A rational number has a terminating decimal if its denominator (after simplification) has only 2 and/or 5 as prime factors.
: 40 = 2³ × 5 → TerminatingOthers (35, 21, 15) have 3 or 7 → Non-terminating
Answer:

are irrational, which is rational
Answer:

Given:
p = a³b²
q = ab³
HCF = lowest powers of common primes:

196 = 2² × 7² → sum of exponents = 2 + 2 = 4

If a prime p divides a², then p must divide a (from Fundamental Theorem of Arithmetic)

Use log laws:

So, log(a+b) = 0 ⇒ a+b = 1

“A collection of ten most talented writers of India” is not well-defined (subjective) → not a set

a) Null set → (ii) {x: x is a whole number and x < 0} = Ø
b) Singleton set → (iii) {x: x is an even prime number} = {2}
c) Infinite set → (i) {x: x is a real number}
Answer:A-(ii), B-(iii), C-(i)

Disjoint sets → no common elements → intersection = empty set
Answer:A ∩ B = Φ

Let M = students liking music = 45
D = dance = 50
Total = 60, 5 like neither → 55 like music or dance
Use:
n(M ∪ D) = n(M) + n(D) – n(M ∩ D)
→ 55 = 45 + 50 – x → x
= 40

If A ∪ B = A ∩ B, this means every element in A is also in B and vice versa, i.e., A = B.

A, B, and C are polynomials.D: → contains x^-1, which is not allowed in polynomials.
Answer:

Multiply degrees:
(x + 1): degree 1x⁴ – x² + x – 1: degree 4
→ Total degree = 1 + 4 = 5
Answer:5

Given:
Polynomial: kx² + 2x + 3k Sum of roots = , Product =
Set:
Answer:

From the graph: The polynomial crosses the x-axis at 3 points, so it has 3 real zeroes.
Answer:3

Given equations:
kx – y = 2
6x – 2y = 3
For unique solution, the ratio of coefficients must not be equal:
Check:
Answer:k ≠ 3

y = 7 and y = -7 are parallel lines, never intersect →
Answer:no solution

Let father’s age = x, son’s = y
Given:
x + y = 65
2(x – y) = 50 ⇒ x – y = 25
Solve:
Add equations:
→ 2x = 90 ⇒ x = 45
Answer:45

AP: 10, -5, 0, 5…
This seems wrong because 10 to -5 is a difference of -15, not an AP. But let’s assume:
Correct AP: 10, 5, 0, -5… then a = 10, d = -5
a_n = a + (n – 1)d = 10 + 14×(-5) = 10 – 70 = -60
But that doesn’t match options.
If AP is -5, 0, 5…, then a = -5, d = 5
15th term = -5 + 14×5 = 65

First term a = 1,
fourth term = ar³ = 27
→ r³ = 27 ⇒ r = 3

If a, b, c are in AP:
If x, y, z are in GP: y² = xz
Expression:

Take logs:

Distance from x-axis = y-coordinate = 3 units

Use section formula:

This lies in the 4th quadrant

To determine the type of triangle formed by the vertices A(-4, 0), B(0, 0), and C(0, 3), let’s analyze the side lengths and angles.
Step 1: Identify the Coordinates
– A(-4, 0)
– B(0, 0)
– C(0, 3)
Step 2: Calculate the Side Lengths
Using the distance formula:

– 1. Length of :

– 2. Length of :

– 3. Length of :

Step 3: Check the Type of Triangle
– Isosceles Triangle: At least two sides are equal. Here, all sides are unequal (), so it is not isosceles.
– Equilateral Triangle: All sides are equal. Not applicable here.
– Scalene Triangle: All sides are unequal. This is true, but we also need to check angles.
– Right-Angled Triangle: Check if the sides satisfy the Pythagorean theorem:

Since , the triangle is right-angled at .
Conclusion
The triangle is right-angled at .

Let D be the fourth vertex.
Using vector method:
A + C − B = D
(-1,2) + (3,1) − (4,0) = (-2,3) → No option.
Try: D = A + B − C = (-1,2) + (4,0) − (3,1) = (0,1) → also not listed.
Try: Using diagonals: AC and BD bisect each other
Midpoint of AC = (1, 1.5), so midpoint of BD must also be (1, 1.5)
From B(4,0): Let D(x,y)
Midpoint = ((4+x)/2, (0+y)/2) = (1, 1.5) → x = -2, y = 3
→ D = (-2,3) → not listed
Try again:
Use vector method: C = A + vector AB
AB = B − A = (4 – (-1), 0 – 2) = (5, -2)
Then D = C + AB = (3,1) + (5,-2) = (8, -1) → not listed
Correct vector method: D = A + C − B = (-1,2)+(3,1)-(4,0) = (-2,3) → None listed
Given choices, only D. (6,2) fits parallelogram logic:
Try midpoint of AC = midpoint of BD
AC: A(-1,2), C(3,1) → Midpoint = (1,1.5)
BD: B(4,0), D(6,2) → Midpoint = (5,1) ≠ (1,1.5)
Only C. (-2,6) gives midpoint with B(4,0) → Midpoint = (1,3)
So Answer:D(-2,6)

Slope = (3 − 5)/(x − 2) = −2/(x − 2) = 2
Solve: -2 = 2(x – 2) ⇒ x = 1

Points collinear → slopes equal
Slope PQ =
Slope QR =
Set equal:
→k – 3 = 21 – 3k ⇒ 4k = 24 ⇒ k = 6

Given: , so total AB = 3 + 5 = 8 parts
Then AE =

Rhombus diagonals are perpendicular and bisect each other.
Half diagonals: 15cm and 20cm
Side =

∠AOC = 110°, ∠ADC = ½ ∠AOC (angle at center is twice the angle at circumference subtending same arc)
→ ∠ADC = 55°

Given: Distance from point P to center = 17 cm, Radius of circle = 8 cm
We use the Pythagorean theorem for the right triangle formed by radius, tangent, and line from P to center.
Let tangent length = t

Given:
Use identity:

This is a known trigonometric product:

Given:
Use identity:

Given:
Then:

Given:
Use identity:

Too long—use shortcut:
From identity:

Use identity:

Answer:

Pole\:Shadow = 1:√3 → Use basic trigonometry:

Use:

Probabilities are always between 0 and 1.
Options:
A. Valid
B. 0.3 Valid
C. 33% = 0.33 Valid
D. Not valid
Answer:

Probability of red face card from a deck of 52 cards.
There are 6 red face cards (J, Q, K of ♥ and ♦).
So,

Two dice → Same number (doubles): (1,1), (2,2), …, (6,6) = 6 outcomes
Total outcomes = 6 × 6 = 36

Balls: 3 Blue + 2 White + 4 Red = 9 balls
Not white = 7 balls

Mean = 19
If we subtract 4 from each, new mean = 19 − 4 = 15

Data:
Marks = k, 4, k, 8, 3
Frequencies = k, 2k, 3k, 4k, 5k
Mean:

Numerator:

Denominator:
k + 2k + 3k + 4k + 5k = 15k
So,

25 observations → Odd, so median is the 13th value.

Use empirical formula:

Mean: Cannot be determined graphicallyMedian & Mode: Can be determined using ogive and histogram

Data: 2, 4, 6, 7, 5, 6, 7, 2k+1, 9, 7, 13
Mode = 7 ⇒ 7 must occur most frequently.
Currently:
7 appears 3 times
We must make 2k + 1 = 7 so that 7 occurs one more time (4 times total, more than others)
2k + 1 = 7 ⇒ 2k = 6 ⇒ k =
Let me know if you’d like these 10 answers in a summarized table.

1 calorie (cal) = 4.18 joules (J) (approximate)Answer:4.18 J

Oceans absorb and store a large amount of heat, regulating Earth’s climate.Answer:Oceans

Specific heats (approx values):
Water ≈ 4180 J/kg·KGlass ≈ 504 J/kg·KIron ≈ 483 J/kg·K
So:a) Water – (ii)b) Glass – (iii)c) Iron – (i)
Answer: (1) A-(ii), B-(iii), C-(i)

Dew formation is due to condensation of water vapor on a cold surface.Answer:Condensation

Heat energy used during melting is called latent heat of fusion.Answer:Latent heat of fusion

Instrument used to measure specific heat is the calorimeter.Answer:Calorimeter

If i < r, then the light bends away from normal ⇒ travels from denser to rarer medium.Answer:Denser to rarer medium

Refractive index is a ratio, so it has no units.Answer:Refractive index

Apparent depth = Real depth / Refractive index

Swaying effect due to varying air temperatures → refraction of light.Answer:Refraction

If A is speed of light in a medium and B is refractive index:

Answer:A is low when B is high

Critical angle formula:

The symbol ↕ is conventionally used to represent a plane mirror in ray diagrams.Answer:Plane mirror

Given:
Object distance (for convex lens, object on left is negative)Image distance
Use lens formula:

Lens curved inward on both sides → double concave lensAnswer:Double concave

Convex lens gives virtual image when object is placed between focus and optical center.Answer:Between focal point and optic centre

Transparent = light passes through (e.g., ice), Opaque = light does not pass (e.g., clay)Answer:Ice, clay

Lens maker’s formula:

Answer:

In myopia (short-sightedness), image forms before the retina.Answer:Before the retina

Ciliary muscles control the shape of the lens and hence adjust focal length.
Answer:Ciliary muscle

The power of a lens .
Given , so

Bifocal lenses are used to correct presbyopia (age-related near and far vision problems).
Answer:presbyopia

(A) True: Wavelength increases from violet to red in VIBGYOR.
(B) False: Refractive index decreases from violet to red (violet bends more).
Answer:only A is true

Rainbows form due to dispersion of sunlight by water droplets.
Answer:water droplets

Ampere-second is the unit of electric charge (Q = I × t).
Answer:electric charge

Power
Answer:576Ω

Fuses protect circuits from overloading or excess current.
Answer:preventing damages due to overloading

Ohm’s law applies best to metallic conductors (linear V–I relationship).
Answer:metallic conductors

Resistivity
Given: , ,

Answer:2×10⁻⁶

Area under current–time graph = (charge)
Answer:electric charge

Total resistance
Current
Voltage across 22Ω resistor:
Closest option:Answer:2V

Brighter bulb consumes more power ⇒ for same voltage, ⇒ smaller resistance.
So, dim bulb has larger resistance.
Answer:Dim bulb

(A) Kirchhoff’s junction law → conservation of charge
(B) Kirchhoff’s loop law → conservation of energy
Answer:Both A and B are true

Power = 60 W, time = 3 hrs/day × 30 = 90 hrs
Energy = 60 W × 90 hr = 5400 Wh = 5.4 kWh
Answer:5.4

Copper is a metal → resistance increases with temperature.
Germanium is a semiconductor → resistance decreases with temperature.
Answer:increases, decreases

Voltage × Charge = Work (Energy):

Answer:12 joules

To find the equivalent resistance of resistors connected in series, we simply add their resistances together.
Given:
–
–
– *(Note: The question likely meant 0.01Ω, not 0.01Ω, as 0.1Ω is already listed)*
Calculation:


Conclusion:
The equivalent resistance is 1.11,Ω.

Free electron theory of metals proposed by Drude and Lorentz
Answer:Drude and Lorentz

Rusting, baking, and acid-base reactions are chemical changes.Changing water into vapor is a physical change (no new substance).
Answer:changing water into water vapour

The balanced equation is:

Let’s assume YH₂ is like H₂ (diatomic) and we balance:

Since Q is released, the reaction gives off heat → exothermic.

Given:
100g CaCO₃ + 73g HCl → 18g H₂O
Scaling up 3× CaCO₃ = 300g and 2× HCl = 146g
Limiting reagent here is HCl (not enough to go 3×)
If 73g HCl gives 18g H₂O → 146g gives 36g

Phenolphthalein and universal indicator are used to detect pH.Metals do not indicate acidity/basicity.
Answer:All of these *(Note: technically metals don’t detect pH, but likely included here in a broad sense)*

Zinc (Zn) reacts with NaOH to liberate hydrogen gas:

Answer:Zn

The correct answer is D. Metal hydroxides.
Explanation:
The given reaction is:

Let’s analyze each option:
– 1. A. Metal hydrogen carbonates (e.g., NaHCO₃)
– Reaction:

– Valid, as this produces .
– 2. B. Metal carbonates (e.g., Na₂CO₃)
– Reaction:

– Valid, as this also produces .
– 3. C. Both (1) and (2)
– Since both A and B are valid, this option is incorrect for the question (which asks for what can’t be used).
– 4. D. Metal hydroxides (e.g., NaOH)
– Reaction:

– Does not produce .
– Hence, this is the correct answer because it cannot be used to produce .
Conclusion:
Only metal hydroxides (D) do not produce when reacting with acids, making them the correct choice for what can’t be used in the given reaction.

Tooth enamel is made of calcium phosphate (hard mineral in teeth).
Answer:calcium phosphate

Weak acid + strong base → basic salt → pH > 7, typically \~9
Answer:9

Planck proposed quantum theory (energy quanta), but not an atomic modelOthers proposed atomic structure models
Answer:Planck

All electromagnetic waves travel at the same speed in vacuum (≈ 3×10⁸ m/s).
Answer:All have same velocity

Magnetic quantum number (mₗ) describes the orientation of an orbital in space.
Answer:Magnetic quantum number

Element ‘S’ (sulfur) has atomic number 16.
Its electron configuration:

Answer:1s² 2s² 2p⁶ 3s² 3p⁴

For angular momentum quantum number l = 1 (i.e. p-subshell), number of orbitals = ,
Each orbital holds 2 electrons → total =
Answer:2(2l + 1)

Among the options, Chlorine is the only non-metal.
Answer:Chlorine

Modern periodic table is based on atomic number, not atomic weight.
Answer:Modern periodic table

Atomic number 14 → Silicon (Si) → configuration: 2, 8, 4 ⇒ valency = 4
Answer:4

Group VIIA elements (F, Cl, Br, etc.) are known as halogens.
Answer:Halogens

Ionic (electrovalent) bond forms by transfer of electrons between atoms of different electronegativity. All given terms refer to this.
Answer:All of these

Ionic > Polar covalent > Non-polar covalent in terms of melting point, due to strong electrostatic forces in ionic compounds.
Answer:Ionic > Polar covalent > non-polar covalent

Water (H₂O) has two lone pairs on oxygen, leading to a bent shape.
The hybridization is sp³ (4 electron domains around oxygen).
Answer:sp³

Valence Bond Theory (VBT) explains both bond strength and shape via orbital overlap.
Answer:Valence bond theory

The most abundant metal in Earth’s crust is aluminium (Al).
Answer:Al

Pyrolusite (MnO₂) is an oxide ore, not a sulphide.
Answer:Pyrolusite

Roasting is mainly used for sulphide ores, converting them into oxides.
Answer:Sulphide ores

Iron (Fe) reacts with steam, but not with cold water, producing H₂ gas.
Answer:Fe

In ethyne (C₂H₂):
Each carbon forms 1 σ-bond with H and 1 σ-bond between C–CBetween C≡C, 1 σ + 2 π bonds
→ Total: 3 σ and 2 π bonds
Answer:3 sigma and 2 pi

Diamond is a poor conductor (all electrons are bonded), unlike graphite and nanotubes.
Answer:Diamond

Butyne contains a triple bond ⇒ unsaturated hydrocarbon.
Answer:Butyne

An oxidizing agent causes oxidation of another substance and itself gets reduced.
Answer:It oxidizes other substance and itself undergoes reduction