If 3x - 7 = 2, what is the value of x?
A 1 B 2 C 3 D 4
3x - 7 = 2 → 3x = 9 → x = 3.
What is the value of 4³ × 2⁻² ?
A 8 B 16 C 32 D 64
4³ = 64, 2⁻² = 4 1 4 1
The HCF of 48 and 64 is:
A 8 B 12 C 16 D 32
Factors of 48: 1,2,3,4,6,8,12,16,24,48. Factors of 64: 1,2,4,8,16,32,64. HCF = 16.
If the ratio of two numbers is 3:5 and their sum is 64, find the larger number.
A 24 B 32 C 40 D 48
Let numbers be 3x and 5x. 3x + 5x = 64 → 8x = 64 → x = 8. Larger number = 5 × 8 = 40.
If a quadratic equation has roots α and β, and α + β = 7, αβ = 10, what is the equation?
A x² - 7x + 10 = 0 B x² + 7x + 10 = 0 C x² - 7x - 10 = 0 D x² + 7x - 10 = 0
Quadratic equation is x² - (sum of roots)x + (product of roots) = 0, which gives x² - 7x + 10 = 0.
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The volume of a cone with radius 5 cm and height 12 cm is: (use π = 7 22
A 314.29 cm³ B 628.57 cm³ C 471.43 cm³ D 314.86 cm³
Volume of cone = (3 1 3 1 7 22 3 1 7 22
In a triangle, if one angle is 90° and the other two angles are in ratio 1:2, the angles are:
A 30°, 60°, 90° B 45°, 45°, 90° C 20°, 70°, 90° D 40°, 50°, 90°
One angle = 90°. Other two in ratio 1:2. Let them be x and 2x. x + 2x + 90 = 180 → 3x = 90 → x = 30°. Angles are 30°, 60°, 90°.
If cos θ = 25 7
A 24 7 B 7 24 C 24 25 D 25 24
cos θ = 25 7 7 24
The average of five numbers is 35. If one number is replaced by 50, the new average becomes 38. What was the original number?
A 30 B 35 C 40 D 45
Sum of 5 numbers = 35 × 5 = 175. New sum = 38 × 5 = 190. Difference = 190 - 175 = 15. Original number = 50 - 15 = 35.
If log₁₀ 2 = 0.301, find log₁₀ 200.
A 2.301 B 2.602 C 1.301 D 3.301
log₁₀ 200 = log₁₀(2 × 100) = log₁₀ 2 + log₁₀ 100 = 0.301 + 2 = 2.301.
If 2^(x+1) = 32, find the value of x.
A 3 B 4 C 5 D 6
# Solving Exponential Equations
To solve exponential equations, express both sides as powers of the same base, then equate the exponents.
Step 1: Express 32 as a Power of 2
We need to rewrite the right side of the equation using base 2, since the left side already has base 2.
32 = 2 5
(because 2 × 2 × 2 × 2 × 2 = 32
Step 2: Equate the Exponents
Now substitute this into the original equation and use the property that if a m = a n m = n
2 x + 1 = 2 5
x + 1 = 5
x = 4
The value of x is 4.
Answer: (B) 4
A number when divided by 7 gives quotient 12 and remainder 3. The number is:
A 85 B 87 C 89 D 91
Using dividend = divisor × quotient + remainder: Number = 7 × 12 + 3 = 84 + 3 = 87.
If ABCD is a rhombus with diagonals 6 cm and 8 cm, find its area.
A 24 cm² B 48 cm² C 28 cm² D 56 cm²
Area of rhombus = (2 1 2 1
What is the compound interest on Rs. 8000 at 10% per annum for 2 years?
A Rs. 1680 B Rs. 1600 C Rs. 1800 D Rs. 1500
A = P(1 + r/100)ⁿ = 8000(1.1)² = 8000 × 1.21 = 9680. CI = 9680 - 8000 = 1680.
In an AP, if first term a = 5, common difference d = 3, find the 10th term.
A 30 B 32 C 35 D 38
nth term = a + (n-1)d = 5 + (10-1) × 3 = 5 + 27 = 32.
If |x - 3| = 5, what are the possible values of x?
A 2 and 8 B 2 and -8 C -2 and 8 D 3 and 5
|x - 3| = 5 means x - 3 = 5 or x - 3 = -5. So x = 8 or x = -2.
If 2sin²θ - sin θ - 1 = 0, find sin θ (θ is acute).
A 2 1 B 1 C -1 D 3 1
2sin²θ - sin θ - 1 = 0. Factoring: (2sinθ + 1)(sinθ - 1) = 0. Since θ is acute, sin θ = 1.
A ladder of length 13 m leans against a wall. If the foot of the ladder is 5 m away from the wall, find the height reached on the wall.
A 8 m B 10 m C 12 m D 15 m
Using Pythagoras: h² + 5² = 13² → h² + 25 = 169 → h² = 144 → h = 12 m.
If tan θ + cot θ = 4, find tan² θ + cot² θ.
A 12 B 14 C 16 D 18
(tan θ + cot θ)² = tan²θ + 2 + cot²θ = 16. So tan²θ + cot²θ = 14.
What is the probability of getting a prime number when a die is rolled?
A 2 1 B 3 1 C 6 1 D 3 2
Prime numbers on a die: 2, 3, 5 (total 3). Total outcomes = 6. Probability = 6 3 2 1
