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 differential equation of the family of circles x^2 + y^2 + gx + fy + c = 0 is
A)
d^2y/dx^2 + (dy/dx)^3 = 0
B)
d^2y/dx^2 + 3(dy/dx)^2 = 0
C)
d^2y/dx^2 - 3(dy/dx) = 0
D)
d^2y/dx^2 + (dy/dx)^2 = 0
General circle x^2+y^2+2gx+2fy+c=0 actually gives third-order ODE (1+y'^2)y'''=3y'y''^2. Among second-order options given, y''+(y')^3=0 (A) is often listed in simplified forms for circles touching axes; question erroneous.
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
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
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
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
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.
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
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
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
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
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
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
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
Objective function of a LPP is
A)
A function to be optimized
B)
A constant
C)
A relation between the variables
D)
A constraint to which the LPP is subjected to
Objective function is function to be optimized (max/min) in LPP.
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
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
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
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
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
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
Shown below are four different positions of the same dice. What is the number on the face opposite the face showing 6?
A)
1
B)
2
C)
3
D)
4
Dice figure missing; typical opposite of 6 is 2 in such patterns. Guess B.
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
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
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.
