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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. |
If x=|alpha+beta|, y=|alpha|+|beta|, z=|alpha-beta|, then : |
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Answer» x = MAX. (y, z) |
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| 2. |
Value of lim_(x to 0^(-))[(sin[x])/x]+ lim_(x to 0^(+))[(sin^(-1)|x|)/|x|]+ lim_(x to 0^(-))[(-2x)/(tanx)] is (where [.] denotes greatest integer function) |
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Answer» Solution :ANS. Of first part =-1 SECOND part =1 third part =-2 `therefore` Ans = -2 |
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| 3. |
In a book with page numbers from 1 to 100, sme pages are torn off. The sum of the numbers on the remaining pages is 4949. How many pages are torn off ? |
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| 4. |
Integrate the following rational functions : int(x^(2)+1)/(x^(2)-5x+6)dx |
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| 5. |
A particle moves in a straight line for 20 seconds with velocity 3 m/s and then moves with velocity 4 m/s for another 20 seconds and finally moves with velocity 5 m/s for next 20 seconds. What is the average velocity of the particle ? |
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Answer» 3 m/s |
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| 6. |
If the normal drawn from the origin to the straight line 2x+7y+6=0 makes an angle theta with the postitive X-axis, then theta= |
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Answer» `TAN^(-1)""(7)/(2)` |
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| 7. |
The range of f(x) = sqrt((a-|x|)/((a+1) - |x|)), (a gt 0) is |
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Answer» [0, a] |
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| 8. |
Let A =((a,b),(c,d)) ,a,b,c,d in R . Suppose there exists x_(1),x_(2) in C x_(1),x_(2) ne 0 such that (A-3.1 il_(2)) ((x_(1)),(x_(2)))=O_(2) then det (A) is equal to ______ . |
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| 9. |
If A^(-1)=[{:(-4,2),(3,-1):}]and B=[{:(0,3),(-2,5):}] then find (AB)^(-1) |
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| 11. |
Um A contains 6 red and 4 black balls and um B contains 4 red and 6 black balls. One ball is drawn at random from um A and placed in um B. Then one ball is drawn at random from um B and placed in um A . If one ball is now drawn from um A, the probability that it is found to be red is |
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Answer» `32//55` |
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| 12. |
U=[2""-3""4],V=[{:(3),(2),(1):}]X=[0" "2" "3]and Y=[{:(2),(2),(4):}] then UV +XY=………. |
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Answer» 20 |
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| 13. |
A company makes 3 model of calculators : A, B and C at factory (I) and factory (II). The company has orders for atleast 6400 calculators of model A, 4000 calculator of model B and 4800calculators of model C. At factory (I), 50 calculators of model A, 50 of model B and 30 of model C are made everyday, at factory (II), 40 calculators of model A, 20 of model B and 40 of model C are made everyday. It costs Rs. 12000 and Rs. 15000 each day to operate factory (I) and (II), respectively. Find the number of days each factory should operate to minimise the opertaing costs and still meet the demand. |
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Answer» 24 |
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| 14. |
int (dx)/(x log x log (logx))=... |
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Answer» `2LOG(logx)+C` |
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| 15. |
A manufacturer produces two models of bikes model X and model Y. Model X takes a 6 man hours to make per unit, while Model Y takes10 man-hours per unit. There is a total of 450 man-hour available per week. Handling and Marketing costs are Rs. 2000 and Rs. 1000 per unit for Models X and Y respectively. The total funds available for these purposes are Rs. 80,000 per week. Profits per unit for Models X and Y are Rs. 1000 and Rs. 500, respectively. How many bikes of each model should the manufacturer produce, so as to yield a maximum profit ? Find the maximum profit. |
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| 16. |
Show that 9^n+1 - 8n - 9 is divisible by 64, whenever n is a positive interger. |
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| 17. |
If the maximum value of ||x-3| - |x-2|| is a and minimum value of |x-1| + |x-2| is b, then find the value of (a+b). |
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Answer» SOLUTION :As the maximum VALUE of `||x - x_(1)| x - x_(2)||` is `|x_(1)x_(2)|` and the minimum value of `|x-x_(3)| + |x - x_(4)|` is `|x_(3) - x_(4)|`. Hence, `a = |3-2| = 1, b = |2-1| = 1` `rArr a+b = 2` |
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| 18. |
Let vec(a)=2hati-hatj+hatk, vec(b)=hati+2hatj-hatk and vec(c)=hati+hatj-2hatk be three vectors. A vector in the plane of vec(b) and vec(c) whose projection on vec(a) is of magnitude sqrt((2)/(3)), is |
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Answer» `2hati+3hatj-3hatk` Projection of `vec(r)" on "vec(c)=(vec(r).vec(a))/(|vec(a)|)=|sqrt((2)/(3))|` `Rightarrow -m-1=pm 2` `Rightarrow m=-3 and 1` `"Hence ", vec(r)=-2hati-5hatj+hatk and vec(r)=2hati+3hatj-3hatk` |
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| 19. |
int (x^(3))/(sqrt(1 + x^(2)) dx is equal to |
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Answer» `SQRT(1 + x^(2)) - (x)/(3) (1 + x^(2))^(3//2) + C` |
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| 20. |
A corner point of a feasible region is a point in the region which is the ……….. of two boundary lines. |
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| 21. |
Which of the following is true ({.} denotes the fractional part of the function)? |
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Answer» ALWAYS 1 `underset(xto2^(+))lim(x)/(x^(2)-x-2)=underset(xto2^(+))lim(x)/((x-2)(x+1))=underset(hto0)lim(2+h)/(h(3+h))=oo` `underset(hto-1)lim(x)/(x^(2)-x-2)=-underset(hto-1)lim(x)/((x-2)(x+1))` `=underset(hto0)lim(-1-h)/((-3-h)(-h))=-underset(hto0)lim(1+h)/((3+h)(h))=-oo` |
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| 22. |
2int_0^(1/sqrt(2))(sin^(- 1)x)/x dx-int_0^1(tan^(- 1)x)/x dx= |
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Answer» `(PI)/(8) ln 2` |
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| 23. |
Express with rational denominator(3sqrt(-2)+2(-5))/(3sqrt(-2)-2sqrt(-2)) |
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Answer» Solution :`(3sqrt(-2)+2(-5))/(3sqrt(-2)-2sqrt(-2))=(3sqrt(2I)-10)/(3sqrt(2i)-2sqrt(2i))=(3sqrt(2i)-10)/SQRT(2i)` ((3sqrt(2i)-10)sqrt(2i))/((sqrt(2i))(sqrt(2i)))=(6i^2-10sqrt(2i))/(2i^2) `=(-6-i10sqrt2)/(-2)=(6+i10sqrt2)/2=3+i5sqrt2` |
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| 25. |
Ifalpha, betaare therootsofx^2- 3x +a =0andgamma, deltaaretheroots ofx^2 -12 x+b=0andalpha, beta ,gamma, deltain thatorderfroma geometricprogressionincreasing orderwith commonrationr gt 1thena+b= |
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Answer» 16 |
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| 26. |
Integrate the following rational functions : int(x^(2)+5x+3)/(x^(2)+3x+2)dx |
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| 27. |
Check the injectivity and surjectivity of the following function . f: N-{1} rarr N,f(n) = Greatest prime factor of n . |
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| 28. |
Check the continuity of the function given by f(x) = 3x -5 t at 5 =1 |
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Answer» SOLUTION :Function is defined at given point x=1 and its value is -2. ` UNDERSET(x to 1) F(x) = underset(x to 1) lim (3X -5) = 3(1) -5 ` =-2 Hence, f is CONTINUOUS at x=1 |
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| 29. |
If the equation 3x^(2)+6xy+my^(2)=0 represents a pair of coincident straight lines, then (3m)/2 is equal to |
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| 30. |
Find the domains of tangent and cotangent functions. |
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Answer» SOLUTION :Domain of tan X is `R-{{(2n+1)pi)/2, N epsilonZ}` as tangent is not DEFINED for `x=((2n+1)pi)/2` Domain of cot x is `R-{npi, n epsilonZ}` as COTANGENT is not defined for `x=npi`. |
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| 31. |
solve x dy - y dx = ((x^(2) + y^(2)) dx |
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| 33. |
Observe the following statements :Statement - I : 1/2 . ""^10C_0 - ""^10C_1 + 2. ""^10C_2 - 2^2. ""^10C_3 + ……+ 2^9. ""^10C_10 = -1/2Statement - II : ""^20C_1 - 2(""^20C_2) + 3.(""^20C_3)-…..-20.(""^20C_20) = 0Then the false statements are: |
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Answer» only I |
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| 34. |
Show that f(x) =(x)/(sqrt(1+x)) =en (1+x) isan increasing function for x gt-1 |
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| 35. |
A,B,C are three mutually exclusive and exhaustive eventsassociated with random experiment .Find P(A), it being given that P(B) = (3)/(2)P(A) and P( C ) =(1)/(2)P(B) |
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Answer» ` (4)/(13)` |
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| 36. |
The origin is translated to (1,2). The point (7,5) in the old system undergoes the following transformations successively. I. Moves to the new point under the given translation of origin. II Translated through 2 units along the negative direction of the new X-axis. III. Rotated through an angle (pi)/(4) about the origin of new system in the clockwise direction. The final position of the point (7,5) is |
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Answer» `((9)/(SQRT(2)),(-1)/(sqrt(2)))` |
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| 37. |
If F(x) = (1)/(x^(2)) underset(4)overset(x)int (4t^(2)- 2F'(t))dt then F'(4) equals to ……… |
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Answer» `(32)/(9)` |
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| 39. |
Find the number of distinct terms in the expansion(x + 2y + 3z + w)^20 |
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| 40. |
Find whether the following function are one-one or many -one & into or onto if f:DtoR where D is its domain f(x)=sin4x:(-(pi)/4,(pi)/4)to(-1,1) |
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| 42. |
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the increasing when the edge is 10 cm long? |
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| 43. |
If(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the value of sumsum_(0 le i lt j le n) (i + j )(C_(i) + C_(i) C_(j)) . |
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| 44. |
Vertices of a variable acute angled triangle ABC lies on a fixed circle. Also a, b, c andA, B, C are lengths of sides and angles of triangle ABC, respectively. If x_(1),x_(2) and x_(3) are distances of orthocentre from A, B and C, respectively, then the maximum value of ((dx_(1))/(da)+(dx_(2))/(db)+(dx_(3))/(dc)) is |
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Answer» `-SQRT3` `(dx_(1))/(dA)=-2R sin A` Also `a=2R sin A rArr (da)/(dA)=2R cos A` `"So,"(dx_(1))/(da)=-tanA,(dx_(2))/(db)=-tanB,(dx_(3))/(dc)=-tanC` `"Now"tanA+tanB+tanCge3sqrt3` `"So,"((dx_(1))/(da)+(dx_(2))/(db)+(dx_(3))/(dc))lt-3sqrt3` |
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| 45. |
Prove that :int_(0)^(pi) (x)/(1 +sin^(2) x) dx =(pi^(2))/(2sqrt(2)) |
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| 46. |
Evaluate : int_(pi/5)^((3pi)/(10))(cosx)/(sinx+cosx)dx |
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| 47. |
If f:RrarrR,f(x) is a differentiable function such that (f(x))^(2)=e^(2)+int_(0)^(x)(f(t)^(2)+(f'(t))^(2))dtAAx inR. The values f(1) can take is/are |
| Answer» ANSWER :A::B | |
| 48. |
If f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3(98))…(x-100)^(100), then the value of (f'(101))/(f(101)) is |
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Answer» 5050 TAKING log both sides, we get `log(f(x))=sum_(i=l)^(100)i(101-i)log(x-i)` Differentiating w.r.t. x, we get `(1)/(f(x)).f'(x)=sum_(i=l)^(100)(i(101-i))/((x-i))` `rArr""(f'(100))/(f(101))=sum_(i=l)^(100)((101-i)/(101-i))=5050` |
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| 49. |
If z=x +iy,x,y in R and if the pointp in theagrand plane represents z then the locus of p satisfying the condition arg (z-1)/(z-3i)=(pi)/(2) is |
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Answer» `{Z in C// |Z-(1+3l)/(2)\|= (SQRT(10))/(2)}` |
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