Mathematics Exam Solutions
Part A - Section I
Group A
1. Write the expanded form of the log 3528.
3528 = 2³ × 3² × 7²
log 3528 = 3 log 2 + 2 log 3 + 2 log 7
2. Draw the venn diagram of A∪B if A = {1, 2, 4, 7, 8, 10} and B = {2, 3, 4, 6, 7, 9}.
3. Check whether (2x + 3)² = x(2x - 4) is a quadratic equation or not?
(2x + 3)² = x(2x - 4)
4x² + 12x + 9 = 2x² - 4x
2x² + 16x + 9 = 0
Yes, this is a quadratic equation.
4. If the angles of a Δ ABC are in Arithmetic progression and the smallest angle is 30°, then find all angles of the Δ ABC.
Let angles be (a - d), a, (a + d)
(a - d) + a + (a + d) = 180°
3a = 180° ⇒ a = 60°
Smallest angle = a - d = 30° ⇒ d = 30°
Angles: 30°, 60°, 90°
5. If p(x) = 5x² - 3x + 7, then find the value of p(2) and p(-3).
p(2) = 5(2)² - 3(2) + 7 = 20 - 6 + 7 = 21
p(-3) = 5(-3)² - 3(-3) + 7 = 45 + 9 + 7 = 61
6. Find the centroid of the Δ ABC whose vertices are A(-2, -6), B(4, 1) and C(7, 8).
Centroid = ((-2+4+7)/3, (-6+1+8)/3) = (9/3, 3/3) = (3, 1)
Group B
7. Perimeters of two similar triangles are in the ratio 1:2. If one side of the first triangle is 6 cm, find the corresponding side of the second triangle.
6/x = 1/2 ⇒ x = 12 cm
8. Draw a rough diagram showing a pair of tangents to a circle from an external point.
9. Express 'cot θ' and 'cos θ' in terms of 'tan θ'.
cot θ = 1/tan θ
cos θ = 1/√(1 + tan²θ)
10. A girl observes the top of a temple of height 90 feet from a point at an angle of elevation 60°. Show that the distance of the point from the foot of the temple is 30√3 feet.
tan 60° = 90/x
√3 = 90/x ⇒ x = 90/√3 = 30√3 feet
11. If two coins are tossed simultaneously find the probability of getting atleast one Head.
Possible outcomes: HH, HT, TH, TT
Favorable outcomes: HH, HT, TH
Probability = 3/4 = 0.75
12. Marks of X class students in a Maths test for 80 marks are as follows: 65, 56, 72, 49, 72, 57, 70, 72, 62 and 68. Find mode of this data.
72 appears 3 times, more than any other value
Mode = 72
Part A - Section II
13. Formulate a pair of linear equations in two variables for given data "4 note books and 7 pens together cost Rs. 184 where as 6 note books and 5 pens together cost Rs. 210."
Let cost of one notebook = x, cost of one pen = y
4x + 7y = 184
6x + 5y = 210
14. If A = {x: x is an even number less than 15}, B = {x: x is a multiple of 3 less than 25}, then find A-B and B-A.
A = {2, 4, 6, 8, 10, 12, 14}
B = {3, 6, 9, 12, 15, 18, 21, 24}
A - B = {2, 4, 8, 10, 14}
B - A = {3, 9, 15, 18, 21, 24}
15. Use division algorithm to show that the square of any positive integer is of the form 4q or 4q + 1.
Let a be any positive integer. By division algorithm, a = 4k, 4k+1, 4k+2, or 4k+3
Case 1: a = 4k ⇒ a² = 16k² = 4(4k²) = 4q
Case 2: a = 4k+1 ⇒ a² = 16k²+8k+1 = 4(4k²+2k)+1 = 4q+1
Case 3: a = 4k+2 ⇒ a² = 16k²+16k+4 = 4(4k²+4k+1) = 4q
Case 4: a = 4k+3 ⇒ a² = 16k²+24k+9 = 4(4k²+6k+2)+1 = 4q+1
Thus, a² is either 4q or 4q+1
16. Find the point on the x-axis which is equidistant from (-3, -10) and (3, 8).
Let the point be (x, 0)
√[(x+3)² + 100] = √[(x-3)² + 64]
(x+3)² + 100 = (x-3)² + 64
x²+6x+9+100 = x²-6x+9+64
12x = -36 ⇒ x = -3
Point is (-3, 0)
17. Write the formulas of finding median for ungrouped data containing odd number of values and even number of values.
For odd n: Median = value at position (n+1)/2
For even n: Median = average of values at positions n/2 and (n/2 + 1)
18. The radius of a conical tent is 5 m and its height is 12 m. Show that the area of the canvas required is 204 2/7 m².
Slant height = √(5² + 12²) = √169 = 13 m
Curved surface area = πrl = (22/7)×5×13 = 1430/7 = 204 2/7 m²
19. Find the value of [(sin²30 × sec²60) + (sec²45 - 2 tan²45)] / [2 cos²90 - cot²90]
sin 30° = 1/2, sec 60° = 2, sec 45° = √2, tan 45° = 1, cos 90° = 0, cot 90° = 0
Numerator = (1/4 × 4) + (2 - 2×1) = 1 + 0 = 1
Denominator = 2×0 - 0 = 0
Expression is undefined (division by zero)
20. Cards numbered from 1 to 50 are placed in a box, and when a card is taken at random, find the probability of the card taken out is a two digit odd composite number.
Two-digit odd composite numbers: 15, 21, 25, 27, 33, 35, 39, 45, 49
Favorable outcomes = 9
Total outcomes = 50
Probability = 9/50
Part A - Section III
Group A
21. Draw the graph of following pair of linear equations, find the intersecting point from graph: 3x + 2y = 4 and 2x + 3y = 11
Solving algebraically:
3x + 2y = 4 ...(1)
2x + 3y = 11 ...(2)
Multiply (1) by 2 and (2) by 3:
6x + 4y = 8
6x + 9y = 33
Subtracting: 5y = 25 ⇒ y = 5
Substitute in (1): 3x + 10 = 4 ⇒ 3x = -6 ⇒ x = -2
Intersection point: (-2, 5)
22. If (0.37)^x = (0.037)^y = 10000, then find the value of 1/x - 1/y.
(0.37)^x = 10000 ⇒ x log(0.37) = 4 ⇒ x = 4/log(0.37)
(0.037)^y = 10000 ⇒ y log(0.037) = 4 ⇒ y = 4/log(0.037)
1/x - 1/y = log(0.37)/4 - log(0.037)/4
= 1/4 [log(0.37) - log(0.037)]
= 1/4 [log(0.37/0.037)] = 1/4 [log(10)] = 1/4 × 1 = 1/4
23. Find the sum of all two digit numbers which are divisible by 3 but not divisible by 2.
Two-digit numbers divisible by 3: 12, 15, 18, ..., 99
Sum = 30/2 × (12 + 99) = 15 × 111 = 1665
Two-digit numbers divisible by both 2 and 3 (divisible by 6): 12, 18, 24, ..., 96
Sum = 15/2 × (12 + 96) = 15/2 × 108 = 15 × 54 = 810
Sum of numbers divisible by 3 but not by 2 = 1665 - 810 = 855
24. Find the points of trisection of the line segment joining (5, 7) and (2, -2).
First point P divides AB in ratio 1:2
P = ((1×2 + 2×5)/(1+2), (1×(-2) + 2×7)/(1+2)) = (12/3, 12/3) = (4, 4)
Second point Q divides AB in ratio 2:1
Q = ((2×2 + 1×5)/(2+1), (2×(-2) + 1×7)/(2+1)) = (9/3, 3/3) = (3, 1)
Points of trisection: (4, 4) and (3, 1)
Part B - Section IV
Multiple Choice Questions
1. The value of log(27/√3) is ---
Correct answer: A) 3
2. If cardinal number of a set A is 3, then the possible number of sub sets of set A is ---
Correct answer: B) 8
3. The degree of the polynomial p(x) = 2x^4 + 5x^3 - 3x^2 + 6x + 7 is ---
Correct answer: A) 4
4. The n-th term of Arithmetic progression a_n = a + (n-1)d, then 'd' indicates.
Correct answer: B) Common difference
5. Which of the following equation is parallel to line 2x-3y+7=0
Correct answer: D) 6x-9y-20=0
6. The discriminant of the equation ax^2+bx+c=0, is---
Correct answer: A) b^2-4ac
7. The slope of the line passing through the points A(5,6) and B(0,-4) is ---
Correct answer: D) 2
8. Which of the following rational number is a terminating decimal.
Correct answer: C) 7/28
9. Which of the following is a zero of the polynomial p(x)=3x^2-x^2-2.
Correct answer: A) -2/3
10. Which of the following is the 9th term of A.P: 2,11,20......
Correct answer: B) 74
11. A tangent is drawn from an external point of a circle of radius 8 cm. If the length of the tangent is 15 cm, then the distance of the point from the centre of the circle is ---
Correct answer: B) 17 cm
12. In Δ PQR, if XY II QR and if PX = 1.5 cm, XQ = 3 cm, PY = 2 cm, then YR =
Correct answer: D) 4 cm
13. If P(E) = 0.59, then P(not E) is ---
Correct answer: A) 0.41
14. If Σfx = 128, Σf = 16, then mean (x̄) is ---
Correct answer: D) 8
15. If sin^2x = 1, then cos^2x is ---
Correct answer: A) 0
16. If the curved surface area of a cylinder of height 8 cm is 176 cm^2, then its radius is ---
Correct answer: A) 3.5 cm
17. If x, x+7, and x+8 are the sides of a right angle triangle where x ∈ N, then the value of 'x' is ---
Correct answer: A) 5 cm
18. Which of the following is not an example of equally likely event.
Correct answer: D) choosing a ball at random from a box containing 5 red balls and 8 blue balls
19. Mode of the values of sin 0°, sin 90°, cos 90°, tan 30° and sec 60° is ---
Correct answer: A) 0
20. If the length of the shadow of a pole is √3 times to its original height, then the angle of elevation of sun rays is ---
Correct answer: D) 30°