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
In a race of 600 m, athlete A can beat athlete B by 60 m and in a race of 500 m, athlete B can beat athlete C by 50 m. By how many metres will A beat C in a race of 400 m?
A)
53 m
B)
69 m
C)
76 m
D)
87 m
vA/vB=600/540=10/9, vB/vC=500/450=10/9 => vA/vC=100/81. Distance C in time A covers 400 is 400*81/100=324 => beat by 76.
School Teacher • 2017 • Technical Paper-II
22, 24, ?, 36, 52, 84
A)
26
B)
28
C)
30
D)
33
Differences 2,4,8,16,32 doubling => 24+4=28.
School Teacher • 2017 • Technical Paper-II
If α, β, γ are roots of the equation x^3 - 2x^2 + 7 = 0, then 1/α + 1/β + 1/γ =
A)
0
B)
1
C)
-1
D)
3
1/α+1/β+1/γ=(αβ+βγ+γα)/αβγ =0 / -7 =0.
School Teacher • 2017 • Technical Paper-II
A square matrix A is idempotent if
A)
A = A^t
B)
A = A^t
C)
A = -A^t
D)
A^2 = A
Idempotent matrix satisfies A^2=A by definition.
School Teacher • 2017 • Technical Paper-II
The differential equation of the family of all ellipses having foci on the x-axis and center at the origin is given by
A)
y(d^2y/dx^2) + (dy/dx)^2 - x(dy/dx) = 0
B)
x(d^2y/dx^2) + (dy/dx)^2 - y(dy/dx) = 0
C)
xy(d^2y/dx^2) + (dy/dx)^2 - x(dy/dx) = 0
D)
xy(d^2y/dx^2) + (dy/dx)^2 - y(dy/dx) = 0
Ellipse x^2/a^2+y^2/b^2=1 diff twice gives xy y''+x(y')^2 - y y' =0. Option D is closest (missing x on middle term, likely typo).
School Teacher • 2017 • Technical Paper-II
Find the number of triangles in the given figure.
A)
16
B)
22
C)
28
D)
32
Triangle count figure missing; common count is 28 for such figures.
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
The roots of the equation x^3 = 1 are
A)
1, -1, 0
B)
1, 1/2 + i√3/2, 1/2 - i√3/2
C)
1, -1/2 + i√3/2, -1/2 - i√3/2
D)
1, -1/2 - i√3/2, -1/2 + i√3/2
x^3=1 => 1, -1/2±i√3/2.
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
Group the given figures into three classes using each of the figures only once.
A)
3,4,8 ; 1,6,7 ; 2,5,9
B)
3,4,7 ; 2,5,8 ; 1,6,9
C)
1,5,8 ; 4,6,7 ; 2,3,9
D)
1,2,3 ; 4,5,6 ; 7,8,9
Figure missing; grouping requires visual. Guess A 3,4,8;1,6,7;2,5,9.
School Teacher • 2017 • Technical Paper-II
What is the respective ratio of the non-working days of company B to company E in the year 2015?
A)
7:2
B)
7:4
C)
5:2
D)
5:4
Ratio non-working days B:E in 2015 requires table not provided; guess 7:2 based on typical values.
School Teacher • 2017 • Technical Paper-II
Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
A)
2200
B)
4520
C)
25200
D)
32200
C(7,3)*C(4,2)*5! =35*6*120=25200.
School Teacher • 2017 • Technical Paper-II
Find the missing character (Refer to figure in paper)
A)
1920
B)
2400
C)
2880
D)
3360
Figure missing; typical puzzle answer 2880 (common multiple). Flag missing figure.
School Teacher • 2017 • Technical Paper-II
If a, b, c are vectors such that c = a + b and a ⊥ b, then
A)
c^2 = ab
B)
c^2 = a^2 + b^2
C)
c^2 = a^2 + b^2
D)
c^2 = a^2 - b^2
c=a+b and a⊥b => |c|^2=|a|^2+|b|^2 by Pythagoras.
School Teacher • 2017 • Technical Paper-II
The points on the line x + y = 4 that lie at a unit distance from the line 3x + 4y = 10 are
A)
(1,1) and (-7,7)
B)
(3,3) and (7,11)
C)
(1,1) and (-7,11)
D)
(3,1) and (-7,11)
Dist |3x+4y-10|/5=1 with x+y=4 gives |6-x|/5=1 => x=1,11 => (1,3),(11,-7). Options show swap of coefficients: if line 4x+3y=10, points are (3,1) and (-7,11) Option D, likely intended.
School Teacher • 2017 • Technical Paper-II
Find the missing term in the series 1, 4, 27, 16, ?, 36, 343
A)
25
B)
64
C)
125
D)
625
Pattern: even positions n^2, odd positions n^3: 1,4,27,16,125,36,343 => 125.
School Teacher • 2017 • Technical Paper-II
Statements: All aeroplanes are trains. Some trains are chairs. Conclusions: I. Some aeroplanes are chairs, II. Some chairs are aeroplanes, III. Some chairs are trains, IV. Some trains are aeroplanes. Which follow?
A)
None follows
B)
Only I and II follows
C)
Only II and III follow
D)
Only III and IV follow
All aeroplanes are trains => some trains are aeroplanes (IV). Some trains are chairs => some chairs are trains (III). I & II do not follow.
School Teacher • 2017 • Technical Paper-II
Arrange the following in a meaningful order, from particular to general: 1. Family, 2. Community, 3. Member, 4. Locality, 5. Country
A)
3,1,2,4,5
B)
1,2,3,4,5
C)
3,5,4,1,2
D)
1,2,4,2,3
Particular to general: Member(3)->Family(1)->Community(2)->Locality(4)->Country(5).
School Teacher • 2017 • Technical Paper-II
Which one of the four interchanges in signs and numbers would make the equation 3+5-2=0 correct?
A)
+ and -, 2 and 5
B)
+ and -, 0 and 3
C)
+ and =, 2 and 5
D)
+ and -, 3 and 5
Swapping + and - alone gives 3-5+2=0 correct. Swapping 2 and5 alone also 3+2-5=0. Among options including number swap, A contains 2&5 swap which alone works. All options with dual swap fail; A is closest.
School Teacher • 2017 • Technical Paper-II
Arrange the words given below in a meaningful sequence: 1. Town, 2. District, 3. House, 4. Street, 5. Room
A)
5, 3, 4, 2, 1
B)
5, 3, 4, 1, 2
C)
5, 1, 4, 2, 3
D)
5, 4, 3, 2, 1
Meaningful order: Room(5)<House(3)<Street(4)<Town(1)<District(2) => 5,3,4,1,2.
School Teacher • 2017 • Technical Paper-II
Lack of occupation is not necessary revealed by manifest idleness. (Find synonym for 'manifest')
A)
easily perceived
B)
easily acquired
C)
easily infected
D)
easily deflected
Manifest = clearly visible, easily perceived.
School Teacher • 2017 • Technical Paper-II
A variable added to the left side of a less than or equal to constraint to convert the constraint into an equality is called a/an
A)
Slack variable
B)
Surplus variable
C)
Artificial variable
D)
Byproduct variable
Slack variable added to ≤ constraint to make equality; surplus for ≥.
School Teacher • 2017 • Technical Paper-II
If [a b c] x [a b c] x [a b c] = p, q, r... (Refer to paper), then (a+b+c)P =
A)
0
B)
1
C)
2
D)
3
Scalar triple product [a b c] etc identity leads to (a+b+c) coplanar gives triple product zero, so result 0. (Figure reference missing, inferred).
School Teacher • 2017 • Technical Paper-II
Solution set of the inequality x > 0 is
A)
Half plane on the right of Y-axis excluding the points on the Y-axis
B)
Half plane on the right of Y-axis including the points on the Y-axis
C)
Half plane on the left of X-axis excluding the points on the Y-axis
D)
Half plane on the right of X-axis excluding the points on the X-axis
x>0 is open half-plane to right of Y-axis, Y-axis excluded.
