18 Answer :B

Discussion

41.	Consider the function f(x) satisfying the identityf(x) +f((x-1)/(x))=1+x AA x in R -{0,1}, and g(x)=2f(x)-x+1. The domain of y=sqrt(g(x)) is
Answer» `(-oo,(1-sqrt(5))/(2)] CUP [1,(1+sqrt(5))/(2)]` `(-oo,(1-sqrt(5))/(2)] cup (0,1)cup [(1+sqrt(5))/(2),oo)` `[(-1-sqrt(5))/(2),0] cup [(-1+sqrt(5))/(2),1)` None of these Solution :`f(X)+f((x-1)/(x))=1+x "(1)" ` In (1), replace x by `(x-1)/(x)`. Then `f((x-1)/(x))+f(((x-1)/(x)-1)/((x-1)/(x)))=1+(x-1)/(x)` `ORF((x-1)/(x))+f((1)/(1-x))=1+(x-1)/(x) "(2)" ` Now, from `(1) -(2)`, we have `f(x)-f((1)/(1-x))=x-(x-1)/(x) "(3)" ` In (3), replace x by `(1)/(x-1)`. Then `f((1)/(1-x))-f((x-1)/(x))=(1)/(1-x)-((1)/(1-x)-1)/((1)/(1-x))` ` orf((1)/(1-x))-f((x-1)/(x))=(1)/(1-x)-x "(4)" ` Now, from `(1)+(3)+(4)`, we have `2f(x)=1+x+x-(x-1)/(x)+(1)/(1-x)-x` ` orf(x)=(x^(3)-x^(2)-1)/(2x(x-1))` `g(x)=(x^(3)-x^(2)-1)/(x(x-1))-x+1=(x^(2)-x-1)/(x(x-1))` Now, for `y=sqrt(g(x)),` we must have `(x^(2)-x-1)/(x(x-1)) GE 0` `or ((x-(1-sqrt(5))/(2))(x-(1+sqrt(5))/(2)))/(x(x-1)) ge 0` `orx in (-oo,(1-sqrt(5))/(2)] cup (0,1)cup [(1+sqrt(5))/(2),oo)`

Discussion

42.	If z = (x + iy) and if the point P in the Argand plane represents z , then discribe geometrically the locus of P satisfying the equations \|2z - 3\| = 7
Answer» ANSWER :A::B::C::D

Discussion

43.	Evaluate: int(sin(a-x))/(sinx)dx
Answer» ANSWER :`sinalogabs(SINX)-COSA*x+c`

Discussion

44.	Evalute the following integrals int (2x + 3)/(x^(3) + x^(2) - 2x)dx
Answer» ANSWER :`-(3)/(2)log\|x\|-(1)/(6)log\|x+2\|+(5)/(3)log\|x-1\|+c`

Discussion

45.	Statement I . The equation 2x^2-3xy-2y^2+5x-5y+3=0 represents a pair of perpendicular straight lines. Statement II A pair of lines given by ax^2+2hxy+by^2+2gx+2fy+c=0 are perpendicular if a+b=0
Answer» ANSWER :d

Discussion

46.	Find the slope of the tangent to the curve y= (x-1)/(x-2), x ne 2at x = 10.
Answer» ANSWER :`(-1)/(64)`

Discussion

47.	int x sin x dx
Answer» ANSWER :`sinx+c`

Discussion

48.	Evaluate int(sqrt(x^(2)+1){log_(e)(x^(2)+1)-2logx}dx)/(x^(4)).
Answer» ANSWER :`= (X^(2) + 1) sqrt((x^2 +1) )/( 9x^3) [2-3 "In" (1+ (1)/(x^2) ) ]+C`.

Discussion

49.	Find the valve of 'a' for which the following equations have equal roots. i) x^(2)+(a+3)x+a+6=0 ii) 2(a+1)x^(2)+2(a+3)x+a+5=0
Answer» ANSWER :i) a= -5, 3; II) `a=(-5+- SQRT(41))/(2)`

Discussion

50.	Three events A, B and C have probabilities (2)/(5), (1)/(3)and (1)/(2) respectively. Given that P(A cap C)=(1)/(5), P(B cap C)=(1)/(4) find the values of P(C//B) and P(A' cap C').
Answer» ANSWER :`(3)/(10)`

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Each side of a square has length 4 units and its center is at (3,4). If one of the diagonals is parallel to the line y=x, then anser the following questions. ,brgt The radius of the circle inscribed in the triangle formed by any two vertices of the square and the center is

Cinnamic acid (Ph-CH=CH-CO OH) is nitrated preferablyat __________ and it is __________ then nitration of benzene.

Draw the rough sketch of y= log x. Using integration find the area bounded by the curve,x-axis and x=2

Let **' be the binary operation on N given by a**'b= L.c.m. of a and b. Is ** commutative ?

If x is positive, the first negative term in the expansion of (1 + x)^(27/5 is

f(x) = (1//2) (x^(2) - 1)

Show that the function f(x) = log cos xis : (i) Strictly decreasing in ]0,pi/2[ (ii) Strictly increasing in ]pi/2, pi [

Area of the triangleformed by the lines x+y =2 and angel bisectors of the pairof st linesx^(2)+2y=1 is

if statements p and r are false and q is true, then trueth value of ~pimplies(q ^^ r)vv ris

If a is a non - zerointerger andb is a postivenumbersuch ab^(2) = log_(10)b,the mediamof the set {0,1,a,b,(1)/(b)} is

From 1^(st) 101 natural numbers, 4 numbers are selected at random. Find the probability that the selected numbers are in A.P. with greatest possible common difference.

Let vec(u),vec(v),vec(w) be such that vec(u)+vec(v)+vec(w)=0. If |vec(u)|=3,|vec(v)|=4,|vec(w)|=5, then vec(u).vec(v)+vec(v).vec(w)+vec(w).vec(u)=

If f (x) is a function such that f''(x)+f(x)=0 and g(x)=[f(x)]^(2)+[f'(x)]^(2)and g(3)=3 then g(8)=

If x and y are positive intergers such that the greastest common factor of x^(2) y^(2) and xy^(3) is 45, then which of the following could y equal?

int_0^4(dx)/(sqrt(x^2+9)

The set of all values of 'a' for which the expression (ax^(2)+2x-3)/(2x-3x^(2)+a) assumes all real values for real vales of x, is

Find all the values of following . (1 + i)^(2//3)

If veca, vecb, vecc are three non-zero vector such that each one of then is perpendicular to the sum of the other two vectors, then the value of |veca+vecb+vecc|^(2) is :

Let f: R rarrR be defined by f(x)=2x+6 which is bijective mapping then f^(-1)(x) is given by

Using differentials, find the approximate value of each of the up to 3 places of decimal. (3.968)^((3)/(2))

Evaluate P(A cup B), if 2P(A) = P(B) =(5)/(13) and P(A|B)=(2)/(5).

Using differentials, find the approximate value of each of the up to 3 places of decimal. (15)^((1)/(4))

{:("Column A","q is an integer greater than 1. Let stand for the smallest positive integer factor of q that is greater than 1.", "Column B"),(,,):}

The sides of a triangle are the straight lines x +y = 1 , 7y = x and y + x = 0. Then which of the following is an interior point of the triangle?

Evaluate (ii) int_(0)^(pi/2)(x)/(sin x + cos x) dx

Find the differential equation that will represented family of all circles havingcentres on the x axisand the radiusis unity

Evaluate the integrals in exercise. overset(2)underset(1)((1)/(x)-(1)/(2x^(2)))e^(2x)+dx

underset(x to 0)(lim) (5^(x) + 4^(x) - 2^(x) -1)/(5x) is equal to

If A and B are two events, then P(A uu B)=P(A nn B) if and only if

The remainder obtained when the polynominal 1+x+x^(3)+x^(9)+x^(27)+x^(81)+x^(243) is divided by x-1 is

For each of the differential equations given in find a particular solution satisfying the given condition : (1+x^(2))(dy)/(dx)+2xy=(1)/(1+x^(2)), y=0 when x=1

If the set A contains 5 elements and the set B contains 6 elements , then the number of one -one and onto mapping from A to B is ....

The maximum and minimum distance of a satellite moving around the earth is an elliptic orbit having the earth at a focus are 82000 km and 24000 km. Find the distance of earth from other focus.

Select the Correct Option If A is an invertible matrix of order 2, then det(A^-1) is equal to

If A=[{:("cos"(2pi)/(3),-"sin"(2pi)/(3)),("sin"(2pi)/(3),"cos"(2pi)/(3)):}] then , A^(3)= …….

Find the minimum integral value of k for which the equation e^(x)=kx^(2) has exactly three real distinct solutions.

If f(x) = |x|^(3), show that f''(x) exists for all real x and find it

Express the following differential equations in the form f(x)dx+g(y)dy = 0 (i) (dy)/(dx) = (2y)/(x) (ii) x+y(dy)/(dx) = 0 (iii) (dy)/(dx) = e^(x-y) + x^(2).e^(-y) (iv) (dy)/(dx) + x^(2) = x^(2)e^(3y)

AFruitsbasketcontains4 organes, 5applesand6 mangoes. Thenumberofwaysapersom makeselectionoffruitsfromamongthefruits in tebasketis

Consider the function f(x) satisfying the identityf(x) +f((x-1)/(x))=1+x AA x in R -{0,1}, and g(x)=2f(x)-x+1. The domain of y=sqrt(g(x)) is

If z = (x + iy) and if the point P in the Argand plane represents z , then discribe geometrically the locus of P satisfying the equations |2z - 3| = 7

Evaluate: int(sin(a-x))/(sinx)dx

Evalute the following integrals int (2x + 3)/(x^(3) + x^(2) - 2x)dx

Statement I . The equation 2x^2-3xy-2y^2+5x-5y+3=0 represents a pair of perpendicular straight lines. Statement II A pair of lines given by ax^2+2hxy+by^2+2gx+2fy+c=0 are perpendicular if a+b=0

Find the slope of the tangent to the curve y= (x-1)/(x-2), x ne 2at x = 10.

int x sin x dx

Evaluate int(sqrt(x^(2)+1){log_(e)(x^(2)+1)-2logx}dx)/(x^(4)).

Find the valve of 'a' for which the following equations have equal roots. i) x^(2)+(a+3)x+a+6=0 ii) 2(a+1)x^(2)+2(a+3)x+a+5=0

Three events A, B and C have probabilities (2)/(5), (1)/(3)and (1)/(2) respectively. Given that P(A cap C)=(1)/(5), P(B cap C)=(1)/(4) find the values of P(C//B) and P(A' cap C').

Cinnamic acid (Ph-CH=CH-CO OH) is nitrated preferablyat and it is then nitration of benzene.

Let ' be the binary operation on N given by a'b= L.c.m. of a and b. Is ** commutative ?