KARNATAKA STATE BOARD CLASS 10TH MATHS 2022-1 Solutions &

Q19 short answer

Find how many terms of the arithmetic progression 3, 6, 9, .... must be added to get the sum 165.

Step-by-Step AI Explanation

Verified Method

Given AP: 3, 6, 9, ... Here, $a = 3$, $d = 3$, and $S_n = 165$. Using the sum formula $S_n = \frac{n}{2} [2a + (n-1)d]$: $165 = \frac{n}{2} [2(3) + (n-1)3]$ $330 = n [6 + 3n - 3]$ $330 = n [3n + 3]$ $330 = 3n^2 + 3n$ Divide by 3: $n^2 + n - 110 = 0$ $(n + 11)(n - 10) = 0$ $n = -11$ (not possible as terms cannot be negative) or $n = 10$. Therefore, 10 terms must be added.

Q19 short answer

Find the sum of arithmetic progression 7, 11, 15, ...... to 16 terms using formula.

Step-by-Step AI Explanation

Verified Method

Given AP: 7, 11, 15, ... First term $a = 7$, common difference $d = 11 - 7 = 4$, and number of terms $n = 16$. The sum of $n$ terms is given by: $S_n = \frac{n}{2} [2a + (n-1)d]$ Substituting the values: $S_{16} = \frac{16}{2} [2(7) + (16-1)4]$ $S_{16} = 8 [14 + (15 \times 4)]$ $S_{16} = 8 [14 + 60]$ $S_{16} = 8 \times 74 = 592$ The sum of the first 16 terms is 592.

Q20 short answer

Find the value of the discriminant of the equation $4x^2 - 12x + 9 = 0$ and hence write the nature of the roots.

Step-by-Step AI Explanation

Verified Method

For the quadratic equation $4x^2 - 12x + 9 = 0$: Here, $a = 4, b = -12, c = 9$. Discriminant $D = b^2 - 4ac$ $D = (-12)^2 - 4(4)(9)$ $D = 144 - 144 = 0$ Since the discriminant is 0, the roots of the equation are real and equal.

Q21 short answer

Find the roots of the equation $x^2 - 3x + 1 = 0$ using quadratic formula.

Step-by-Step AI Explanation

Verified Method

For $x^2 - 3x + 1 = 0$, $a = 1, b = -3, c = 1$. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(1)}}{2(1)}$ $x = \frac{3 \pm \sqrt{9 - 4}}{2}$ $x = \frac{3 \pm \sqrt{5}}{2}$ So, the roots are $x = \frac{3 + \sqrt{5}}{2}$ and $x = \frac{3 - \sqrt{5}}{2}$.

Q22 short answer

In the figure ABC is a right angled triangle. If AB = 24 cm, BC = 7 cm and AC = 25 cm, then write the value of $\sin \alpha$ and $\cos \alpha$.

Step-by-Step AI Explanation

Verified Method

In triangle ABC, for angle $\alpha$ (at vertex C): Opposite side = $AB = 24$ cm Adjacent side = $BC = 7$ cm Hypotenuse = $AC = 25$ cm $\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{24}{25}$ $\cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{7}{25}$

Q23 short answer

Find the distance between the points P ( 2, 3 ) and Q ( 4, 1 ) using distance formula.

Step-by-Step AI Explanation

Verified Method

Given $P(2, 3)$ and $Q(4, 1)$. Distance $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $d = \sqrt{(4 - 2)^2 + (1 - 3)^2}$ $d = \sqrt{(2)^2 + (-2)^2}$ $d = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$ units.

Q23 short answer

Find in what ratio the point P ( – 4, 6 ) divides the line segment joining the points A ( – 6, 10 ) and B ( 3, – 8 ).

Step-by-Step AI Explanation

Verified Method

Let the ratio be $m:n$. Using the section formula for $x = -4$: $-4 = \frac{m(3) + n(-6)}{m+n}$ $-4(m + n) = 3m - 6n$ $-4m - 4n = 3m - 6n$ $2n = 7m$ $\frac{m}{n} = \frac{2}{7}$ The point P divides the line segment in the ratio $2:7$.

Q24 short answer

Draw a line segment of length 8·4 cm and divide it in the ratio 1 : 3 by geometric construction.

Step-by-Step AI Explanation

Verified Method

1. Draw a line segment $AB = 8.4$ cm using a scale. 2. Draw an acute angle $\angle BAX$ at vertex A. 3. Mark $1 + 3 = 4$ equal points on $AX$ (say $A_1, A_2, A_3, A_4$) using a compass. 4. Join $A_4$ to $B$. 5. Draw a line parallel to $A_4B$ through point $A_1$ (using corresponding angles) to meet $AB$ at $P$. 6. The point $P$ divides $AB$ in the ratio $1:3$.

Q25 short answer

Find the arithmetic progression whose third term is 16 and its 7th term exceeds the 5th term by 12.

Step-by-Step AI Explanation

Verified Method

Let the first term be $a$ and common difference be $d$. The $n$-th term is given by $a_n = a + (n-1)d$. Given $a_3 = 16 \implies a + 2d = 16$ (Eq. 1). Given $a_7 = a_5 + 12 \implies (a + 6d) = (a + 4d) + 12 \implies 2d = 12 \implies d = 6$. Substituting $d = 6$ into Eq. 1: $a + 2(6) = 16 \implies a + 12 = 16 \implies a = 4$. The progression is $a, a+d, a+2d, \dots$, which is $4, 10, 16, 22, \dots$

Q26 short answer

The sum of the reciprocals of Rehman’s age ( in years ) 3 years ago and his age 5 years from now is $1/3$. Find his present age.

Step-by-Step AI Explanation

Verified Method

Let Rehman's present age be $x$ years. Three years ago, age was $(x-3)$; five years from now, age will be $(x+5)$. The condition is: $\frac{1}{x-3} + \frac{1}{x+5} = \frac{1}{3}$. $\frac{x+5+x-3}{(x-3)(x+5)} = \frac{1}{3} \implies \frac{2x+2}{x^2+2x-15} = \frac{1}{3}$. $x^2+2x-15 = 6x+6 \implies x^2-4x-21 = 0$. Factoring: $(x-7)(x+3) = 0$. Since age cannot be negative, $x=7$. Rehman's present age is 7 years.

Q26 short answer

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.