Class 10 Maths Basic

Long Answer practice across CBSE Class 10 Maths Basic chapters.

Practice coverage:155 published questions across 6 formats.

ChapterTopic

Base

Find the HCF and LCM of $252$ and $198$ by prime factorisation. Verify that HCF $\times$ LCM equals the product of the two numbers. (4 marks)

Answer

HCF $= 18$ , LCM $= 2772$ ; $18 \times 2772 = 252 \times 198 = 49896$ .

Worked solution

$252 = 2^{2} \times 3^{2} \times 7$ and $198 = 2 \times 3^{2} \times 11$ . The HCF is $2 \times 3^{2} = 18$ . The LCM is $2^{2} \times 3^{2} \times 7 \times 11 = 2772$ . Hence $18 \times 2772 = 49896 = 252 \times 198$ .

FAQ

How is HCF related to LCM for two positive integers?

Their product equals the product of the two integers.

Base

Prove that $2 + 35$ is irrational, using the fact that $5$ is irrational. (4 marks)

Answer

$2 + 35$ is irrational.

Worked solution

Suppose $2 + 35 = r$ for some rational number $r$ . Then $5 = \frac{r - 2}{3}$ , which would be rational. This contradicts the fact that $5$ is irrational. Therefore $2 + 35$ is irrational.

FAQ

Why does the proof isolate the square root?

It turns the assumption into a claim that an already-known irrational number is rational.

Base

The zeroes of $2 x^{2} - 7 x + 3$ are $α$ and $β$ . Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Answer

$α = \frac{1}{2}$ , $β = 3$ ; sum $= \frac{7}{2}$ and product $= \frac{3}{2}$ .

Worked solution

$2 x^{2} - 7 x + 3 = (2 x - 1) (x - 3)$ . Thus the zeroes are $\frac{1}{2}$ and $3$ . Their sum is $\frac{1}{2} + 3 = \frac{7}{2} = - \frac{b}{a}$ , and their product is $\frac{1}{2} \times 3 = \frac{3}{2} = \frac{c}{a}$ .

FAQ

What are the zero-coefficient relations for a quadratic?

The sum is $- b / a$ and the product is $c / a$ .

Base

Find a quadratic polynomial with zeroes $3 + 2$ and $3 - 2$ . If $p (x)$ is the monic polynomial you obtain, evaluate $p (4)$ . (4 marks)

Answer

$p (x) = x^{2} - 6 x + 7$ and $p (4) = - 1$ .

Worked solution

The sum of the zeroes is $6$ and their product is $9 - 2 = 7$ . Therefore the monic polynomial is $p (x) = x^{2} - 6 x + 7$ . Hence $p (4) = 16 - 24 + 7 = - 1$ .

FAQ

Why do conjugate surds give rational coefficients?

Their sum and product are rational.

Base

A school sells adult tickets for Rs. $120$ and student tickets for Rs. $80$ . It sells $45$ tickets and collects Rs. $4400$ . Find how many tickets of each type it sold using simultaneous equations. (4 marks)

Answer

$20$ adult tickets and $25$ student tickets.

Worked solution

Let adult tickets be $a$ and student tickets be $s$ . Then $a + s = 45$ and $120 a + 80 s = 4400$ . Dividing the second by $40$ gives $3 a + 2 s = 110$ . Subtracting $2 (a + s) = 90$ gives $a = 20$ , so $s = 25$ .

FAQ

Which two quantities become variables here?

The numbers of adult tickets and student tickets.

Base

For each value of $k$ , classify $2 x + 3 y = 7$ and $4 x + k y = 14$ as having a unique solution or infinitely many solutions. For $k = 8$ , find the solution. (4 marks)

Answer

$k = 6$ : infinitely many solutions; $k \neq = 6$ : unique solution. At $k = 8$ , $(x, y) = (\frac{7}{2}, 0)$ .

Worked solution

Twice the first equation is $4 x + 6 y = 14$ . Thus $k = 6$ gives the same line and infinitely many solutions; any $k \neq = 6$ gives a unique solution. At $k = 8$ , subtracting gives $2 y = 0$ , so $y = 0$ and $x = \frac{7}{2}$ .

FAQ

When do two linear equations have infinitely many solutions?

When they represent the same line.

Base

The product of two consecutive positive integers is $156$ . Form a quadratic equation and find the integers. (4 marks)

Answer

The equation is $n^{2} + n - 156 = 0$ ; the integers are $12$ and $13$ .

Worked solution

Let the smaller integer be $n$ . Then $n (n + 1) = 156$ , so $n^{2} + n - 156 = 0$ . Factoring gives $(n + 13) (n - 12) = 0$ . Since the integers are positive, $n = 12$ , so the next integer is $13$ .

FAQ

Why is the next consecutive integer written as $n+1$?

Consecutive integers differ by one.

Base

Solve $2 x^{2} - 5 x - 3 = 0$ by the quadratic formula. Check your answer by factorisation. (4 marks)

Answer

$x = 3$ or $x = - \frac{1}{2}$ .

Worked solution

Here $a = 2$ , $b = - 5$ , $c = - 3$ , so $D = b^{2} - 4 a c = 49$ . Thus $x = \frac{5 \pm 7}{4}$ , giving $3$ and $- \frac{1}{2}$ . Also $2 x^{2} - 5 x - 3 = (2 x + 1) (x - 3)$ , which gives the same roots.

FAQ

What does a positive discriminant indicate?

It indicates two distinct real roots.

Base

A student saves Rs. $200$ in the first week and increases the weekly saving by Rs. $25$ each week. How much is saved in week $12$ , and what is the total saved in the first $12$ weeks? (4 marks)

Answer

Week $12$ : Rs. $475$ ; total: Rs. $4050$ .

Worked solution

This AP has $a = 200$ , $d = 25$ . $a_{12} = 200 + 11 (25) = 475$ . Then $S_{12} = \frac{12}{2} (200 + 475) = 6 (675) = 4050$ .

FAQ

Which formula gives the sum of first n AP terms?

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Base

The 7th term of an AP is $22$ , and the 15th term is $46$ . Find the first term, the common difference and the sum of the first $20$ terms. (4 marks)

Answer

$a = 4$ , $d = 3$ , $S_{20} = 650$ .

Worked solution

$a + 6 d = 22$ and $a + 14 d = 46$ . Subtracting gives $8 d = 24$ , so $d = 3$ and $a = 4$ . Then $S_{20} = \frac{20}{2} [2 (4) + 19 (3)] = 10 (65) = 650$ .

FAQ

How can two AP terms determine a and d?

Write two linear equations using $a_{n} = a + (n - 1) d$ and solve them.

Base

In triangle $A B C$ , $D$ lies on $A B$ and $E$ lies on $A C$ , with $D E ∥ B C$ . Given $A D = 6$ cm, $D B = 9$ cm and $A C = 20$ cm, find $A E$ and $E C$ . State the theorem used. (4 marks)

Answer

$A E = 8$ cm and $E C = 12$ cm; Basic Proportionality Theorem.

Worked solution

By BPT, $\frac{A D}{D B} = \frac{A E}{E C} = \frac{6}{9} = \frac{2}{3}$ . Since $A E + E C = 20$ , divide $20$ in the ratio $2 : 3$ . Therefore $A E = 8$ cm and $E C = 12$ cm.

FAQ

What condition permits use of BPT?

A line inside the triangle must be parallel to one side.

Base

In triangles $A B C$ and $P QR$ , $∠ A = ∠ P$ and $∠ B = ∠ Q$ . If $A B = 8$ cm, $B C = 12$ cm and $P Q = 10$ cm, prove that the triangles are similar and find $QR$ . If $A C = 14$ cm, also find $P R$ . (4 marks)

Answer

$△ A B C \sim △ P QR$ ; $QR = 15$ cm and $P R = 17.5$ cm.

Worked solution

The two given equal angles prove AA similarity. $\frac{A B}{P Q} = \frac{8}{10} = \frac{4}{5}$ . Hence $\frac{B C}{QR} = \frac{4}{5}$ , so $QR = 15$ . Also $\frac{A C}{P R} = \frac{4}{5}$ , so $P R = 17.5$ cm.

FAQ

How many equal angles are enough for triangle similarity?

Two equal corresponding angles are enough for AA similarity.

Base

Points $A (- 3, 2)$ , $B (3, 10)$ and $C (9, 2)$ are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Answer

$A B = 10$ , $B C = 10$ , $A C = 12$ ; the triangle is isosceles.

Worked solution

$A B = 6^{2} + 8^{2} = 10$ , $B C = 6^{2} + (- 8)^{2} = 10$ , and $A C = 1 2^{2} = 12$ . Since $A B = B C$ , triangle $A B C$ is isosceles.

FAQ

How can side lengths classify a triangle?

Two equal side lengths show that it is isosceles.

Base

Point $P$ divides the segment joining $A (- 2, 5)$ and $B (10, - 1)$ internally in the ratio $A P : P B = 1 : 2$ . Find $P$ using the section formula, then find $A P$ using the distance formula. (4 marks)

Answer

$P = (2, 3)$ and $A P = 25$ units.

Worked solution

$P = (\frac{1 ( 10 ) + 2 ( - 2 )}{3}, \frac{1 ( - 1 ) + 2 ( 5 )}{3}) = (2, 3)$ . Then $A P = (2 + 2)^{2} + (3 - 5)^{2} = 20 = 25$ units.

FAQ

What does internal division mean?

The dividing point lies between the two endpoints.

Base

If $tan θ = \frac{5}{12}$ and $θ$ is acute, find $sin θ$ , $cos θ$ , $sec θ$ and the value of $\frac{1 - s i n ^{2} θ}{c o s ^{2} θ}$ . (4 marks)

Answer

$sin θ = \frac{5}{13}$ , $cos θ = \frac{12}{13}$ , $sec θ = \frac{13}{12}$ , expression $= 1$ .

Worked solution

Take opposite $= 5$ , adjacent $= 12$ , so the hypotenuse is $13$ . Thus $sin θ = 5/13$ , $cos θ = 12/13$ , and $sec θ = 13/12$ . Since $1 - sin^{2} θ = cos^{2} θ$ , the given expression is $1$ .

FAQ

Which identity simplifies the final expression?

$sin^{2} θ + cos^{2} θ = 1$ .

Base

Evaluate $E = 2 sin 3 0^{\circ} cos 6 0^{\circ} + tan^{2} 4 5^{\circ} - sin^{2} 6 0^{\circ}$ . Show the standard values used. (4 marks)

Answer

$E = \frac{3}{4}$ .

Worked solution

$sin 3 0^{\circ} = \frac{1}{2}$ , $cos 6 0^{\circ} = \frac{1}{2}$ , $tan 4 5^{\circ} = 1$ , and $sin 6 0^{\circ} = \frac{3}{2}$ . Therefore $E = 2 (\frac{1}{2}) (\frac{1}{2}) + 1 - \frac{3}{4} = \frac{3}{4}$ .

FAQ

What is $\tan45^\circ$?

$1$ .

Base

A student stands $30$ m from the foot of a vertical tower and sees its top at an angle of elevation of $3 0^{\circ}$ . Find the height. If the student walks $20$ m towards the tower, find the tangent of the new angle. (4 marks)

Answer

Tower height $= 103$ m; new tangent $= 3$ .

Worked solution

$tan 3 0^{\circ} = \frac{h}{30} = \frac{1}{3}$ , so $h = 103$ m. After moving forward, the distance is $10$ m. The new tangent is $\frac{10 3}{10} = 3$ .

FAQ

Which ratio relates a vertical height and horizontal distance?

Tangent.

Base

From the top of a $24$ m building, the angle of depression of a car on level ground is $6 0^{\circ}$ . Find the horizontal distance of the car from the foot of the building. (4 marks)

Answer

$83$ m.

Worked solution

The angle of depression equals the angle of elevation from the car. If distance is $d$ , then $tan 6 0^{\circ} = \frac{24}{d}$ . Hence $3 = \frac{24}{d}$ and $d = 83$ m.

FAQ

Why are elevation and depression angles equal here?

They are alternate interior angles between parallel horizontals.

Base

Point $P$ is outside a circle with centre $O$ and radius $7$ cm. $P A$ is a tangent at $A$ and $O P = 25$ cm. Prove that triangle $O A P$ is right angled at $A$ and find $P A$ . (4 marks)

Answer

$∠ O A P = 9 0^{\circ}$ and $P A = 24$ cm.

Worked solution

A radius to the point of contact is perpendicular to the tangent, so $O A ⊥ P A$ . Thus triangle $O A P$ is right-angled at $A$ . By Pythagoras, $P A^{2} = 2 5^{2} - 7^{2} = 576$ , giving $P A = 24$ cm.

FAQ

What theorem links a radius and tangent?

The radius at the point of contact is perpendicular to the tangent.

Base

$P A$ and $P B$ are tangents from an external point $P$ to a circle. If $P A = 4 x - 3$ cm and $P B = 2 x + 9$ cm, find $x$ and the length of each tangent. State $∠ O A P$ . (4 marks)

Answer

$x = 6$ , each tangent $= 21$ cm, and $∠ O A P = 9 0^{\circ}$ .

Worked solution

Tangents from an external point are equal: $4 x - 3 = 2 x + 9$ . Hence $2 x = 12$ , so $x = 6$ . Both tangent lengths are $21$ cm. A radius to a tangent is perpendicular, so $∠ O A P = 9 0^{\circ}$ .

FAQ

Why are $PA$ and $PB$ equal?

They are tangent segments drawn from the same external point.

Base

A circular garden has radius $14$ m. A $9 0^{\circ}$ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use $π = \frac{22}{7}$ . (4 marks)

Answer

Area $= 154$ m $^{2}$ , arc length $= 22$ m, perimeter $= 50$ m.

Worked solution

Area $= \frac{1}{4} \times \frac{22}{7} \times 1 4^{2} = 154$ m $^{2}$ . Arc length $= \frac{1}{4} \times 2 \times \frac{22}{7} \times 14 = 22$ m. Perimeter $= 22 + 2 (14) = 50$ m.

FAQ

What is included in a sector perimeter?

The arc length plus two radii.

Base

A chord of a circle of radius $14$ cm subtends an angle of $9 0^{\circ}$ at its centre. Find the area of the corresponding minor segment. Use $π = \frac{22}{7}$ . (4 marks)

Answer

Minor segment area $= 56$ cm $^{2}$ .

Worked solution

The $9 0^{\circ}$ sector area is $\frac{1}{4} \times \frac{22}{7} \times 1 4^{2} = 154$ cm $^{2}$ . The triangle made by the radii has area $\frac{1}{2} (14) (14) sin 9 0^{\circ} = 98$ cm $^{2}$ . Segment area $= 154 - 98 = 56$ cm $^{2}$ .

FAQ

What creates a minor segment?

A chord and its corresponding minor arc.

Base

A hemisphere of radius $3$ cm is attached to one flat end of a cylinder of radius $3$ cm and height $10$ cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of $π$ . (4 marks)

Answer

Total volume $= 108 π$ cm $^{3}$ ; external surface area $= 87 π$ cm $^{2}$ .

Worked solution

Cylinder volume is $π (3)^{2} (10) = 90 π$ and hemisphere volume is $\frac{2}{3} π (3)^{3} = 18 π$ , so total volume is $108 π$ . External area is cylinder CSA $60 π$ , hemisphere CSA $18 π$ , and uncovered base $9 π$ : total $87 π$ cm $^{2}$ .

FAQ

Why is one cylinder base excluded?

It is covered by the hemisphere and is not exposed.

Base

A toy is made by attaching a cone of radius $3$ cm and height $4$ cm to a cylinder of radius $3$ cm and height $7$ cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of $π$ . (4 marks)

Answer

Volume $= 75 π$ cm $^{3}$ ; external surface area $= 66 π$ cm $^{2}$ .

Worked solution

Cone volume is $\frac{1}{3} π (3)^{2} (4) = 12 π$ and cylinder volume is $π (3)^{2} (7) = 63 π$ , so total is $75 π$ . Cone slant height is $5$ cm. External area $= π (3) (5) + 2 π (3) (7) + π (3)^{2} = 15 π + 42 π + 9 π = 66 π$ cm $^{2}$ .

FAQ

How is cone slant height found here?

By Pythagoras: $l = r^{2} + h^{2}$ .

Base

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $3, 5, 8, 4$ , find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

Answer

Mean $= 21.5$ minutes; mode $= \frac{170}{7} \approx 24.29$ minutes.

Worked solution

Class marks are $5, 15, 25, 35$ and $N = 20$ . $Σ f x = 3 (5) + 5 (15) + 8 (25) + 4 (35) = 430$ , so mean $= 430/20 = 21.5$ . Modal class is $20$ - $30$ : $l = 20$ , $h = 10$ , $f_{1} = 8$ , $f_{0} = 5$ , $f_{2} = 4$ . Mode $= 20 + \frac{3}{7} (10) = \frac{170}{7}$ .

FAQ

Which class is modal here?

$20$ - $30$ , because it has the highest frequency.

Base

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $4, 6, 10, 5$ , find the median and mean. Show cumulative frequencies and class marks. (4 marks)

Answer

Median $= 22.5$ ; mean $= 21.4$ .

Worked solution

Cumulative frequencies are $4, 10, 20, 25$ , so $N = 25$ and $N /2 = 12.5$ . The median class is $20$ - $30$ . Thus median $= 20 + \frac{12.5 - 10}{10} (10) = 22.5$ . Class marks are $5, 15, 25, 35$ and $Σ f x = 535$ , so mean $= 535/25 = 21.4$ .

FAQ

Why is $20$-$30$ the median class?

Its cumulative frequency is the first one greater than $N /2 = 12.5$ .

Base

A bag contains $5$ red, $4$ blue and $3$ green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Answer

(a) $\frac{5}{12}$ , (b) $\frac{2}{3}$ , (c) $\frac{2}{3}$ .

Worked solution

There are $12$ balls. $P (red) = 5/12$ . Not blue means red or green, giving $(5 + 3) /12 = 8/12 = 2/3$ . Red or green is the same event, so it also has probability $2/3$ .

FAQ

Why are not blue and red or green the same event?

Every ball is red, blue, or green.

Base

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is $9$ , then find the probability that the sum is $9$ and the probability that the sum is not $9$ . (4 marks)

Answer

Outcomes: $(3, 6), (4, 5), (5, 4), (6, 3)$ ; $P (sum = 9) = \frac{1}{9}$ , $P (sum \neq = 9) = \frac{8}{9}$ .

Worked solution

There are $6 \times 6 = 36$ equally likely ordered outcomes. Four give a sum of $9$ : $(3, 6)$ , $(4, 5)$ , $(5, 4)$ , $(6, 3)$ . Therefore $P (sum = 9) = 4/36 = 1/9$ , and its complement is $1 - 1/9 = 8/9$ .

FAQ

Why are there 36 outcomes when two dice are rolled?

Each of six outcomes on one die can pair with each of six outcomes on the other.

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Class 10 Maths Basic

Long Answer practice across CBSE Class 10 Maths Basic chapters.

Practice coverage:155 published questions across 6 formats.

ChapterTopic

Base

Find the HCF and LCM of $252$ and $198$ by prime factorisation. Verify that HCF $\times$ LCM equals the product of the two numbers. (4 marks)

Answer

HCF $= 18$ , LCM $= 2772$ ; $18 \times 2772 = 252 \times 198 = 49896$ .

Worked solution

FAQ

How is HCF related to LCM for two positive integers?

Their product equals the product of the two integers.

Base

Prove that $2 + 35$ is irrational, using the fact that $5$ is irrational. (4 marks)

Answer

$2 + 35$ is irrational.

Worked solution

Suppose $2 + 35 = r$ for some rational number $r$ . Then $5 = \frac{r - 2}{3}$ , which would be rational. This contradicts the fact that $5$ is irrational. Therefore $2 + 35$ is irrational.

FAQ

Why does the proof isolate the square root?

It turns the assumption into a claim that an already-known irrational number is rational.

Base

The zeroes of $2 x^{2} - 7 x + 3$ are $α$ and $β$ . Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Answer

$α = \frac{1}{2}$ , $β = 3$ ; sum $= \frac{7}{2}$ and product $= \frac{3}{2}$ .

Worked solution

FAQ

What are the zero-coefficient relations for a quadratic?

The sum is $- b / a$ and the product is $c / a$ .

Base

Find a quadratic polynomial with zeroes $3 + 2$ and $3 - 2$ . If $p (x)$ is the monic polynomial you obtain, evaluate $p (4)$ . (4 marks)

Answer

$p (x) = x^{2} - 6 x + 7$ and $p (4) = - 1$ .

Worked solution

The sum of the zeroes is $6$ and their product is $9 - 2 = 7$ . Therefore the monic polynomial is $p (x) = x^{2} - 6 x + 7$ . Hence $p (4) = 16 - 24 + 7 = - 1$ .

FAQ

Why do conjugate surds give rational coefficients?

Their sum and product are rational.

Base

A school sells adult tickets for Rs. $120$ and student tickets for Rs. $80$ . It sells $45$ tickets and collects Rs. $4400$ . Find how many tickets of each type it sold using simultaneous equations. (4 marks)

Answer

$20$ adult tickets and $25$ student tickets.

Worked solution

FAQ

Which two quantities become variables here?

The numbers of adult tickets and student tickets.

Base

For each value of $k$ , classify $2 x + 3 y = 7$ and $4 x + k y = 14$ as having a unique solution or infinitely many solutions. For $k = 8$ , find the solution. (4 marks)

Answer

$k = 6$ : infinitely many solutions; $k \neq = 6$ : unique solution. At $k = 8$ , $(x, y) = (\frac{7}{2}, 0)$ .

Worked solution

FAQ

When do two linear equations have infinitely many solutions?

When they represent the same line.

Base

The product of two consecutive positive integers is $156$ . Form a quadratic equation and find the integers. (4 marks)

Answer

The equation is $n^{2} + n - 156 = 0$ ; the integers are $12$ and $13$ .

Worked solution

FAQ

Why is the next consecutive integer written as $n+1$?

Consecutive integers differ by one.

Base

Solve $2 x^{2} - 5 x - 3 = 0$ by the quadratic formula. Check your answer by factorisation. (4 marks)

Answer

$x = 3$ or $x = - \frac{1}{2}$ .

Worked solution

FAQ

What does a positive discriminant indicate?

It indicates two distinct real roots.

Base

A student saves Rs. $200$ in the first week and increases the weekly saving by Rs. $25$ each week. How much is saved in week $12$ , and what is the total saved in the first $12$ weeks? (4 marks)

Answer

Week $12$ : Rs. $475$ ; total: Rs. $4050$ .

Worked solution

This AP has $a = 200$ , $d = 25$ . $a_{12} = 200 + 11 (25) = 475$ . Then $S_{12} = \frac{12}{2} (200 + 475) = 6 (675) = 4050$ .

FAQ

Which formula gives the sum of first n AP terms?

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Base

The 7th term of an AP is $22$ , and the 15th term is $46$ . Find the first term, the common difference and the sum of the first $20$ terms. (4 marks)

Answer

$a = 4$ , $d = 3$ , $S_{20} = 650$ .

Worked solution

$a + 6 d = 22$ and $a + 14 d = 46$ . Subtracting gives $8 d = 24$ , so $d = 3$ and $a = 4$ . Then $S_{20} = \frac{20}{2} [2 (4) + 19 (3)] = 10 (65) = 650$ .

FAQ

How can two AP terms determine a and d?

Write two linear equations using $a_{n} = a + (n - 1) d$ and solve them.

Base

In triangle $A B C$ , $D$ lies on $A B$ and $E$ lies on $A C$ , with $D E ∥ B C$ . Given $A D = 6$ cm, $D B = 9$ cm and $A C = 20$ cm, find $A E$ and $E C$ . State the theorem used. (4 marks)

Answer

$A E = 8$ cm and $E C = 12$ cm; Basic Proportionality Theorem.

Worked solution

By BPT, $\frac{A D}{D B} = \frac{A E}{E C} = \frac{6}{9} = \frac{2}{3}$ . Since $A E + E C = 20$ , divide $20$ in the ratio $2 : 3$ . Therefore $A E = 8$ cm and $E C = 12$ cm.

FAQ

What condition permits use of BPT?

A line inside the triangle must be parallel to one side.

Base

In triangles $A B C$ and $P QR$ , $∠ A = ∠ P$ and $∠ B = ∠ Q$ . If $A B = 8$ cm, $B C = 12$ cm and $P Q = 10$ cm, prove that the triangles are similar and find $QR$ . If $A C = 14$ cm, also find $P R$ . (4 marks)

Answer

$△ A B C \sim △ P QR$ ; $QR = 15$ cm and $P R = 17.5$ cm.

Worked solution

FAQ

How many equal angles are enough for triangle similarity?

Two equal corresponding angles are enough for AA similarity.

Base

Points $A (- 3, 2)$ , $B (3, 10)$ and $C (9, 2)$ are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Answer

$A B = 10$ , $B C = 10$ , $A C = 12$ ; the triangle is isosceles.

Worked solution

$A B = 6^{2} + 8^{2} = 10$ , $B C = 6^{2} + (- 8)^{2} = 10$ , and $A C = 1 2^{2} = 12$ . Since $A B = B C$ , triangle $A B C$ is isosceles.

FAQ

How can side lengths classify a triangle?

Two equal side lengths show that it is isosceles.

Base

Point $P$ divides the segment joining $A (- 2, 5)$ and $B (10, - 1)$ internally in the ratio $A P : P B = 1 : 2$ . Find $P$ using the section formula, then find $A P$ using the distance formula. (4 marks)

Answer

$P = (2, 3)$ and $A P = 25$ units.

Worked solution

$P = (\frac{1 ( 10 ) + 2 ( - 2 )}{3}, \frac{1 ( - 1 ) + 2 ( 5 )}{3}) = (2, 3)$ . Then $A P = (2 + 2)^{2} + (3 - 5)^{2} = 20 = 25$ units.

FAQ

What does internal division mean?

The dividing point lies between the two endpoints.

Base

If $tan θ = \frac{5}{12}$ and $θ$ is acute, find $sin θ$ , $cos θ$ , $sec θ$ and the value of $\frac{1 - s i n ^{2} θ}{c o s ^{2} θ}$ . (4 marks)

Answer

$sin θ = \frac{5}{13}$ , $cos θ = \frac{12}{13}$ , $sec θ = \frac{13}{12}$ , expression $= 1$ .

Worked solution

FAQ

Which identity simplifies the final expression?

$sin^{2} θ + cos^{2} θ = 1$ .

Base

Evaluate $E = 2 sin 3 0^{\circ} cos 6 0^{\circ} + tan^{2} 4 5^{\circ} - sin^{2} 6 0^{\circ}$ . Show the standard values used. (4 marks)

Answer

$E = \frac{3}{4}$ .

Worked solution

FAQ

What is $\tan45^\circ$?

$1$ .

Base

A student stands $30$ m from the foot of a vertical tower and sees its top at an angle of elevation of $3 0^{\circ}$ . Find the height. If the student walks $20$ m towards the tower, find the tangent of the new angle. (4 marks)

Answer

Tower height $= 103$ m; new tangent $= 3$ .

Worked solution

$tan 3 0^{\circ} = \frac{h}{30} = \frac{1}{3}$ , so $h = 103$ m. After moving forward, the distance is $10$ m. The new tangent is $\frac{10 3}{10} = 3$ .

FAQ

Which ratio relates a vertical height and horizontal distance?

Tangent.

Base

From the top of a $24$ m building, the angle of depression of a car on level ground is $6 0^{\circ}$ . Find the horizontal distance of the car from the foot of the building. (4 marks)

Answer

$83$ m.

Worked solution

The angle of depression equals the angle of elevation from the car. If distance is $d$ , then $tan 6 0^{\circ} = \frac{24}{d}$ . Hence $3 = \frac{24}{d}$ and $d = 83$ m.

FAQ

Why are elevation and depression angles equal here?

They are alternate interior angles between parallel horizontals.

Base

Point $P$ is outside a circle with centre $O$ and radius $7$ cm. $P A$ is a tangent at $A$ and $O P = 25$ cm. Prove that triangle $O A P$ is right angled at $A$ and find $P A$ . (4 marks)

Answer

$∠ O A P = 9 0^{\circ}$ and $P A = 24$ cm.

Worked solution

FAQ

What theorem links a radius and tangent?

The radius at the point of contact is perpendicular to the tangent.

Base

$P A$ and $P B$ are tangents from an external point $P$ to a circle. If $P A = 4 x - 3$ cm and $P B = 2 x + 9$ cm, find $x$ and the length of each tangent. State $∠ O A P$ . (4 marks)

Answer

$x = 6$ , each tangent $= 21$ cm, and $∠ O A P = 9 0^{\circ}$ .

Worked solution

FAQ

Why are $PA$ and $PB$ equal?

They are tangent segments drawn from the same external point.

Base

A circular garden has radius $14$ m. A $9 0^{\circ}$ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use $π = \frac{22}{7}$ . (4 marks)

Answer

Area $= 154$ m $^{2}$ , arc length $= 22$ m, perimeter $= 50$ m.

Worked solution

Area $= \frac{1}{4} \times \frac{22}{7} \times 1 4^{2} = 154$ m $^{2}$ . Arc length $= \frac{1}{4} \times 2 \times \frac{22}{7} \times 14 = 22$ m. Perimeter $= 22 + 2 (14) = 50$ m.

FAQ

What is included in a sector perimeter?

The arc length plus two radii.

Base

A chord of a circle of radius $14$ cm subtends an angle of $9 0^{\circ}$ at its centre. Find the area of the corresponding minor segment. Use $π = \frac{22}{7}$ . (4 marks)

Answer

Minor segment area $= 56$ cm $^{2}$ .

Worked solution

FAQ

What creates a minor segment?

A chord and its corresponding minor arc.

Base

A hemisphere of radius $3$ cm is attached to one flat end of a cylinder of radius $3$ cm and height $10$ cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of $π$ . (4 marks)

Answer

Total volume $= 108 π$ cm $^{3}$ ; external surface area $= 87 π$ cm $^{2}$ .

Worked solution

FAQ

Why is one cylinder base excluded?

It is covered by the hemisphere and is not exposed.

Base

A toy is made by attaching a cone of radius $3$ cm and height $4$ cm to a cylinder of radius $3$ cm and height $7$ cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of $π$ . (4 marks)

Answer

Volume $= 75 π$ cm $^{3}$ ; external surface area $= 66 π$ cm $^{2}$ .

Worked solution

FAQ

How is cone slant height found here?

By Pythagoras: $l = r^{2} + h^{2}$ .

Base

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $3, 5, 8, 4$ , find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

Answer

Mean $= 21.5$ minutes; mode $= \frac{170}{7} \approx 24.29$ minutes.

Worked solution

FAQ

Which class is modal here?

$20$ - $30$ , because it has the highest frequency.

Base

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $4, 6, 10, 5$ , find the median and mean. Show cumulative frequencies and class marks. (4 marks)

Answer

Median $= 22.5$ ; mean $= 21.4$ .

Worked solution

FAQ

Why is $20$-$30$ the median class?

Its cumulative frequency is the first one greater than $N /2 = 12.5$ .

Base

A bag contains $5$ red, $4$ blue and $3$ green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Answer

(a) $\frac{5}{12}$ , (b) $\frac{2}{3}$ , (c) $\frac{2}{3}$ .

Worked solution

There are $12$ balls. $P (red) = 5/12$ . Not blue means red or green, giving $(5 + 3) /12 = 8/12 = 2/3$ . Red or green is the same event, so it also has probability $2/3$ .

FAQ

Why are not blue and red or green the same event?

Every ball is red, blue, or green.

Base

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is $9$ , then find the probability that the sum is $9$ and the probability that the sum is not $9$ . (4 marks)

Answer

Outcomes: $(3, 6), (4, 5), (5, 4), (6, 3)$ ; $P (sum = 9) = \frac{1}{9}$ , $P (sum \neq = 9) = \frac{8}{9}$ .

Worked solution

FAQ

Why are there 36 outcomes when two dice are rolled?

Each of six outcomes on one die can pair with each of six outcomes on the other.

Class 10 Maths Basic

Find the HCF and LCM of 252 and 198 by prime factorisation. Verify that HCF × LCM equals the product of the two numbers. (4 marks)

Prove that 2+35​ is irrational, using the fact that 5​ is irrational. (4 marks)

The zeroes of 2x2−7x+3 are α and β. Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Find a quadratic polynomial with zeroes 3+2​ and 3−2​. If p(x) is the monic polynomial you obtain, evaluate p(4). (4 marks)

A school sells adult tickets for Rs. 120 and student tickets for Rs. 80. It sells 45 tickets and collects Rs. 4400. Find how many tickets of each type it sold using simultaneous equations. (4 marks)

For each value of k, classify 2x+3y=7 and 4x+ky=14 as having a unique solution or infinitely many solutions. For k=8, find the solution. (4 marks)

The product of two consecutive positive integers is 156. Form a quadratic equation and find the integers. (4 marks)

Solve 2x2−5x−3=0 by the quadratic formula. Check your answer by factorisation. (4 marks)

A student saves Rs. 200 in the first week and increases the weekly saving by Rs. 25 each week. How much is saved in week 12, and what is the total saved in the first 12 weeks? (4 marks)

The 7th term of an AP is 22, and the 15th term is 46. Find the first term, the common difference and the sum of the first 20 terms. (4 marks)

In triangle ABC, D lies on AB and E lies on AC, with DE∥BC. Given AD=6 cm, DB=9 cm and AC=20 cm, find AE and EC. State the theorem used. (4 marks)

In triangles ABC and PQR, ∠A=∠P and ∠B=∠Q. If AB=8 cm, BC=12 cm and PQ=10 cm, prove that the triangles are similar and find QR. If AC=14 cm, also find PR. (4 marks)

Points A(−3,2), B(3,10) and C(9,2) are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Point P divides the segment joining A(−2,5) and B(10,−1) internally in the ratio AP:PB=1:2. Find P using the section formula, then find AP using the distance formula. (4 marks)

If tanθ=125​ and θ is acute, find sinθ, cosθ, secθ and the value of cos2θ1−sin2θ​. (4 marks)

Evaluate E=2sin30∘cos60∘+tan245∘−sin260∘. Show the standard values used. (4 marks)

A student stands 30 m from the foot of a vertical tower and sees its top at an angle of elevation of 30∘. Find the height. If the student walks 20 m towards the tower, find the tangent of the new angle. (4 marks)

From the top of a 24 m building, the angle of depression of a car on level ground is 60∘. Find the horizontal distance of the car from the foot of the building. (4 marks)

Point P is outside a circle with centre O and radius 7 cm. PA is a tangent at A and OP=25 cm. Prove that triangle OAP is right angled at A and find PA. (4 marks)

PA and PB are tangents from an external point P to a circle. If PA=4x−3 cm and PB=2x+9 cm, find x and the length of each tangent. State ∠OAP. (4 marks)

A circular garden has radius 14 m. A 90∘ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use π=722​. (4 marks)

A chord of a circle of radius 14 cm subtends an angle of 90∘ at its centre. Find the area of the corresponding minor segment. Use π=722​. (4 marks)

A hemisphere of radius 3 cm is attached to one flat end of a cylinder of radius 3 cm and height 10 cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of π. (4 marks)

A toy is made by attaching a cone of radius 3 cm and height 4 cm to a cylinder of radius 3 cm and height 7 cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of π. (4 marks)

For class intervals 0-10, 10-20, 20-30, 30-40 with frequencies 3,5,8,4, find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

For class intervals 0-10, 10-20, 20-30, 30-40 with frequencies 4,6,10,5, find the median and mean. Show cumulative frequencies and class marks. (4 marks)

A bag contains 5 red, 4 blue and 3 green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is 9, then find the probability that the sum is 9 and the probability that the sum is not 9. (4 marks)

Class 10 Maths Basic

Find the HCF and LCM of 252 and 198 by prime factorisation. Verify that HCF × LCM equals the product of the two numbers. (4 marks)

Prove that 2+35​ is irrational, using the fact that 5​ is irrational. (4 marks)

The zeroes of 2x2−7x+3 are α and β. Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Find a quadratic polynomial with zeroes 3+2​ and 3−2​. If p(x) is the monic polynomial you obtain, evaluate p(4). (4 marks)

A school sells adult tickets for Rs. 120 and student tickets for Rs. 80. It sells 45 tickets and collects Rs. 4400. Find how many tickets of each type it sold using simultaneous equations. (4 marks)

For each value of k, classify 2x+3y=7 and 4x+ky=14 as having a unique solution or infinitely many solutions. For k=8, find the solution. (4 marks)

The product of two consecutive positive integers is 156. Form a quadratic equation and find the integers. (4 marks)

Solve 2x2−5x−3=0 by the quadratic formula. Check your answer by factorisation. (4 marks)

A student saves Rs. 200 in the first week and increases the weekly saving by Rs. 25 each week. How much is saved in week 12, and what is the total saved in the first 12 weeks? (4 marks)

The 7th term of an AP is 22, and the 15th term is 46. Find the first term, the common difference and the sum of the first 20 terms. (4 marks)

In triangle ABC, D lies on AB and E lies on AC, with DE∥BC. Given AD=6 cm, DB=9 cm and AC=20 cm, find AE and EC. State the theorem used. (4 marks)

In triangles ABC and PQR, ∠A=∠P and ∠B=∠Q. If AB=8 cm, BC=12 cm and PQ=10 cm, prove that the triangles are similar and find QR. If AC=14 cm, also find PR. (4 marks)

Points A(−3,2), B(3,10) and C(9,2) are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Point P divides the segment joining A(−2,5) and B(10,−1) internally in the ratio AP:PB=1:2. Find P using the section formula, then find AP using the distance formula. (4 marks)

If tanθ=125​ and θ is acute, find sinθ, cosθ, secθ and the value of cos2θ1−sin2θ​. (4 marks)

Evaluate E=2sin30∘cos60∘+tan245∘−sin260∘. Show the standard values used. (4 marks)

A student stands 30 m from the foot of a vertical tower and sees its top at an angle of elevation of 30∘. Find the height. If the student walks 20 m towards the tower, find the tangent of the new angle. (4 marks)

From the top of a 24 m building, the angle of depression of a car on level ground is 60∘. Find the horizontal distance of the car from the foot of the building. (4 marks)

Point P is outside a circle with centre O and radius 7 cm. PA is a tangent at A and OP=25 cm. Prove that triangle OAP is right angled at A and find PA. (4 marks)

PA and PB are tangents from an external point P to a circle. If PA=4x−3 cm and PB=2x+9 cm, find x and the length of each tangent. State ∠OAP. (4 marks)

A circular garden has radius 14 m. A 90∘ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use π=722​. (4 marks)

A chord of a circle of radius 14 cm subtends an angle of 90∘ at its centre. Find the area of the corresponding minor segment. Use π=722​. (4 marks)

A hemisphere of radius 3 cm is attached to one flat end of a cylinder of radius 3 cm and height 10 cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of π. (4 marks)

A toy is made by attaching a cone of radius 3 cm and height 4 cm to a cylinder of radius 3 cm and height 7 cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of π. (4 marks)

For class intervals 0-10, 10-20, 20-30, 30-40 with frequencies 3,5,8,4, find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

For class intervals 0-10, 10-20, 20-30, 30-40 with frequencies 4,6,10,5, find the median and mean. Show cumulative frequencies and class marks. (4 marks)

A bag contains 5 red, 4 blue and 3 green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is 9, then find the probability that the sum is 9 and the probability that the sum is not 9. (4 marks)

Find the HCF and LCM of $252$ and $198$ by prime factorisation. Verify that HCF $\times$ LCM equals the product of the two numbers. (4 marks)

Prove that $2 + 35$ is irrational, using the fact that $5$ is irrational. (4 marks)

The zeroes of $2 x^{2} - 7 x + 3$ are $α$ and $β$ . Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Find a quadratic polynomial with zeroes $3 + 2$ and $3 - 2$ . If $p (x)$ is the monic polynomial you obtain, evaluate $p (4)$ . (4 marks)

A school sells adult tickets for Rs. $120$ and student tickets for Rs. $80$ . It sells $45$ tickets and collects Rs. $4400$ . Find how many tickets of each type it sold using simultaneous equations. (4 marks)

For each value of $k$ , classify $2 x + 3 y = 7$ and $4 x + k y = 14$ as having a unique solution or infinitely many solutions. For $k = 8$ , find the solution. (4 marks)

The product of two consecutive positive integers is $156$ . Form a quadratic equation and find the integers. (4 marks)

Solve $2 x^{2} - 5 x - 3 = 0$ by the quadratic formula. Check your answer by factorisation. (4 marks)

A student saves Rs. $200$ in the first week and increases the weekly saving by Rs. $25$ each week. How much is saved in week $12$ , and what is the total saved in the first $12$ weeks? (4 marks)

The 7th term of an AP is $22$ , and the 15th term is $46$ . Find the first term, the common difference and the sum of the first $20$ terms. (4 marks)

In triangle $A B C$ , $D$ lies on $A B$ and $E$ lies on $A C$ , with $D E ∥ B C$ . Given $A D = 6$ cm, $D B = 9$ cm and $A C = 20$ cm, find $A E$ and $E C$ . State the theorem used. (4 marks)

In triangles $A B C$ and $P QR$ , $∠ A = ∠ P$ and $∠ B = ∠ Q$ . If $A B = 8$ cm, $B C = 12$ cm and $P Q = 10$ cm, prove that the triangles are similar and find $QR$ . If $A C = 14$ cm, also find $P R$ . (4 marks)

Points $A (- 3, 2)$ , $B (3, 10)$ and $C (9, 2)$ are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Point $P$ divides the segment joining $A (- 2, 5)$ and $B (10, - 1)$ internally in the ratio $A P : P B = 1 : 2$ . Find $P$ using the section formula, then find $A P$ using the distance formula. (4 marks)

If $tan θ = \frac{5}{12}$ and $θ$ is acute, find $sin θ$ , $cos θ$ , $sec θ$ and the value of $\frac{1 - s i n ^{2} θ}{c o s ^{2} θ}$ . (4 marks)

Evaluate $E = 2 sin 3 0^{\circ} cos 6 0^{\circ} + tan^{2} 4 5^{\circ} - sin^{2} 6 0^{\circ}$ . Show the standard values used. (4 marks)

A student stands $30$ m from the foot of a vertical tower and sees its top at an angle of elevation of $3 0^{\circ}$ . Find the height. If the student walks $20$ m towards the tower, find the tangent of the new angle. (4 marks)

From the top of a $24$ m building, the angle of depression of a car on level ground is $6 0^{\circ}$ . Find the horizontal distance of the car from the foot of the building. (4 marks)

Point $P$ is outside a circle with centre $O$ and radius $7$ cm. $P A$ is a tangent at $A$ and $O P = 25$ cm. Prove that triangle $O A P$ is right angled at $A$ and find $P A$ . (4 marks)

$P A$ and $P B$ are tangents from an external point $P$ to a circle. If $P A = 4 x - 3$ cm and $P B = 2 x + 9$ cm, find $x$ and the length of each tangent. State $∠ O A P$ . (4 marks)

A circular garden has radius $14$ m. A $9 0^{\circ}$ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use $π = \frac{22}{7}$ . (4 marks)

A chord of a circle of radius $14$ cm subtends an angle of $9 0^{\circ}$ at its centre. Find the area of the corresponding minor segment. Use $π = \frac{22}{7}$ . (4 marks)

A hemisphere of radius $3$ cm is attached to one flat end of a cylinder of radius $3$ cm and height $10$ cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of $π$ . (4 marks)

A toy is made by attaching a cone of radius $3$ cm and height $4$ cm to a cylinder of radius $3$ cm and height $7$ cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of $π$ . (4 marks)

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $3, 5, 8, 4$ , find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $4, 6, 10, 5$ , find the median and mean. Show cumulative frequencies and class marks. (4 marks)

A bag contains $5$ red, $4$ blue and $3$ green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is $9$ , then find the probability that the sum is $9$ and the probability that the sum is not $9$ . (4 marks)

Find the HCF and LCM of $252$ and $198$ by prime factorisation. Verify that HCF $\times$ LCM equals the product of the two numbers. (4 marks)

Prove that $2 + 35$ is irrational, using the fact that $5$ is irrational. (4 marks)

The zeroes of $2 x^{2} - 7 x + 3$ are $α$ and $β$ . Find the zeroes and verify both the sum and product relations with its coefficients. (4 marks)

Find a quadratic polynomial with zeroes $3 + 2$ and $3 - 2$ . If $p (x)$ is the monic polynomial you obtain, evaluate $p (4)$ . (4 marks)

A school sells adult tickets for Rs. $120$ and student tickets for Rs. $80$ . It sells $45$ tickets and collects Rs. $4400$ . Find how many tickets of each type it sold using simultaneous equations. (4 marks)

For each value of $k$ , classify $2 x + 3 y = 7$ and $4 x + k y = 14$ as having a unique solution or infinitely many solutions. For $k = 8$ , find the solution. (4 marks)

The product of two consecutive positive integers is $156$ . Form a quadratic equation and find the integers. (4 marks)

Solve $2 x^{2} - 5 x - 3 = 0$ by the quadratic formula. Check your answer by factorisation. (4 marks)

A student saves Rs. $200$ in the first week and increases the weekly saving by Rs. $25$ each week. How much is saved in week $12$ , and what is the total saved in the first $12$ weeks? (4 marks)

The 7th term of an AP is $22$ , and the 15th term is $46$ . Find the first term, the common difference and the sum of the first $20$ terms. (4 marks)

In triangle $A B C$ , $D$ lies on $A B$ and $E$ lies on $A C$ , with $D E ∥ B C$ . Given $A D = 6$ cm, $D B = 9$ cm and $A C = 20$ cm, find $A E$ and $E C$ . State the theorem used. (4 marks)

In triangles $A B C$ and $P QR$ , $∠ A = ∠ P$ and $∠ B = ∠ Q$ . If $A B = 8$ cm, $B C = 12$ cm and $P Q = 10$ cm, prove that the triangles are similar and find $QR$ . If $A C = 14$ cm, also find $P R$ . (4 marks)

Points $A (- 3, 2)$ , $B (3, 10)$ and $C (9, 2)$ are vertices of a triangle. Use the distance formula to find its three sides and decide whether the triangle is isosceles. (4 marks)

Point $P$ divides the segment joining $A (- 2, 5)$ and $B (10, - 1)$ internally in the ratio $A P : P B = 1 : 2$ . Find $P$ using the section formula, then find $A P$ using the distance formula. (4 marks)

If $tan θ = \frac{5}{12}$ and $θ$ is acute, find $sin θ$ , $cos θ$ , $sec θ$ and the value of $\frac{1 - s i n ^{2} θ}{c o s ^{2} θ}$ . (4 marks)

Evaluate $E = 2 sin 3 0^{\circ} cos 6 0^{\circ} + tan^{2} 4 5^{\circ} - sin^{2} 6 0^{\circ}$ . Show the standard values used. (4 marks)

A student stands $30$ m from the foot of a vertical tower and sees its top at an angle of elevation of $3 0^{\circ}$ . Find the height. If the student walks $20$ m towards the tower, find the tangent of the new angle. (4 marks)

From the top of a $24$ m building, the angle of depression of a car on level ground is $6 0^{\circ}$ . Find the horizontal distance of the car from the foot of the building. (4 marks)

Point $P$ is outside a circle with centre $O$ and radius $7$ cm. $P A$ is a tangent at $A$ and $O P = 25$ cm. Prove that triangle $O A P$ is right angled at $A$ and find $P A$ . (4 marks)

$P A$ and $P B$ are tangents from an external point $P$ to a circle. If $P A = 4 x - 3$ cm and $P B = 2 x + 9$ cm, find $x$ and the length of each tangent. State $∠ O A P$ . (4 marks)

A circular garden has radius $14$ m. A $9 0^{\circ}$ sector is set aside for flowers. Find the area of the sector, its arc length and its perimeter. Use $π = \frac{22}{7}$ . (4 marks)

A chord of a circle of radius $14$ cm subtends an angle of $9 0^{\circ}$ at its centre. Find the area of the corresponding minor segment. Use $π = \frac{22}{7}$ . (4 marks)

A hemisphere of radius $3$ cm is attached to one flat end of a cylinder of radius $3$ cm and height $10$ cm. Find its total volume and external surface area, including the uncovered circular base. Give answers in terms of $π$ . (4 marks)

A toy is made by attaching a cone of radius $3$ cm and height $4$ cm to a cylinder of radius $3$ cm and height $7$ cm. Find its volume and external surface area, including the cylinder bottom. Give answers in terms of $π$ . (4 marks)

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $3, 5, 8, 4$ , find the mean by the direct method and the mode using the grouped-data formula. (4 marks)

For class intervals $0$ - $10$ , $10$ - $20$ , $20$ - $30$ , $30$ - $40$ with frequencies $4, 6, 10, 5$ , find the median and mean. Show cumulative frequencies and class marks. (4 marks)

A bag contains $5$ red, $4$ blue and $3$ green balls. One ball is drawn at random. Find the probabilities that it is (a) red, (b) not blue and (c) either red or green. Explain why (b) and (c) agree. (4 marks)

Two ordinary six-sided dice are rolled once. List the outcomes whose sum is $9$ , then find the probability that the sum is $9$ and the probability that the sum is not $9$ . (4 marks)