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Q1
short answer
1 mark
If HCF (336, 54) = 6, find LCM (336, 54).
Solution
Using the relationship: LCM × HCF = Product of the two numbers
LCM(336, 54) × HCF(336, 54) = 336 × 54
LCM × 6 = 18144
LCM = 18144 ÷ 6 = 3024
Therefore, LCM(336, 54) = 3024.
Q2
short answer
1 mark
Find the nature of roots of the quadratic equation 2x2 – 4x + 3 = 0.
Solution
For the quadratic equation 2x² – 4x + 3 = 0, we calculate the discriminant:
D = b² – 4ac = (–4)² – 4(2)(3) = 16 – 24 = –8
Since D < 0, the equation has no real roots (the roots are imaginary/complex).
Q3
short answer
1 mark
Find the common difference of the Arithmetic Progression (A.P.) 1/a, (3-a)/3a, (3-2a)/3a, ... (a 0)
Solution
The given A.P. is: 1/a, (3–a)/3a, (3–2a)/3a, ...
First, rewrite the first term: 1/a = 3/3a
Common difference d = second term – first term
d = (3–a)/3a – 3/3a = (3–a–3)/3a = (–a)/3a = –1/3
Verification: (3–2a)/3a – (3–a)/3a = (3–2a–3+a)/3a = (–a)/3a = –1/3 ✓
Therefore, the common difference is –1/3.
Q4
short answer
1 mark
Evaluate : sin2 60 + 2 tan 45 – cos2 30 OR If sin A = 4/3, calculate sec A.
Solution
Main question: Evaluate sin²60° + 2tan45° – cos²30°
sin60° = √3/2, so sin²60° = 3/4
tan45° = 1, so 2tan45° = 2
cos30° = √3/2, so cos²30° = 3/4
Expression = 3/4 + 2 – 3/4 = 2
Answer: 2
OR
If sinA = 4/3: Note – this appears to be a misprint; sinA cannot exceed 1. It is likely sinA = 3/4 (common exam question) or the intended value gives a valid triangle. However, taking the question as printed with sinA = 4/3 is impossible for a real angle. The standard version of this question is sinA = 3/4:
If sinA = 3/4, then in a right triangle, opposite = 3, hypotenuse = 4
adjacent = √(16–9) = √7
secA = hypotenuse/adjacent = 4/√7 = 4√7/7
Q5
short answer
1 mark
Write the coordinates of a point P on x-axis which is equidistant from the points A(– 2, 0) and B(6, 0).
Solution
Let P = (x, 0) be a point on the x-axis equidistant from A(–2, 0) and B(6, 0).
PA = PB
√[(x–(–2))² + 0²] = √[(x–6)² + 0²]
(x+2)² = (x–6)²
x² + 4x + 4 = x² – 12x + 36
16x = 32
x = 2
Therefore, the coordinates of point P are (2, 0).
Q6
short answer
1 mark
In Figure 1, ABC is an isosceles triangle right angled at C with AC = 4 cm. Find the length of AB. OR In Figure 2, DE || BC. Find the length of side AD, given that AE = 1.8 cm, BD = 7.2 cm and CE = 5.4 cm.
Q7
short answer
2 marks
Write the smallest number which is divisible by both 306 and 657.
Q8
short answer
2 marks
Find a relation between x and y if the points A(x, y), B(– 4, 6) and C(– 2, 3) are collinear. OR Find the area of a triangle whose vertices are given as (1, – 1) (– 4, 6) and (– 3, – 5).
Q8
short answer
2 marks
Find a relation between x and y if the points A(x, y), B(– 4, 6) and C(– 2, 3) are collinear.
Q8
short answer
2 marks
Find the area of a triangle whose vertices are given as (1, – 1) (– 4, 6) and (– 3, – 5).
Q9
short answer
2 marks
The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is 1/5. The probability of selecting a black marble at random from the same jar is 1/4. If the jar contains 11 green marbles, find the total number of marbles in the jar.
Q10
short answer
2 marks
Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.
Q11
short answer
2 marks
The larger of two supplementary angles exceeds the smaller by 18. Find the angles.
Q11
short answer
2 marks
Sumit is 3 times as old as his son. Five years later, he shall be two and a half times as old as his son. How old is Sumit at present ?
Q12
short answer
2 marks
Find the mode of the following frequency distribution : Class Interval : 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50 50 – 55 Frequency : 25 34 50 42 38 14
Q16
long answer
In Figure 3, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting PQ at A and RS at B. Prove that AOB = 90.
Q16
long answer
Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (– 2, – 5) and (6, 3). Find the coordinates of the point of intersection.
Q17
short answer
Evaluate : (3 sin^2 43° + cos^2 47° - (cos 37° / sin 53°)) / (tan 5° tan 25° tan 45° tan 65° tan 85°)
Q18
long answer
In Figure 4, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of the shaded region. (Use = 3·14)
Q18
long answer
OR In Figure 5, ABCD is a square with side 22 cm and inscribed in a circle. Find the area of the shaded region. (Use = 3·14)
Q19
long answer
A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use = 22/7)
Q20
long answer
Find the mean marks of the students given the distribution of marks for 100 students: 30-35 (14), 35-40 (16), 40-45 (28), 45-50 (23), 50-55 (18), 55-60 (8), 60-65 (3).
Q21
long answer
For what value of k, is the polynomial f(x) = 3x^4 – 9x^3 + x^2 + 15x + k completely divisible by 3x^2 – 5?
Q21
long answer
OR Find the zeroes of the quadratic polynomial 7y^2 - (11/3)y - (2/3) and verify the relationship between the zeroes and the coefficients.
Q22
long answer
Write all the values of p for which the quadratic equation x^2 + px + 16 = 0 has equal roots. Find the roots of the equation so obtained.
Q23
long answer
4 marks
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Q24
long answer
4 marks
Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30°. Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45°. Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.
Q25
long answer
4 marks
A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm^3 of iron has approximately 8 gm mass. (Use = 3.14)
Q26
long answer
Construct an equilateral Δ ABC with each side 5 cm. Then construct another triangle whose sides are 3/2 times the corresponding sides of Δ ABC. OR Draw two concentric circles of radii 2 cm and 5 cm. Take a point P on the outer circle and construct a pair of tangents PA and PB to the smaller circle. Measure PA.
Q27
long answer
Change the following data into ‘less than type’ distribution and draw its ogive : Class Interval : 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100 Frequency : 7 5 8 10 6 6 8
Q28
long answer
Prove that : tan / (1 - cot) + cot / (1 - tan) = 1 + sec cosec. OR Prove that : sin^2 / (cosec - cot) = 1 + cosec cot
Q29
long answer
Which term of the Arithmetic Progression –7, –12, –17, –22, ... will be –82 ? Is –100 any term of the A.P. ? Give reason for your answer. OR How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ? Explain the double answer.
Q30
long answer
In a class test, the sum of Arun’s marks in Hindi and English is 30. Had he got 2 marks more in Hindi and 3 marks less in English, the product of the marks would have been 210. Find his marks in the two subjects.
