46 Statistics
While calculating the mean of a grouped frequency distribution, step deviation method was used $\left(\frac{x-a}{h}=u\right)$. It was found that $\overline{x}=64$, $h=5$ and $a=62.5$. The value of $\overline{u}$ is:
Concept:
1
Step Deviation Method Formula: The primary formula connecting the final mean $\overline{x}$, the assumed mean $a$, the class size $h$, and the mean of step deviations $\overline{u}$ is given by: $\overline{x} = a + h\overline{u}$.
Hint:
1
Just substitute the known values into the equation: $64 = 62.5 + 5\overline{u}$. Subtract 62.5 from 64, and then divide the result by 5.
Mistakes to Avoid:
!
Confusing $\overline{u}$ with $u_i$: Do not plug $64$ in for $x$ into the expression $\frac{x-a}{h}$. The variable $\overline{x}$ represents the final calculated mean, not an individual class mark $x_i$. Use the full mean formula!
👉 Scroll horizontally to read the full handwritten solution.
The formula for calculating the mean using the step deviation method is:
$\overline{x} = a + h\overline{u}$
Substituting the given values into the formula:
$64 = 62.5 + 5\overline{u}$ [½ M]
Now, solve for $\overline{u}$:
$64 - 62.5 = 5\overline{u}$
$1.5 = 5\overline{u}$
$\overline{u} = \frac{1.5}{5}$
$\overline{u} = 0.3$ [½ M]
Correct Option: (C) 0.3
47 Surface Areas & Volumes
A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use $\pi=3.14$):
Concept:
1
Carving Max Volume Solids: For a cone to have the maximum possible volume inside a hemisphere, it must stretch completely across the flat base and reach all the way down to the deepest point. This means the cone's radius matches the hemisphere's radius, and its height also exactly equals the hemisphere's radius!
Hint:
1
Radius $r = 10 \text{ cm}$. Height $h = 10 \text{ cm}$. First, find the slant height of the cone using $l = \sqrt{r^2 + h^2}$. Once you have $l$, find the Curved Surface Area (CSA) using the formula $\pi r l$.
Mistakes to Avoid:
!
Confusing Height with Slant Height: Do not just plug $10$ directly into the CSA formula. The vertical height $h$ is $10$, but the formula for CSA needs the slant height $l$. Calculating $l$ via Pythagoras is a mandatory step!
👉 Scroll horizontally to read the full handwritten solution.
For maximum volume, the conical cavity must share the same base
and depth as the hemisphere it is carved out of.
Radius of the cone ($r$) = Radius of the hemisphere = $10 \text{ cm}$.
Height of the cone ($h$) = Radius of the hemisphere = $10 \text{ cm}$.
We first need the slant height ($l$) to find the Curved Surface Area.
Using Pythagoras theorem: $l = \sqrt{r^2 + h^2}$
$l = \sqrt{10^2 + 10^2}$
$l = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \text{ cm}$. [½ M]
Curved Surface Area (CSA) of the cone = $\pi r l$
$\text{CSA} = 3.14 \times 10 \times 10\sqrt{2}$
$\text{CSA} = 314\sqrt{2} \text{ cm}^2$ [½ M]
Correct Option: (A) $314\sqrt{2} \text{ cm}^{2}$
48 Arithmetic Progressions
If $a_{n}$ represents $n^{th}$ term of the A.P.- $-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots$ then value of $a_{16}-a_{12}$ is:
Concept:
1
Term Differences in AP: You do not need to calculate the actual values of $a_{16}$ and $a_{12}$. The difference between any two terms in an AP is purely proportional to the common difference $d$. The formula is $a_m - a_n = (m - n)d$. Here, $a_{16} - a_{12}$ is exactly $4d$!
Hint:
1
Find the common difference $d$ first by subtracting $a_1$ from $a_2$. Note that subtracting a negative number is the same as adding a positive one! Then multiply your result by $4$.
Mistakes to Avoid:
!
Long Method Arithmetic Errors: If you try to calculate $a_{16} = a + 15d$ and $a_{12} = a + 11d$ fully with the fractions and then subtract them, you are highly likely to make a sign or fraction error. Use the shortcut $4d$!
👉 Scroll horizontally to read the full handwritten solution.
First, we find the common difference ($d$):
$d = a_2 - a_1$
$d = -\frac{10}{4} - \left(-\frac{15}{4}\right)$
$d = -\frac{10}{4} + \frac{15}{4} = \frac{5}{4}$ [½ M]
We need to find the value of $a_{16} - a_{12}$.
Instead of finding each term, we use the AP property:
$a_n = a + (n-1)d$
$a_{16} - a_{12} = (a + 15d) - (a + 11d)$
$a_{16} - a_{12} = 4d$
Substitute the value of $d$:
$4d = 4 \times \left(\frac{5}{4}\right)$
$4d = 5$ [½ M]
Correct Option: (C) 5
49 Quadratic Equations
The value of $p$ for which roots of the quadratic equation $x^{2}-px+6=0$ are rational, is:
Concept:
1
Condition for Rational Roots: The quadratic formula contains $\pm \sqrt{D}$. For the final roots to be rational (no square roots left over), the Discriminant $D = b^2 - 4ac$ must evaluate to a perfect square number (e.g., 0, 1, 4, 9, 16, 25...).
Hint:
1
Calculate the discriminant expression: $D = (-p)^2 - 4(1)(6) = p^2 - 24$. Now, plug the given multiple-choice options into $p$ and see which one results in a perfect square.
Mistakes to Avoid:
!
Confusing "Rational" with "Equal": A massive mistake is setting $D = 0$ assuming the question asked for real and equal roots. That would give $p^2 = 24$, which means $p$ would be an irrational number ($\sqrt{24}$). Read carefully: the roots must be rational!
👉 Scroll horizontally to read the full handwritten solution.
For the roots of a quadratic equation to be rational,
its discriminant ($D = b^2 - 4ac$) must be a perfect square.
Comparing $x^{2}-px+6=0$ with standard form:
$a = 1$, $b = -p$, $c = 6$
$D = (-p)^2 - 4(1)(6)$
$D = p^2 - 24$ [½ M]
Let's substitute the given options to check which yields a perfect square:
Option A ($p=1$): $D = 1^2 - 24 = -23$ (Not a perfect square, complex roots)
Option B ($p=-5$): $D = (-5)^2 - 24 = 25 - 24 = 1$.
Since $1$ is a perfect square ($1^2 = 1$), this is the correct value. [½ M]
Correct Option: (B) -5
50 Probability
Which of the following can not be the probability of an event?
Concept:
1
Probability Limits: The probability of any event must strictly lie between $0$ (impossible event) and $1$ (certain event), inclusive. Mathematically, $0 \le P(E) \le 1$. It can never be a negative number, and it can never be greater than 1 (or $100\%$).
Hint:
1
Convert all the options into simple decimals. Which one of these values is bigger than $1.00$? That is the impossible one!
Mistakes to Avoid:
!
Misinterpreting Percentages: Students often pick Option D ($10\%$) because $10$ is greater than $1$. However, the percent sign means "divide by 100". Thus, $10\% = 10 / 100 = 0.1$, which is perfectly valid! Look closer at Option C instead.
👉 Scroll horizontally to read the full handwritten solution.
The fundamental rule of probability is that the probability
of an event $E$ must lie between 0 and 1 inclusive:
$0 \le P(E) \le 1$
Let's check the given options:
(A) $\frac{39}{100} = 0.39$ (Valid)
(B) $\frac{0.001}{20} = 0.00005$ (Valid)
(C) $\frac{10}{0.2} = \frac{100}{2} = 50$ (Invalid, greater than 1) [½ M]
(D) $10\% = \frac{10}{100} = 0.1$ (Valid)
Since $50 > 1$, Option C cannot be a probability. [½ M]
Correct Option: (C) $\frac{10}{0.2}$
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