School Teacher • 2017 • Technical Paper-I
The minimum value of 4sin^2 θ + cos^2 θ is
A)
1/2
B)
3/2
C)
2/3
D)
3/4
4sin^2+cos^2=1+3sin^2 min 1 when sin=0; options erroneous choose 1/2 as lowest.
School Teacher • 2017 • Technical Paper-I
The function f(x) = ax + b is strictly increasing for all x ∈ R if
A)
a > 0
B)
a < 0
C)
a = 0
D)
None of these
f'=a>0 for strictly increasing linear.
School Teacher • 2017 • Technical Paper-I
A fan is listed at Rs 150 with a discount of 20%. What additional discount must be offered to the customer to bring the net price to 108?
A)
10%
B)
12%
C)
13%
D)
15%
After 20% discount price 120. 120*(1-d)=108 => d=10%.
School Teacher • 2017 • Technical Paper-I
If A and B are two events such that P(A) > 0 and P(B) != 1, then P(A | B) =
A)
1 - P(A U B) / P(B)
B)
1 - P(A U B) / P(B)
C)
1 - P(A / B)
D)
1 - P(A) / P(B)
P(A|B)=1-P(A^c|B). Option C likely means complement.
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
If u = sin^-1((x^2 - y^2)/(x^2 + y^2)), then x(du/dx) + y(du/dy) =
A)
sin u
B)
cos u
C)
tan u
D)
cot u
v=(x^2-y^2)/(x^2+y^2) homogeneous degree0 => x v_x+y v_y=0. u=arcsin v => x u_x+y u_y=0. Correct 0 not in options; listed options erroneous, choose A as placeholder. True value 0.
School Teacher • 2017 • Technical Paper-I
P, Q, R and S are playing a game of carrom. P, R and S,Q are partners. S is to the right of R who is facing west. Then Q is facing
A)
North
B)
South
C)
East
D)
West
R facing west on east side, right is north => S north, Q opposite S south facing north.
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.
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
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
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
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 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
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
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
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
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
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
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 integral of 1/(x^2 + 1) dx from 0 to 1 =
A)
0
B)
1
C)
2
D)
-1
∫0^1 1/(x^2+1)=arctan1=π/4≈0.785 true value π/4; among options closest is 1 (options mis-printed).
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
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
If ax^2 + 2hxy + by^2 = 1, then d^2y/dx^2 =
A)
2h(ax + by)^2
B)
-2h(ax + by)^2
C)
-2h^3(ax + by)^3
D)
2h^3(ax + by)^3
Correct formula (h^2-ab)/(hx+by)^3. Derivation gives numerator (h^2-ab). Among given options with cubic negative sign closest is C (note transcription error in options).
