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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Show that the line through the points (1,-1,2), (3,4,-2) is perpendicular to the line through the points (0,3,2) and (3,5,6). |
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Answer» The direction ratios of the line through `(x_1, y_1,z_1)` and `(x_2,y_2,z_2)` are `x_2-x_1, y_2-y_1, z_2-z_1` `THEREFORE` The sum of products of the direction ratios = `2xx3+5xx2+(-4)xx4 = 6+10-16 = 0 ` HENCE, the two lines are perpendicular. |
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| 2. |
Find the points of local maxima and local minima for the following functions. Also find respective maximum and minimum values. (i) y=x^(4)-14x^(2)+24x-3 (ii) y=((x+1)(x+4))/((x-1)(x-4)) (iii) y=x (x-1)^(2) (x+1)^(2) (iv) y=(log x)//x y=e^(x) sin x" in "[-pi,pi] y=sin x+cos x" in "[0,2pi] |
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Answer» (i) max : (2,-9) and min (-2,-1/9) (iii) `max (1/SQRT5): (16sqrt5)/(125) and (-1,0): " min "(-1)/sqrt5, -(16sqrt5)/(125), (1,0)` (iv) `max (E 1/e) ` v `max ((3pi)/(4), (1)/sqrt2 e^(3pi//4)) min ((-pi)/(4), (-1)/(sqrt2), e^(-pi//4))` |
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| 3. |
Solve the following linear programming problem graphically: Minimise Z=200x+500y……………1 subject to the constraints: x+2yge10……………2 3x+4yle24……………3 xge0,yge0…………..4 |
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| 4. |
{:("Column A","", "Column B"),("The perimeter of quadrilateral",,"The The circumference of the circle"):} |
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Answer» If column A is larger |
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| 5. |
An aqueous .............. |
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Answer» `Na_(2)B_(4)O_(7). 10H_(2)O+HCl RARR 2NACl+4H_(3) BO_(3)+4H_(2)O` |
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| 6. |
By using the properties of definite integrals, evaluate the integrals int_(0)^(4)abs(x-1)dx |
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| 7. |
Construct truth tables for the following and indicate which of these are tautologiesp vv q rarr ~~(p^^q) |
Answer» SOLUTION :
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| 8. |
If 1,alpha_1,alpha_2,…..alpha_(n-1) are the n^(th) roots of unity and n is an odd natural number then(1+alpha_1)(1+alpha_2)…..(1+alpha_(n-1))= |
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Answer» 1 |
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| 9. |
Two coins are tossed . The probability of getting 2 tails if it is known that there is atleast one tail on the coins is |
| Answer» Answer :A | |
| 10. |
If the vectors of a trianlge are A(hati+hatj+2hatk), B(3hati-hatj+2hati) and C(2hati-hatj+hatk) the area of triangle is |
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Answer» `2sqrt3` SQ UNITS and `AC=(2hat(i)-hat(j)+hat(k))-(hat(i)+hat(j)+2hat(k))=hat(i)-2hat(j)-hat(k)` `(ABxxAC)=|{:(hat(i),hat(j),hat(k)),(2,-2,0),(1,-2,-1):}|` `=hat(i)(2-0)-hat(j)(-2-0)+hat(k)(-4+2)` `=2hat(i)+2hat(j)+2hat(k)` `therefore` Area of TRIANGLE `AB=(1)/(2)|ABxxAC|` `=(1)/(2)sqrt((2)^2+(2)^2+(-2)^2)` `=(2)/(2)sqrt(1+1+1)=sqrt(3)` sq units. |
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| 11. |
int (6x^(2) - 17 x - 5)/((x - 3)(x - 2)^(2))dx = |
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Answer» `LOG"" ((X-2)^(2))/((x - 3)^(4)) + (3)/(x - 2)+ C ` |
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| 12. |
Examine the continuity of the following function at the indicated pionts. f(x)={{:(,(x-cos (sin^(-1)x))/(1-tan(sin^(-1)x))x ne 1/2),(,(-1)/(sqrt2)x =1/sqrt2):}" at x="1/sqrt2 |
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| 13. |
If x=cos alpha+i sin alpha,y=cos beta+isinbeta, then (x-y)/(x+y)= |
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Answer» `i.cos((alpha-beta)/(2))` |
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| 14. |
Choose the correct answer. Area of the region bounded by the curvey^2=4x, y-axis and the line y=3 is : |
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Answer» 2 |
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| 15. |
int (dx)/(x^(1//5)(1 +x^(4//5))^(1//2)) = |
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Answer» `sqrt(1 + x^(4//5)) ` + c |
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| 16. |
Does the function f(x) =3x^(2)-1 satisfy the condition of the Fermat theorem in the interval [(1,2)] ? |
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| 17. |
If intsqrt(2)(sqrt(1+sinx)dx=4cos(ax+b)+c, then the value of a,b are |
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Answer» `(1)/(2),(pi)/(4)` `=sqrt(2)int("sin"(x)/(2)+"COS"(x)/(2))dx` `{because+-sqrt(1+sinA)=sin((A)/(2))+cos((A)/(2))]` `=2intsin((pi)/(4)+(x)/(2))dx` `=-4cos((x)/(2)+(pi)/(4))+C` But `l=-4cos(ax+b)+c` On comparing, we get `a=(1)/(2),b=(pi)/(4)`. |
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| 18. |
Human being forms two sets of teeth during life, a set of _____teeth replaced by a set of _____teeth. Correct options in the blanks are - |
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Answer» MILKY , Deciduous |
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| 19. |
Point of contact of y=1-x w.r.t. y^(2)-y+x=0 is |
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Answer» (1,1) |
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| 20. |
Let f: [0,1] to R be such that f(xy). F(y) AA x, y in (0,1) and f(0) != 0. If y = y(x) satisfies the differential equation, (dy)/(dx) = f(x) with y(9) = 1 they y((1)/(5))+y((4)/(5)) is equal to: |
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Answer» 4 `f(o) = 1 as f(o) != 0 implies f(x) =1` `(DY)/(DX) = f(x) = 1 implies y = x + C` At `x = 0, y = 1 implies c = 1 implies y = x +1` `implies y((1)/(5)) + y((4)/(5)) = 1/5 + 1 + 4/5 + 1 = (1+1+1) = 3`. |
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| 21. |
X and Y are independent binomial variables. B(5,(1)/(2))and B(7,(1)/(2)) then P(X + Y =3)= ……… |
| Answer» Answer :A | |
| 22. |
The point on y^(2)=x where tangent makes angle of measure (pi)/(4) with the positive X-axis is ……….. |
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Answer» `((1)/(4),(1)/(2))` |
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| 24. |
Find the maximum and minimum values, if any, of thefunctions given by f(x) = |x + 2| – 1 |
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| 25. |
Let n!, the factorial of a positive integer n, be defined as the product of the integers 1, 2 ...., n. In words, n! = 1xx2xx...xxn What is the number of zeros at the end of the integer 10^(2)! +11^(2)!+12^(2)!+99^2! |
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| 26. |
Which of the following is//are correct order of bond angle: |
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Answer» `NH_(3)gtNF_(3)` |
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| 27. |
If n letters are placed at random in n addressed envelops then the probability that all the letters are placed in correct envelops is |
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| 28. |
Sum of the coefficients of terms of degree 13 in the expansion of(1+x)^(11) (1+y^(2)-z)^(10) is |
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Answer» `.^(10)C_(3)` `=` coefficient of `t^(15)` in `(1+t)^(11) (1+t^(2) - t)^(100)` `=` coefficeint of `t^(13)` in `(1+t)(1+t^(3))^(10)` `=` coefficient of `t^(13)` in `((1+t^(3))^(10) + t(1+t^(3))^(10))` `=` coefficient of `t^(12)` in `(1+t^(3))^(10)` `= .^(10)C_(4)` |
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| 29. |
Let a function f:[0,5] to R be continuous, f(1)=3 and F be defined as : F(x)=int_(1)^(x)t^(2)g(t)dt, where g(t)=int_(1)^(t)f(u)"du". Then for the function F, the point x=1 is : |
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Answer» a POINT of INFLECTION. |
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| 30. |
Foot of the perpendicular drawn from the origin to the plane passing through (1,0,0),(0,1,0)and(0,0,1) is |
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Answer» `(3,3,3)` |
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| 31. |
Which of the following function are continuous at x=0 ? [Note : sgn x denotes signum dunction od x.] |
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Answer» `COS((pi)/(2)sgn|X|)+sgn|x|` `"As,"sgn|x|={{:(0","x!=0),(1","x=0):}rArrcos((pi)/(2)sgn|x|)+sgn|x|=1AAx INR` `"Also,"cos((pi)/(2)sgn|x|)-sgn|x|={{:(-1","x!=0),(1","x=0):}` `"As,"sin((pi)/(2)sgn|x|)-sgn|x|={{:(1","x!=0),(0","x=0):}rArrsin((pi)/(2)sgn|x|)+sgn|x|={{:(2","x!=0),(0","x=0):}` `andsin((pi)/(2)sgn|x|)-sgn|x|=0AAx inR`. |
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| 32. |
A radioactive substance 'A' converts to stable nuclei D by following series of reaction : AtoBtoCtoD Given : t_(1//2)"for"'A'=0.0693 days t_(1//2)"for"'B'=6930 days t_(1//2)"for"'C'=6.93 days Number of nuclei of 'D' present after 6930 days are, if initially 10^(20) nuclei of A is taken |
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Answer» `10^(10)` |
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| 33. |
A radioactive substance 'A' converts to stable nuclei D by following series of reaction : AtoBtoCtoD Given : t_(1//2)"for"'A'=0.0693 days t_(1//2)"for"'B'=6930 days t_(1//2)"for"'C'=6.93 days Number of nuclei of 'C' formed in the 10 days are, if initially 10^(20) nuclei of A is taken |
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Answer» `10^(18)` `"SINCE "lamda_(1)gtgtlamda_(2) ltltlamda_(3)` we can assume that all the 'A' has been converetd into 'B' in small duration Number of moles of C formed = number of number of moles of 'B' dissociated `DeltaB=lamda_(2)"N t"` `=(ln2)/(6930)xx10^(20)xx10=10^(17)]` |
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| 34. |
Statement-1: The period of sinx , cos x is 2pi and period of f(x)+g(x) is the LCM of the periods of f(x) and g(x) |
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Answer» Statement-1 is TRUE, Statement-2 is True, statement-2 is a correct explanation for the statement-1 . Now, ` sin""(pi)/(4)[x]=sin(2pi+(pi)/(4)[x])=sin""(pi)/(4)(8+[x])=sin""(pi)/(4)[x+8]` `implies sin""(pi)/(4)[x]` is periodicwith period 3. `cot""(pi)/(3)[x]=cot(pi+(pi)/(3)[x])=cot""(pi)/(3)[3+[x])=cot""(pi)/(3)[x+3]` `implies cot""(pi)/(3)[x]` is PERIODIC with period 3. and , ` cos""(pix)/(2)` is periodic with period `(2pi)/(pi//2)=4` Using statement-2, period of F(x) is LCM of (8,3,4)=24 |
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| 36. |
A coin is tossed three times. Find the probability of getting all heads. |
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Answer» `|S|=8` Let A be the EVENT of GETTING all heads `thereforeA={HHH}impliesO.(A)=1` `therefore P(A)=|A|/|S|=1/8` |
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| 37. |
If P(A)=3/5andP(B)=1/5 find P(AcapB). If A and B are independent events |
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| 38. |
The position vector of the points A and B are respectively vec(a) and vec(b). Find the position vectors of the points which divide AB in trisection. |
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| 39. |
int_(0)^(2) (2x-2)/(2x-x) dx is equal to |
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Answer» 0 |
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| 40. |
A biased coin with probability p, 0 lt p lt 1 of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2//5, then q equals |
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Answer» `(1)/(3)` |
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| 41. |
If ""^(n)C_(r-1):""^(n)C_(r):""^(n)C_(r+1)=2:4:5 then (n,r) is |
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Answer» (10,4) |
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| 42. |
If the normal of the plane makes angles (pi)/(4),(pi)/(4)" and "(pi)/(2) with positive X-axis, Y-axis and Z-axis respectively and the length of the perpendicular line segment form origin to the plane is sqrt(2), then the equation of the plane is ……...... |
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Answer» `x+y+z=sqrt(2)` |
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| 43. |
If for a unit vector a.(x-a).(x+a)=12, then |x| is equal to |
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Answer» 4 [` :'`Itis a unitvector, mangnitude of unitvector is 1 ] ` therefore(x-a).(x+a)=12` `IMPLIES x.x+x.a-a.x-a.a=12` `[ :'a.a =|a|^(2) anda.b=b.a]` `implies |x|^(2)-|a|^(2) = 12implies |x|^(2) -1^(2)=12` `[ :'|a|=1 ` asit is unitvector ] ` | x|^(2)= 13gt |x|= sqrt(13)` |
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| 44. |
Coefficient of x^5 in ( 1+ x)^21 + (1 + x)^22 + …….+ (1 + x)^30 is |
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Answer» `""^51C_5` |
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| 45. |
Evaluate the following integrals (ii) int_(0)^(1)x^(2) sin^(-1) x dx |
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| 46. |
Resolve the following into factor. a^(3) - 1/a^(3) + 4 |
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| 47. |
Statement 1: Equation of the pair of lines bisecting the angle between the pair of lines ax ^(2)+by ^(2) + 2hxy+ 2gx+2fy+c =0 can be written as x ^(2)-y ^(2) - ((a-b))/(h) xy + lamda x + lamda x + mu y + c' =0.because Statement 2: Equatin of any pair of lines parallel to the lines ax ^(2)+by ^(2) +2hxy =0 is ax^(2) + 2hxy+ by^(2) +Ax+By+c^(n) =0, |
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Answer» Stateme-1 is TRUE, Statement-2 is True, Statemetn-2 is correct explanation for Statement-1 |
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| 48. |
Evaluate int_(0)^(2pi){cos(x)cos(2x)cos(2^(2)x)…cos(2^(n-1)x)cos(2^(n)-1)x}dx. |
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| 49. |
Three distinct numbers a, b and c are chosen at random from the numbers 1, 2, …, 100. The probability that{:("List I","List II"),("a.a, b, c are in AP is","p. "(53)/(161700)),("b.a, b, c are in GP is","q. "(1)/(66)),("c. "(1)/(a). (1)/(b). (1)/(c) " are in GP is ","r. "(1)/(22)),("d.a + b + c is divisible by 2 is","s. "(1)/(2)):} |
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Answer» `{:("a","B","c","d"),("q","s","s","R"):}` a. 2b = a + c = even This means that a and c are both even or both odd. `n(E) = .^(50)C_(2) + .^(50)C_(2) = 50 XX 49` b. Taking r = 2, 3, …, 10, a, b, c can be in GP in 53 ways. c.`(1)/(a), (1)/(b), (1)/(c)` are in GP = a, b, c are in GP d. P(a + b + c is even) = `((.^(50)C_(3)+.^(50)C_(1) xx .^(50)C_(2))/(.^(100)C_(3))) = (1)/(2)` |
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| 50. |
Consider oversetrarra=overset^^i+2overset^^j-3overset^^k,"" oversetrarrb=3overset^^i-overset^^j+2overset^^k "and " oversetrarrc=11 overset^^i+overset^^j:Find oversetrarra+oversetrarrb"and"oversetrarra.oversetrarrb |
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Answer» SOLUTION :`veca+vecb=4overset^^i+overset^^j-overset^^k` `veca.vecb=1xx3+2xx(-1)+(-3)xx2` |
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