Live practice

Formula or theorem used: Write the equation in the form \(ax^2+bx+c=0\). Factorise where possible; otherwise use \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).

Working: Display the standard form, the factorisation or discriminant calculation, both roots, and reject any value that is impossible in the stated context.

After substitution and simplification:

(i) k = ± 2 6 (ii) k = 6

Formula or theorem used: Translate the information into two linear equations. Use the method named in the exercise, keeping equivalent equations aligned.

Working: Eliminate or substitute one variable, obtain the other variable, substitute back, and verify the ordered pair in both original equations.

After substitution and simplification:

(i) x – y = 26, x = 3y, where x and y are two numbers (x > y); x = 39, y = 13.
(ii) x – y = 18, x + y = 180, where x and y are the measures of the two angles in degrees;
x = 99, y = 81.
(iii) 7x + 6y = 3800, 3x + 5y = 1750, where x and y are the costs (in ₹) of one bat and one
ball respectively; x = 500, y = 50.
(iv) x + 10y = 105, x + 15y = 155, where x is the fixed charge (in ₹) and y is the charge (in
₹ per km); x =5, y = 10; ₹ 255.
(v) 11x – 9y + 4 = 0, 6x – 5y + 3 = 0, where x and y are numerator and denominator of the
fraction;
7 ( 7, 9).9 x y= =
(vi) x – 3y – 10 = 0, x – 7y + 30 = 0, where x and y are the ages in years of Jacob and his
son; x = 40, y = 10.

There are \(5+8=13\) equally likely fish, of which 5 are male. Therefore,

Formula or theorem used: For grouped data use class marks and frequencies systematically. Apply the direct, assumed-mean, or step-deviation formula and show the working table totals.

Working: Write the relevant mean, median, or mode formula, identify the modal/median class, substitute the table totals, and interpret the numerical result in the question's context.

After substitution and simplification:

65.625 hours

The radius is \(1.5\) cm and the cylinder height is \(12-2-2=8\) cm.

Using \(\pi=\frac{22}{7}\), \(V=66\text{ cm}^3\).

Formula or theorem used: Use \(C=2\pi r\), \(A=\pi r^2\), sector area \(=\frac{\theta}{360^\circ}\pi r^2\), and arc length \(=\frac{\theta}{360^\circ}2\pi r\).

Working: Split the figure into sectors, segments, triangles, or rectangles; calculate each labelled area and combine them with the correct addition or subtraction.

After substitution and simplification:

\(2158125\,\mathrm{cm}\)
126

Formula or theorem used: Use that a tangent is perpendicular to the radius at the point of contact and that tangents drawn from an external point are equal.

Working: Name the radii and tangents, state the theorem, establish congruent right triangles when required, and then obtain the requested angle or length.

After substitution and simplification:

\(8\,\mathrm{cm}\)

Let the width be \(x\) and height be \(h\).

Equating \(1\) and \(2\): \(3x=x+20\), so \(x=10\) m and \(h=10\sqrt3\) m.

Formula or theorem used: In a right triangle use \(\sin\theta=\frac{P}{H}\), \(\cos\theta=\frac{B}{H}\), and \(\tan\theta=\frac{P}{B}\), together with the standard-angle values and identities.

Working: Write the relevant ratio or identity, substitute the known value, simplify one equality at a time, and state restrictions when a denominator could be zero.

Write each reason beside the corresponding equality. The conclusion follows only after the stated theorem or congruence/similarity criterion has been established.

After substitution and simplification:

(i) False (ii) True (iii) False (iv) False (v) False

Formula or theorem used: Use \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) or the section formula \(\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)\).

Working: Substitute the coordinates before simplifying. For a shape, compare the required side lengths or diagonals and state the conclusion.

After substitution and simplification:

(1, 3)