We are given:

Shortcut Method:
Recall:

Now, write as an exponent:

So:

Correct Answer:

To find the **cardinal numberof a set, simply **count the number of distinct elementsin it.
Set A = {−2, −1, 0, 1, 2}
Number of elements = 5
Correct Answer: A. 5
Want a shortcut trick for identifying cardinal numbers in big sets quickly?

We're given the quadratic polynomial:

To find the nature of its zeroes, we can factor it.
Try splitting the middle term:
We need two numbers that add up to 24and multiply to 119.
Check factors of 119:
119 = 17 × 7
And:
17 + 7 = 24
So we can write:

The zeroes are:

Both are negative.
Correct Answer: both negative
Want a shortcut method to factor quadratics quickly using a table?

Shortcut: The degreeof a polynomial is the highest power of the variable.
Here, the highest exponent of is:

To find the HCF (Highest Common Factor)of 8, 9, and 25:
8= 9= 25=
These numbers have no common prime factor.
Shortcut:
If no prime factor is common, then HCF = 1.

We are given the number:

Step-by-step shortcut:
is irrationalThe reciprocal of an irrational number is also irrational
(unless it simplifies to a rational, which does not)
Final Answer: Irrational number

We are given:

Shortcut method:
Rewrite as a power of 2:

So,

We are given two sets:
A = {C, O, V, I, D, 19, 2020}
B = {C, O, V, I, D, 19, 2021}
We are asked to find B − A, which means the elements that are in Bbut not in A.
Shortcut:
Compare both sets:
Common elements: C, O, V, I, D, 19Bhas 2021which Adoes not.
So:

Final Answer: (2021)

We are asked to find:

Shortcut Method:
Write both numbers as powers of 10:

So the expression becomes:

Use change of base formula:

We are given the quadratic equation:
Shortcut Method: Splitting the middle term
We need two numbers that multiply to -6and add to +1(coefficient of x):
Numbers: 3 and -2
So we factor:
x² + 3x – 2x – 6 = 0
x(x + 3) -2(x + 3) = 0
(x – 2)(x + 3) = 0
Roots are: x = 2, -3

We are given that α and β are roots of the quadratic equation:
ax² + bx + c = 0, a ≠ 0
We need to find the value of:

Shortcut Trick:

From the standard formulas for a quadratic:


So,

Final Answer:
Want a similar shortcut for ?

We are given the arithmetic progression:
2,-1,-4,…
Step 1: Identify first term and common difference
First term a = 2
Common difference d = -1 – 2 = -3
Step 2: Use formula for nth term of AP:
T_n = a + (n-1)d
For the 10th term:
TT₁₀ = 2 + (10-1)(-3) = 2 + 9(-3) = 2 – 27 = -25

To find how many two-digit numbers are divisible by 7, follow this shortcut method:
Step 1: Two-digit numbers range from 10 to 99.
Step 2: Find the first two-digit number divisible by 7:

Step 3: Find the last two-digit number divisible by 7:

Step 4: Use AP count formula:

We are given an A.P.:3, 6, 9, …
This is a common A.P. where:
First term Common difference d = 6 – 3 = 3
Number of terms n = 15
Use shortcut formula for sum of n terms of A.P.:

Plug in values:

Solve:
2x – (4 – x) = 5 – x
Simplify LHS:
2x – 4 + x = 5 – x ⇒ 3x – 4 = 5 – x
Bring all terms to one side:
3x + x = 5 + 4 ⇒ 4x = 9 ⇒ x = = 2.25

Eliminate incorrect solution:
You are given equations:
⇒1. 3x – 4y = 7
⇒2. 2x + 3y = -1
Now test the options by substituting values.
⇒A.
1st:
So, this is incorrect.
Correct Answer:
(does notsatisfy the equations)

If Σn = 45, then n = ?
We assume this means the sum of first n natural numbers:

Try n = 9:
9 × 10 = 90

Midpoint of diameter endpoints (1, 2) and (7, -4)
Use midpoint formula:

Correct Answer: (4, -1)

Area of triangle formed by line with axes
Equation: xcosα + ysinα = p
Find x- and y-intercepts:
Put y = 0:

Put x = 0:

Area of triangle =

Correct Answer:

Solve:
1) } x + 7y = 7
2) } 7x – 3y = -3
From (1):x = 7 – 7y
Substitute into (2):
7(7 – 7y) – 3y = -3
49 – 49y – 3y = -3 ⇒ -52y = -52 ⇒ y = 1

Which of the following equations is not a linear equation?
(2 + 3x = y – 5) → Linear(Degree 1)
(3 – x = y² + 7) → Not linear(Degree 2 in (y))
(x + 3y = 2y – x) → Linear(Degree 1)
(5x + 2y = 0) → Linear(Degree 1)

If (x² + kx + 1 = 0) has a root (x = 1), then (k = ?)
Substitute (x = 1):
1² + k(1) + 1 = 0 ⇒ 1 + k + 1 = 0 ⇒ k = -2

If the roots of (ax^2 + bx + c = 0) are (sin α) and (cos α), then (1 + 2 = ?)
Given roots: (sin α + cos α = –) and (sin α cos α = $\frac{c}{a}).
Using the identity:
sin² α + cos² α = 1 ⇒ (sin α + cos α)² – 2 sin α cos α = 1
Substitute:
\[ \left(-\frac{b}{a}\right)^2 – 2\left(\frac{c}{a}\right) = 1 ⇒ \frac{b^2}{a^2} – \frac{2c}{a} = 1 \]
Rearrange:
\[ 1 + 2\frac{c}{a} = \frac{b^2}{a^2} \]

If the system (3x – 2y – 7 = 0) and (kx + 2y + 11 = 0) has a unique solution, then:
For a unique solution, the lines must not be parallel:
\[ \frac{a_1}{a_2} ≠q \frac{b_1}{b_2} ⇒ \frac{3}{k} ≠q \frac{-2}{2} ⇒ \frac{3}{k} ≠q -1 ⇒ k ≠q -3 \]

If (7x – 5y = 2) and (3x + y = 4), then (x = ?)
From the second equation: (y = 4 – 3x).
Substitute into the first equation:
7x – 5(4 – 3x) = 2
⇒ 7x – 20 + 15x = 2
⇒ 22x = 22
⇒ x = 1

Distance between (0, 0) and (5, 12) is:
\[ \sqrt{(5-0)^2 + (12-0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

Slope of the line through (x, 5) and (5, 2) is 3. Find (x):
\[ \text{Slope} = \frac{2 – 5}{5 – x} = 3 ⇒ \frac{-3}{5 – x} = 3 ⇒ -3 = 15 – 3x ⇒ x = 6 \]

If (ΔABC ~ ΔPQR), (∠A = 32°), (∠R = 65°), then (∠B = ?)
Corresponding angles are equal, so (∠C = ∠R = 65°).
Sum of angles in (ΔABC):
∠B = 180° – ∠A – ∠C = 180° – 32° – 65° = 83°

The angle in the minor segment is:
An angle in the minor segment is obtuse(greater than (90°).

In the figure, (∠BAO = 30°), (∠BCO = 40°). Then (∠AOC = ?)
From the figure (assuming (O) is the center of the circle):
– (∠AOC = 2(∠ABC) (Central angle theorem).
– (∠ABC = ∠BAO + ∠BCO = 30° + 40° = 70°).
– Thus, (∠AOC = 2 × 70° = 140°).

Given OB = 13 cm, OP ⊥ AB, OP = 12 cm. Find AB.
In right triangle OPB:
OB² = OP² + PB² ⇒ 13² = 12² + PB² ⇒ 169 = 144 + PB² ⇒ PB = 5
Since OP ⊥ AB and lies at midpoint, AB = 2 × PB = 2 × 5 = 10 cm

D, E, F are midpoints of triangle ABC. Then triangle DEF is the midpoint triangle.
Area of triangle formed by joining midpoints is one-fourthof original triangle:
Correct Answer: 1:4

∠BAC = 90°, AD ⊥ BC, BD = 9 cm, CD = 16 cm → AC = ?
This is a right triangle with altitude from right angle to hypotenuse:

Use Pythagorean relation:
In right triangle with perpendicular from right angle, AB² = BD × BC

Correct Answer: 20 cm

Similar triangles: base ratio = 24:18 = 4:3
So corresponding sides are in same ratio:

Correct Answer: 6 cm

A cyclic parallelogram has all angles = 90°, because opposite angles in a cyclic quadrilateral are supplementary.
Only parallelogram with all 90° angles is a rectangle.
Correct Answer: A. Rectangle

Perimeter of rhombus = 52 → side = 52/4 = 13 cm
One diagonal = 24 cm → use Pythagoras to find the other:

Correct Answer: 10 cm

Cone radius r = 7 m, height h = 10 m → slant height = √(r² + h²)

Correct Answer: 12.2 m

If , then
Then,
Correct Answer: √2

Angle between tangent and radius at point of contact is always
Correct Answer: 90°

Volume of cone =
Given:
Let height be .

Correct Answer: 9:5
Let me know if you'd like a visual explanation for any geometry question.
Let's solve Questions 41 to 50step by step:

Three cubes with sides 6 cm, 8 cm, and 1 cm are melted to form a new cube.
Volume of cube = a³

Total volume:
= 6³ + 8³ + 1³
= 216 + 512 + 1
= 729 cm³
New cube's edge:

Correct Answer: 9 cm

If A + B = 90° and , find
Since A = 90° – B,
Then
Correct Answer:

Diameter = d → radius = d/2
Volume of sphere:

Correct Answer:

Volume of a frustum:

Given:
Top radius R = 4, bottom radius r = 2, height h = 6, π = 3.14

Closest option:
Correct Answer: 170 m³

Dice is thrown twice. Probability of getting at least one '5'
Total outcomes = 36
Probability of not getting 5in a single throw = 5/6
So, probability of not getting 5 at allin two throws:

Correct Answer:

Mean of 6, 7, x, 8, y, 14 = 9
Total sum = 6 + 7 + x + 8 + y + 14 = 35 + x + y

Correct Answer: 19

From the figure, if opposite side is √3 and adjacent side is 1, then

Correct Answer: 60°

Two poles 20 m and 14 m → vertical difference = 6 m
Wire makes 30° with ground:

Correct Answer: 12 m

Three coins tossed → total outcomes = 8
Favorable outcomes for at least 2 heads:
HHTHTHTHHHHH → 4 outcomes

Correct Answer:

A.M. of 30 students is 42
⇒ Total marks = 30 × 42 = 1260
2 students got zero ⇒ remaining 28 students got 1260 – 0 – 0 = 1260
New A.M. = 1260 / 28 = 45Answer: B. 45

Data: 17, 31, 12, 27, 15, 19, 23
Sorted: 12, 15, 17, 19, 23, 27, 31 → Median is the middle = 19

Mode = Most frequent value
A to Z are all distinct letters
⇒ No repetition
⇒ No mode
Answer: No mode

From the figure: triangle with sides 5, 12, 13 → right-angled at B
cosec A = hypotenuse / opposite = 13 / 12
Answer:

Two-digit numbers: 10 to 99 → Total = 90
Multiples of 3 in this range = from 12 to 99 = 12, 15, …, 99
This is an AP with a = 12, d = 3
Use formula: nth term = a + (n-1)d
99 = 12 + (n-1)×3 ⇒ n = 30
Probability = 30 / 90 = 1/3
Answer:

P(x) + P(not x) = 1 (law of total probability)

Given cotθ = b/a ⇒ cosθ = a/√(a² + b²), sinθ = b/√(a² + b²)
Then,
Answer:

Height = 100 m, Distance = 100 m
Right triangle ⇒ tan θ = 100/100 = 1 ⇒ θ = 45°

Normal to a curved surface at a point is drawn by joining that point to the centre of curvature (C).

Parallel rays inclined at 15° converge to a point on the focal plane, not directly at F.
Answer: Converge to a point on focal plane

Focal plane is perpendicular to the principal axis at the focus.
Answer: Plane perpendicular to the principal axis at focus (F)

Dentists use concave mirrorsto get a magnified, erect image.
Answer: Concave mirror

If both object and image are on same side, it must be a concave mirrorforming a real or magnified virtual image.
Answer: Concave

If the object–screen distance is more than 4f, a convex lens can produce a sharp image in twopositions (lens formula).
Answer: 2

A ray passing through Cstrikes the mirror perpendicularly and retraces its path.
Answer: Retraces the path And Through centre of curvature C.

Magnification = 1 ⇒ Image size = Object size
Answer: Image is of the same size as that of object

Use mirror formula:


Answer: 15 cm

Image formation happens on the retina, but among options, the crystalline lensfocuses light on the retina.
Answer: Crystalline lens

optic centre (P)
(In optics, object distance (u), image distance (v), and focal length (f) are all measured from the optic centre (P) of the lens/mirror.)

colour
(Cones in the human eye are responsible for color vision, while rods detect light intensity.)

Ciliary muscle
(The ciliary muscles help adjust the curvature of the eye-lens, changing its focal length for accommodation.)

Further away than actual distance
(Due to refraction at the water-air interface, the bird appears higher (farther away) than its actual position to the fish.)

( f = )
(The focal length of a plano-convex lens with refractive index ( n ) and radius of curvature ( R ) is given by ( f = ).)

focus (F)
(For a convex lens, an object at infinity forms an image at the focal point (F).)

8Ω
(In series, resistances add up: ( 4Ω + 8Ω + R = 20Ω ) → ( R = 8Ω ).)

20 A
(A typical fuse melts when the current exceeds its rated value, often around 20 A for household applications.)

Manganin
(Manganin is an alloy made of copper (Cu), manganese (Mn), and nickel (Ni), known for its stable resistance.)

Conductivity = D. Resistivity^-1
(Conductivity (σ) is the reciprocal of resistivity (ρ), i.e., ( ).)

Least distance of distinct vision is …… 25 cm
(The least distance of distinct vision for a normal human eye is about 25 cm, the closest distance at which an object can be seen clearly.)

Human body resistance varies widely depending on conditions: 100Ω to 500,000Ω
Answer: 100Ω to 5,00,000Ω

Focal length of a lens is 25 cm. Its power is ……. 4D
(Power of a lens (P) is given by ( P = ) (in meters). Here, ( f = 0.25 m ), so ( P = = 4 D ).)

When a person is suffering from both Myopia and Hypermetropia, which should be used?
Bi-focal lens
(A bi-focal lens has two parts: the upper part corrects myopia (short-sightedness), and the lower part corrects hypermetropia (long-sightedness).)

The current in a conductor is directly proportional to the potential difference between its ends. This is called …… Ohm's law
(Ohm's law states ( V = IR ), meaning current (I) is directly proportional to voltage (V) for a constant resistance (R).)

When a copper wire is connected to a battery, the electric field in the wire is
uniform in the wire
(In a uniform conductor like copper, the electric field is constant (uniform) along the wire when connected to a steady voltage source.)

Units for specific resistance ….
Ωm
(Specific resistance (resistivity) has units of ohm-meter (Ωm).)

( ) = which quantity?
Potential
(Electric potential (V) is defined as work done per unit charge, and since work = force × distance, ( V = ).)

A current through a conductor is 2A. When there is a potential difference of 10V between its ends, its resistance is
5Ω
(Using Ohm's law, ( R = ).)

( )
(This is the formula for refraction at a curved spherical surface separating two media with refractive indices ( n₁ ) and ( n₁ ).)

When an electron jumps from ground to excited state, it absorbs energy.
Answer: Energy is absorbed

After 3p, the next orbital filled (according to the Aufbau principle) is 4s.
Answer: 4s

For n = 4, the number of subshells = n⇒ 4 subshells: 4s, 4p, 4d, 4f
Answer: 4

In EM spectrum, radio waveshave the longest wavelength.
Answer: Radio waves

Magnetic quantum number (m₁)explains orientationof orbitals in space.
Answer: m₁

Sodium's valence electron: atomic number 11 → electron in 3s¹ ⇒n = 3, l = 0, m = 0, s = +½
Answer: n=3, l=0, m=0, s=+½

Impurities in ore = gangue
Answer: Gangue

In smelting haematite, CaCO₃acts as a fluxto remove SiO₂.
Answer: CaCO₃

Potassium (K) and Calcium (Ca) are in Group 1 and 2 respectively → s-block
Answer: s-block elements

Haematite (Fe₂O₃)is an important ore of iron.
Answer: Haematite

In electrolytic refining, pure metalis deposited on the cathode.
Answer: Cathode

Thermite processinvolves reductionof metal oxides (e.g., Fe₂O₃) using aluminum.
Answer: Reduction

Symbol Δ(A) in a chemical equation indicates heating.
Answer: Heating

Quick lime = CaO, reacts with water to form Ca(OH)₂ (slaked lime).
Answer: Ca(OH)₂

Electrolysis of aqueous NaCl gives NaOH, Cl₂, H₂. Not O₂.
Answer: O₂

One mole of H₂Ohas highest mass (≈18g).
Answer: One mole H₂O

Balanced equation:
2Na + 2H₂O → 2NaOH + H₂↑
Coefficient of Na = 2
Answer: 2

Reactions absorbing heatare endothermic.
Answer: Endothermic reaction

Na (Sodium)is notan inert gas (it's an alkali metal).
Answer: Na

In 2nd period:
s-block: 2 elements (Li, Be)p-block: 6 elements (B to Ne)
Answer: 2, 6

Elements with similar chemical properties belong to the same group.9 (F), 17 (Cl), 35 (Br)→ Halogens(Group 17)
Answer: 9, 17, 35

Group 2 (IIA) elements like Be, Mg, Caare called alkaline earth metals.
Answer: Alkaline earth metals

Calcium (Ca) has atomic number 20 and electronic configuration: 2, 8, 8, 2 → valency = 2
Answer: 2

Element with atomic number 17 is Cl, and it is in 3rd period.
Answer: 3

HClis a strong acid, while CH₃COOH is weak, NH₄OH is weak base.
Answer: HCl

CO₂forms carbonic acid in water → it is an acidic oxide.
Answer: CO₂

Metallic oxideslike Na₂O, CaO, MgO are generally basic.
Answer: Basic

Antacidsneutralize excess acid → they are basicin nature.
Answer: Basic

NaClis a neutral salt → aqueous solution has pH = 7
Answer: 7

Blue copper sulphatecrystals = CuSO₄·5H₂O(pentahydrate)
Answer: CuSO₄·5H₂O