Unit I: NUMBER SYSTEMS
Chapter 1: Real Numbers
Topic: Concepts of Real Numbers
Subtopic: Rational & Irrational Numbers
Q. Which of the following is an irrational number?
A)
2
B)
0.5
C)
√3
✓ Correct
D)
4
💡 Hint:Note that √3 cannot be expressed as p/q.
✅
Solution:Irrational numbers are those which cannot be written in p/q form where p and q are integers and q is not zero. √3 is non-terminating and non-repeating.
Subtopic: Euclid’s Division Lemma
Q. The HCF of 135 and 225 using Euclid’s Division Lemma is:
A)
45
✓ Correct
B)
15
C)
5
D)
75
💡 Hint:Apply Euclid’s Division Lemma: 225 = 135 * 1 + 90.
✅
Solution:225 = 135 * 1 + 90; 135 = 90 * 1 + 45; 90 = 45 * 2 + 0. Since the remainder is 0, HCF is 45.
Subtopic: Decimal Expansions
Q. The decimal expansion of 17/8 is:
A)
Terminating
✓ Correct
B)
Non-terminating repeating
C)
Non-terminating non-repeating
D)
None of the above
💡 Hint:Check the prime factorization of the denominator.
✅
Solution:Denominator is 8 = 2^3. Since the denominator is in the form 2^n * 5^m, it is a terminating decimal.
Q. If the HCF of two numbers is 1, then the numbers are called:
A)
Co-prime
✓ Correct
B)
Twin prime
C)
Composite
D)
Perfect
💡 Hint:Think about the definition of numbers with no common factor.
✅
Solution:Co-prime numbers are pairs of numbers that do not have any common divisor other than 1.
Subtopic: Rational & Irrational Numbers
Q. Every point on the number line represents:
A)
A unique real number
✓ Correct
B)
A natural number
C)
A rational number
D)
An irrational number
💡 Hint:Number line covers both rational and irrational numbers.
✅
Solution:The set of real numbers has a one-to-one correspondence with the set of points on a number line.
Subtopic: Euclid’s Division Lemma
Q. For two positive integers a and b, if HCF(a, b) = 1, then LCM(a, b) is equal to:
A)
a / b
B)
a * b
✓ Correct
C)
1
D)
a + b
💡 Hint:Recall the relationship: HCF(a, b) * LCM(a, b) = a * b.
✅
Solution:Since HCF(a, b) * LCM(a, b) = a * b, if HCF is 1, then LCM(a, b) = a * b.
Q. For two positive integers a and b, if HCF(a, b) = 1, then LCM(a, b) is equal to:
A)
a / b
B)
a * b
✓ Correct
C)
1
D)
a + b
💡 Hint:Recall the relationship: HCF(a, b) * LCM(a, b) = a * b.
✅
Solution:Since HCF(a, b) * LCM(a, b) = a * b, if HCF is 1, then LCM(a, b) = a * b.
Subtopic: Fundamental Theorem of Arithmetic
Q. The prime factorization of 140 is:
A)
2^2 * 5 * 7
✓ Correct
B)
2 * 5^2 * 7
C)
2 * 5 * 7^2
D)
2^3 * 5 * 7
💡 Hint:Divide 140 by the smallest prime numbers sequentially.
✅
Solution:140 = 2 * 70 = 2 * 2 * 35 = 2 * 2 * 5 * 7 = 2^2 * 5 * 7.
Q. The sum of exponents of prime factors in the prime factorization of 196 is:
A)
3
B)
4
✓ Correct
C)
5
D)
2
💡 Hint:First, find the prime factorization of 196.
✅
Solution:196 = 2^2 * 7^2. The exponents are 2 and 2. Their sum is 2 + 2 = 4.
Subtopic: Rational & Irrational Numbers
Q. Which of the following numbers is rational?
A)
√2
B)
π
C)
√4
✓ Correct
D)
√7
💡 Hint:A rational number can be expressed as an integer or a terminating decimal.
✅
Solution:√4 = 2, which is an integer and thus a rational number. √2, π, and √7 are all irrational.
Subtopic: Decimal Expansions
Q. After how many decimal places will the decimal expansion of 23 / (2^2 * 5) terminate?
A)
1
B)
2
✓ Correct
C)
3
D)
4
💡 Hint:The number of decimal places is determined by the highest power of 2 or 5 in the denominator.
✅
Solution:The denominator is 2^2 * 5^1. The highest power is 2. Therefore, it will terminate after 2 decimal places.
Unit II: ALGEBRA
Chapter 2: Polynomials
Topic: Concepts of Polynomials
Q. The zeros of the quadratic polynomial x² + 7x + 10 are:
A)
-2 and -5
✓ Correct
B)
2 and 5
C)
-2 and 5
D)
2 and -5
💡 Hint:Factorize the polynomial by splitting the middle term.
✅
Solution:x² + 7x + 10 = x² + 5x + 2x + 10 = x(x+5) + 2(x+5) = (x+2)(x+5). Thus zeros are -2 and -5.
Chapter 3: Pair of Linear Equations in Two Variables
Topic: Pair of Linear Equations in Two Variables
Q. For which value of k, the system of equations x + 2y = 3 and 5x + ky + 7 = 0 have no solution?
A)
10
✓ Correct
B)
6
C)
3
D)
1
💡 Hint:For no solution, a1/a2 = b1/b2 != c1/c2.
✅
Solution:1/5 = 2/k => k = 10.
Q. The pair of equations y = 0 and y = -7 has:
A)
No solution
✓ Correct
B)
One solution
C)
Two solutions
D)
Infinitely many solutions
💡 Hint:Check if these lines intersect.
✅
Solution:Lines y = 0 and y = -7 are parallel as they are both horizontal. Parallel lines never intersect, so no solution.
Chapter 4: Quadratic Equations
Topic: Quadratic Equations
Q. The roots of the quadratic equation x² - 0.04 = 0 are:
A)
± 0.2
✓ Correct
B)
± 0.02
C)
0.4
D)
2
💡 Hint:x² = 0.04
✅
Solution:x² = 0.04 => x = ±√0.04 = ±0.2.
Chapter 5: Arithmetic Progressions
Topic: Concepts of Arithmetic Progressions
Q. The 10th term of the AP: 2, 7, 12, ... is:
A)
47
✓ Correct
B)
52
C)
42
D)
37
💡 Hint:Use formula an = a + (n-1)d.
✅
Solution:a = 2, d = 5, n = 10. a10 = 2 + (9)*5 = 2 + 45 = 47.