InterviewSolution
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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.
| 301. |
`sum_(k=1)^6``(sin,(2pik)/7 -icos, (2pik)/7)=?` |
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Answer» Let ` S = sum_ (k=1)^6 (sin((2pik)/7) - icos((2pik)/7))` `=>S = -i sum_ (k=1)^6 (cos((2pik)/7) + isin((2pik)/7))` ...(As `i^2 = -1`) `=>S = -i sum_(k=1)^6 (e^((i2pik)/7))` Now, this summation of terms happens to be the sum of the seventh roots of unity; and, the sum of the `n` roots of a number is zero. So, `1+sum_(k=1)^6 (e^((i2pik)/7)) = 0` `=>sum_(k=1)^6 (e^((i2pik)/7)) = -1` `:. S = -i(-1) = i` |
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| 302. |
`If x_1, x_2, .....x_n` are n observations such that `sum_(i=1)^n (x_i)^2=400` and `sum_(i=1)^n x_i=100` then possible values of n among the following is (1) 18 (2) 20 (3) 24 (4) 27 |
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Answer» AM of `m^(th)` power>`m^(th)`power of AM `((x_1^m+x_2^m...+x_n^m)/n)>((x_1+x_2+x_3+..+x_n)/n)^m` Lets take m=2 `(x_1^2+x_2^2+...+x_n^2)/n>(x_1+x_2+x_3+...+x_n)/n^2` `sumx_i^2/n>[sumx_i]^2/n^x` `n>100^2/400` `n>25` option 4 is correct. |
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| 303. |
There are two beakers (I) having pure volatile solvent and (II) having volatile solvent and non-volatile solute. If both beakers are placed together in a closed container then:A. Volume of solvent beaker will decrease and solution beaker will increaseB. Volume of solvent beaker will increase and solution beaker will also increaseC. Volume of solvent beaker will decrease and solution beaker will also decreaseD. Volume of solvent beaker will increase and solution beaker will decrease |
| Answer» Correct Answer - A | |
| 304. |
The correct statement(s) among I to III with respect to potassium ions that are abundant within the cell fluids is/ are : I. They activate many enzymes II. They participate in the oxidation of glucose to produce ATP III. Along with sodium ions, they are responsible for the transmission of nerve signalsA. I and III onlyB. I and III onlyC. I,II and IIID. III only |
| Answer» Correct Answer - A | |
| 305. |
The number of four digit natural numbers which contains exactly two distinct digits, is (A) 567 (B) 576 (C) 657 (D) 675 |
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Answer» The first (non-zero) digit of the number `F` (thousands digit) can be any one of nine. The second digit `S` which is used if there are two digits can be any one of the nine digits different from the first. Now consider the Hundreds Tens and Units digits in the case that there are two digits used in the number. We have two possibilities `F` or `S` to fill each place but we exclude `FFF` as not involving two digits, so there are `2^3-1=7` possible patterns with exactly two different digits and `9**9` ways of choosing the pair of digits in the first place. Then there are `9` possibilities with just one digit. So the required total is `9**9**7+9=576`. |
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| 306. |
If `P=[[sqrt(3)/2,1/2],[-1/2,sqrt(3)/2]], A=[[1,1],[0,1]]` and `Q=PAP^T`, then `P^TQ^(2015)P` is |
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Answer» `P = [[sqrt3/2,1/2],[-1/2, sqrt3/2]]` `:. P^T = [[sqrt3/2,-1/2],[1/2, sqrt3/2]]` Now, `P^TP = [[1,0],[0,1]] = I` Now, `P^TQ^2015P = P^T(PAP^T)^2015P = P^T(PAP^T)(PAP^T)(PAP^T)...P = A^2015` Here, `A = [[1,1],[0,1]]` `A^2 = A.A = [[1,1],[0,1]][[1,1],[0,1]] = [[1,2],[0,1]]` `A^3 = A^2.A = [[1,2],[0,1]][[1,1],[0,1]] = [[1,3],[0,1]]` `:. A^2015 = [[1,2015],[0,1]]` `:. P^TQ^2015P = [[1,2015],[0,1]]` |
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| 307. |
`f(x)= int_0^1 |t-x|t dt` |
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Answer» `f(x) = int_0^1|t-x|t dt ` When, `x le t`, then, `f(x) = int_0^1 (t-x)t dt = int_0^1 (t^2-xt) dt` `f(x) = [t^3/3-(xt^2)/2]_0^1 = [(1/3-x/2) -0 +0] = (2-3x)/6` When, `x gt t`, then, `f(x) = int_0^1 (x-t)t dt = int_0^1 (xt-t^2) dt` `f(x) = [(xt^2)/2-t^3/3]_0^1 = [(x/2-1/3) -0 +0] = (3x-2)/6` |
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| 308. |
If m is a non-zero number and `int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx=f(x)+c`, then f(x) is: |
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Answer» Let `I = int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx` `=int (x^(5m-1)+2x^(4m-1))/(x^(6m)(1+x^-m+x^(-2m))^3 dx` `=int (x^(-m-1)+2x^(-2m-1))/((1+x^-m+x^(-2m))^3 dx` Let `1+x^-m+x^(-2m) = t` Then, `-m(x^(-m-1))-2m(x^(-2m-1))dx = dt``=>-m(x^(-m-1)+2x^(-2m-1))dx = dt` Then, `I = -1/m int t^-3dt` `=>I = -1/m (t^-2/(-2))+c` `=>I = 1/(2m(1+x^-m+x^(-2m))^2) +c` Comparing it with given equation, `I = f(x) +c` `f(x) = 1/(2m(1+x^-m+x^(-2m))^2)` |
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| 309. |
If `f(x)=e^(x)-x` and `g(x)=x^(2)-x`. The the interval in which fog(x) is increasing isA. `(0,1//2)cup(1,infty)`B. `(-1//2,0)cup(1,infty)`C. `(-1,infty)`D. `(-1//2,0)` |
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Answer» Correct Answer - A |
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| 310. |
The marks of 20 students in an examination are given in the following table. Average marks of these student is:A. 2.6B. 2.7C. 2.8D. 2.9 |
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Answer» Correct Answer - C |
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| 311. |
The value of `(1+I z +z^(5) + z^(8))^(9)` when `z = (sqrt(3) + i)/(2)` is |
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Answer» Correct Answer - C |
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| 312. |
One of the focus of ellipse `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1` is `(0.5 sqrt(3))` and difference in lengths of major and mirror axis is `10` units. Then length of latus rectum isA. 3B. 5C. 10D. 15 |
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Answer» Correct Answer - B |
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| 313. |
If two points `B` and `C` at distance of `5` units from each other, lie on the line `(x+2)/(3)=(y-1)/(0)=(2)/(4)`. If point `A` has coordinates `(1,-1,2)`. Then the area of `DeltaABC` isA. `sqrt(34)`B. `2sqrt(34)`C. `sqrt(41)`D. `3sqrt(34)` |
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Answer» Correct Answer - A |
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| 314. |
If points `P(2,3,1), R(4,y,z), 4Q(2,-1,3)` lie on the same line, then distance of point R from origin is :A. `sqrt(2)1`B. `2sqrt(20)`C. `sqrt(42)`D. `sqrt(31)` |
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Answer» Correct Answer - C |
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| 315. |
If points `(h,k) (1,2)` and `(-3, 4)` lie on line `L_(1)` and points `(h,k)` and `(4,3)` lie on `L_(2)`. If `L_(2)` is perpendicular to `L_(1)`, then value of `(h)/(k)` isA. `-(1)/(7)`B. `(1)/(3)`C. 3D. 7 |
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Answer» Correct Answer - C |
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| 316. |
Two radioactive substances are having same initial number of nuclei. Disintegration constant of one substance is `10lamda`, other one `lamda` after how much time ratio of number of nuclei becomes `(1)/(e)`?A. `(1)/(9lamda)`B. `(1)/(10lamda)`C. `(1)/(11lamda)`D. `(1)/(lamda)` |
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Answer» Correct Answer - A |
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| 317. |
Growth of a bacteria is represented as `N(t)=N_(0)e^(lamdat)` After one hour a drug is given which decreases bacterial growth as `(dN)/(dt)=-5N^(2)` Which of the following graph is correct.A. B. C. D. |
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Answer» Correct Answer - B |
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| 318. |
A position of particle is `x=`at`+bt^(2)-ct^(3)` find out velocity when acceleration is zero (1) `v=a+b^2/(3c)` (2) `v=a-b^2/(3c)` (3) `v=2a-b^2/(3c)` (4) None of theseA. `v-a+(b^(2))/(3c)`B. `v-a-(b^(2))/(3c)`C. None of theseD. 1 |
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Answer» Correct Answer - A |
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| 319. |
`S_(1)to` canductivity increases on decreasing concentration of electrolyte `S_(2)to` molar conductivity increases on decreasing concentration of electrolyte.A. `S_(1)` is true, `S_(2)` is falseB. both `S_(1)` & `S_(2)` are trueC. both `S_(1)` & `S_(2)` are falseD. `S_(1)` is false & `S_(2)` is true |
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Answer» Correct Answer - D |
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| 320. |
10 ml of hydrocarbon requries 55 ml of oxygen for complete combustion producing 40 ml of `CO_(2)`. The formula of the hydrocarbon is:A. `C_(4)H_(6)`B. `C_(5)H_(10)`C. `C_(4)H_(8)`D. `C_(4))H_(10)` |
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Answer» Correct Answer - A |
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| 321. |
A plane electromagnetic wave of wavelength `lambda` has an intensity I. It is propagating along the positive Y-direction. The allowed expression for the electric and magnetic fields ar given by : `vecE=sqrt((2I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hatk`,A. `vecB=+(1)/(c)E " veci` `vecE=sqrt((2I)/(in_(0)c))cos[(2pi)/(lambda)(y+ct)]hatk`,B. `vecB=(1)/(c)E " veci` `vecE=sqrt((I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hatk`,C. `vecB=(1)/(c)E " veci` `vecE=sqrt((I)/(in_(0)c))cos[(2pi)/(lambda)(y-ct)]hati`,D. `vecB=(1)/(c)E " veci` |
| Answer» Correct Answer - A | |
| 322. |
The magnetic field of a plane electromagnetic wave is given by: `vec(B)=B_(0)hat(i)-[cos(kz- omegat)]+B_(1)hat(j)cos(kz+omegat)` where `B_(0)=3xx10^(-5)T` and `B_(1)=2xx10^(-6)T`. The rms value of the force experienced by a stationary charge `Q=10^(-4)C` at `z=0` is close to:A. `3xx10^(-2)`B. `0.6`C. `0.9`D. `0.1` |
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Answer» Correct Answer - B |
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| 323. |
An oscillator of mass M is at rest in its equilibrium position in a potential `V=(1)/(2)k(x-X)^(2)`. A particle of massm comes from right with speed u and collides completely inelastically with M and sticks to it. The process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscilllations after 13 collisions is : (M=10, m=5, u=1 ,k=1)A. `sqrt((3)/(5))`B. `(2)/(3)`C. `(1)/(sqrt3)`D. `(1)/(2)` |
| Answer» Correct Answer - C | |
| 324. |
Light of intensity `50W//m^(2)` is incident on a arae of `1m^(2)` in such a way that `25%` of light is reflected back. Find the force exerted by light on surface if light incident perpendicularlyA. `10.8xx10^(-8)N`B. `15.8xx10^(-8)N`C. `20.8xx10^(-8)N`D. `25.8xx10^(-8)N` |
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Answer» Correct Answer - C |
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| 325. |
Two conductors of same cross-section and conductivities K, 3K and length 3 d and d respectively are connected end to end as shown in figure. Temperature of end of first conductor is `theta_(1)` and that of second conductor is `theta_(2)` .the temperature of junction is steady state is `(theta_(2)gttheta_(1))`A. `(10theta_(2)+9theta_(1))/(19)`B. `(theta_(2)+9theta)/(10)`C. `(9theta_(2)+theta_(1))/(10)`D. `(9theta_(2)+10theta_(1))/(19)` |
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Answer» Correct Answer - C |
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| 326. |
Let `A=[[a,b],[c,d]]` be a `2xx2` real matrix. If `A-alphaI` is invertible for every real number `alpha` , then |
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Answer» `A-alpha*I=[[a,b],[c,d]]-[[alpha,0],[0,alpha]]` `=[[a-alpha,b],[c,d-alpha]]` `=(a-alpha)(d-alpha)-bc!=0` `alpha^2-alpha(a+d)+ad-bc` `(a+d)^2-4(ad-bc)<0` |
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| 327. |
For the matrix `A= [[3,2],[1,1]]`, find the numbers a and b such that `A^2+aA+bI=0.` |
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Answer» `A = [[3,2],[1,1]]` `:. A^2 = [[3,2],[1,1]] [[3,2],[1,1]] = [[9+2,6+2],[3+1,2+1]] = [[11,8],[4,3]]` `aA = a [[3,2],[1,1]] = [[3a,2a],[a,a]]` `bI = b[[1,0],[0,1]] = [[b,0],[0,b]]` `:. A^2-aA+bI = [[11,8],[4,3]]+[[3a,2a],[a,a]]+ [[b,0],[0,b]] ` It is given that `A^2-aA+bI = 0` `:. [[11,8],[4,3]]+[[3a,2a],[a,a]]+ [[b,0],[0,b]] = [[0,0],[0,0]]` `=>11+3a+b = 0 => 3a+b = -11->(1)` `=>3+a+b = 0 => a = -b - 3` Putting this value of `a` in (1), `3(-b-3)+b = -11` `=> - 3b -9 +b = -11` `=>-2b = -2 => b = 1` `:. a = -1-3 = -4` `:. a = -4 and b = 1` |
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| 328. |
if B is skew symmetric matrix then find matrix `ABA^T` is symmetric or skew symmetric |
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Answer» `C=ABA^T` `C^T=(ABA^T)^T` `=(A^T)^TB^TA^T` `=A(-B)A^T` `=-ABA^T=-C` `C^T=-C` Skew symmetric metrics. |
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| 329. |
If 50% of a reaction occurs in 100 second and 75% of the reaction occurs in 200 second, the order of this reaction is :A. 1B. 3C. ZeroD. 2 |
| Answer» Correct Answer - A | |
| 330. |
The correct match between items of List - I and List -II is : `{:(,"List-I",,"List-II"),((A)," Phenelzine ",(P)," Pyrimidine"),((B)," Chloroxylenol ",(Q)," Furan"),((C)," Uracil ",(R), " Hydrazine "),((D)," Ranitidine ",(S)," Phenol "):}`A. (A)-(S), (B)-(R),(C)-(P), (D)-(Q)B. (A)-(R), (B)-(S),(C)-(P), (D)-(Q)C. (A)-(R), (B)-(S),(C)-(Q), (D)-(P)D. (A)-(S), (B)-(R),(C)-(Q), (D)-(P) |
| Answer» Correct Answer - B | |
| 331. |
At 320 K, a gas `A_(2)` is 20% dissociated to A(g). The standard free energy change at 320 K and 1 atm in `J mol^(-1)` is approximately : `(R =8.314 Jk^(-1)"mol"^(-1), "In" 2=0.693, "In" 3 = 1.098)`A. 1844B. 4763C. 2068D. 4281 |
| Answer» Correct Answer - B | |
| 332. |
if `(2sqrt(sin^(2)x-2sinx+5))/(4^(sin^(2))y)le1` then which option is correct.A. `2sinx=siny`B. `|sinx|=siny`C. `sinx=|siny|`D. `sinx=2siny` |
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Answer» Correct Answer - C |
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| 333. |
Let `2f(x^2)+3f(11x^2)=x^2-1` for all `x in R=[0],` Then `f(x)` equals |
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Answer» `2f(x^2)+3(f(1/x^2)=x^-1` replacing `x^2->x` `2f(x)+3f(1/x)=x-1-(1)` replacing`1/x->x` `2f(1/x)+3f(x)=1/x-1-(2)` from equation 1 and 2 `-5f(x)=(2x^2-3+x)/x` `5f(x)=(3-x-2x^2)/x` `f(x)=(3-x-2x^2)/(5x)`. |
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| 334. |
`cos^(- 1)sqrt((sqrt(1+x^2)+1)/(2sqrt(1+x^2)))` |
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Answer» `cos^(-1)sqrt(((1+x^2)+1)/(2sqrt(1+x^2))` Let `x=tantheta` `1+x^2=1+tan^2theta=secc^2theta` `=cos^(-1)sqrt((sqrt(sec^2theta)+1)/(2sqrtsec^2theta))` `=cos^(-1)sqrt((sectheta+1)/(2sectheta))` `=cos^(-1)sqrt((1+costheta)/2)` `cos2theta=2cos^2theta-1` `1+cos2theta=2cos^2theta` `1+costheta=2cos^2(theta/2)` `cos^(-1)sqrt(1/2(2cos^2(theta/2)))` `cos^(-1)sqrt(cos^2(theta/2))` `cos^(-1)(cos(theta/2))` `theta/2` `1/2tan^(-1)x`. |
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| 335. |
Three identical Urns I, ll and III contain 4 white and 3 red balls, 3 white and 7 red balls and 2 white and 3 red balls respectively. An Urn is chosen at random and a ball is drawn out of it at random. If the ball is found to be white, what is the probability that Urn ll was selected? |
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Answer» `P(U_1)=1/3` `P(U_2)=1/3` `P(U_3)=1/3` `P(W/U_1)=4/7` `P(W/U_2)=3/10` `P(W/U_3)=2/5` `P(U_2/W)=(P(U_2)*P(W/U_2))/(P(U_1)*P(W/U_1)+P(U_2)*P(W/U_2)+P(U_3)*P(W/U_3))` `=(1/3*3/10)/(1/3*4/7+1/3*3/10+1/3*2/5)` `=21/89`. |
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| 336. |
From a rectangular sheet of uniform density a reactangle is cut out as shown in figure. Find new co-ordinate of center of mass `w.r.t` orign A. `((1)/(12),(b)/(12))`B. `((5a)/(12),(5b)/(12))`C. `((7a)/(12),(7b)/(12))`D. none of these |
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Answer» Correct Answer - B |
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| 337. |
`+HCHOunderset((ii).H_(2)O)overset((i). 50%NaOH)to` productA. `+HCOOH`B. `+HCOOH`C. `CH_(3)OH+HCOOH`D. `+CH_(3)OH` |
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Answer» Correct Answer - B |
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| 338. |
The correct rate of reaction of given compounds towards electrophilic aromatic substitution reaction isA. `bgtdgtagtc`B. `bgtagtdgtc`C. `cgtagtdgtb`D. `agtbgtdgtc` |
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Answer» Correct Answer - A |
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| 339. |
The major product formed in the following reaction is: A. B. C. D. |
| Answer» Correct Answer - A | |
| 340. |
Which among the following is the most reactive gtA. `Br_(2)`B. `I_(2)`C. `Icl`D. `Cl_(2)` |
| Answer» Correct Answer - C | |
| 341. |
An ideal gas has molecules with 5 degrees of freedom. The ratio of specific heats at constant pressure `(C_(p))` and at constant volume `(C_(v))` isA. `7/5`B. 6C. `7/2`D. `5/2` |
| Answer» Correct Answer - A | |
| 342. |
Metal with low melting point containing impurities of high melting point can be purified byA. Zone refiningB. Vapor phase refiningC. DistillationD. Liquation |
| Answer» Correct Answer - D | |
| 343. |
Two sides of a rhombus ABCD are parallel to the lines y = x + 2 and y = 7x + 3 If the diagonals of the rhombus intersect at the point (1, 2) and the vertex A is on the y-axis, then vertex A can be |
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Answer» y=x+2 slope=1 y=7x+3 slope=7 O(1,2)=`((0+m)/2,(n+alpha)/2)=((x+a)/2,(b+y)/2)` m=2 x+a=2 `n+alpha=4 b+y=4` `m_(ab)=(b-alpha)/(a-0)=1=b-alpha=a` `m_(bc)=(b-n)/(a-2)=7=b-n=7(a-2)` AB=BC `sqrt((b-alpha)^2+(a-0)^2)=sqrt((b-n^2+(a-2)^2)` solving this `a=5/2,5/3` `b=a+alpha=5/2+alpha,5/3+alpha` `B(5/2,5/2+alpha),B(5/3,5/3+alpha)` A(0,alpha) O(1,2) `m_(OA)*m_(OB)=-1` `(2-alpha)/(1-0)*((5/2)+alpha-2)/((5/2)-1)=-1` solving this `alpha=(0,3)` A=`(0,alpha)` A=(0,0),(0,3) |
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| 344. |
let `theta in[-2pi,2pi]` and `2cos^(2)theta+3sintheta=0` then sum of all solution isA. `2pi`B. `3pi`C. `pi`D. `(pi)/(3)` |
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Answer» Correct Answer - A |
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| 345. |
Magnetic field in a plane electromagnetic wave is given by `bar(B) = B_(0)"sin"(kx + omegat)hat(j)T` Expression for corresponding electric field will beA. `vec(E) oo -B_(0)c"sin" (kx +omegat)hat(k)V//m`B. `vec(E) = B_(0)c"sin"(kx - omegat)hat(k)V//m`C. `vec(E) = B_(0)c"sin"(kx + omegat)hat(k) V//m`D. `vec(E) = (B_(0))/(c)"sin" (kx + omegat) hat(k)V//m` |
| Answer» Correct Answer - C | |
| 346. |
Magnetic field at the center (at nucleus) of the hydrogen like atom `("atomic number" = z)` due to the motion of electron in nth orbit is proporional toA. `n^(-3)`B. `n^(-2)`C. `n^(-4)`D. `n^(-5)` |
| Answer» Correct Answer - B | |
| 347. |
Which of the following metal is purified by using moonds process.A. `Ni`B. `Zr`C. `Ti`D. `Cu` |
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Answer» Correct Answer - A |
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| 348. |
Find out `%` strength of 11.2V `H_(2)O_(2)`A. `34%`B. `3.4%`C. `1.7%`D. `13.8%` |
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Answer» Correct Answer - B |
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| 349. |
When radius of central atom is double with respect to corner atoms, then find out `%` packing efficiencyA. `75%`B. `90%`C. `60%`D. `65%` |
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Answer» Correct Answer - B |
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| 350. |
Which compound to use for treatment of tumor?A. `Cis[PdCl_(2)(NH_(3))_(2)]`B. Trans `[PdCl_(2)(NH_(3))_(2)]`C. `Cis[PtCl_(2)(NH_(3))_(2)]`D. Trans `[PtCl_(2)(NH_(3))_(2)]` |
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Answer» Correct Answer - C |
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