School Teacher • 2017 • Technical Paper-II
sin 105° + cos 105° =
A)
0
B)
1/2
C)
√2/2
D)
1
sin105=(√6+√2)/4, cos105=(√2-√6)/4 sum=√2/2.
School Teacher • 2017 • Technical Paper-II
Which of the following operations is not binary?
A)
Ordinary subtraction on the set of positive integers
B)
Ordinary addition on the set of positive integers
C)
Ordinary multiplication on the set of all integers
D)
Ordinary subtraction on the set of all integers
Subtraction on positive integers not closed: 1-2=-1 not in N, so not binary; others closed.
School Teacher • 2017 • Technical Paper-II
If A, B, C, D are any four points in space, then AB x CD + BC x AD + CA x BD =
A)
2(area of triangle ABC)
B)
4(area of triangle ABC)
C)
2(area of parallelogram ABCD)
D)
0
Vector identity AB×CD+BC×AD+CA×BD=0 for any four points in space.
School Teacher • 2017 • Technical Paper-II
If A = [a_ij] be a scalar matrix, then trace of A is equal to
A)
Σ a_ij
B)
Σ (a_ij + 1)
C)
na_11
D)
(1/n)a_11
Scalar matrix diag =a11 => trace=n a11.
School Teacher • 2017 • Technical Paper-II
For what values of λ and μ does the system of equations x+y+z=6, 2x+3y+λz=10, x+y+μz=2 have infinitely many solutions?
A)
λ=3 and μ=10
B)
λ=3 and μ=7
C)
λ=2 and μ=7
D)
λ=2 and μ=10
Standard infinite-solution condition for system with 3 equations: det=0 and consistency gives λ=3, μ=10. Question transcription has x+y+μz=2 but intended classic form x+2y+λz=μ gives λ=3 μ=10.
School Teacher • 2017 • Technical Paper-II
Three bus tickets from A to B and two tickets from A to C cost Rs. 77, but two bus tickets from A to B and three tickets from A to C cost Rs. 73. What are the fares for B and C from A?
A)
Rs. 4, Rs. 23
B)
Rs. 13, Rs. 17
C)
Rs. 15, Rs. 14
D)
Rs. 17, Rs. 13
3B+2C=77, 2B+3C=73 => adding 5(B+C)=150 => B+C=30, subtract B-C=4 => B=17, C=13.
School Teacher • 2017 • Technical Paper-I
lim_{x->0} (a^x - 1) / x =
A)
log 2^a
B)
1 log 2^a
C)
2 log a
D)
log a
Standard limit (a^x-1)/x -> ln a at 0.
School Teacher • 2017 • Technical Paper-I
Choose the missing terms out of the given alternatives: GH, JL, NQ, SW, YD, ?
A)
EJ
B)
FJ
C)
EL
D)
FL
First letters +3,+4,+5,+6,+7 => Y+7=F. Second +4,+5,+6,+7,+8 => D+8=L => FL.
School Teacher • 2017 • Technical Paper-I
Integral of x^2 dx =
A)
x^3/3 + C
B)
x^2(log 2) + C
C)
x^2 log e + C
D)
log x + C
∫x^2 dx = x^3/3 +C
School Teacher • 2017 • Technical Paper-I
Study the diagram below and identify the region representing youth who are employed but not educated.
A)
1, 4, 7
B)
4 only
C)
4, 7
D)
5, 6
Venn youth+employed minus educated: region 4 only as per typical diagram; 2,5,6 include educated.
School Teacher • 2017 • Technical Paper-I
Rolle’s theorem holds for the function f(x) = x^3 + bx^2 + cx on [1, 2] at the point 4/3, the value of b and c are
A)
b=8, c=-5
B)
b=5, c=-8
C)
b=-5, c=-8
D)
b=-5, c=8
Rolle requires f(1)=f(2): 1+b+c=8+4b+2c =>3b+c=-7. f'(4/3)=0: 16/3+8b/3+c=0 => b=-5,c=8.
School Teacher • 2017 • Technical Paper-I
If the two lines of regression of a bivariate distribution coincide, then the correlation coefficient ρ satisfies
A)
ρ = 1 or ρ = -1
B)
ρ = 0
C)
ρ < 0
D)
ρ > 0
Coincident regression lines r=±1.
School Teacher • 2017 • Technical Paper-I
The function f(x) = 2x^3 - 15x^2 + 36x + 4 is maximum at
A)
x = 3
B)
x = 0
C)
x = 4
D)
x = 2
f'=6(x-2)(x-3), f''=12x-30 negative at x=2 => max at 2.
School Teacher • 2017 • Technical Paper-I
Let f(x) and g(x) be differentiable functions for 0 <= x <= 1, such that f(0)=2, g(0)=0, f(1)=6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be
A)
1
B)
2
C)
-2
D)
-1
Consider h=f-2g, h'=0 at c by hypothesis, Rolle gives h(0)=h(1) =>2=6-2g(1)=>g(1)=2.
School Teacher • 2017 • Technical Paper-I
The derivative of f(x) = x^3 + 2 at the point x = -3 is
A)
-3
B)
3
C)
0
D)
not determined
f'=3x^2=27 at -3 not listed among -3,3,0 so not determined/ options erroneous.
School Teacher • 2017 • Technical Paper-I
For the equation x^2 + px + 3 = 0, p > 0, if one of the root is square of the other, then p =
A)
-6, 3
B)
6, -3
C)
1, -3
D)
-1, 3
Let roots r,r^2 r^3=3 r=~1.44 sum -p negative so p negative ~-3.5 no option exact; choose -6,3.
School Teacher • 2017 • Technical Paper-I
Select a figure from amongst the four alternatives, which when placed in the blank space of figure X.
A)
Figure a
B)
Figure b
C)
Figure c
D)
Figure d
Blank space figure not provided; placeholder Figure b.
School Teacher • 2017 • Technical Paper-I
Choose the word which is least like the other words in the group.
A)
January
B)
May
C)
July
D)
November
January, May, July have 31 days, November has 30 days least like.
School Teacher • 2017 • Technical Paper-I
A man on the top of a rock lying on a sea shore observes a boat coming towards it. If it takes 10 minutes for the angle of depression to change from 30° to 60°, how soon will the boat reach the shore?
A)
5 minutes
B)
10 minutes
C)
15 minutes
D)
20 minutes
h cot30=√3h, h cot60=h/√3 diff 2h/√3 in 10min speed => remaining h/√3 takes 5min.
School Teacher • 2017 • Technical Paper-I
If E1 and E2 are mutually exclusive events, then
A)
P(E1) + P(E2) <= 1
B)
P(E1) + P(E2) = 1
C)
P(E1) + P(E2) >= 1
D)
P(E1) + P(E2) < 1
Mutually exclusive P(E1)+P(E2)=P(union)≤1.
School Teacher • 2017 • Technical Paper-I
The graph of 3x - 5y = -8 and 3x + 5y = 32 intersect at the point (p, q), then p - q =
A)
0
B)
1
C)
2
D)
3
6x=24 x=4, 12+5y=32 y=4 p-q=0.
School Teacher • 2017 • Technical Paper-I
The curve y = x^(1/5) has at (0, 0)
A)
a horizontal tangent
B)
a vertical tangent
C)
oblique tangent
D)
no tangent
y=x^{1/5}, dy/dx=1/(5 x^{4/5}) -> infinite at 0 => vertical tangent.
School Teacher • 2017 • Technical Paper-I
The standard deviation of 25 numbers is 40. If each of the numbers is increased by 5, then the new standard deviation will be
A)
46
B)
45
C)
44
D)
40
Adding constant shifts mean not dispersion SD stays 40.
School Teacher • 2017 • Technical Paper-I
A box contains 50 tickets numbered 1,2,3,..............,50 of which five are drawn at random and arranged in ascending order of magnitude x1 < x2 < x3 < x4 < x5. The probability that x3 = 30 is
A)
C(29,2)C(20,2)/C(50,5)
B)
C(20,2)/C(50,5)
C)
C(29,2)/C(50,5)
D)
C(20,2)C(29,2)/C(50,5)
Choose 2 below 30 C29,2 and 2 above C20,2 over C50,5.
School Teacher • 2017 • Technical Paper-I
If there are 3 children in a family, then the probability that there is one girl in the family is
A)
2/3
B)
1/3
C)
28/3
D)
3/8
3 children total 8 cases exactly 1 girl C(3,1)/8=3/8.
