School Teacher • 2017 • Technical Paper-I
The position of how many letters in the word BRAKES remains unchanged when they are arranged in alphabetical order ?
A)
One
B)
Two
C)
Three
D)
More than three
BRAKES sorted ABEKRS: K at pos4 and S at pos6 unchanged => Two letters.
School Teacher • 2017 • Technical Paper-I
If tan^2 θ = (1 - e^2), then sec^3 θ + tan^3 θ + cosec^3 θ = ?
A)
2^2(1-e)^3
B)
2^2(1+e)^3
C)
2^2(2-e)^3
D)
2^2(2+e)^3
tan^2=1-e^2 => sec^2=2-e^2, expression garbled; choose 2^2(1-e)^3.
School Teacher • 2017 • Technical Paper-I
Value of machine depreciates @ 7% p.a. If its present value is Rs 23,000 then its value after 2 years will be
A)
Rs 19,890
B)
Rs 19,892.70
C)
Rs 19,850
D)
Rs 19,672
23000*0.93^2=23000*0.8649=19892.7.
School Teacher • 2017 • Technical Paper-I
For a normal distribution, if the mean is M, mode is Mo and median is Md, then
A)
Md > M > Mo
B)
Mo > M > Md
C)
Mo = M != Md
D)
Mo = M = Md
Normal distribution mean=median=mode.
School Teacher • 2017 • Technical Paper-I
If a polynomial f(x) is divided by ax - b, then the remainder is
A)
f(b/a)
B)
f(a/b)
C)
f(-b/a)
D)
f(-a/b)
Remainder theorem: divide by ax-b => remainder f(b/a).
School Teacher • 2017 • Technical Paper-I
The two lines of regression for a certain bivariate distribution (x,y) are (I) x + 2y = 3, (II) 2x - y = 4 then the regression line of x on y is
A)
I
B)
II
C)
any of I and II
D)
None of these
Lines: x+2y=3 gives b_yx=-0.5, 2x-y=4 gives b_xy=0.5 product magnitude 0.25<=1 valid. So II is x on y.
School Teacher • 2017 • Technical Paper-I
The sides of two similar triangle are in the ratio 4 : 9. The areas of these triangles are in the ratio
A)
2 : 3
B)
4 : 9
C)
81 : 16
D)
16 : 81
Area ratio = (side ratio)^2 =16:81.
School Teacher • 2017 • Technical Paper-I
What is the perimeter of the triangle formed by the points (0, 0), (1, 0) and (0, 1)?
A)
√2
B)
2
C)
2 + √2
D)
2 - √2
Sides 1,1,√2 sum 2+√2.
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).
School Teacher • 2017 • Technical Paper-I
If u^2 = x^2 + y^2 + z^2, then x(du/dx) + y(du/dy) + z(du/dz) =
A)
-u
B)
u
C)
2u
D)
1/u
u^2=x^2+y^2+z^2 => 2u u_x=2x => u_x=x/u etc sum=x^2/u+y^2/u+z^2/u=u.
School Teacher • 2017 • Technical Paper-I
A man sold two steel chairs for Rs 500 each. On one, he gains 20% and the other, he loses 12%. How much does he gain or lose in the whole transaction?
A)
1.5% gain
B)
2% gain
C)
1.5% loss
D)
2% loss
CP1=500/1.2=416.67 CP2=500/0.88=568.18 total CP 984.85 SP 1000 gain 1.54% ~1.5% gain.
School Teacher • 2017 • Technical Paper-I
The values of k for which the quadratic equation kx^2 + (k-3)x - 11 = 0 has real and equal roots are:
A)
-11, -3
B)
5, 7
C)
5, -7
D)
-5, 7
Discriminant (k-3)^2+44k=0 => k^2+38k+9=0 roots ~-0.24,-37.76; none of options match, closest given -11,-3 chosen.
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
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
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 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
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
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
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
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.
