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Mathematics Mock Test 01 (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.
The LCM of smallest two digit composite number and smallest composite number is:
(a) 12
(b) 4
(c) 20
(d) 44
2.
If the lines given by \(3x + 2ky = 2\) and \(2x + 5y + 1 = 0\) are parallel, then the value of \(k\) is:
(a) \(\frac{-5}{4}\)
(b) \(\frac{2}{5}\)
(c) \(\frac{15}{4}\)
(d) \(\frac{3}{2}\)
3.
The zeroes of the quadratic polynomial \(x^2 - 2x - 8\) are:
(a) 2, -4
(b) 4, -2
(c) -2, -2
(d) -4, -4
4.
The pair of linear equations \(x = 0\) and \(x = 5\) has:
(a) No solution
(b) Unique solution
(c) Two solutions
(d) Infinitely many solutions
5.
The nature of roots of the quadratic equation \(2x^2 - 4x + 3 = 0\) is:
(a) Real and equal
(b) Real and distinct
(c) No real roots
(d) Cannot be determined
6.
The 10th term of the AP: 2, 7, 12, ... is:
(a) 47
(b) 42
(c) 37
(d) 52
7.
If \(\triangle ABC \sim \triangle DEF\) such that \(AB = 1.2\) cm and \(DE = 1.4\) cm, the ratio of the areas of \(\triangle ABC\) and \(\triangle DEF\) is:
(a) 16:25
(b) 49:36
(c) 36:49
(d) 25:16
8.
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
9.
The value of \(\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}\) is:
(a) \(\tan 90^\circ\)
(b) 1
(c) \(\sin 45^\circ\)
(d) 0
10.
If the length of the shadow of a tree is \(\sqrt{3}\) times the height of the tree, then the angle of elevation of the sun is:
(a) \(45^\circ\)
(b) \(30^\circ\)
(c) \(60^\circ\)
(d) \(90^\circ\)
11.
Given that HCF (306, 657) = 9, the LCM (306, 657) is:
(a) 22338
(b) 23328
(c) 22833
(d) 33228
12.
If one zero of the polynomial \(f(x) = (k^2+4)x^2 + 13x + 4k\) is reciprocal of the other, then \(k =\)
(a) 2
(b) -2
(c) 1
(d) -1
13.
The next term of the A.P. \(\sqrt{8}, \sqrt{18}, \sqrt{32} \dots\) is:
(a) \(\sqrt{40}\)
(b) \(\sqrt{48}\)
(c) \(\sqrt{50}\)
(d) \(\sqrt{54}\)
14.
In \(\triangle ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = x\), \(DB = x-2\), \(AE = x+2\) and \(EC = x-1\), the value of \(x\) is:
(a) 1
(b) 2
(c) 3
(d) 4
15.
If the mid-point of the line segment joining the points \(A(3, 4)\) and \(B(k, 6)\) is \(P(x, y)\) and \(x + y - 10 = 0\), the value of \(k\) is:
(a) 5
(b) 6
(c) 7
(d) 8
16.
If \(\sin A = \frac{1}{2}\) and \(A\) is an acute angle, then the value of \(\cot A\) is:
(a) \(\frac{1}{\sqrt{3}}\)
(b) \(\sqrt{3}\)
(c) \(\frac{\sqrt{3}}{2}\)
(d) 1
17.
The roots of the quadratic equation \(x^2 - p^2 = 0\) are:
(a) \(p, p\)
(b) \(-p, -p\)
(c) \(p, -p\)
(d) \(p^2, -p^2\)
18.
The value of \(9\sec^2 A - 9\tan^2 A\) is:
(a) 1
(b) 9
(c) 8
(d) 0
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 corresponding sides of two triangles are proportional, then they are similar.
Reason (R): Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional (SSS similarity criterion).
20.
Assertion (A): \(\sin(A+B) = \sin A + \sin B\) for all acute angles \(A\) and \(B\).
Reason (R): For \(A=30^\circ\) and \(B=60^\circ\), \(\sin(30^\circ+60^\circ) = \sin 90^\circ = 1\), which is not equal to \(\sin 30^\circ + \sin 60^\circ\).
SECTION B (10 Marks)(Very Short Answer Type - 2 Marks Each)
21.
Explain why \(7 \times 11 \times 13 + 13\) is a composite number.
22.
Find the ratio in which the y-axis divides the line segment joining the points \(A(5, -6)\) and \(B(-1, -4)\).
23.
Find a quadratic polynomial whose zeroes are \(3 + \sqrt{2}\) and \(3 - \sqrt{2}\).
24.
In \(\triangle ABC\), \(D\) and \(E\) are points on the sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = 4x - 3\), \(AE = 8x - 7\), \(BD = 3x - 1\) and \(CE = 5x - 3\), find the value of \(x\).
25.
Prove the following trigonometric identity: \[ \sec A (1 - \sin A) (\sec A + \tan A) = 1 \]
SECTION C (18 Marks)(Short Answer Type - 3 Marks Each)
26.
Prove that \(\sqrt{3}\) is an irrational number.
27.
Find the zeroes of the quadratic polynomial \(4\sqrt{3}x^2 + 5x - 2\sqrt{3}\) and verify the relationship between the zeroes and the coefficients.
28.
The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
29.
Find the sum of all three-digit natural numbers which are divisible by 7.
30.
Prove that: \[ (\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A \]
31.
If 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 the value of \(p\).
SECTION D (20 Marks)(Long Answer Type - 5 Marks Each)
32.
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. Formulate the quadratic equation and find the original speed of the train.
33.
State and prove the Basic Proportionality Theorem (Thales Theorem).
OR
In a right triangle \(ABC\), right angled at \(B\), \(D\) is a point on hypotenuse \(AC\) such that \(BD \perp AC\). Prove that \(\triangle ADB \sim \triangle BDC\) and hence prove that \(BD^2 = AD \times DC\).
34.
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is \(60^\circ\) and the angle of depression of its foot is \(45^\circ\). Determine the height of the tower. (Use \(\sqrt{3} = 1.732\))
35.
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.
SECTION E (12 Marks)(Case-Based Questions - 4 Marks Each)
36.
Case Study 1: Car Production (Arithmetic Progressions)
The production of cars in a factory increases uniformly by a fixed number every year. The factory produced 6000 cars in the 3rd year and 7000 cars in the 7th year.
(i) Find the production of cars in the 1st year. (1 Mark)
(ii) Find the production of cars in the 10th year. (1 Mark)
(iii) Find the total production of cars in the first 7 years. (2 Marks)
OR
In which year will the production of cars reach 10,000? (2 Marks)
37.
Case Study 2: Hot Air Balloon (Applications of Trigonometry)
A boy flying a kite observes a hot air balloon moving horizontally at a constant height of \(1500\sqrt{3}\) meters from the ground. The angle of elevation of the balloon from the boy's eyes is initially \(60^\circ\). After a few seconds, as the balloon moves away horizontally, the angle of elevation reduces to \(30^\circ\).
(i) Draw a neat labeled rough diagram illustrating the given situation. (1 Mark)
(ii) Find the horizontal distance traveled by the balloon during this interval. (2 Marks)
OR
If the time taken for this change in angle is 15 seconds, find the speed of the balloon in meters per second. (2 Marks)
(iii) What is the line-of-sight distance between the boy and the balloon at its initial position? (1 Mark)
38.
Case Study 3: Highway Underpass (Polynomials)
A highway underpass is parabolic in shape. A parabola is the graph that results from \(p(x) = ax^2 + bx + c\). The mathematical representation of the shape of the underpass is given by the polynomial \(p(x) = x^2 - 2x - 8\).
(i) Find the zeroes of the polynomial (the points where the parabola hits the ground). (1 Mark)
(ii) What is the value of the polynomial when \(x = 3\)? (1 Mark)
(iii) If a new underpass is designed such that its zeroes are exactly double the zeroes of the original underpass, write the expression for the new quadratic polynomial. (2 Marks)
OR
Calculate the distance between the two zeroes of the original underpass. (2 Marks)
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