Question 1
Every rational number is:
✅ Correct Answer: a real number
Explanation: Real numbers consist of all rational and irrational numbers. Therefore, every rational number is a real number, whereas it may not be a whole number, natural number, or integer (e.g., $2/3$).
Question 2
If $p(x) = x + 3$ then $p(x) + p(-x)$ is equal to:
✅ Correct Answer: $6$
Explanation: Given $p(x) = x + 3$, we find $p(-x) = -x + 3$.
Adding both equations: $p(x) + p(-x) = (x + 3) + (-x + 3) = 6$.
Question 3
The point whose ordinate is $4$ and which lies on the y-axis is:
✅ Correct Answer: $(0,4)$
Explanation: Any point lying on the y-axis has its x-coordinate (abscissa) equal to $0$. Since the ordinate (y-coordinate) is given as $4$, the coordinates of the point are $(0,4)$.
Question 4
If a linear equation has solutions $(-2,2)$, $(0,0)$, and $(2,-2)$ then it is of the form:
✅ Correct Answer: $x+y=0$
Explanation: By substituting the given points into the equations, we see that for $(-2,2)$ we get $-2+2=0$. For $(2,-2)$, we get $2+(-2)=0$. Thus, the points satisfy the equation $x+y=0$.
Question 5
If two lines intersect each other, then the vertically opposite angles are:
✅ Correct Answer: Equal
Explanation: According to Euclid's axioms and structural theorems of geometry, when two straight lines intersect, the vertically opposite angles formed are always equal to each other.
Question 6
In $\triangle ABC$ and $\triangle PQR$, $AB=QR$, $BC=PR$ and $CA=PQ$. Then which of the following holds true?
✅ Correct Answer: $\triangle BAC \cong \triangle QRP$
Explanation: We need to match the corresponding vertices based on the given side equalities. Since $AB=QR$, $BC=PR$, and $CA=PQ$, the correct symbolic congruence correspondence is $\triangle BAC \cong \triangle QRP$ by SSS criteria.
Question 7
The total surface area of a cone whose radius is $\frac{r}{2}$ and slant height is $2l$ is:
✅ Correct Answer: $\pi r\left(l+\frac{r}{4}\right)$
Explanation: Total Surface Area of a cone $= \pi \times \text{radius} \times (\text{slant height} + \text{radius})$.
Substituting the values: $\text{TSA} = \pi \left(\frac{r}{2}\right) \left(2l + \frac{r}{2}\right) = \pi r l + \frac{\pi r^2}{4} = \pi r\left(l+\frac{r}{4}\right)$.
Question 8
In a sample of $500$ electric bulbs, the dynamic breakdown analysis shows that the probability of getting a defective bulb is $0.04$. Find the number of non-defective bulbs in the lot.
✅ Correct Answer: $480$
Explanation: Number of defective bulbs $= 500 \times 0.04 = 20$.
Therefore, the number of non-defective bulbs $= 500 - 20 = 480$.
Question 9
In the given figure, $O$ is the center of the circle. If $\angle OBC=40^\circ$, then the value of $\\angle BAC$ is:
✅ Correct Answer: $50^\circ$
Explanation: In $\triangle OBC$, $OB=OC$ (radii), so $\angle OCB = \angle OBC = 40^\circ$.
$\angle BOC = 180^\circ - (40^\circ + 40^\circ) = 100^\circ$.
Since the angle subtended by an arc at the center is double the angle subtended at the remaining part of the circle, $\angle BAC=\frac{100^\circ}{2}=50^\circ$.
Question 10
The class marks of a frequency distribution are given as $15, 20, 25, 30, \dots$ The class interval corresponding to the class mark $20$ is:
✅ Correct Answer: $17.5-22.5$
Explanation: The class size $h$ is the difference between two consecutive class marks: $20-15=5$.
Lower limit $= \text{Class mark} - \frac{h}{2} = 20 - 2.5 = 17.5$.
Upper limit $= \text{Class mark} + \frac{h}{2} = 20 + 2.5 = 22.5$. Thus, the interval is $17.5 - 22.5$.
Question 11
A corporate park wants to create a triangular green zone. The sides of this triangular plot are $50\text{ m}$, $65\text{ m}$, and $65\text{ m}$. Find the cost of turfing this lawn at the rate of $7$ per $\text{m}^2$.
✅ Correct Answer: $10,500$
Explanation: Using Heron's Formula:
$s = \frac{50+65+65}{2} = 90\text{ m}$.
$\text{Area} = \sqrt{90 \times (90-50) \times (90-65) \times (90-65)} = \sqrt{90 \times 40 \times 25 \times 25} = 1500\text{ m}^2$.
$\text{Total Cost} = 1500 \times 7 = \$10,500$.
Question 12
Rahul has a cylindrical water storage tank at his home. The diameter of the base is $1.4\text{ m}$ and its height is $2\text{ m}$. Due to wear and tear, he wants to paint the outer curved surface area of this open tank. Find the area to be painted.
✅ Correct Answer: $8.8\text{ m}^2$
Explanation: Given radius $r=\frac{1.4}{2}=0.7\text{ m}$ and height $h=2\text{ m}$.
Curved Surface Area ($\text{CSA}$) of a cylinder $= 2\pi rh = 2 \times \frac{22}{7} \times 0.7 \times 2 = 8.8\text{ m}^2$.
Question 13
In a residential colony, a linear water pipeline passes through the paths mapped out by the equation $2x+3y=12$. If a maintenance valve needs to be installed on the y-axis, what will be its mapping coordinates?
✅ Correct Answer: $(0,4)$
Explanation: Since the valve needs to be installed on the y-axis, its x-coordinate must be $0$.
Substituting $x=0$ into the pipeline equation: $2(0)+3y=12 \implies 3y=12 \implies y=4$.
Thus, the coordinates are $(0,4)$.
Question 14
Assertion (A): The construction of a triangle $ABC$ with given dimensions $AB=5\text{ cm}$, $BC=3\text{ cm}$ and $AC=9\text{ cm}$ is completely impossible.
Reason (R): In any triangle, the sum of the lengths of any two sides must always be strictly greater than the length of the third side.
✅ Correct Answer: Both (A) and (R) are true and (R) is the correct explanation of (A)
Explanation: Here, check the side sum property: $AB+BC=5+3=8\text{ cm}$. Since $8\text{ cm} < 9\text{ cm}$ ($AC$), a triangle cannot be formed. Both Assertion and Reason are true, and Reason is the correct explanation.
Question 15
Assertion (A): The degree of a non-zero constant polynomial like $f(x)=-7$ is zero.
Reason (R): The degree of the zero polynomial is not defined.
✅ Correct Answer: Both (A) and (R) are true but (R) is not the correct explanation of (A)
Explanation: The assertion is true because $-7$ can be written as $-7x^0$ making its degree $0$. The reason is also a true mathematical definition statement (degree of zero polynomial is undefined). However, (R) does not explain why the degree of a non-zero constant is zero.
Question 16
Two batch-mates, Piyush and Karan, marked three critical hub points for their model at positions $A(2, 3)$, $B(-2, 3)$, and $C(-2,-3)$. If they join $A$, $B$, and $C$ in sequence, what are the coordinates of the fourth point $D$ required to complete a rectangle $ABCD$?
✅ Correct Answer: $(2,-3)$
Explanation: Plotting the points shows $AB$ is a horizontal line segment from $x=-2$ to $x=2$ at height $y=3$ (length $=4$).
$BC$ is a vertical line segment from $y=3$ to $y=-3$ at $x=-2$ (length $=6$).
To complete the rectangle $ABCD$, point $D$ must share the x-coordinate of $A$ ($2$) and the y-coordinate of $C$ ($-3$). Thus, $D=(2,-3)$.
Question 17
What is the total perimeter of the completed rectangle $ABCD$ on the grid?
✅ Correct Answer: $20\text{ units}$
Explanation: Length of side $AB = 2-(-2) = 4\text{ units}$.
Breadth of side $BC=3-(-3)=6\text{ units}$.
$\text{Perimeter of rectangle} = 2 \times (\text{Length} + \text{Breadth}) = 2 \times (4+6) = 20\text{ units}$.
Question 18
[Tricky Question] If $x+\frac{1}{x}=5$, then the calculated value of $x^4+\frac{1}{x^4}$ is:
✅ Correct Answer: $527$
Explanation: Squaring both sides of $x+\frac{1}{x}=5$:
$x^2+\frac{1}{x^2}+2=25 \implies x^2+\frac{1}{x^2}=23$.
Squaring once again:
$\left(x^2+\frac{1}{x^2}\right)^2=23^2 \implies x^4+\frac{1}{x^4}+2=529 \implies x^4+\frac{1}{x^4}=527$.
Question 19
[Tricky Question] The value of $\sqrt{10} \times \sqrt{15}$ is equal to:
✅ Correct Answer: $5\sqrt{6}$
Explanation: $\sqrt{10} \times \sqrt{15} = \sqrt{10 \times 15} = \sqrt{150}$.
Simplifying the radical into prime factors: $\sqrt{2 \times 3 \times 5 \times 5} = 5\sqrt{2 \times 3} = 5\sqrt{6}$.
Question 20
[Tricky Question] In a quadrilateral $ABCD$, the bisectors of $\angle A$ and $\angle B$ meet at a point $P$. Then $2\angle APB$ is always equal to:
✅ Correct Answer: $\angle C + \angle D$
Explanation: In $\triangle APB$, $\angle APB = 180^\circ - \frac{1}{2}(\angle A + \angle B)$.
In quad $ABCD$, $\angle A + \angle B + \angle C + \angle D = 360^\circ \implies \angle A + \angle B = 360^\circ - (\angle C + \angle D)$.
Substituting this: $\angle APB = 180^\circ - \frac{1}{2}[360^\circ - (\angle C + \angle D)] = \frac{1}{2}(\angle C + \angle D)$.
Multiplying by $2$: $2\angle APB = \angle C + \angle D$.
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