28278 + Interview Questions in Maths in Class 12 Page 243 InterviewSolution

12101.	R be a relation on Q (set of all rational numbers) defined by R={(x, y):1+xy gt 0}. Then the relation R is
Answer» Only TRANSITIVE Only REFLEXIVE Reflexive and SYMMETRIC but not transitive EQUIVALENCE relation Answer :C

Discussion

12102.	Obtain the following integrals : inttan^(2)x sec^(4)x dx
Answer» Answer :`:.I=(TAN^(5)X)/(5)+(tan^(3)+x)/(3)+C`

Discussion

12103.	Let a= e^(i(2pi)/(3)). Then the equation whose roots are a+a^(-2)" and "a^(2)+a^(-4) is
Answer» `X^(2)-2x+4=0` `x^(2)-x+1=0` `x^(2)+x+4=0` `x^(2)+2x+4=0` ANSWER :D

Discussion

12104.	Evaluate the following integrals. int(3x-5)/(x(x^(2)+2x+4))dx
Answer» ANSWER :`-(5)/(4)log\|x\|+(5)/(8)log\|x^(2)+2x+4\|+(17)/(4sqrt(3))tan^(-1)((x+1)/(sqrt(3)))+C`

Discussion

12105.	A family has two children. What is the probability that both the children are boys given that at least one of them is a boy ?
Answer» ANSWER :`(1)/(3)`

Discussion

12106.	A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.
Answer» ANSWER :`0.320 PI CM^(2)//SEC`

Discussion

12107.	vec(r ) =(6hat(i) +3hat(k) ) + lambda(2hat(i) -hat(j) +4hat(k)) vec(r )=(-9hat(i) +hat(j) -10hat(k)) + mu (4hat(i) +hat(j) +6hat(k))
Answer» ANSWER :`SQRT(38)` UNITS

Discussion

12108.	For a parabolay^(2)-2y-4x+9=0 the tangent at some point B is 3y=x+10, where the normal at some other point K is 27y-9x+10=0. Let alpha and betaare the segments of the chord BK cut by the axes of the parabola. Find the number of integral values of ‘a’ for which the equation3x^(2)-(alpha+beta)x+(a^(2)-5a+ (353)/(27)) alpha beta=0 has its roots real and distinct
Answer» Solution :Clearly BK is a FOCAL chord so`alpha+ beta= alpha beta rArr alpha, beta =(100)/(9)` For real and distinct ROOTS `rArr a^(2)-5a-14 lt 0 rArr -2 lt a lt 7` number of INTEGRAL values of a are 8

Discussion

12109.	Let Oa,Ob,OC be the co-terminal edges of a rectangular parallelopiped of volume V and let P be the vertex opposite to O. Then [APBPCP] =
Answer» 2V 12V `3 SQRT(3)` V 0 Answer :A

Discussion

12110.	If x gt 1, y gt 1, z gt 1 are in G.P then 1/(1 + log x),1/(1 + log y),1/(1 + log z) are in
Answer» A.P H.P G.P None of these Answer :A

Discussion

12111.	Determine order and Degree(if defined) of differential equations given y''+(y')^(2) + 2y = 0
Answer» ANSWER :ORDER 2; DEGREE 1

Discussion

12112.	One diagonal of a square is the portion of the line 3x+2y=12 intercepted between the axes. The cordinates of the extremity of the other diagonals not lying in the first quadrant are
Answer» `(1,-1)` `(-1,-1)` `(-1,1)` NONE of these ANSWER :C

Discussion

12113.	Verify Rolle's theorem for the following functions in the given intervals. (i) f(x) = (x-2) (x-3)^(2) in the interval [2,3] . (ii) f(x) = x^(3) (x-1)^(2) in the interval [0,1].
Answer»

Discussion

12114.	If20sin^2theta+21costheta -24= 0" & "(7pi)/4
Answer» 3 `(sqrt15)/3` `-(sqrt15)/3` -3 Answer :D

Discussion

12115.	int(1)/(sinxsin(x+a))dx=
Answer» `cos a LOG[(sinx)/(SIN(x+a))]+c` `cos ALOG[(sinx)/(sin(x+a))]+c` `COSX alog [(sec(x+a))/(secx)]+c` `cos a log[(secx)/(sec(x+a))]+c` Answer :B

Discussion

12116.	The values of k, for which \|k.bar(a)\|lt \|bar(a)\| and k.bar(a)+(1)/(2)bar(a) is parallel to bar(a) holds true are ………….. where k in [-1,1]-{-(1)/(2)} i.e. k in [-1,1]k ne -(1)/(2)
Answer» ANSWER :`K in [-1,1],k NE -(1)/(2)`

Discussion

12117.	Evalute the following integrals int (x + 1)/(sqrt(x^(2) - x+ 1))dx
Answer» Answer :`sqrt(x^(2) - x + 1) + (3)/(2)" sinh^(-1) ((2x - 1)/(sqrt(3))) + C `

Discussion

12118.	If f(x) = 2[x] + cosx, then f : R to R is (where [ * ] denotes greatest integer function)
Answer» one–one and ONTO one–one and into many–one and into many–one and onto Answer :C

Discussion

12119.	When designing a stairway, an architect can use the riser-tread formula 2h + d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. Some building codes require that, for indoor stairways, the tread depth must be at least 9 inches and the riser height must be at least 5 inches. According to the riser-tread formula, which of the following inequalities represents the set of all possible values for the riser height that meets this code requirement?
Answer» `0 leh le 5` `h ge 5` `5 le h le 8` `8 le h le 16` Solution : Since the tread depth, d, MUST be at least 9 inches, and the riser height, h, must be at least 5 inches, it FOLLOWS that d `ge` 9 and h `ge` 5, respectively. Solving for d in the risertread formula 2H + d = 25 gives d = 25 − 2h. Thus the first inequality, d `ge` 9, is equivalent to 25 − 2h `ge` 9. This inequality can be solved for h as follows: `-2h ge 9-25` `2h le 25-9` `2h le 16` `h le 8` Therefore, the inequality 5 `le` h `le` 8, DERIVED from combining the inequalities h `ge` 5 and h `le` 8, represents the set of all possible values for the riser height that meets the code requirement. Choice A is incorrect because the riser height, h, cannot be less than 5 inches. Choices B and D are incorrect because the riser height, h, cannot be greater than 8. For example, if h = 10, then according to the riser-tread formula 2h + d = 25, it follows that d = 5 inches. However, d must be at least 9 inches according to the building CODES, so h cannot be 10.

Discussion

12120.	When designing a stairway, an architect can use the riser-tread formula 2h + d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. Which of the following expresses the riser height in terms of the tread depth?
Answer» `h=1/2(25+d)` `h=1/2(25-d)` `h=-1/2(25+d)` `h=-1/2(25-d)` Solution :Isolating the term that contains the RISER HEIGHT, h, in the FORMULA 2h + d =25 gives 2h = 25 − d. Dividing both sides of this equation by 2 yields `h=(25-d)/2` , or `h=1/2(25-d)`. CHOICES A, C, and D are incorrect and may RESULT from incorrect transformations of the risertread formula 2h + d = 25 when expressing h in terms of d.

Discussion

12121.	When designing a stairway, an architect can use the riser-tread formula 2h + d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. An architect wants to use the riser-tread formula to design a stairway with a total rise of 9 feet, a riser height between 7 and 8 inches, and an odd number of steps. With the architect’s constraints, which of the following must be the tread depth, in inches, of the stairway? (1 foot = 12 inches)
Answer» 7.2 9.5 10.6 15 Solution :Let h be the riser height, in inches, and n be the number of the steps in the stairway. According to the architect’s design, the total RISE of the stairway is 9 feet, or 9 × 12 =108 inches. Hence, nh = 108, and solving for n gives `n=108/h`. It is given that 7 lt h lt 8. it FOLLOWS that `108/8 lt 108/h lt 108/7` , or equivalently , `108/8 lt n lt 108/7`. Since `108/8 lt 14` and `108/7 gt 15` and n is an integer , it follows that `14 le n le 15`.Since n can be an odd number, n can only be 15, therefore, `h=108/15=7.2` inches .Substituting 7.2 for h in the riser-tread formula 2H + d = 25 gives 14.4 + d =25. Solving for d gives d = 10.6 inches. Choice A is INCORRECT because 7.2 inches is the riser height, not the tread depth of the stairs. Choice B is incorrect and may be the result of CALCULATION errors. Choice D is incorrect because 15 is the number of steps, not the tread depth of the stairs.

Discussion

12122.	Integrationof certainirrational expressions I=int(sqrt(x)+3sqrt(x))/(4sqrtx^(5)-6sqrt(x^(7)))dx.
Answer» Answer :`4ROOT(4)(x)+6root(6)(x)+24root(12)(x)+24 In\|root(12)(x)-1\|+C.`

Discussion

12123.	If intsqrt(1+sec x) dx = K sin^(-1)(f(x)) + C then
Answer» `f(X) = SQRT(2)sin (x//2)` `f(x) = sqrt(2) cos (x//2), K = 2` `f(x) = sqrt(2) tan (x//2), K = 2` `f(x) = sqrt(2) sin (x//2), K = sqrt(2)` Answer :A

Discussion

12124.	Let vec(a),vec(b) and vec( c ) be three unit vectors such that vec(a),+vec(b)+vec( c )=vec(0). If lambda=vec(a)vec(b)+vec(b)vec( c )+vec( c )*vec(a) and vec(d)=vec(a)xx vec(b)+vec(b)xx vec( c )+vec( c )xx vec(a) then the ordered pair (lambda, vec(d)) is equal to :
Answer» `((3)/(2),3vec(a)XX vec( C ))` `(-(3)/(2),3vec( c )xx vec(b))` `(-(3)/(2),3vec(a)xx vec(b))` `((3)/(2),3vec(b)xx vec( c ))` ANSWER :C

Discussion

12125.	The sum and sum of squares corresponding to length x (in cm) and weighty (k. gm) of 50 plant products are given below: sum_(i=1)^(50) x_(i) = 212 , sum_(i=1)^(50) x_(i)^(2)=902.8 , sum_(i=1)^(50) y^(i)=261 , sum_(i=1)^(50) y_(i)^(2)=1457.6 Which is morevarying , the length or weight ?
Answer» ANSWER :WEIGHT

Discussion

12126.	If sin^6 theta+sin^4 theta+cos^6 theta+cos^4 theta is written in the form (a+b cos ctheta)/d (where a,b,c,d in N, a,b,d are coprime numbers) then (a+b+c+d) is equal to
Answer» 22 26 28 30

Discussion

12127.	Given the function f(x)=(5x^(2)+1)/(2-x) Find f(3x), f(x^(3)), 3f(x), [f(x)]^(3).
Answer» Answer :`f(3x)=(45X^(2)+1)/(2-3x); f(X^(3))=(5x^(6)+1)/(2-x^(3)); 3f(x)=(15x^(2)+3)/(2-x), [f(x)]^(3)=(125x^(6)+75X^(4)+15x^(2)+1)/(8-12x+6x^(2)-x^(3)`

Discussion

12128.	If A and Bare mutuallyexclusive eventsP(A)=0.35and P(B) = 0.45 , then P(A nn B)is
Answer» `0.45` ` 0.35` ` 0.8 ` ` 0` ANSWER :B

Discussion

12129.	The solution of (3y -7x + 7) dx + (7y -3x + 3) dy = 0 is
Answer» `(y-x' + 1)^(2) (y+ x+1)^(5) =c` `(y-x + 1)^(2) (y-x -1)^(5) =c` `(y-x-1)^(2) (y + x -1)^(5) = c` `(y -x +1)^(2) (y + x -1)^(5) = c` ANSWER :D

Discussion

12130.	Find the condition for the line x cos alpha + y sin alpha = pto be tangent to the hyperbole (x^(2))/(a^(2))-(y^(2))/(b^(2))=1
Answer» <P> Answer :`rArra^(2) COS^(2) ALPHA - b^(2) SIN ^(2) alpha = p^(2)`

Discussion

12131.	A pair of dice is thrown. If the two numbers appearing are different , find the probability thet 6 appears on one die.
Answer» Solution :LET C be the event that 6 appears on one DIE . `therefore` C={ (1,6),(2,6),(3,6),(4,6),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5) } `therefore P(C)=absC/absS=10/30=1/3`

Discussion

12132.	Find the mean number of heads in three tosses of a fair coin.
Answer» ANSWER :1.5

Discussion

12133.	If a, b, c are integers not all equal and w is a cube root of unity (w ne 1), then the minimum value of\|a+ bw+ cw^2\| is
Answer» `0` `1` `1//2 LT \|z\| lt 3//4` `1//2` ANSWER :B

Discussion

12134.	Show that the function f defined by f(x)={(1 if "x is rational"),(0if "x is irrational"):} is discontinuous AAne0inalpha
Answer» Solution :Consider any real point x=a If a is rational then f(a)=1. Again `lim_(xtoa+)f(x)=lim_(hto0)f(a+h)` which does not EXIST because a+h may be rational or IRRATIONAL. SIMILARLY `lim_(xtoa-)f(x)` does not exist. Thus f(x) is discontinuous at any rational point.Similarly we can show that f(x) is discontinuous at any rational point . Similarly we can show that f(x) is discontinuous at any irrational point . Hence f(x) is discontinuous for all `XINR`

Discussion

12135.	Evaluate (i) int_(0)^(pi) ( x sin x)/(1+ sin x) dx
Answer» ANSWER :`(PI)/(2)(pi-2)`

Discussion

12136.	Vertify mean value theorem for the following functions: f(x) = tan^(-1)x, x in [0, 1]
Answer» ANSWER :`SQRT((4)/(PI)-1)`

Discussion

12137.	Find the second order derivatives of the following functions: tan^(-1) 3x
Answer» ANSWER :`(-54X)/((1+ 9X^(2))^(2))`

Discussion

12138.	log_(3)/sqrt(2)32 =
Answer» 3 5 15 20 Answer :C

Discussion

12139.	The value of least term in the expansion is -
Answer» 16 160 32 81 Answer :C

Discussion

12140.	If sec alpha" and cosec "alpha are the roots of the equation x^(2)-px+q=0, then
Answer» <P>`p^(2)=p+2q` `Q^(2)=p+2q` `p^(2)=q(q+2)` `q^(2)=p(p+2)` ANSWER :C

Discussion

12141.	intdx/(1-cos^2x)
Answer» SOLUTION :`INT(DX)/(1-cos^2x)=INTDX/(sin^2x)=int(cosec^2xdx)=-cotx+C`

Discussion

12142.	Let S={a_(1),a_(2),………a_(12)}. Find the total numebr sets which contain one or more of the elements of set S (including the possibility of using all the elements of S) so that the element in a specific set must be an integral multiple of the smallest number in the set. Also generalise the result.
Answer» ANSWER :`2^(n-1)+2^((n//2)-1)+2^((n//3)-1)+……+2^((n//n)-1)`

Discussion

12143.	The points (-a, -b), (0, 0), (a, b) and (a^(2), ab) are
Answer» Collinear Vertices of a parallelogram Vertices of a RECTANGLE but not a square Vertices of square Answer :A

Discussion

12144.	The roots of the quadratic equation x^(2)-2sqrt(3)x-22=0 are
Answer» imaginary REAL, RATIONAL and equal real, IRRATIONAL and UNEQUAL real, rational and unequal Answer :c

Discussion

12145.	The number of ways of dividing 15 cards into 3 groups of 5 cards each is
Answer» `(20!)/(2!(10!)^(3))` `(10!)/(2!(6!)^(3))` `(15!)/(3!(5!)^(3))` `(12!)/(2!(7!)^(3))` ANSWER :C

Discussion

12146.	int (cos (x - a))/(cos (x - b))dx =
Answer» cos (B - a) - sin (b - a) LOG \| sec ( x - b)\| +C x cos (b- a) + sin (b - a) log \| sec( x - b)\| + c x cos (b + a ) - sin (b +a) log \| sec (x + b) \| + c x cos ( b + a) - sin ( b + a ) log \| sec (x - b )\| + c Answer :A

Discussion

12147.	Translate "Life is short, but virtue is lasting" propositions into symbolic form, stating the prime components
Answer» Solution :LET p :LIFE is short. Q :VIRTUE is lasting `:.` ANSWER is `p ^^q`.

Discussion

12148.	If the difference between the roots of x^(2) + ax + b = 0 is equal to the difference between the roots of x^(2)+px+q= 0 then a^(2)-p^(2)=
Answer» b-q 2(p-q) 4(b-q) 8(b+q) ANSWER :C

Discussion

12149.	The sumof n terms of the series 5/(1.2). 1/3 +7/(2.3). 1/(3^(2))+9/(3.4). 1/(3^(3))+11/(4.5). 1/(3^(4))+….is
Answer» `1 - 1/(N+1). 1/(3^(n))` `1+1/(n+1). 1/(3^(n))` `1-1/(n-1). 1/(3^(n+1))` NONE of these ANSWER :A

Discussion

12150.	The sum of numbers formed by taking all the digits 2, 4, 6, 8 is
Answer» 123320 13220 133320 156670 Answer :C

Discussion

Explore topic-wise InterviewSolutions in .

R be a relation on Q (set of all rational numbers) defined by R={(x, y):1+xy gt 0}. Then the relation R is

Obtain the following integrals : inttan^(2)x sec^(4)x dx

Let a= e^(i(2pi)/(3)). Then the equation whose roots are a+a^(-2)" and "a^(2)+a^(-4) is

Evaluate the following integrals. int(3x-5)/(x(x^(2)+2x+4))dx

A family has two children. What is the probability that both the children are boys given that at least one of them is a boy ?

A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.

vec(r ) =(6hat(i) +3hat(k) ) + lambda(2hat(i) -hat(j) +4hat(k)) vec(r )=(-9hat(i) +hat(j) -10hat(k)) + mu (4hat(i) +hat(j) +6hat(k))

Let Oa,Ob,OC be the co-terminal edges of a rectangular parallelopiped of volume V and let P be the vertex opposite to O. Then [APBPCP] =

If x gt 1, y gt 1, z gt 1 are in G.P then 1/(1 + log x),1/(1 + log y),1/(1 + log z) are in

Determine order and Degree(if defined) of differential equations given y''+(y')^(2) + 2y = 0

One diagonal of a square is the portion of the line 3x+2y=12 intercepted between the axes. The cordinates of the extremity of the other diagonals not lying in the first quadrant are

Verify Rolle's theorem for the following functions in the given intervals. (i) f(x) = (x-2) (x-3)^(2) in the interval [2,3] . (ii) f(x) = x^(3) (x-1)^(2) in the interval [0,1].

If20sin^2theta+21costheta -24= 0" & "(7pi)/4

int(1)/(sinxsin(x+a))dx=

The values of k, for which |k.bar(a)|lt |bar(a)| and k.bar(a)+(1)/(2)bar(a) is parallel to bar(a) holds true are ………….. where k in [-1,1]-{-(1)/(2)} i.e. k in [-1,1]k ne -(1)/(2)

Evalute the following integrals int (x + 1)/(sqrt(x^(2) - x+ 1))dx

If f(x) = 2[x] + cosx, then f : R to R is (where [ * ] denotes greatest integer function)

Integrationof certainirrational expressions I=int(sqrt(x)+3sqrt(x))/(4sqrtx^(5)-6sqrt(x^(7)))dx.

If intsqrt(1+sec x) dx = K sin^(-1)(f(x)) + C then

Let vec(a),vec(b) and vec( c ) be three unit vectors such that vec(a),+vec(b)+vec( c )=vec(0). If lambda=vec(a)*vec(b)+vec(b)*vec( c )+vec( c )*vec(a) and vec(d)=vec(a)xx vec(b)+vec(b)xx vec( c )+vec( c )xx vec(a) then the ordered pair (lambda, vec(d)) is equal to :

The sum and sum of squares corresponding to length x (in cm) and weighty (k. gm) of 50 plant products are given below: sum_(i=1)^(50) x_(i) = 212 , sum_(i=1)^(50) x_(i)^(2)=902.8 , sum_(i=1)^(50) y^(i)=261 , sum_(i=1)^(50) y_(i)^(2)=1457.6 Which is morevarying , the length or weight ?

If sin^6 theta+sin^4 theta+cos^6 theta+cos^4 theta is written in the form (a+b cos ctheta)/d (where a,b,c,d in N, a,b,d are coprime numbers) then (a+b+c+d) is equal to

Given the function f(x)=(5x^(2)+1)/(2-x) Find f(3x), f(x^(3)), 3f(x), [f(x)]^(3).

If A and Bare mutuallyexclusive eventsP(A)=0.35and P(B) = 0.45 , then P(A nn B)is

The solution of (3y -7x + 7) dx + (7y -3x + 3) dy = 0 is

Find the condition for the line x cos alpha + y sin alpha = pto be tangent to the hyperbole (x^(2))/(a^(2))-(y^(2))/(b^(2))=1

A pair of dice is thrown. If the two numbers appearing are different , find the probability thet 6 appears on one die.

Find the mean number of heads in three tosses of a fair coin.

If a, b, c are integers not all equal and w is a cube root of unity (w ne 1), then the minimum value of|a+ bw+ cw^2| is

Show that the function f defined by f(x)={(1 if "x is rational"),(0if "x is irrational"):} is discontinuous AAne0inalpha

Evaluate (i) int_(0)^(pi) ( x sin x)/(1+ sin x) dx

Vertify mean value theorem for the following functions: f(x) = tan^(-1)x, x in [0, 1]

Find the second order derivatives of the following functions: tan^(-1) 3x

log_(3)/sqrt(2)32 =

The value of least term in the expansion is -

If sec alpha" and cosec "alpha are the roots of the equation x^(2)-px+q=0, then

intdx/(1-cos^2x)

The points (-a, -b), (0, 0), (a, b) and﻿ (a^(2), ab) are﻿

The roots of the quadratic equation x^(2)-2sqrt(3)x-22=0 are

The number of ways of dividing 15 cards into 3 groups of 5 cards each is

int (cos (x - a))/(cos (x - b))dx =

Translate "Life is short, but virtue is lasting" propositions into symbolic form, stating the prime components

If the difference between the roots of x^(2) + ax + b = 0 is equal to the difference between the roots of x^(2)+px+q= 0 then a^(2)-p^(2)=

The sumof n terms of the series 5/(1.2). 1/3 +7/(2.3). 1/(3^(2))+9/(3.4). 1/(3^(3))+11/(4.5). 1/(3^(4))+….is

The sum of numbers formed by taking all the digits 2, 4, 6, 8 is

Let vec(a),vec(b) and vec( c ) be three unit vectors such that vec(a),+vec(b)+vec( c )=vec(0). If lambda=vec(a)vec(b)+vec(b)vec( c )+vec( c )*vec(a) and vec(d)=vec(a)xx vec(b)+vec(b)xx vec( c )+vec( c )xx vec(a) then the ordered pair (lambda, vec(d)) is equal to :

The points (-a, -b), (0, 0), (a, b) and (a^(2), ab) are