TAMIL NADU STATE BOARD CLASS 10 MATHS 2023 Solutions &

Q9 mcq 1 mark

The slope of the straight line perpendicular to x-axis is :

a.

1

b.

0

c.

∞ Correct Answer

d.

-1

Step-by-Step AI Explanation

Verified Method

The line perpendicular to the x-axis is a vertical line. A vertical line has undefined slope, often written as $\infty$ in such school-level MCQs. So the correct choice is (c) $\infty$. The other options are not correct because slope $0$ is for a line parallel to the x-axis, and $1$ or $-1$ are slopes of slanting lines.

Q13 mcq

If the sum and mean of a data are 407 and 11 respectively, then the number of observations in the data are:

a.

37 Correct Answer

b.

4477

c.

396

d.

418

Step-by-Step AI Explanation

Verified Method

Mean = $\dfrac{\text{sum}}{\text{number of observations}}$. Given sum = 407 and mean = 11, $$11 = \frac{407}{n}$$ So, $$n = \frac{407}{11} = 37$$ Hence, the number of observations is 37. Therefore, the correct option is (a).

Q14 mcq

If a letter is chosen at random from the English alphabets {a, b, ..., z}, then the probability that the letter chosen precedes x:

a.

$\frac{12}{13}$

b.

$\frac{1}{13}$ Correct Answer

c.

$\frac{23}{26}$

d.

$\frac{3}{26}$

Step-by-Step AI Explanation

Verified Method

There are 26 English letters in total: $a, b, c, \dots, z$. Letters that precede $x$ are $a$ through $w$, which is 23 letters. So the probability is $$\frac{23}{26}$$ Wait: the question asks for the probability that the chosen letter precedes $x$, which means any letter before $x$ in the alphabet. That is 23 out of 26. Therefore the correct answer is (c) $\frac{23}{26}$.

Q15 short answer

If $B \times A = \{(-2, 3), (-2, 4), (0, 3), (0, 4), (3, 3), (3, 4)\}$, find $A$ and $B$.

Step-by-Step AI Explanation

Verified Method

Given $$B \times A = \{(-2,3),(-2,4),(0,3),(0,4),(3,3),(3,4)\}$$ In a Cartesian product $B \times A$, the first components come from $B$ and the second components come from $A$. So, $$B = \{-2,0,3\}$$ and $$A = \{3,4\}$$ Hence, $A = \{3,4\}$ and $B = \{-2,0,3\}$.

Q16 short answer

Find $k$ if $f(k)=5$ where $f(k)=2k-1$.

Step-by-Step AI Explanation

Verified Method

Given $f(k)=2k-1$ and $f(k)=5$. So, $$2k-1=5$$ $$2k=6$$ $$k=3$$ Therefore, $k=3$.

Q17 short answer

Find $x$ so that $x^{16}$, $x^{112}$ and $x^{115}$ are consecutive terms of a Geometric Progression.

Step-by-Step AI Explanation

Verified Method

For three consecutive terms of a G.P., we have $$\text{(middle term)}^2 = (\text{first term})(\text{third term})$$ Here the terms are $x^{16}, x^{112}, x^{115}$. So, $$\left(x^{112}\right)^2 = x^{16}\cdot x^{115}$$ $$x^{224} = x^{131}$$ Thus, $$x^{224-131}=1$$ $$x^{93}=1$$ So the required value of $x$ is any number satisfying $$x^{93}=1.$$ For real values, this gives $x=1$.

Q18 short answer

Simplify: $$\frac{2x^2+2x-6}{4x^2-12y}$$

Step-by-Step AI Explanation

Verified Method

Simplify: $$\frac{2x^2+2x-6}{4x^2-12y}$$ Factorise numerator and denominator: $$2x^2+2x-6 = 2(x^2+x-3)$$ $$4x^2-12y = 4(x^2-3y)$$ So, $$\frac{2(x^2+x-3)}{4(x^2-3y)}$$ Cancel the common factor $2$: $$\frac{x^2+x-3}{2(x^2-3y)}$$ Hence, the simplified form is $$\boxed{\frac{x^2+x-3}{2(x^2-3y)}}$$

Q19 short answer

Determine the nature of roots for the following quadratic equation: $$2x^2-x-1=0$$

Step-by-Step AI Explanation

Verified Method

Given quadratic equation: $$2x^2-x-1=0$$ Here, $a=2$, $b=-1$, $c=-1$. Discriminant: $$D=b^2-4ac$$ $$D=(-1)^2-4(2)(-1)=1+8=9$$ Since $D>0$ and is a perfect square, the equation has two distinct real and rational roots. Therefore, the nature of the roots is: two distinct real rational roots.

Q20 short answer

In the figure AD is the bisector of $\angle BAC$, if $AB=10$ cm, $AC=14$ cm and $BC=6$ cm. Find $BD$ and $DC$.

Step-by-Step AI Explanation

Verified Method

Since $AD$ bisects $\angle BAC$, by the Angle Bisector Theorem, $$\frac{BD}{DC}=\frac{AB}{AC}=\frac{10}{14}=\frac{5}{7}.$$ Also, $BD+DC=BC=6$. Let $BD=5x$ and $DC=7x$. Then $$5x+7x=6\implies 12x=6\implies x=\frac12.$$ Therefore, $$BD=5\left(\frac12\right)=2.5\text{ cm},$$ $$DC=7\left(\frac12\right)=3.5\text{ cm}.$$

Q21 short answer

A cat is located at the point $(-6,-4)$ in $xy$ plane. A bottle of milk is kept at $(5,11)$. The cat wishes to consume the milk travelling through shortest possible distance. Find the equation of the path it needs to take the milk.

Step-by-Step AI Explanation

Verified Method

The shortest path between two points is the straight line joining them. The points are $(-6,-4)$ and $(5,11)$. Slope of the line: $$m=\frac{11-(-4)}{5-(-6)}=\frac{15}{11}.$$ Using point-slope form with $(-6,-4)$: $$y+4=\frac{15}{11}(x+6).$$ So the equation of the path is $$11y+44=15x+90$$ $$\Rightarrow 15x-11y+46=0.$$

Q22 short answer

If the straight lines $12y=-(P13)x112$, $12x-7y=16$ are perpendicular then find $P$.