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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 8151. |
Find derivatives of the following functionssqrt log x |
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Answer» SOLUTION :`y = SQRT log x dy/dx =1/(2sqrt log x).d/dx (log x) = 1/(2 sqrt log x).1/x` |
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| 8152. |
If A'=[{:(-2,3),(1,2):}]andB=[{:(-1,0),(1,2):}]then find (A+2B)'. |
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| 8154. |
By using elementary row operations, find the inverese of the martrix A=[{:(1,3,-2),(-3,0,-5),(2,5,0):}]. |
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Answer» Solution :We have `[{:(1,3,-2),(-3,0,-5),(2,5,0):}]=[{:(1,0,0),(0,1,0),(0,0,1):}].A` `implies""[{:(1,3,-2),(0,9,-11),(0,-1," "4):}]=[{:(1,0,0),(3,1,0),(-2,0,1):}].A""[{:(R_(2)toR_(2)+3R_(1)),(R_(3)toR_(3)-2R_(1)):}]` `implies""[{:(1,3,-2),(0,-1," "4),(0,9,-11):}]=[{:(1,0,0),(-2,0,1),(3,1,0):}].A""[R_(2)harrR_(3)]` `implies""[{:(1,0,10),(0,-1,4),(0,0,25):}]=[{:(-5,0,3),(-2,0,1),(-15,1,9):}].A""[{:(R_(1)toR_(1)+3R_(2)),(R_(3)toR_(3)+9R_(2)):}]` `implies""[{:(1,0,10),(0,1,-4),(0,0,25):}]=[{:(-5,0,3),(" "2,0,-1),(-15,1,9):}].A""[{:(R_(2)tp(-1).R_(2)]` `implies""[{:(1,0,10),(0,1,-4),(0,0,1):}]=[{:(-5,0,3),(2,0,-1),((-3)/(5),(1)/(25),(9)/(25)):}].A" "[R_(3)to(1)/(25)R_(3)]` `implies""[{:(1,0,0),(0,1,0),(0,0,1):}]=[{:(1,(-2)/(5),(-3)/(5)),((-2)/(5),(4)/(25),(11)/(25)),((-3)/(5),(1)/(25),(9)/(25)):}].A""[{:(R_(1)toR_(1)-10R_(3)),(R_(2)toR_(2)+4R_(3)):}].` Hence, `A^(-1)=[{:(1,(-2)/(5),(-3)/(5)),((-2)/(5),(4)/(25),(11)/(25)),((-3)/(5),(1)/(25),(9)/(25)):}].` |
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| 8155. |
int(1+sinx)/(1-sinx)dx |
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Answer» SOLUTION :`I=INT(1+sinx)/(1-sinx)DX` =`int(1+sinx)^2/cos^2xdx` =`int(sec^2x+tan^2x+2secxtanx)dx` =`int(2sec^2x-1+2secxtanx)dx` =2tanx-x+2secx+c |
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| 8156. |
Find the area of the region bounded by y= cos x and y= sin 2x between x=0 and x=(pi)/(2) |
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| 8157. |
Evaluate :tan^(-1){2cos(2sin^(-1)""(1)/2)} |
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| 8158. |
Let n and k be positive integers such that n ge(k(k+1))/2. Find the number of solutions (x_(1),x_(2),x_(3),……………x_(k)), x_(1)ge1,x_(2)ge2,……………..,x_(k)gek, all integers satisfying the condition x_(1)+x_(2)+x_(3)+…………….1x_(k)=n. |
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| 8159. |
If ""^(n)C_(r)+3""^(n)C_(r+1)+3""^(n)C_(r+2)+""^(n)C_(r+3)=""^(15)C_(9) ,then which of the following is true? |
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Answer» n=12 |
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| 8160. |
Find the slope of the lines whose inclinations are given.45^@. |
| Answer» SOLUTION :SLOPE = `TAN45^@`=+1. | |
| 8161. |
The tangent to the curve y=e^(2x) at the point (0, 1) meets X-axis at : |
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Answer» `(-1,0)` |
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| 8162. |
Evaluate the following integrals (vii) int_(0)^(pi//2) (1)/( 2+3 sin x) dx |
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| 8163. |
Integrate the following functions : intcosxsqrt(9-sin^(2)x)dx |
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| 8164. |
(4)/(1.3)-(6)/(2.4)+(12)/(5.7)-(14)/(6.8)+….= |
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Answer» `log_(E )((2)/(e ))` |
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| 8165. |
Find the principal value of cos^-1(frac(sqrt3)(2)) |
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Answer» SOLUTION :`COS (pi/6)=sqrt3/2`and `pi/6 in [0,pi]` `therefore`The principal VALUE of `cos^(-1)(sqrt3/2)=pi/6` |
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| 8166. |
The mean deviation of the data 3, 5, 11, 13, 17, 19, 23, 29 about its arithmetic mean is |
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Answer» A.8.5 |
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| 8167. |
If A(bara), B(barb), C(barc) and D(bard) are the four points such that 3bara + 8barb = 6barc + 5bard, then the lines AB and CD are |
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Answer» SKEW |
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| 8168. |
int(3x+4)/sqrt(2x-3)dx |
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Answer» Solution :`INT(3x+4)/sqrt(2x-3)DX` [Put 2x-3=t^2 Then 2dx=2tdt or dx=tdt] =`int(3((t^2+3)/2)+4)/t .tdt` =`3/2intt^2dt+17/2intdt=3/2.t^3/3+17/2t+C` =`1/2t^3+17/2t+C` =`1/2(2x-3)^(2/3) +17/2(2x-3)^(1/2) +C` =`(x+7)sqrt(2x-3)dx` |
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| 8169. |
If int_(0)^(4pi)ln|13sinx+3sqrt3cosx|dx=kpiln7, thenthe value of k is |
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Answer» 2 `=4int_(0)^(4pi)ln|14sin(x+tan^(-1).(3sqrt3)/(13))|dx` `=8int_(0)^(pi//2)(ln14+log|sinx|)dx` `=4pi(ln14-ln2)=4piln7.` |
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| 8170. |
A rod PQ of length 2a sides with its ends on the axes the locus of the circumcentre of triangleOPQ is |
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Answer» `X^(2)+y^(2)=2a^(2)` |
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| 8171. |
Let f(x)=int((4x^(4)-2x^(3)-3x^(2)-4)dx)/((x^(4)+x^(3)+1)^(3//2)). If f(0)=2, then the value of f(-1) is |
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Answer» 6 |
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| 8172. |
A bag contains a total of 20 books on physics and mathematics, Anypossible combination of books is equally likely. Ten books are chosen from the bag and it is found that it contains 6 books of mathematics. Find out the probability that the remaining books in the bag contains 3 books on mathematics. |
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Answer» Solution :Let `E_(i)(i=0,1,2,...,20)` be the event that the bag contains I books on MATHEMATICS. Since all these events are equally likely and mutually EXCLUSIVE and exhaustive, so `P(E_(1))=1//21(i=0,1,2,....,20)`and let A be the event that a drw of 10 books contains 6 books on matematics Then So, USING Bay's THEOREM, `P(A)=underset(i=0)overset(20)sumP(E_(i)).P(A//E_(i))` `=1/21[underset(i=0)overset(20)sumP(A//E_(i))]` `=1/21[underset(i=6)overset(16)sum(""^(i)C_(6)xx""^(20-i)C_(4))/(""^(20)C_(10))]` Now, we want that the bag should contain 2 more books on mathematics, i.e., `E_(8)` must occur. Using Baye's theorem, `P(E_(8)//A)=(P(E_(8))P(A//E_(8)))/(P(A))` `=((""^(8)C_(6)xx""^(12)C_(4))/(""^(20)C_(10)))/(underset(i=6)overset(16)sum((""^(i)C_(6)xx""^(20-i)C_(4))/(""^(20)C_(10))))` `=(""^(8)C_(6)xx""^(12)C_(4))/(underset(i=6)overset(16)sum(""^(i)C_(6)xx""^(20-i)C_(4)))` |
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| 8173. |
Integrate the following functions (3x)/(1+2x^4) |
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Answer» Solution :Let t = `sqrt2 x^2`. Then DT = `2 sqrt2 x dx` `gt x dx = 1/(2 sqrt2) dt` therefore` INT (3x)/(1+2x^4) dx = int 3/(1+t^2) 1/(2sqrt2) dt` =`3/(2 sqrt2) tan^-1 t+C` `3/(2 sqrt2) tan^-1 (sqrt2 x^2)+c` |
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| 8174. |
Resolve (2x^(2)+1)/(x^(3)-1) into partial fractions. |
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| 8175. |
Find the principal value of cosec^-1(-sqrt2) |
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| 8176. |
Determine the number of subsets of {1,2,3…70} whose sum is larger than 1243. |
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| 8177. |
Find the number of solutions in ordered paris of positive integers (x,y) of the equation 1/x+1/y=1/n where n is positive integer. |
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| 8178. |
If an infinite G.P. has 2nd term x and its sum is 4, then prove that xin(-8,1]-{0} |
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Answer» Solution :Let the common ratio of G.P be r. Since SECOND TERM is X,first terms is x/r Now, sum of infinite terms of G.P.=4 `RARR(x/r)/(1-r)=4` `thereforex=4(r-r^(2))` For `rin(-1,-1)-{0}`, we have `r-r^(2)=1/4-(r-1/2)^(2)in(-2,1/4]-{0}` `rArrxin(-8,1]-{0}` |
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| 8179. |
Two coins are tossed find the conditional probability that two tails result, given that there is atleast one tail. |
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| 8180. |
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails. |
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| 8181. |
Determine the integral values of k for which the system (tan^(-1)x)^(2)+(cos^(-1)y)^(2)=pi^(2)k and tan^(-1)x+cos^(-1)y=(pi)/2 possess solution and find all the solutions. |
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| 8182. |
Consider the regions A={(x,y)|x^(2)+y^(2)le100} and B=|(x,y)|sin(x+y)gt0} in the plane. Then the area of the region AnnB is |
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Answer» `10pi` `rArr "Points lies inside CIRCLE or on the circle."` `SIN(x+y)gt0` `x+y in (0,pi)uu(2pi,3pi)..` `x+y=c` is equation of a line Required area is shown in the following FIGURE :  `"Required area = shaded region"` `"= HALF of the area of the circle "` `=(1)/(2)pi(10)^(2)=50pi` |
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| 8183. |
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs (i) None(ii) Not more than one (iii) more than one(iv) at least one will fuse after 150 days of use? |
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| 8184. |
The range of the function is f(x) =""^(7-x)P_((x)-(x-3)) is |
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Answer» {1,2,3,4} |
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| 8185. |
If [vec(a),vec(b),vec(c)]=3andabsvec(c)=1" then "|(vec(b)xxvec(c))xx(vec(c)xxvec(a))| is |
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Answer» 1 |
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| 8186. |
If int(2x^2)/((2x^2+alpha)(x^2+5))dx =sqrt(5)/3 tan^(-1)"" x/sqrt2 +c ,"then alpha= |
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Answer» 1 |
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| 8188. |
If P, Q, R are angles of an isosceles triangle and sqrtP=pi//2, then the value of (cosP/3-isinP/3)^(3)+(cosQ+isinQ)(cosR-isinR)+(cosP-isinQ)(coisR-isinR) is equal to |
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Answer» i |
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| 8189. |
(dx)/((x+1)^(3)sqrt(x^(2)+3x+2)). |
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| 8190. |
lf L and L^(1) are ends or latusrectum of the parabola 9y^(2)= 4x then the combined equation of OL and OL^(1) is |
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Answer» `4X^(2)=9Y^(2)` |
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| 8191. |
Two paralel straight lines indined at an angle theta to x-axis where (90^(@)lt theta lt 180^(@))meet the x-axis at points A, B and y -axis at points C and D respectively. If the equation of the locus of the point of intercectin of AD and BC is x -y =0, te slope of the lines is |
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Answer» `-1` |
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| 8192. |
int sqrt(cos 2x)/(sin x) dx = |
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Answer» `1/(2sqrt(2))log |(SQRT(2)+sqrt(1-tan^(2)X))/(sqrt(2)-sqrt(1-tan^(2) x))|-1/2 log |(1-sqrt(1-tan^(2)x))/(1+sqrt(1-tan^(2)x))|+c` |
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| 8193. |
Compute the integral int_(1//2)^(sqrt(3//2)) (dx)/(x sqrt(1 - x^(2))) |
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| 8194. |
If sales of potatoes were to increase by $173 next month and sales of all other items were held constant, approximately what percentage of the total sales would be be potatoes ? |
| Answer» ANSWER :B | |
| 8195. |
Let A=(2,3,7,9} , f:A rarr B is a function defined as f(x)=x^(2). Then find the range of f(x). |
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| 8196. |
Assertion (A) : The probability of getting exactly 2 heads in tossing a coind thrice is ((1)/(2))^(3) Reason (R ) : The probability of getting exactly r heads in tossing n coins is (.^(n)C_(r ))/(2^(n)). The correct answer is |
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Answer» Both A and R are TRUE, R is the CORRECT EXPLANATION of A |
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| 8197. |
Examine the continuity of the following functions at indicated points.f(x)={(1/(e^(1/2)-1)ifxgt0 at x=0),(0 if xle0):} |
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Answer» Solution :F(0)=0 R.H.L.=`lim_(xto0+)f(x)=lim_(hto0)(0+h)` `=lim_(hto0)1/(E^(1/x)-1)=0` `L.H.L.=lim_(xto0)f(x)=lim_(hto0)f(0-h)` `=lim_(hto0)0=0` |
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| 8198. |
Identify the constant function if any. {(a,1),(b,1),(c,1)} |
| Answer» SOLUTION :The FUNCTION {(a,1),(b,1),(c,1)} is a CONSTANT function. | |
| 8199. |
((x+1))/((2x-1)(3x+1))=A/((2x-1))+B/((3x+1)) rArr 16A+9B= |
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Answer» 4 |
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| 8200. |
If a parallelogram ABCD, |AB|=a, |AD|=b and |AC|=C, then DA. AB is equal to |
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Answer» `(1)/(2)(a^(2)+b^(2)+c^(2))`  In `Delta ABC,""AB + BC = AC ` `rArr AB+AD=AC "" [ because BC=AD]` `rArr AB-DA=AC "" [ because AD=-DA]` `rArr |AB-DA|=|AC|` `rArr |AB-DA|^(2)=|AC|^(2)` `rArr |AB|^(2)+|DA|^(2)-2AB*DA=(AC)^(2)` `rArr a^(2)+b^(2)-2AB*DA=c^(2) "" [ because |AD|=|DA|=b]` `rArr 2DA *AB=a^(2)+b^(2)-c^(2)` `rArr DA*AB=(1)/(2)[a^(2)+b^(2)-c^(2)]` |
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