Vardaan Learning Institute
VARDAAN COMET
MEGA PRACTICE SHEET
AP · Quadratic Equations · Coordinate Geometry — Class 10 (CBSE)
Arithmetic Progressions
Topic A — Finding the nth Term & Checking AP
1.
Which term of the AP: 3, 15, 27, 39, … will be 132 more than its 54th term? Moderate
[2]
2.
The 17th term of an AP exceeds its 10th term by 7. Find the common difference. Moderate
[2]
3.
Find the number of terms of the AP: 18, 15½, 13, … such that their sum is 45. Explain the double answer. Hard
[3]
4.
If the $m$th term of an AP is $\dfrac{1}{n}$ and the $n$th term is $\dfrac{1}{m}$, find the $(mn)$th term. Hard
[3]
5.
The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP. Moderate
[3]
6.
For what value of $n$ are the $n$th terms of two APs: 63, 65, 67, … and 3, 10, 17, … equal? Moderate
[2]
7.
Find the first negative term of the AP: 116, 111, 106, 101, … Moderate
[2]
8.
Determine the AP whose 3rd term is 16 and the 7th term exceeds the 5th term by 12. Moderate
[3]
9.
The 26th, 11th and the last term of an AP are 0, 3 and $-\dfrac{1}{5}$ respectively. Find the common difference, first term and the number of terms. Hard
[4]
10.
Find the middle term(s) of the AP: –11, –7, –3, …, 49. Moderate
[3]
Topic B — Counting Terms Divisible by a Number
11.
How many three-digit numbers are divisible by 7? Moderate
[2]
12.
How many integers between 200 and 500 are divisible by 8? Moderate
[2]
13.
Find the sum of all two-digit natural numbers which are divisible by 4. Moderate
[3]
14.
How many multiples of 4 lie between 10 and 250? Also find their sum. Moderate
[3]
15.
Find the sum of all natural numbers between 100 and 200 that are divisible by neither 2 nor 3. (Hint: Use inclusion-exclusion on AP sums.) Hard
[5]
16.
How many terms of the AP: 9, 17, 25, … must be taken to give a sum of 636? Moderate
[3]
Topic C — Sum of n Terms & Related Problems
17.
If the sum of first $n$ terms of an AP is $4n – n^2$, find the first term, second term, 3rd, 10th and $n$th terms. Also state what happens when $n = 5$ and explain. Hard
[4]
18.
The sum of first $n$ terms of an AP is given by $S_n = 3n^2 + 5n$. Find the 25th term of the AP. Moderate
[3]
19.
The ratio of the sums of $m$ and $n$ terms of an AP is $m^2 : n^2$. Show that the ratio of $m$th and $n$th terms is $(2m-1) : (2n-1)$. Hard
[5]
20.
Find the sum of all integers between 84 and 719 that are multiples of 5. Moderate
[3]
21.
The ratio of 10th to 5th term of an AP is 2.5 : 1. Find the ratio of the sum of first 10 terms to the sum of first 5 terms. Hard
[4]
22.
In an AP, if $S_5 + S_7 = 167$ and $S_{10} = 235$, find the AP. Hard
[4]
23.
The sum of the first 7 terms of an AP is 49 and that of 17 terms is 289. Find the sum of first $n$ terms. Moderate
[3]
24.
Show that the sum of an AP whose first term is $a$, second term $b$ and last term $c$ equals $\dfrac{(a+c)(b+c-2a)}{2(b-a)}$. Hard
[5]
Topic D — Finding $a$ and $d$ from Given Conditions / Proofs
25.
The 4th term of an AP is zero. Prove that its 25th term is triple its 11th term. Hard
[4]
26.
If the $p$th term of an AP is $q$ and the $q$th term is $p$, prove that its $n$th term is $(p+q-n)$. Hard
[5]
27.
In an AP, the sum of first ten terms is –150 and the sum of its next ten terms is –550. Find the AP. Hard
[4]
28.
The 5th term and 15th term of an AP are 13 and –17 respectively. Find the sum of first 21 terms. Moderate
[3]
29.
If $S_n$ denotes the sum of the first $n$ terms of an AP, prove that $S_{12} = 3(S_8 - S_4)$. Hard
[4]
Topic E — Three/Four Numbers in AP & Splitting Problems
30.
The sum of three numbers in AP is 24 and their product is 440. Find the numbers. Moderate
[3]
31.
Divide 32 into four parts that are in AP such that the product of extremes is to the product of means in the ratio 7 : 15. Hard
[4]
32.
If the sum of first four terms of an AP is 40 and that of first eight terms is 120, find the sum of first 20 terms. Moderate
[3]
33.
If the sum of $n$ terms of an AP is $pn + qn^2$, where $p$ and $q$ are constants, find the common difference. Moderate
[2]
34.
The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. Moderate
[3]
Topic F — Real-life Application & Word Problems
35.
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, … What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take $\pi = \frac{22}{7}$) Hard
[4]
36.
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the next, and so on. In how many rows are the 200 logs placed, and how many logs are in the top row? Moderate
[4]
37.
A man repays a loan of ₹3250 by paying ₹20 in the first month and then increases the payment by ₹15 every month. How long will it take him to clear the loan? Moderate
[3]
38.
The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find $x$. Hard
[5]
39.
A factory produces 500 articles in the first week. Each week it produces 50 more than the week before. In how many weeks will the total production first exceed 28000? Hard
[4]
Topic G — Higher Order & MCQ (AP)
40.
If the first term of an AP is 2 and the sum of first five terms is one-fourth of the sum of the next five terms, find the sum of the first 30 terms. Hard
[1]
41.
The common difference of the AP $\dfrac{1}{3}, \dfrac{1-3b}{3}, \dfrac{1-6b}{3}, …$ is: Moderate
[1]
42.
A thief runs with a uniform speed of 100 m/minute. After 1 minute, a policeman runs after the thief. He goes 100 m in the first minute and increases his speed by 10 m/minute each subsequent minute. After how many minutes from the start of the chase will the policeman catch the thief? Hard
[5]
43.
If the sum of $n$ terms of two arithmetic progressions are in the ratio $(3n+8):(7n+15)$, find the ratio of their 12th terms. Hard
[4]
44.
The first and the last terms of an AP are 8 and 65 respectively. If the sum of all its terms is 730, find its common difference. Moderate
[3]
45.
Balls are arranged in rows to form an equilateral triangle: 1 ball in the first row, 2 in the second, and so on. If 669 more balls are added, all balls can be arranged in a square shape, each side of the square containing 8 balls less than each side of the triangle. Find the initial number of balls. Hard
[5]
46.
The 9th term of an AP is equal to 7 times the 2nd term. If its 13th term is 97, find the AP. Moderate
[3]
47.
Find the sum: $5 + (-41) + 9 + (-39) + 13 + (-37) + 17 + \cdots + (-5) + 81 + (-3)$. Hard
[4]
48.
In a school, each section of each class plants trees equal to its class number (Section of Class I plants 1, Class II plants 2, and so on till Class XII). There are three sections per class. How many trees will be planted in total? Moderate
[3]
49.
If $\dfrac{a^n + b^n}{a^{n-1}+b^{n-1}}$ is the arithmetic mean of $a$ and $b$, find the value of $n$. Hard
[3]
50.
The sum of three terms of an AP is 21 and the sum of the squares of the first and the third terms is 58. Find the three terms. Hard
[4]
Quadratic Equations
Topic A — Nature of Roots & Discriminant
51.
Find the value of $k$ for which the roots of $kx^2 + 4x + k = 0$ are real and equal. Moderate
[2]
52.
If the equation $(1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0$ has equal roots, prove that $c^2 = a^2(1+m^2)$. Hard
[5]
53.
For what values of $k$ does the equation $4x^2 - 12x - k = 0$ have no real roots? Moderate
[2]
54.
Find the value(s) of $p$ so that $px(x-3) + 9 = 0$ has two equal roots. Moderate
[2]
55.
If the roots of $(a-b)x^2 + (b-c)x + (c-a) = 0$ are equal, prove that $2a = b + c$. Hard
[5]
56.
Find all values of $k$ for which the equation $9x^2 + 3kx + 4 = 0$ has real roots. Moderate
[2]
57.
If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + k = 0$ such that $\alpha - \beta = 1$, find $k$. Hard
[3]
58.
Find the value of $k$ for which the quadratic equation $(3k+1)x^2 + 2(k+1)x + 1 = 0$ has equal roots. Also find its roots. Hard
[4]
Topic B — Solving by Factorisation (Non-trivial)
59.
Solve: $4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0$. Moderate
[2]
60.
Solve: $\dfrac{1}{x+1} + \dfrac{2}{x+2} = \dfrac{5}{x+4}$. Moderate
[3]
61.
Solve: $\dfrac{x-3}{x+3} - \dfrac{x+3}{x-3} = \dfrac{48}{7}$, $x \ne \pm 3$. Hard
[4]
62.
Solve: $x^2 - (\sqrt{3}+1)x + \sqrt{3} = 0$. Moderate
[2]
63.
Solve: $abx^2 + (b^2 - ac)x - bc = 0$. Hard
[3]
64.
Solve: $\dfrac{a}{ax-1} + \dfrac{b}{bx-1} = a+b$, where $a+b \neq 0$, $ab \neq 0$. Hard
[4]
Topic C — Completing the Square & Quadratic Formula
65.
Find the roots of $5x^2 - 6x - 2 = 0$ using the completing-the-square method. Moderate
[3]
66.
Solve: $x^2 + \dfrac{3x}{4} - \dfrac{1}{2} = 0$ by completing the square. Moderate
[3]
67.
Find the roots of $3x^2 - 5x + 2 = 0$ using the quadratic formula. Verify that the sum and product of roots match $-b/a$ and $c/a$. Moderate
[3]
68.
Solve: $2x^2 - 7x + 3 = 0$ using the quadratic formula. Moderate
[2]
Topic D — Forming Quadratic Equations from Word Problems
69.
The product of two consecutive positive integers is 306. Find the integers. Moderate
[3]
70.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers. Hard
[4]
71.
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. Find the speed of the train. Moderate
[4]
72.
Two water taps together can fill a tank in $9\dfrac{3}{8}$ hours. The larger tap takes 10 hours less than the smaller to fill it separately. Find the time for each tap alone. Hard
[5]
73.
A cottage industry produces pottery. The cost per article (in ₹) is 3 more than twice the number of articles produced that day. If the total cost was ₹90, find the number of articles produced. Moderate
[4]
74.
In a class test, the sum of Shefali's marks in Maths and English is 30. Had she got 2 marks more in Maths and 3 less in English, the product of marks would have been 210. Find her marks. Moderate
[4]
75.
The sum of ages of a father and son is 45 years. Five years ago, the product of their ages was 4 times the father's age at that time. Find their present ages. Hard
[4]
Topic E — Speed, Time & Distance Problems (Quadratic)
76.
A motor boat, whose speed is 18 km/h in still water, takes 1 hour more to go 24 km upstream than to return downstream. Find the speed of the stream. Hard
[5]
77.
A car travels 400 km at some speed. Had the speed been 12 km/h more, the time taken would have been 1 hour 40 minutes less. Find the original speed. Hard
[5]
78.
A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/h from its usual speed. Find the usual speed of the train. Moderate
[4]
Topic F — Area / Geometry Based Quadratic Problems
79.
The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares. Moderate
[4]
80.
The hypotenuse of a right triangle is 6 m more than twice its shortest side. If the third side is 2 m less than the hypotenuse, find the sides. Hard
[5]
81.
A rectangular park is 70 m long and 30 m wide. A path of equal width runs around the inside of the park. If the area of the path is 600 m², find the width of the path. Hard
[5]
82.
A right triangle has hypotenuse of length $p$ cm and one side of length $q$ cm. If $p - q = 1$, find the length of the third side in terms of $p$ and $q$. Moderate
[3]
83.
A rectangular field is 70 m long. A farmer uses 600 m of fencing to fence three sides of the field (the fourth side is a wall). Find the width of the field using a quadratic equation. Hard
[4]
Topic G — Sum & Product of Roots / Relationship Problems
84.
If $\alpha, \beta$ are roots of $2x^2 - 3x + 1 = 0$, find the equation whose roots are $\dfrac{1}{\alpha}$ and $\dfrac{1}{\beta}$. Hard
[3]
85.
If one root of $x^2 + px + q = 0$ is three times the other, prove that $3p^2 = 16q$. Hard
[4]
86.
If $\alpha$ and $\beta$ are the roots of the equation $3x^2 + 8x + 2 = 0$, find $\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2}$. Hard
[4]
87.
The roots $\alpha$ and $\beta$ of $x^2 - 5x + 3(k-1) = 0$ are such that $\alpha - \beta = 11$. Find $k$. Hard
[3]
88.
If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, form an equation whose roots are $2\alpha + 3$ and $2\beta + 3$. Hard
[4]
Topic H — Equations Reducible to Quadratic Form
89.
Solve: $\sqrt{\dfrac{x}{x-3}} + \sqrt{\dfrac{x-3}{x}} = \dfrac{5}{2}$, $x \ne 0, x \ne 3$. Hard
[5]
90.
Solve: $4x^2 - 4a^2x + (a^4 - b^4) = 0$. Hard
[4]
91.
Solve: $9\left(x^2 + \dfrac{1}{x^2}\right) - 9\left(x + \dfrac{1}{x}\right) - 52 = 0$. Hard
[5]
92.
Solve: $\dfrac{1}{x} - \dfrac{1}{x-2} = 3$, $x \ne 0, 2$. Moderate
[3]
93.
Solve: $(x^2 + 5x)^2 - 2(x^2 + 5x) - 24 = 0$. Hard
[4]
94.
Two pipes running together can fill a cistern in $3\dfrac{1}{13}$ minutes. If one pipe takes 3 minutes more than the other to fill it alone, find the time each pipe would take separately. Hard
[5]
Topic I — MCQ / Assertion-Reason (Quadratic Equations)
95.
The equation $x^2 + bx + c = 0$ has equal roots if: Moderate
[1]
96.
If $x = -\dfrac{1}{2}$ is a root of $2x^2 + px - 4 = 0$, the value of $p$ is: Moderate
[1]
97.
Which of the following is NOT a quadratic equation? Moderate
[1]
98.
Assertion (A): The equation $x^2 + 2\sqrt{2}x + 2 = 0$ has no real roots.
Reason (R): A quadratic equation has no real roots when discriminant $< 0$. Hard
Reason (R): A quadratic equation has no real roots when discriminant $< 0$. Hard
[1]
99.
If $r$ and $s$ are the roots of $ax^2 + bx + c = 0$, the value of $r^2 + s^2$ is: Hard
[1]
100.
Case Study: A ball is thrown vertically upward. Its height $h$ (in metres) after $t$ seconds is $h = 25t - 5t^2$.
(i) Find the time(s) when the ball is at 30 m height.
(ii) Find the maximum height and time to reach it.
(iii) When does the ball hit the ground? Hard
(i) Find the time(s) when the ball is at 30 m height.
(ii) Find the maximum height and time to reach it.
(iii) When does the ball hit the ground? Hard
[5]
Coordinate Geometry
Topic A — Distance Formula (Non-trivial Applications)
101.
If the distance between $(4, p)$ and $(1, 0)$ is 5, find the value(s) of $p$. Moderate
[2]
102.
Find the values of $y$ for which the distance between $P(2, -3)$ and $Q(10, y)$ is 10. Moderate
[2]
103.
If $A(5, 2)$, $B(2, -2)$ and $C(-2, t)$ are the vertices of a right angled triangle with $\angle B = 90°$, find $t$. Hard
[4]
104.
Show that the points $A(1, 2)$, $B(5, 4)$, $C(3, 8)$ and $D(-1, 6)$ are the vertices of a square. Moderate
[4]
105.
Show that $(-2, 3)$, $(8, 3)$ and $(6, 7)$ are the vertices of a right triangle. Find its area. Moderate
[4]
106.
Find the point on the $y$-axis equidistant from $(5, -2)$ and $(-3, 2)$. Moderate
[3]
107.
The point $P(x, y)$ is equidistant from $A(5, 1)$ and $B(-1, 5)$. Prove that $3x = 2y$. Hard
[3]
108.
Name the type of quadrilateral formed by $(4, 5)$, $(7, 6)$, $(4, 3)$, $(1, 2)$ and give reasons. Hard
[4]
109.
Show that $A(3, 2)$, $B(0, 5)$, $C(-3, 2)$ and $D(0, -1)$ form a rhombus. Is it a square? Justify. Hard
[5]
Topic B — Section Formula (Internal Division) & Midpoint
110.
Find the coordinates of the point that divides $(-1, 3)$ and $(4, -7)$ internally in the ratio $3:4$. Moderate
[2]
111.
The points $A(6, 1)$, $B(8, 2)$, $C(9, 4)$ and $D(p, 3)$ are the vertices of a parallelogram (taken in order). Find $p$. Moderate
[3]
112.
Find the ratio in which the $y$-axis divides the line segment joining $(5, -6)$ and $(-1, -4)$. Also find the point of intersection. Hard
[4]
113.
The mid-point of $AB$ where $A(3, 4)$ and $B(k, 6)$ is $P(x, y)$. If $x + y - 10 = 0$, find $k$. Moderate
[3]
114.
The vertices of a triangle are $A(-1, 3)$, $B(1, -1)$ and $C(5, 1)$. Find the length of the median through vertex $A$. Moderate
[3]
115.
In what ratio does the point $(-4, 6)$ divide the line segment joining $A(-6, 10)$ and $B(3, -8)$? Hard
[4]
116.
Three vertices of a parallelogram taken in order are $(-1, 0)$, $(3, 1)$ and $(2, 2)$. Find the fourth vertex. Moderate
[3]
117.
Find the point P on the $x$-axis equidistant from $A(-2, 5)$ and $B(2, -3)$. Also find the ratio in which $P$ divides $AB$. Hard
[4]
118.
The line segment joining $A(3, -4)$ and $B(1, 2)$ is trisected at $P$ and $Q$. Find the coordinates of $P$ and $Q$. Hard
[4]
Topic C — Area of Triangle & Collinearity
119.
Find the area of the triangle with vertices $(2, 3)$, $(-1, 0)$ and $(2, -4)$. Moderate
[2]
120.
For what value of $k$ are the points $(k, 2-2k)$, $(-k+1, 2k)$ and $(-4-k, 6-2k)$ collinear? Hard
[4]
121.
Find the area of the quadrilateral $ABCD$ with vertices $A(-5, 7)$, $B(-4, -5)$, $C(-1, -6)$ and $D(4, 5)$. Hard
[5]
122.
The area of a triangle is 5 sq. units. Two vertices are $(2, 1)$ and $(3, -2)$. If the third vertex is $\left(\dfrac{7}{2}, y\right)$, find $y$. Hard
[4]
123.
Show that $A(a, b+c)$, $B(b, c+a)$ and $C(c, a+b)$ are collinear. Hard
[4]
124.
Find the area of the triangle formed by joining the mid-points of the sides of the triangle with vertices $(0, -1)$, $(2, 1)$ and $(0, 3)$. Find the ratio of this area to the original triangle's area. Hard
[5]
125.
A point $P$ divides $R(-1, 3)$ and $S(9, 8)$ such that $PR:RS = k:1$. If $P$ lies on $x - y + 2 = 0$, find $k$. Hard
[4]
Topic D — Hard Mixed: Distance + Section + Area Combined
126.
The three vertices of a parallelogram $ABCD$ are $A(3, -4)$, $B(-1, -3)$ and $C(-6, 2)$. Find the coordinates of vertex $D$. Hence verify that the diagonals bisect each other using the midpoint formula. Hard
[4]
127.
Find the ratio in which the line segment joining $A(3, -3)$ and $B(-2, 7)$ is divided by the $x$-axis. Also find the coordinates of the point of division. Hard
[4]
128.
If $A(4, 7)$, $B(p, 3)$ and $C(7, 3)$ are collinear, find $p$. Also find the area of the triangle formed by $A$, $C$ and the point that divides $BC$ in ratio $1:2$ internally. Hard
[4]
129.
Using coordinate geometry and the distance formula, show that the mid-point $M$ of the hypotenuse of a right triangle is equidistant from all three vertices. [Use $O(0,0)$, $A(2a, 0)$, $B(0, 2b)$.] Hard
[5]
Topic E — MCQ / Assertion-Reason (Coordinate Geometry)
130.
The distance between $A(0, 6)$ and $B(0, -2)$ is: Moderate
[1]
131.
$A(1, -2)$, $B(3, 4)$, $C(7, 2)$ and $D(3, -4)$ form a: Hard
[1]
132.
Assertion (A): $(-1, 6)$ divides the segment joining $(-3, 10)$ and $(6, -8)$ in ratio $2:7$ internally.
Reason (R): Section formula: $P = \left(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\right)$. Hard
Reason (R): Section formula: $P = \left(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\right)$. Hard
[1]
Topic F — Case Study Based Problems (Coordinate Geometry)
133.
Case Study: Three schools — St. Mary's at $A(3, 4)$, DPS at $B(6, 7)$, Government School at $C(-2, 3)$ — plan a new bus stop. The bus stop $P$ is to be placed on segment $AB$ such that $AP:PB = 2:1$.
(i) Find the coordinates of bus stop $P$.
(ii) Find the distance $PC$ from the bus stop to the Government School.
(iii) Find the area of $\triangle ABC$ formed by the three schools.
(iv) A road team verifies that $A$, $B$ and $C$ are NOT collinear. Confirm this using the area method. Hard
(i) Find the coordinates of bus stop $P$.
(ii) Find the distance $PC$ from the bus stop to the Government School.
(iii) Find the area of $\triangle ABC$ formed by the three schools.
(iv) A road team verifies that $A$, $B$ and $C$ are NOT collinear. Confirm this using the area method. Hard
[5]
134.
Case Study: A farmer has a triangular plot with vertices $P(2, 1)$, $Q(4, 5)$ and $R(6, 3)$.
(i) Find the perimeter of the plot.
(ii) The farmer's well is at the midpoint of $QR$. Find its coordinates.
(iii) Find the area of the plot.
(iv) Is the triangle equilateral, isosceles or scalene? Hard
(i) Find the perimeter of the plot.
(ii) The farmer's well is at the midpoint of $QR$. Find its coordinates.
(iii) Find the area of the plot.
(iv) Is the triangle equilateral, isosceles or scalene? Hard
[5]
Topic G — Mixed / Multi-concept Hard Problems
135.
If $P(2, p)$ is the midpoint of $A(6, -5)$ and $B(-2, 11)$, find $p$. Then find $AP$. Moderate
[3]
136.
Find the ratio in which the line $3x + y - 9 = 0$ divides the segment joining $(1, 3)$ and $(2, 7)$. Hard
[4]
137.
ABCD is a rectangle. Three vertices are $B(4, 0)$, $C(4, 3)$ and $D(0, 3)$. Find the length of the diagonal and the coordinates of vertex $A$. Moderate
[4]
138.
Prove that the diagonals of the quadrilateral with vertices $A(1, 0)$, $B(5, 3)$, $C(2, 7)$ and $D(-2, 4)$ bisect each other. Name the type of quadrilateral. Hard
[5]
139.
The coordinates of one end of a diameter of a circle are $(4, -1)$ and the centre is $(1, -3)$. Find the other end of the diameter. Moderate
[2]
140.
If the points $(a, 0)$, $(0, b)$ and $(1, 1)$ are collinear, show that $\dfrac{1}{a} + \dfrac{1}{b} = 1$. Hard
[4]
141.
The vertices of $\triangle ABC$ are $A(4, 6)$, $B(1, 5)$ and $C(7, 2)$. A line intersects $AB$ and $AC$ at $D$ and $E$ respectively such that $\dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{1}{4}$. Calculate the area of $\triangle ADE$ and compare with area of $\triangle ABC$. Hard
[5]
142.
$P(-2, 5)$ and $Q(3, 2)$ are two points. Find the point $R$ on $PQ$ such that $PR = 2QR$. Hard
[3]
143.
The vertices of $\triangle PQR$ are $P(2, 1)$, $Q(-2, 3)$ and $R(4, 5)$. Show that $\triangle PQR$ is a right-angled triangle using the converse of the Pythagorean theorem (distance formula). Also find the area of the triangle. Hard
[5]
144.
The base $QR$ of an equilateral triangle $PQR$ lies on the $x$-axis. $Q$ is at $(-2, 0)$ and $R$ at $(2, 0)$. Find the coordinates of vertex $P$. (Two answers possible.) Hard
[3]
145.
The two opposite vertices of a square are $(-1, 2)$ and $(3, 2)$. Find the coordinates of the other two vertices. Hard
[5]
146.
If $A(-2, -1)$, $B(1, 0)$, $C(4, 3)$ and $D(1, 2)$ are vertices of a parallelogram, verify this using both the midpoint and distance formula approaches. Is it a rhombus? Hard
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147.
The area of a triangle formed by $O$ (origin), $A(a, 0)$ and $B(0, b)$ is 6 sq. units. If $a, b > 0$ and $a = 2b$, find $a$ and $b$. Moderate
[3]
148.
Find the area of pentagon with vertices $A(1, 1)$, $B(7, 21)$, $C(10, 2)$, $D(3, -3)$ and $E(-3, 2)$ taken in order. Hard
[5]
149.
Case Study: A mobile tower is to be installed at a point $T$ that divides the line segment joining sub-station $A(-3, 2)$ and sub-station $B(9, -4)$ in ratio $2:1$ internally.
(i) Find the coordinates of tower $T$.
(ii) A third sub-station $C(3, 5)$ is also connected. Find the area of $\triangle ACT$.
(iii) A technician wants to confirm $A$, $T$ and $B$ are on the same straight line. Verify this. Hard
(i) Find the coordinates of tower $T$.
(ii) A third sub-station $C(3, 5)$ is also connected. Find the area of $\triangle ACT$.
(iii) A technician wants to confirm $A$, $T$ and $B$ are on the same straight line. Verify this. Hard
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150.
If $A(1, 2)$, $B(-2, 3)$, $C(-3, -4)$ and $D(2, -5)$ are the vertices of quadrilateral $ABCD$, find its area. Also verify that the diagonals are NOT equal (hence it is not a rectangle). Hard
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🔥 Bonus — Cross-chapter Challenge Problems
B1.
The $n$th term of a sequence is $2n^2 + 1$. Find the sum of the first 10 terms. Also verify whether the sequence is an AP. Hard
[4]
B2.
Using the quadratic formula, solve: $\dfrac{1}{a+b+x} = \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{x}$, where $a + b \ne 0$. Hard
[5]
B3.
Three vertices of a rhombus taken in order are $(-2, -1)$, $(3, 4)$ and $(-2, 9)$. Find the fourth vertex. Also find the area of the rhombus using coordinate geometry. Hard
[5]
B4.
The sum of first $n$ terms of an AP is $\dfrac{n(3n-1)}{2}$. Find the AP. Also check whether 68 is a term of this AP. Hard
[4]
B5.
A line intersects the $y$-axis and $x$-axis at $P$ and $Q$ respectively. If $(2, -5)$ is the mid-point of $PQ$, find the coordinates of $P$ and $Q$ and the length $PQ$. Hard
[4]