28278 + Interview Questions in Maths in Class 12

1.	If n ge 1 is a positive integer, then prove that 3^(n) ge 2^(n) + n . 6^((n - 1)/(2))
Answer» Solution :We KNOW that , `a^(n) - B^(n) = (a-b) (a^(n-1) + a^(n-2) b + a^(n-3) b^(2) + .. . + b^(n-1))` `THEREFORE 3^(n) - 2^(n)= 3^(n-1) + 3^(n-2) 2 + 3^(n-3) 2^(2) + … + 2^(n-1)` Using `A.M. ge G.M` , we GET `(3^(n-1)+ 3^(n-2) . 2 + … + 2^(n-1))/(n) ge [(3 * 3^(2) * ... * 3^(n-1)) (2* 2^(2) * ... * 2^(n-1))]^(1//n)` `3^((n-1)/(2)) * 2((n-1)/(2)) = 6^((n-1)/(2))` `implies 3^(n) ge 2^(n) + n* 6 ((n-1)/(2))`.

Discussion

2.	Prove that {1,7,5,7,9,…….} set are equivalent.
Answer» SOLUTION :Let h: `A RARR D` defined as `h(x) = x^2` CLEARLY h is bijective. `implies` There is a one-to-one CORRESPONDENCE between A to D.

Discussion

3.	Let x^2 + y^2 = r^2 and xy = 1 intersect at A & B in first quadrant, If AB = sqrt14 then find the value of r.
Answer» ANSWER :5.5

Discussion

4.	If[[1,1,1],[1,1+x,1],[1,1,1+y]]=0 what are x and y?
Answer» SOLUTION :`[[1,1,1],[1,1+X,1],[1,1,1+y]]=0` or,`[[1,0,0],[1,x,-x],[1,0,y]]=0` `(C_2=C_2-C_1,C_3=C_3-C_2)` or, `1[[x,-x],[0,-y]]=0` or, xy-0=0 `RARR` x=0,or y=0

Discussion

5.	Find the equation of the circle passing through the points of contact of the direct commontangent of x^2+y^2=16 and x^2+y^2-12x+32=0
Answer» ANSWER :B

Discussion

6.	Cards are drawn one by one from pack of 52 cards without replacement until 3 aces are obtained for the first time. Find the probability of drawing 3rd ace first time in the 10th draw.
Answer» Answer :`(12)/(43) xx (.^(48)C_(7))/(.^(52)C_(9))`

Discussion

7.	(a,0) and (b, 0) are centres of two circles belonging to a co-axial system of which y-axis is the radical axis. If radius of one of the circles 'r', then the radius of the other circle is
Answer» `(r^2+B^2+a^2)^(1//2)` `(r^2+b^2-a^2)^(1//2)` `(r^2+b^2-a^2)^(1//3)` `(r^2+b^2+a^2)^(1//3)` ANSWER :B

Discussion

8.	Which of the following complex numbers is equivalent to (8-4i)/(5+3i)? (Note: i=sqrt(-1))
Answer» `(14)/(17)+(22)/(17)i` `(14)/(17)-(22)/(17)i` `(13)/(17)-(24)/(17)i` `(13)/(8)+(14)/(17)i` ANSWER :B

Discussion

9.	If overline(a), overline(b), overline(c) are non-coplanar vectors and the vectors overline(p)=2overline(a)-5overline(b)+2overline(c), overline(q)= overline(a)+5overline(b)-6overline(c) andoverline(r)= 3overline(a)-4overline(c)are coplanar such that overline(p)=moverline(q)+noverline(r), then
Answer» `m=1, n=1` `m=1, n=-1` `m=-1, n=1` `m=-1, n=-1` ANSWER :C

Discussion

10.	Ifalpha, beta, gammaare therootsofx^3 +px^2 +qx +r=0then find sum alpha^2
Answer» ANSWER :`p^2 -2Q`

Discussion

11.	If the mean deviation from the median is 15 and median is 450, then find the coefficient of mean deviation.
Answer» ANSWER :`1//3`

Discussion

12.	IFsintheta= sin15^@+ sin45^@, where0^@lt thetalt 90 ^@, thenthetaisequalto
Answer» `45^@` `54^@` `60^@` `75^@` ANSWER :D

Discussion

13.	Find the probability distribution of number of doublets in three throws of a pair of dice.
Answer» ANSWER :`(##NCERT_TEL_MAT_XII_P2_C13_SLV_025_A01##)`

Discussion

14.	Each side of an equilateral triangle subtends an angle of 60^(@) at the top of a tower h m high located at the centre of the triangle. If a is the length of each side of the triangle, then
Answer» `3a^(2)=2h^(2)` `2A^(2)=2h^(2)` `a^(2)=3H^(2)` `3a^(2)=h^(2)` Answer :A::B::C

Discussion

15.	One of the foci of thehyperbola is originand the corresponding directrix is3x + 4y + 1=0.The eccentricity of the hyperbola issqrt 5.The equation of the hyperbola is
Answer» only I is TRUE only II is true both I and II are TURE NEITHER I nor II true ANSWER :C

Discussion

16.	Integrate the following functions : sec^(3)x
Answer» Answer :`(1)/(2)[secx.tanx+log\|tan""(pi)/(4)+(X)/(2)\|]+c`

Discussion

17.	(Diet problem): A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units oif calcium, 4 units iof iron, 6 units of cholesteroa and 6 units of vitamiin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholestero. How many packets of each food should be used to minimise the maount of vitamin A in the diet? What is the minimum amount of vitamin A?
Answer» ANSWER :150 UNITS.

Discussion

18.	Show that the vectors veca=overset^^i-2overset^^j+3overset^^k,vecb=-2overset^^i+3overset^^j-4overset^^kand vecc=overset^i-3overset^j+5overset^^k are coplanar.
Answer» SOLUTION :`VECA.(vecbxxvecc)=\|(1,-2,3),(-2,3,-4),(1,-3,5)\|=0` HENCE, in view of THEOREM `1,veca,vecb"and"vecc` are coplanar vectors.

Discussion

19.	A company manufactures two types of telephone sets A and B . The A type telephone requires 2 hour and B type telephone requires 2 hourand B type telephone requires 4 hours to make . The company has 800 work hours per day . 300 telephones can pack in a day . The selling prices of A and B type telephones are Rs.300 and 400 respectively . For maximum profits company produces x telephones of a type and y telephones of B types . Then exceptxge0 and yge0 , linear constraints and the probable region of the LPP is of the type .
Answer» `x+2yle400,x+yle300,` Max z = 300X +400y, BOUNDED `2x+yle400,x+yge300,` Max z = 400X +300y, unbounded `2x+yge400,x+yge300,` Max z = 300x +400y, PARALLELOGRAM `2x+yle400,x+yge300,` Max z = 300x +400y, square Answer :A

Discussion

20.	Decide whether each statement Must Be True, Could Be True, or Will Never Be True. p^(2) gt s^(4)
Answer» ANSWER :COULD Be TRUE

Discussion

21.	How many four digit numbers abcd exist such that a is odd, b is divisible by 3, c is even and d prime?
Answer» 380 360 400 520 Answer :C

Discussion

22.	Which of the following statement is correct ? Statement I if dy+2xydx=2e^(-x^(2))dx, Then ye^(x^(2))=2x+c Statement II If ye^(x^(2))-2x=c, Then dx=(2e^(-x^(2))-2xy)dy
Answer» Both I and II are TRUE NEITHER I nor II is true I is true, II is false I is false, II is true Answer :C

Discussion

23.	If f(x) = (2k + 1) x- 3 - ke^(-x) + 2e^(x)is monotonically increasing for all x in Rthen the least value of k is
Answer» 1 0 `-1/2` `-1` ANSWER :B

Discussion

24.	Solve system of linear equations, using matrix method in examples 7 to 14 2x-y=-2 3x+4y=3
Answer» ANSWER :`x=-5/(11),y=(12)/(11)`

Discussion

25.	Find the number of common tangents ofx^(2)+y^(2)-8x-6y+21=0, x^(2)+y^(2)-2y-16=0 also find point of intersection of tangents.
Answer» ANSWER :2,(8,5)

Discussion

26.	R is the set of real numbers. If a^2+ b^2 + c^2 ne ab +bc +ca and a+b+c ne 0then the set of points satisfying the equations ax+by+z=0,bx + by + az = 0, cx+ay+bz = 0is
Answer» `R XX R xx R` `[(0, 0, 0)]` `R xx R xx [0]` `R xx [0] xx R` ANSWER :B

Discussion

27.	{a. (b xx i)} + {a. (b xx j)} j + {a . (b xx k)} k =
Answer» `2 (a XX B)` `3 (a xx b)` `(a xx b)` none Answer :C

Discussion

28.	Statement 1 : If x , y , z are positiveand x+y + z=1 , then (1/x - 1) (1/y - 1) (1/z - 1) ge 8because Statement 2 : A.MgeG.Mfor positivereal numbers .
Answer» STATEMENT - 1 is TRUE , Statement- 2 is True , Statement- 2 is a correctexplanation for Statement- 1 Statement - 1 is True, Statement- 2 is True , Statement- 2 is Not a correctexplanation for statement -1 Statement - 1 is True , Statement - 2 is FALSE Statement -1 is False , Statement - 2 is True ANSWER :A

Discussion

29.	If the equation x^(2)+y^(2) -10x +21=0has real rootsx = alphaand y = beta, then
Answer» `3 LE x le 7` `3 le y le 7` `-2 le y le 2 ` `-2 le x le 2` ANSWER :a,c

Discussion

30.	Show that ,the area of the coodinates axes is(a^(2))/( 6) square units (ii)Using the method of integration find the area bounded by the curve\|x\| + \|y\| =1
Answer» ANSWER :2 SQUARE UNITS

Discussion

31.	If a, b c in R are such that 4a+2b+c gt0and ax^(2)+bx + c=0 has no real roots, then the value of (c+a)(c+b) is
Answer» GREATER than AB LESS than BC greater than CA less than ab + bc + ca Answer :A

Discussion

32.	Find the equation of a curve passing through the point (0, -2) given that at anypoint (x,y) on the curve, the product of the slope of itstangent and y coordinate of the point is equal to the x coordinate of the point.
Answer» ANSWER :`y^(2) - X^(2) = 4`

Discussion

33.	A determinant of second order is made with the elements 0 or 1. What is the probability that the determinant made is (i) non-negative (ii) non-zero
Answer» ANSWER :`(13)/(16), (3)/(8)`

Discussion

34.	int_(0)^(1//sqrt2)(Sin^(-1))/((1-x^(2))^(3//2))dx=
Answer» `pi/4 + LN 2` `pi/4 - ln 2` `pi/4 - 1/2 ln 2` `ln 2` ANSWER :C

Discussion

35.	Show that the lines(x+3)/(-3)=(y-1)/(1)=(z-5)/(5)and(x+1)/(-1)=(y-2)/(2)=(z-5)/(5)are coplanar .Al,so find the equation of the plane containing these two lines.
Answer» ANSWER :`x-2y +Z =0`

Discussion

36.	If \|z_1\|=\|z_2\|=\|z_3\|"…."=\|z_n\|=1 then \|z_(1)+z_(2)+"….."+z_(n)\|=
Answer» `\|z_(1)z_(2)z_(3)"………."z_(N)\|` `(1)/(\|z_(1)z_(2)"………"z_(n)\|)` `\|(1)/(z_(1))+(1)/(z_(2))+"………"+(1)/(z_(n))\|` `\|(1)/(z_(1)z_(2))+(1)/(z_(2)z_(3))+"………"(+1)/(z_(n-1)z_(n))\|` ANSWER :C

Discussion

37.	int(x-1)/(x+2)^(3/4)dx
Answer» Solution :`int(x-1)/(x+2)^(3/4)DX` [Put`x+2=t^4` Then `dx=4T^3dt`] =`(t^4-3)/t^3 .4t^3dt` =`4int(t^4-3)dt=4{t^5/5-3t}+C` =`(4t)/5(t^4-15)+C` =`4/5(x+2)^(1/4) (x-13)+C`

Discussion

38.	A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by cutting off squares from the corners and folding up the flaps. What should be the side of the square in order the volume of the box is maximum.
Answer» ANSWER :5CM

Discussion

39.	If the subnormal at every point of a curve is a constant k, then its equation is
Answer» `y^(2) = KX` `y^(2) = 2KX` `y^(2) = 2kx +C` none Answer :C

Discussion

40.	Let f (x) = {{:( -3 "," , - 3 le x lt 0 ), ( x ^(2) - 3 "," , 0 lt x le 3 ):} and g (x) = \|f (x) \|+f (\|x\|), then which of the following is true ?
Answer» at `x =0 , g (x)` is CONTINUOUS as WELL as DIFFERENTIABLE `at x = sqrt3, g (x)` is continuous but not differentiable `at x =2, g (x)` is neither continuous nor differentiable None of these ANSWER :A::B

Discussion

41.	A die is thrown 6 time if getting an odd numbers is a success. What is the probability of a. 5 successes b. at least 5 successes c. at most 5 successes
Answer» ANSWER :`63/64`

Discussion

42.	Find (d)/(dx) ["cosec"^(-1)x]_(x= -2)
Answer» ANSWER :`-(1)/(2 SQRT3)`

Discussion

43.	inte^(x/sqrt2)cos((x)/(sqrt2))dx=......+c
Answer» `E^(X/sqrt2)SIN((x)/(sqrt2)-(PI)/(4))` `e^(x/sqrt2)sin((x)/(sqrt2)+(pi)/(4))` `e^(x/(sqrt2))COS((x)/(sqrt2)+(pi)/(4))` `e^(x/sqrt2)/(sqrt2)cos((x)/(sqrt2)+(pi)/(4))` Answer :B

Discussion

44.	According to the table above, 100-Watt bulbs made up what fraction of the working lightbulbs?
Answer» `(1,230)/(3,614)` `(2,384)/(3,614)` `271/398` `(2,384)/(2,655)` ANSWER :B

Discussion

45.	I: In a Delta ABC, if a= R tan A then b^2+c^2=bc+a^2 II: In a Delta ABC if sin A+ sin B+ sin C is maximum then triangle is equilateral
Answer» only I is true only II is true both I, II are true neither I nor II is true Answer :C

Discussion

46.	A and B are independent events. P(A' cap B)= (2)/(15) and P(A cap B')=(1)/(6) then find P(B).
Answer» ANSWER :`(1)/(6)` OR `(4)/(5)`

Discussion

47.	The value of a for which the function f(x)={{:(tan^(-1)a -3x^2" , " 0ltxlt1),(-6x","xge1):} has a maximum at x=1 , is
Answer» 0 1 2 -1 Answer :D

Discussion

48.	Let there is an A.P. with common difference 4 such that the squares of the first term plus the sum of all other terms is at most 100. If the number of terms is maximum then the range of 'a' is
Answer» `[-4,-3]` `[(-7)/(2),-(5)/(2)]` `[-3,-2]` none of these Answer :A

Discussion

49.	If int (log(t+sqrt(1+t^(2))))/(sqrt(1+t^(2)))dt=(1)/(2)(g(t))^(2)+C, where C is a constant, then g(2) is equal to:
Answer» A) `(1)/(SQRT(5)) LOG(2+sqrt(5))` B) `(1)/(2)log(2+sqrt(5))` C) `2LOG(2+sqrt(5))` D) `log(2+sqrt(5))` ANSWER :4

Discussion

50.	Assertion (A): The locus of the point ((e^(2t)+e^(-2t))/(2), (e^(2t)-e^(-2t))/(2)) when 't' is a parameter represents a rectangular hyperbola. Reason (R ) : The eccentricity of a rectangular hyperbola is 2.
Answer» Both A and R are true and R is the correct EXPLANATION of A Both A and R are true and R is not the correct explanation of A A is true, R is FALSE A is false, R is true Answer :C

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

If n ge 1 is a positive integer, then prove that 3^(n) ge 2^(n) + n . 6^((n - 1)/(2))

Prove that {1,7,5,7,9,…….} set are equivalent.

Let x^2 + y^2 = r^2 and xy = 1 intersect at A & B in first quadrant, If AB = sqrt14 then find the value of r.

If[[1,1,1],[1,1+x,1],[1,1,1+y]]=0 what are x and y?

Find the equation of the circle passing through the points of contact of the direct commontangent of x^2+y^2=16 and x^2+y^2-12x+32=0

Cards are drawn one by one from pack of 52 cards without replacement until 3 aces are obtained for the first time. Find the probability of drawing 3rd ace first time in the 10th draw.

(a,0) and (b, 0) are centres of two circles belonging to a co-axial system of which y-axis is the radical axis. If radius of one of the circles 'r', then the radius of the other circle is

Which of the following complex numbers is equivalent to (8-4i)/(5+3i)? (Note: i=sqrt(-1))

If overline(a), overline(b), overline(c) are non-coplanar vectors and the vectors overline(p)=2overline(a)-5overline(b)+2overline(c), overline(q)= overline(a)+5overline(b)-6overline(c) andoverline(r)= 3overline(a)-4overline(c)are coplanar such that overline(p)=moverline(q)+noverline(r), then

Ifalpha, beta, gammaare therootsofx^3 +px^2 +qx +r=0then find sum alpha^2

If the mean deviation from the median is 15 and median is 450, then find the coefficient of mean deviation.

IFsintheta= sin15^@+ sin45^@, where0^@lt thetalt 90 ^@, thenthetaisequalto

Find the probability distribution of number of doublets in three throws of a pair of dice.﻿

Each side of an equilateral triangle subtends an angle of 60^(@) at the top of a tower h m high located at the centre of the triangle. If a is the length of each side of the triangle, then

One of the foci of thehyperbola is originand the corresponding directrix is3x + 4y + 1=0.The eccentricity of the hyperbola issqrt 5.The equation of the hyperbola is

Integrate the following functions : sec^(3)x

Show that the vectors veca=overset^^i-2overset^^j+3overset^^k,vecb=-2overset^^i+3overset^^j-4overset^^kand vecc=overset^i-3overset^j+5overset^^k are coplanar.

Decide whether each statement Must Be True, Could Be True, or Will Never Be True. p^(2) gt s^(4)

How many four digit numbers abcd exist such that a is odd, b is divisible by 3, c is even and d prime?

Which of the following statement is correct ? Statement I if dy+2xydx=2e^(-x^(2))dx, Then ye^(x^(2))=2x+c Statement II If ye^(x^(2))-2x=c, Then dx=(2e^(-x^(2))-2xy)dy

If f(x) = (2k + 1) x- 3 - ke^(-x) + 2e^(x)is monotonically increasing for all x in Rthen the least value of k is

Solve system of linear equations, using matrix method in examples 7 to 14 2x-y=-2 3x+4y=3

Find the number of common tangents ofx^(2)+y^(2)-8x-6y+21=0, x^(2)+y^(2)-2y-16=0 also find point of intersection of tangents.

R is the set of real numbers. If a^2+ b^2 + c^2 ne ab +bc +ca and a+b+c ne 0then the set of points satisfying the equations ax+by+z=0,bx + by + az = 0, cx+ay+bz = 0is

{a. (b xx i)} + {a. (b xx j)} j + {a . (b xx k)} k =

Statement 1 : If x , y , z are positiveand x+y + z=1 , then (1/x - 1) (1/y - 1) (1/z - 1) ge 8because Statement 2 : A.MgeG.Mfor positivereal numbers .

If the equation x^(2)+y^(2) -10x +21=0has real rootsx = alphaand y = beta, then

Show that ,the area of the coodinates axes is(a^(2))/( 6) square units (ii)Using the method of integration find the area bounded by the curve|x| + |y| =1

If a, b c in R are such that 4a+2b+c gt0and ax^(2)+bx + c=0 has no real roots, then the value of (c+a)(c+b) is

Find the equation of a curve passing through the point (0, -2) given that at anypoint (x,y) on the curve, the product of the slope of itstangent and y coordinate of the point is equal to the x coordinate of the point.

A determinant of second order is made with the elements 0 or 1. What is the probability that the determinant made is (i) non-negative (ii) non-zero

int_(0)^(1//sqrt2)(Sin^(-1))/((1-x^(2))^(3//2))dx=

Show that the lines(x+3)/(-3)=(y-1)/(1)=(z-5)/(5)and(x+1)/(-1)=(y-2)/(2)=(z-5)/(5)are coplanar .Al,so find the equation of the plane containing these two lines.

If |z_1|=|z_2|=|z_3|"…."=|z_n|=1 then |z_(1)+z_(2)+"….."+z_(n)|=

int(x-1)/(x+2)^(3/4)dx

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by cutting off squares from the corners and folding up the flaps. What should be the side of the square in order the volume of the box is maximum.

If the subnormal at every point of a curve is a constant k, then its equation is

Let f (x) = {{:( -3 "," , - 3 le x lt 0 ), ( x ^(2) - 3 "," , 0 lt x le 3 ):} and g (x) = |f (x) |+f (|x|), then which of the following is true ?

A die is thrown 6 time if getting an odd numbers is a success. What is the probability of a. 5 successes b. at least 5 successes c. at most 5 successes

Find (d)/(dx) ["cosec"^(-1)x]_(x= -2)

inte^(x/sqrt2)cos((x)/(sqrt2))dx=......+c

According to the table above, 100-Watt bulbs made up what fraction of the working lightbulbs?

I: In a Delta ABC, if a= R tan A then b^2+c^2=bc+a^2 II: In a Delta ABC if sin A+ sin B+ sin C is maximum then triangle is equilateral

A and B are independent events. P(A' cap B)= (2)/(15) and P(A cap B')=(1)/(6) then find P(B).

The value of a for which the function f(x)={{:(tan^(-1)a -3x^2" , " 0ltxlt1),(-6x","xge1):} has a maximum at x=1 , is

Let there is an A.P. with common difference 4 such that the squares of the first term plus the sum of all other terms is at most 100. If the number of terms is maximum then the range of 'a' is

If int (log(t+sqrt(1+t^(2))))/(sqrt(1+t^(2)))dt=(1)/(2)(g(t))^(2)+C, where C is a constant, then g(2) is equal to:

Assertion (A): The locus of the point ((e^(2t)+e^(-2t))/(2), (e^(2t)-e^(-2t))/(2)) when 't' is a parameter represents a rectangular hyperbola. Reason (R ) : The eccentricity of a rectangular hyperbola is 2.

Find the probability distribution of number of doublets in three throws of a pair of dice.