OAVS TGT Math Mock Test - 6 - OTET MCQ

Let one root be a, other root is a².
(x-a)(x-a²)=x²+(-a²-a)x+a³
p³=(-a²-a)³=-a⁶-3a⁵-3a⁴-a³
-q(3p-1)= -a³(3(-a²-a)-1)=3a⁵+3a⁴+a³
q²=a⁶
Substituting the values
-a⁶-3a⁵-3a⁴-a³+3a⁵+3a⁴+a³+a⁶=0

Let Sharma’s present age be x years then his age 3 years ago is (x−3) years and 5 years from now is (x+5) years. According to question,

But x = 3 does not satisfy the given condition.
Therefore, Sharma's present age is 7 years. 

Smallest prime number = 2 and smallest composite number = 4 
∴ HCF (2, 4) = 2

= 1 / √3 x 1 / √3

= 1 / 3.

clearly first term is 4/3
for common difference 1/3-4/3=-1

Area of the sector

⇒ Area of the sector 
= 462 sq. cm

tan 1.tan 2.tan 3...tan (90 - 3 ).tan ( 90 - 2 ).tan ( 90 - 1) 
=tan 1.tan 2 .tan 3...cot 3.cot 2.cot 1 
=tan 1.cot 1.tan 2.cot 2.tan 3.cot 3 ... tan 89.cot 89 
1 x 1 x 1 x 1 x ... x 1 =1

Solution :- sin^2 30° − cos^2 30°

= (1/2)² −((3)^1/2/2)²

= 1/4 - 3/4

= -1/2

The side BC is congruent to side EF.

∠LQ = ∠MNR,

∠Q = ∠R

QM = MR,

Hence, ΔQLM ≅ ΔMNR (by AAS)

According to the question, p (2) = 3x² + mx - 14 = 0
⇒ 3(2)² + m x 2 - 14 = 0
⇒ 12 + 2m - 14 = 0 ⇒ m = 1
 Also p(2) = 2x³ + nx² + x - 2 = 0

(a) The given equation is x² - 4x + 3√2 = 0.
On comparing with ax² + bx + c = 0, we get
a = 1, b = -4 and c = 3√2
The discriminant of x² - 4x + 3√2 = 0 is
D = b² - 4ac
= (-4)² - 4(1)(3√2) = 16 - 12√2 = 16 - 12 x (1.41)
= 16 - 16.92 = -0.92
⇒ b² - 4ac < 0
(b) The given equation is x² + 4x - 3√2 = 0
On comparing the equation with ax² + bx + c = 0, we get
a = 1, b = 4 and c = -3√2
Then, D = b² - 4ac = (-4)² - 4(1)(-3√2)
= 16 + 12√2 > 0
Hence, the equation has real roots.
(c) Given equation is x² - 4x - 3√2 = 0
On comparing the equation with ax² + bx + c = 0, we get
a = 1, b = -4 and c = -3√2
Then, D = b² - 4ac = (-4)² - 4(1) (-3√2)
= 16 + 12√2 > 0
Hence, the equation has real roots.
(d) Given equation is 3x^​2 + 4√3x + 4 = 0.
On comparing the equation with ax² + bx + c = 0, we get
a = 3, b = 4√3 and c = 4
Then, D = b² - 4ac = (4√3)² - 4(3)(4) = 48 - 48 = 0
Hence, the equation has real roots.
Hence, x² - 4x + 3√2 = 0 has no real roots.

Taking different values of n we find that A and B are coprime

e.g. put n=1

we get A= 2(1)+13=15

B=1+7=8
∴ HCF = 1
 

put n=2

we get A= 2(2)+13=17

B=2+7=9
∴ HCF = 1

Given, x³ + 3x² + 3x + a = 0

and its factor is x + 1 = 0 

⇒ x = -1

Put the value in equation i.e. x³ + 3x² + 3x + a = 0

it becomes = (-1) + 3 (1) + 3(-1) + a = 0

⇒ -1 + 3 - 3 + a = 0

⇒ a = 1

In the thirteen books that comprise Euclid's Elements there is a total of 465 propositions.

We have 
Taking LCM,

Multiplying LHS and RHS by x
x²+1=4x
x²-4x+1=0
Which is the required equation.

Let x = the length of a side of the square

So area of square = x² = 50cm²

The correct answer is a.

From the polynomial we get,

α + β = - b/a = -(-5/15) = 1/3

αβ = c/a = (6/15) = 2/5

To find: (1+1/α) (1+1/β)...... (taking L.C.M) 

= (α+1/α) (β+1/β) ....(simplifying brackets)

= αβ + α + β + 1 / αβ

=( 2/5 + 1/3+ 1)  /  2/5 

=26/15 X 5/2 = 13/3

In a quadratic equation with rational coefficients has an irrational root  α + √β, then it has a conjugate root α - √β.
So if the root is m+ √n the other root will be m- √n

If the graph of a polynomial intersects the x-axis at three points, then the number of zeroes are 3 because number of zeroes of the polynomial are the number of the coordinates of the points where its graph intersects the x-axis.

The quadratic equation is of the form  x² - (sum of zeros) x + (product of zeros)
=x² - 5x - 14

 Probability facts.

a = pq²

b = p³q

LCM (a,b) = p³q²

For the relation x = qy+r, 0 ⩽ r < y So, here r lies between 0 ⩽ r < 6. Hence r = 0, 1, 2, 3, 4, 5