1. Which of the following is NOT a rational number?
A) 3/7
B) 0.333...
C) √9
D) √7 ✓
Explanation:
D) √7:- The square root of 7 is a non-terminating and non-repeating decimal. It cannot be expressed in the form of p/q. Therefore, it is an irrational number, and thus not a rational number.
2.Which of the following is not a polynomial?
A) x2+3x+5
B) 4x3−7x+2
C) 1/x +2x ✓
D) x4+x2+1
Correct Answer:- C) 1/x +2x
Explanation:-
A polynomial is an expression where the powers (exponents) of the variable(s) are all non-negative integers (i.e., 0, 1, 2, 3, ...).
Options A, B, and D are all polynomials because the powers of xxx in each term are whole numbers (non-negative).
Option C contains 1/x, which is equivalent to x^{-1}, and this is a negative power. Hence, it is not a polynomial.
3. In a pair of linear equations, what does the "Elimination Method" mean?
A. Eliminating one variable to find the other ✓
B. Drawing a graph
C. Adding the values
D. Classifying the two equations
Correct Answer:-
A. Eliminating one variable to find the other
Explanation :-
Eliminate" means to remove or get rid of something.
In this method, we eliminate one variable from the equations.
By doing so, we can find the value of the other variable.
Then, we substitute that value into one of the original equations to find the second variable as well.
Example
x + y = 10 ---(1)
x – y = 4 ---(2)
---------
2x=14.
--------
x=14/2
x=7
x+y=10
7+y=10
y=10-7
y =3
x=7, y=3
4. If the sum and product of roots are 5 and 6 respectively, then the equation is:
A) x² – 5x + 6 = 0 ✓
B) x² + 5x + 6 = 0
C) x² – 6x + 5 = 0
D) x² – 11x + 30 = 0
Correct Answer:-
x² – 5x + 6 = 0
Sum of roots (α + β) = 5
Product of roots (α × β) = 6
Quadratic equation with roots α and β is always formed as:
x^2 - (α+β)x + (αβ) = 0
α + β = 5
α × β = 6
quation:-
x^2 - (5)x + 6 = 0
x^2 - 5x + 6 = 0
Verification :-
Let's see what the roots of this equation are:
x^2 - 5x + 6 = 0
Factoring:
(x - 2)(x - 3) = 0
x = 2, 3
Sum = 2 + 3 = 5
Product = 2 × 3 = 6
5. In an arithmetic progression, the 4th term is 18 and the 7th term is 30. What is the first term (a)?
A. 6 ✓
B. 8
C. 10
D. 12
Solution:-
We know that:
A₄ = a + 3d = 18
A₇ = a + 6d = 30
Subtracting the two equations:
(a + 6d) – (a + 3d) = 30 – 18
⇒ 3d = 12
⇒ d = 4
Now,
A₄ = a + 3×4 = 18 ⇒ a + 12 = 18 ⇒ a = 6
Correct answer: A) 6
6. A product costs ₹600. First, a 10% discount is given. After that, another 20% discount is applied. What is the final price?
A) Rs.420
B) Rs.432
C) Rs.440
D) Rs.450
Original price = ₹600
First 10% discount on ₹600 =
600 × 10/100 = ₹60
Price after the first discount =
600 – 60 = ₹540
Now, on the remaining ₹540, a 20% discount =
540 × 20/100 = ₹108
Price after the second discount =
540 – 108 = ₹432
So, after both discounts, the final price of the product = ₹432.
7. For an angle, if cosθ= 3/5 (in the first quadrant), what is tanθ=?
A)3/4
B)4/3
C)5/3
D)3/5
Correct Answer:- B) 4/3
sin²θ + cos²θ = 1
sin²θ=1-cos²θ
(cosθ=3/5 => cos²θ=9/25)
sin²θ=1-9/25
sin²θ=16/25
sinθ=√16/25
sinθ=4/5
tanθ=sinθ/cosθ
tanθ=(4/5)/(3/5)
tanθ=4/5*5/3
tanθ=4/3
8.Sita walks 4 km east, then 3 km north. How far is she from the starting point?
సీత 4 కి.మీ తూర్పు వైపు నడిచింది, తర్వాత 3 కి.మీ ఉత్తర దిశలో నడిచింది. ఇప్పుడు ఆరంభ స్థానం నుంచి ఆమె ఎంత దూరంలో ఉంది?
A) 5 km (5 కి.మీ)
B) 6 km (6 కి.మీ)
C) 7 km (7 కి.మీ)
D) 4 km (4 కి.మీ)
సమాధానం/Answer:- A) 5 km (5 కి.మీ)
Explanation:–
Pythagoras Theorem
Hypotenuse² = Side² + Side²
= 4² + 3²
= 16 + 9
Hypotenuse² = 25
Hypotenuse = √25
Hypotenuse = 5 km
వివరణ:-
పైథాగరస్ సూత్రం:-
కర్ణము^2= భుజము^2+భుజము^2
=4^2+3^2
=16+9
కర్ణము^2=25
కర్ణము =√25
కర్ణము = 5
దూరం= 5 కిలో మీటర్లు
9.If sin θ + cos θ = √2, then the value of sin θ × cos θ is:
sin θ + cos θ = √2 అయితే, sin θ × cos θ విలువ ఎంత?
A) 0
B) 1/2
C) 1/4
D) 1
Answer:- B) 1/2
Explanation:-
(sin θ + cos θ)² = 2
⇒ sin² θ+cos² θ+2 sin θ cos θ=2
⇒ 1 + 2 sin θ cos θ = 2
⇒ 2 sin θ cos θ = 1
⇒ sin θ cos θ = 1/2
10.
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