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VARDAAN LEARNING INSTITUTE

Mathematics Mock Test 02 (CBSE Class 10)
Class: 10 CBSE Subject: Mathematics Time: 3 Hours Max. Marks: 80
General Instructions:
SECTION A (20 Marks)(Multiple Choice Questions - 1 Mark Each)
1.
If \(a = x^3 y^2\) and \(b = x y^3\), where \(x, y\) are prime numbers, then \(\text{HCF}(a, b)\) is:
(a) \(xy\)
(b) \(xy^2\)
(c) \(x^3y^3\)
(d) \(x^2y^2\)
2.
A quadratic polynomial, whose zeroes are \(-3\) and \(4\), is:
(a) \(x^2 - x + 12\)
(b) \(x^2 + x + 12\)
(c) \(\frac{x^2}{2} - \frac{x}{2} - 6\)
(d) \(2x^2 + 2x - 24\)
3.
The pair of equations \(5x - 15y = 8\) and \(3x - 9y = \frac{24}{5}\) has:
(a) one solution
(b) two solutions
(c) infinitely many solutions
(d) no solution
4.
The values of \(k\) for which the quadratic equation \(2x^2 - kx + k = 0\) has equal roots is:
(a) 0 only
(b) 4
(c) 8 only
(d) 0, 8
5.
The distance of the point \(P(2, 3)\) from the x-axis is:
(a) 2 units
(b) 3 units
(c) 1 unit
(d) 5 units
6.
In an A.P., if \(d = -4\), \(n = 7\), \(a_n = 4\), then \(a\) is:
(a) 6
(b) 7
(c) 20
(d) 28
7.
If \(\triangle ABC \sim \triangle PQR\) such that \(AB = 1.2\) cm and \(PQ = 1.4\) cm, the ratio of the areas of \(\triangle ABC\) and \(\triangle PQR\) is:
(a) 16:25
(b) 49:36
(c) 36:49
(d) 25:16
8.
The value of \(\sin 30^\circ \cos 60^\circ + \cos 30^\circ \sin 60^\circ\) is:
(a) 0
(b) 1
(c) 2
(d) \( \frac{\sqrt{3}}{2} \)
9.
The shadow of a tower is equal to its height at:
(a) 10:45 a.m.
(b) angle of elevation 45°
(c) angle of elevation 60°
(d) angle of elevation 30°
10.
If the distance between the points \((4, p)\) and \((1, 0)\) is 5, then the value of \(p\) is:
(a) 4 only
(b) \(\pm 4\)
(c) -4 only
(d) 0
11.
The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is:
(a) 13
(b) 65
(c) 875
(d) 1750
12.
If one root of the quadratic equation \(2x^2 + kx - 6 = 0\) is 2, the value of \(k\) is:
(a) 1
(b) -1
(c) 2
(d) -2
13.
The sum of the first 100 natural numbers is:
(a) 5050
(b) 5000
(c) 10000
(d) 10050
14.
In \(\triangle ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = 2\) cm, \(DB = 3\) cm, and \(AC = 10\) cm, then \(AE\) is:
(a) 2 cm
(b) 3 cm
(c) 4 cm
(d) 5 cm
15.
The midpoint of the line segment joining \(A(-2, 8)\) and \(B(-6, -4)\) is:
(a) \((-4, -6)\)
(b) \((2, 6)\)
(c) \((-4, 2)\)
(d) \((4, 2)\)
16.
Given that \(\sin \theta = \frac{a}{b}\), then \(\cos \theta\) is equal to:
(a) \(\frac{b}{\sqrt{b^2 - a^2}}\)
(b) \(\frac{b}{a}\)
(c) \(\frac{\sqrt{b^2 - a^2}}{b}\)
(d) \(\frac{a}{\sqrt{b^2 - a^2}}\)
17.
If the sum of zeroes of the quadratic polynomial \(3x^2 - kx + 6\) is 3, then the value of \(k\) is:
(a) 3
(b) -3
(c) 9
(d) -9
18.
The value of \(\tan 30^\circ \times \tan 60^\circ\) is:
(a) 1
(b) \(\sqrt{3}\)
(c) \(\frac{1}{\sqrt{3}}\)
(d) 3
DIRECTION for Q19 and Q20: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option:
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is NOT the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
19.
Assertion (A): If the points \(A(4, 3)\) and \(B(x, 5)\) lie on a circle with center \(O(2, 3)\), then the value of \(x\) is 2.
Reason (R): The distance from the center to any point on the circle is equal to the radius.
20.
Assertion (A): The equation \(x^2 + 3x + 1 = (x - 2)^2\) is a quadratic equation.
Reason (R): Any equation of the form \(ax^2 + bx + c = 0\) where \(a \neq 0\), is a quadratic equation.
SECTION B (10 Marks)(Very Short Answer Type - 2 Marks Each)
21.
Given that \(\text{HCF}(306, 657) = 9\), find \(\text{LCM}(306, 657)\).
22.
In \(\triangle ABC\), right-angled at \(B\), \(AB = 5\) cm and \(\sin C = \frac{1}{2}\). Determine the length of the side \(AC\) and \(BC\).
23.
Find the value of \(p\) for which the pair of linear equations \(px + 3y - (p-3) = 0\) and \(12x + py - p = 0\) has infinitely many solutions.
24.
Find the ratio in which the y-axis divides the line segment joining the points \(A(5, -6)\) and \(B(-1, -4)\).
25.
In the given figure, if \(DE \parallel BC\), \(AD = x\), \(DB = x - 2\), \(AE = x + 2\) and \(EC = x - 1\). Find the value of \(x\).
SECTION C (18 Marks)(Short Answer Type - 3 Marks Each)
26.
Prove that \(3 + 2\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is an irrational number.
27.
Find the zeroes of the quadratic polynomial \(6x^2 - 3 - 7x\) and verify the relationship between the zeroes and the coefficients.
28.
A fraction becomes \(\frac{1}{3}\) when 1 is subtracted from the numerator and it becomes \(\frac{1}{4}\) when 8 is added to its denominator. Find the fraction.
29.
Prove the following identity: \[ \frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A \]
30.
How many terms of the A.P. 9, 17, 25... must be taken to give a sum of 636?
31.
Show that the points \(A(1, 7)\), \(B(4, 2)\), \(C(-1, -1)\) and \(D(-4, 4)\) are the vertices of a square.
SECTION D (20 Marks)(Long Answer Type - 5 Marks Each)
32.
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 to the same spot. Find the speed of the stream.
33.
State and prove the Basic Proportionality Theorem (Thales Theorem).
Using the theorem, prove that the line drawn from the mid-point of one side of a triangle parallel to another side bisects the third side.
34.
The angles of depression of the top and bottom of an 8 m tall building from the top of a multi-storeyed building are \(30^\circ\) and \(45^\circ\), respectively. Find the height of the multi-storeyed building and the distance between the two buildings. (Use \(\sqrt{3} = 1.732\))
35.
The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P.
SECTION E (12 Marks)(Case-Based Questions - 4 Marks Each)
36.
Case Study 1: Sports Day Grid (Coordinate Geometry)
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD. Niharika runs \(\frac{1}{4}\) th the distance AD on the 2nd line and posts a green flag. Preet runs \(\frac{1}{5}\) th the distance AD on the 8th line and posts a red flag.
(i) Find the coordinates of the green flag. (1 Mark)
(ii) Find the coordinates of the red flag. (1 Mark)
(iii) What is the distance between both the flags? (2 Marks)
OR
If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag? (2 Marks)
37.
Case Study 2: Saving for a Trip (Arithmetic Progressions)
Karan is planning a school trip and decides to save money from his pocket allowance. He saves ₹ 100 in the first month and then increases his monthly savings by ₹ 20 every subsequent month.
(i) How much money will Karan save in the 10th month? (1 Mark)
(ii) In which month will his monthly saving be ₹ 340? (1 Mark)
(iii) What is the total amount saved by Karan in 12 months? (2 Marks)
OR
If the total cost of the trip is ₹ 3200, will he be able to go on the trip after saving for 15 months? Justify. (2 Marks)
38.
Case Study 3: The Lighthouse (Applications of Trigonometry)
A lighthouse is a tower with a bright light at the top, located at an important or dangerous place regarding navigation (travel over water). A lighthouse keeper is observing two ships from the top of the 75m high lighthouse. The angles of depression of the two ships, approaching the lighthouse in the same straight line, are \(30^\circ\) and \(45^\circ\).
(i) Draw a neat labeled figure to illustrate the situation. (1 Mark)
(ii) Find the distance of the closer ship from the base of the lighthouse. (1 Mark)
(iii) Calculate the distance between the two ships. (Use \(\sqrt{3} = 1.732\)) (2 Marks)
OR
If the ship originally at \(30^\circ\) moves towards the lighthouse and reaches the position of the first ship, how much distance did it cover? (2 Marks)
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