School Teacher • 2017 • Technical Paper-II
A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has?
A)
30
B)
45
C)
60
D)
90
16n=480 => n=30 each => total 90.
School Teacher • 2017 • Technical Paper-II
Select a suitable figure from the Answer Figures that would replace the question mark (?)
A)
1
B)
2
C)
3
D)
4
Figure-based question: image missing. Typical pattern suggests option 2. (No figure to verify).
School Teacher • 2017 • Technical Paper-II
The order and degree of the differential equation (dy/dx)^5 - x(y dy/dx) = 0 is respectively
A)
1 and 2
B)
2 and 2
C)
2 and 3
D)
1 and 5
Order is highest derivative =1, degree is power 5 of dy/dx.
School Teacher • 2017 • Technical Paper-II
Which of the following sets is convex?
A)
{ (x,y) : x^2 + y^2 > 6 }
B)
{ (x,y) : 2x^2 + 3y^2 ≤ 6 }
C)
{ (x,y) : 4/9 ≤ x^2 + y^2 ≤ 1 }
D)
{ (x,y) : x^2 + y^2 ≥ 3/4 }
Ellipse interior 2x^2+3y^2≤6 is convex; exterior and annulus are non-convex.
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
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
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
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 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
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
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
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-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
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
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
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
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
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
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
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 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
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.
