Given:
△ABC \~ △PQR and ∠A = 50°
In similar triangles, corresponding angles are equal, and the sum of angles in any triangle is 180°.
So:
∠A corresponds to ∠P → ∠P = 50°
Then, ∠Q + ∠R = 180° – ∠P = 180° – 50° = 130°
Correct answer: 130°

We are given:
Radius r = 7 cm
Height h = 10 cm
We need to find the lateral height (also called slant height), denoted by l
Shortcut Formula for slant height of a cone:


Final Answer: D. 12.2 cm

We are given:

Shortcut Trick:
Use the identity:

Substitute:


Final Answer: 2

We are given:

Shortcut Method:
Recall the standard value:

Now, use:

Final Answer:

We are given:

Shortcut Method: Use Pythagorean Triplet
If ,
then use the Pythagorean identity:

Now,

Final Answer:

We are asked to evaluate:

Shortcut Observation:
Use the identity:

So,

Therefore:

Final Answer: 1

We are given:

Shortcut Step 1: Recall standard values

Step 2: Find

Final Answer:

We are given:

Shortcut Method: Use standard angles

Final Answer: 45°

We are given:
Height of the pole = h
Length of the shadow =
Shortcut Method: Use basic trigonometry
From the triangle formed by the pole and its shadow:

Final Answer: 30°

To solve this quickly, let's apply a shortcut method using the concept of distribution of days in a non-leap year:
Step 1: Total days in a non-leap year = 365
That's 52 weeks + 1 extra day
So, every non-leap year has 52 Thursdays for sure, and whether it has 53 Thursdays depends on that 1 extra day.
Step 2: What can the extra day be?
The extra day could be: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday (7 possibilities).
Out of these 7, only if the extra day is a Thursday, then we will get 53 Thursdays.
Probability =

Final Answer:

Let's solve this using the shortcut method:
Step 1: Total number of balls
\= 4 (black) + 6 (red) = 10 balls
Step 2: Number of favorable outcomes (red ball)
\= 6 red balls
Probability =

Final Answer:

If events E₁ and E₂ are mutually exclusive, it means they cannot happen at the same time.
Shortcut Concept:
Mutually exclusive ⇒ No overlap ⇒

Final Answer: Φ (null set)

Shortcut Method:
When n coins are tossed, the total number of outcomes =

Here, n = 3 ⇒

Final Answer: 8

Shortcut for Mode:
Mode = Number that appears most frequently
Given numbers:
1, 2, 3, 8, 10, 11, 16
Each number appears once → no repetition.
Final Answer: none of these
(Since no number repeats, there's no mode)

To find the arithmetic mean, use the formula:

Given terms:a – 3d, a – d, a + d, a + 3d
Step 1: Add them all
(a – 3d) + (a – d) + (a + d) + (a + 3d) = 4a
Step 2: Divide by number of terms (4)

Final Answer: a

Use the shortcut formula:

Given:
Product = 30
HCF = 5
So:

Final Answer: 6

A composite number has more than two factors (not prime), and an odd number is not divisible by 2.
Check options:
3 → Prime
5 → Prime
7 → Prime
9 → 3 × 3 → More than 2 factors → Composite and odd
Final Answer: 9

To solve the equation , let's follow these steps:
Step 1: Rewrite the Logarithmic Equation in Exponential Form
Given:

This means:


Step 2: Solve for x
Take the square root of both sides:


Step 3: Verify the Solutions
Both x = 2 and x = -2 satisfy the original equation because:

Final Answer
Both A. 2 and B. -2 are correct solutions. However, if the question expects a single answer, either or is acceptable.
If the options are to be interpreted as allowing multiple correct answers, both A and B are correct.
But typically, such questions expect one correct choice, so you can select either A. 2 or B. -2 based on the context.
However, looking at the options, both A and B are correct. If only one option is to be selected, the primary answer is: A. 2
(and B. -2 is also correct)
If the question implies selecting all correct options, then both A and B are correct.
But in most standard cases, the simplest answer is:
A. 2
*(Note: Since logarithms of squared numbers are defined for both positive and negative values, both solutions are valid.)
Conclusion
The correct options are A and B, but if only one answer is expected, A. 2 is the standard choice.
Final Answer: A. 2 (and B. -2 is also correct)

We are given:
A ∩ B = B
This means:
All elements of B are also in A, because the intersection gives back B itself.
So, B is a subset of A, i.e.,B ⊆ A
Which matches Option B: B ⊂ A
(Note: In some contexts, ⊂ is used to mean ⊆ — subset. If strict subset is intended, the wording would be different.)
Final Answer: B ⊂ A

A biquadratic polynomial is a polynomial of degree 4, typically of the form:

Even though it's written in terms of , its highest power is 4.
Shortcut:
A polynomial of degree n can have at most n real or complex zeroes.
So, a biquadratic polynomial (degree 4) can have:
At most 4 zeroes
Final Answer: 4

We are given the quadratic polynomial:

Rewriting it in standard form:

Shortcut to find product of zeroes of a quadratic:
For a quadratic ,
Product of zeroes =
Here:


Final Answer: D. ½

We are given the cubic polynomial:

Step 1: Factor the polynomial
Factor out the common term:

Step 2: Find the zeroes
Set :

So the zeroes are 0, 0, and 1.
Final Answer: 0, 0, 1

To form a quadratic polynomial with given zeroes and , use the standard form:

Final Answer:

To determine the nature of the solutions for the equation , let's analyze it step-by-step.
Step 1: Understand the Equation
The given equation is:
x – 4y = 5
This is a linear equation in two variables ( and ).
Step 2: Express in Slope-Intercept Form
Rewriting the equation to solve for :
x – 4y = 5
-4y = -x + 5

This is now in the slope-intercept form y = mx + b , where:
– Slope (m):
– Y-intercept (b):
Step 3: Analyze the Solutions
A linear equation in two variables represents a straight line on the Cartesian plane. For such equations:
– Every point (x, y) on the line is a solution to the equation.
– Since a line extends infinitely in both directions, there are infinitely many solutions.
Step 4: Verify with Examples
Let's find a few solutions to confirm:
→1. If :

So, is a solution.
→2. If :

So, is another solution.
→3. If :

So, is yet another solution.
This pattern continues infinitely, demonstrating that there are infinitely many solutions.
Step 5: Conclusion
The equation represents a straight line with infinitely many points (solutions).
Final Answer: Infinitely many solutions

We are given the equation:
ax + b = 0
To solve for :

Final Answer:

To determine which is not a linear equation, we check for variables raised to powers other than 1 (i.e., squares, roots, etc.).
Definitions:
A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable, and no variable is raised to a power other than 1.
Options:
A. 3x – 2y = y + x
Simplifies to 2x – 3y = 0 — Linear
B. x + y = 1
Already linear
C. 1 + 2x = y – 5
Simplifies to — Linear
D.
Has , which is non-linear
Correct Answer:

We are given:
Siri bought 5 apples and 6 oranges → Cost: 5x + 6y
Laxmi bought 2 apples and 15 oranges → Cost: 2x + 15y
Total cost is same for both → So, equate both expressions:
5x + 6y = 2x + 15y
Correct Answer: 5x + 6y = 2x + 15y

Let's check each option to see which represents a quadratic equation (i.e., an equation of the form ):
A. x(x + 4) = 12
→ Expand LHS:
→ Rearranged: Quadratic!
B.
→ LHS: , RHS:
→ Subtract RHS:
Linear, not quadratic
C. x(x + 4) – x(x – 2) = 0
→ Expand:
→ Simplify:
Linear
D. x(x + 4) = x(x + 5) – x
→ LHS: , RHS:
→ Both sides equal ⇒ 0 = 0
Identically true, not a quadratic equation
Correct Answer:

To determine the degree of the given equation:

Step 1: Expand the Left Side

So, the equation becomes:

Step 2: Simplify the Equation
Subtract from both sides:

Bring all terms to one side:

Combine like terms:

Step 3: Determine the Degree
The simplified equation is:

The degree of an equation is the highest power of present. Here, the highest power is , so the degree is 2.
Final Answer:

Correct Option: 2

We are given that 18, x, 36 are in an arithmetic progression (A.P.)
Shortcut method:
In an A.P., the middle term is the average of the first and third terms:

Final Answer: 27

If a, b, c are in arithmetic progression (A.P.), then:

Multiply both sides by 2:

Final Answer: 2b

To find the common difference (d) in an Arithmetic Progression (A.P.), subtract any term from the next term:
d = 806 – 781 = 25
Final Answer: 25

We are given:
Product = 91
Arithmetic Mean = 10 → So,

We now solve using sum and product shortcut:
We need two numbers whose:
Sum = 20
Product = 91
Try options:
Option C: 13 and 7
13 + 7 = 20 and 13 × 7 = 91
Both conditions are satisfied.
Final Answer: 13, 7

We are given:
Vertices of triangle:
A(a, b), B(b, c), C(c, a)
Centroid is
Step 1: Use centroid formula
The coordinates of the centroid of triangle with vertices is:

Plug in:


Given centroid is , so:

Step 2: Use identity
We want to find:

Use identity:

From (1), , so:

Final Answer: 3abc

We are given the vertices of a parallelogram in order:

Step 1: Diagonals intersect at their midpoint
We can pick opposite vertices (diagonal endpoints):
Diagonal AC: Points and
Find midpoint of AC:

Check diagonal BD just to verify:
Points and
Midpoint:

Same point! So diagonals intersect at (0, -1)
Final Answer: (0, -1)

We are given two points:

Step-by-step shortcut:
These two points lie on the y-axis, so the distance between them is just the difference in y-coordinates.
Distance = |a – (-a)| = |a + a| = 2a
Final Answer: 2a

We are given:
Height of first pole = 6 m
Height of second pole = 11 m
Distance between their feet = 12 m
Step-by-step shortcut (Apply Pythagoras theorem):
Difference in height = 11 – 6 = 5 m
This forms a right triangle with:
vertical side = 5 m
horizontal side = 12 m
hypotenuse (distance between tops) = ?

Final Answer: 13

Materials with resistivity in the order of to -m\~ -m\~ to -m
Insulators: Resistivity 10^{11}10^{16}\ \Omega$-m
Final Answer: Insulators

We use Ohm's Law:
V = I × R
Given:
I = 1.5\ \text{A}


Final Answer: 30 V

To find the refractive index , use the shortcut formula:

Step-by-step shortcut:

Final Answer: 1.5