UP TGT Mathematics Mock Test - 5 - UPTET MCQ

• The Kingdom of Anga was indeed a powerful entity before being conquered by Bimbisara, making statement 1 correct.
• Avanti, with its capital at Ujjain, was a significant competitor, partly due to its access to iron deposits, which made it a rival in military capabilities, so statement 2 is correct.
• Magadha's economic prosperity was significantly boosted by its agricultural productivity due to the fertile alluvial soil of the middle Gangetic plain, contrary to statement 3, which underestimates agriculture's role.

• The area between the Tropic of Cancer and the Tropic of Capricorn is known as the Torrid Zone.
• The mid-day Sun is exactly overhead at least once a year on all the latitudes in this area; hence, this area receives maximum amount of heat.

The New York Times (NYT) has taken legal action against OpenAI and Microsoft, accusing them of unauthorized usage of its copyrighted content for AI model training, notably ChatGPT. This legal dispute highlights the broader debate on intellectual property rights in the generative AI era.

The correct answer is Sutlej.

Key Points

• The Sutlej river originates from Lake Langa with an elevation above 15000 feet located at the southwestern Tibet.

Additional InformationSatluj River:

• The Satluj rises from the Manasarovar-Rakas Lakes in western Tibet at a height of 4,570 m within 80 km of the source of the Indus.
• Like the Indus, it takes a north-westerly course upto the Shipki La on the Tibet-Himachal Pradesh boundary.
• It cuts deep gorges where it pierces the Great Himalaya and the other Himalayan ranges.
• Before entering the Punjab plain, it cuts a gorge in Naina Devi Dhar, where the famous Bhakra dam has been constructed.
• After entering the plain at Rupnagar (Ropar), it turns westwards and is joined by the Beas at Harike.

Beas River:

• The Beas originates near the Rohtang Pass, at a height of 4,062 m above sea level, on the southern end of the Pir Panjal Range, close to the source of the Ravi.
• It crosses the Dhaola Dhar range and it takes a south-westerly direction and meets the Satluj river at Harike in Punjab.
• It is a comparatively small river which is only 460 km long but lies entirely within the Indian territory.

Jhelum River

• The Jhelum has its source in a spring at Verinag in the south-eastern part of the Kashmir Valley.
• It flows northwards into Wular Lake (north-western part of Kashmir Valley).
• From Wular Lake, it changes its course southwards. At Baramulla the river enters a gorge in the hills.
• The river forms steep-sided narrow gorge through Pir Panjal Range below Baramula.
• At Muzaffarabad, the river takes a sharp hairpin bend southward.

Ravi River:

• The Ravi has its source in Kullu hills near the Rohtang Pass in Himachal Pradesh.
• It drains the area between the Pir Panjal and the Dhaola Dhar ranges.
• After crossing Chamba, it takes a south-westerly turn and cuts a deep gorge in the Dhaola Dhar range.
• It enters Punjab Plains near Madhopur and later enters Pakistan below Amritsar.
• It debouches into the Chenab a little above Rangpur in Pakistani Punjab.

Given: 

Mean of the observations 25, 29, 25, 32, 24 and x is 27.

Concept used: 

Mean = sum of all observations / total number of observations

Median:

The median formula of a given set of numbers, say having 'n' odd number of observations, can be expressed as:
 term

The median formula of a given set of numbers say having 'n' even number of observations, can be expressed as:
term

Calculation: 

Mean of the observations =  25 + 29 + 25 + 32 + 24 + x / 6 = 27

x = 27 × 6 - 135

⇒ 162 - 135 

⇒ 27

Arranging the terms in ascending order,

24, 25, 25, 27,29, 32 

Here, number of terms are even, 

∴ Median = (25 + 27) / 2 

⇒ 26

 ∴ Option 3 is correct.

Given:

For any two real number, an operation defined
by a * b = 1 + ab

Concept used:

a * b = b * a for all a, b ∈ G (commutative property)

a * (b * c) = (a * b) * c for all a, b, c ∈ G (associative property)

Calculation:

1) Commutive property:

a * b = 1 + ab = 1 + ba = b * a 

⇒ commutativity satisfied 

2) Associative property:

(a * b) * c = (1 + ab) * c = 1 + (1 + ab)c = 1 + c + abc

a * (b * c) = a * (1 + bc) = 1 +  a(1 + bc) = 1 + a + abc 

a * (b * c) ≠ (a * b) * c

⇒ associativity not satisfied  

∴ option 3 is correct 

Calculation:

Given I = ∫(xx)2 (1 + ln x)dx

⇒ I = ∫x2x (1 + ln x)dx  __(1)

Consider x2x = u 

Taking log both sides, 

⇒ 2x ln x = ln u

Given:

65 x3y4z5, 117 x2y3z3, 156 x6y5z⁷

Calculation:

65 = 13 × 5,

117 = 13 × 9

156 = 13 × 12

So, the HCF of 65, 117, and 156 is 13.

Again,

The HCF of x², x^3, and x⁶ = x²

The HCF of y³, y⁴, and y⁵ = y³

The HCF of z³, z⁵, and z⁷ = z³

The  HCF of  x3y4z5,  x2y3z3, and x6y5z⁷ = x²y³z³

So, the HCF of 65 x3y4z5, 117 x2y3z3, 156 x6y5z⁷ is 13x2y3z3

∴ The HCF is 13x2y3z3

Concept:

Continuity:- A function f(x) is said to be continuous at the point x = a if right hand limit and left hand limit of f(x) as x → a are equal and finite and both limits coincide with the value of function f(x) at x = a

f(x) is said to be continuous at x = a if

The above definition can be stated in short form as: 

Discontinuous Function:- If a function f(x) is not continuous at the point x = a, then f(x) is said to be discontinuous at x = a

Formula Used:

Calculation:

We have function,

f(x) = 

Right hand limit

⇒ f(0 + 0) = 

⇒ f(0 + 0) = 

Given:

4x5 + 9x4 - 7x3 - 5x2 - 4kx + 3k2 

Concept:

If (x - a) is a factor of P(x) then P(a) = 0

Calculation:

We have,

⇒ 4x5 + 9x4 - 7x3 - 5x2 - 4kx + 3k2 

⇒ x - 1 = 0 

⇒ x = 1 

Put x = 1 in the given equation

⇒ 4(1)5 + 9(1)4 - 7(1)3 - 5(1)2 - 4k(1) + 3k2 = 0

⇒ 4 + 9 - 7 - 5 - 4k + 3k² = 0

⇒ 1 - 4k + 3k² = 0

⇒ 3k² - 4k + 1 = 0

⇒ 3k² - 3k - k + 1 = 0

⇒ 3k(k - 1) -1(k - 1) = 0

⇒ (3k - 1)(k - 1) = 0

⇒ 3k - 1 = 0 or k - 1 = 0

⇒ 3k = 1 or k = 1

⇒ k =  or k = 1

∴ The value of k is 1 and 

Concept used:

Factor theorem:

If f(x) be a polynomial of degree n ≥ 1, a belong to Real number then  x - a is the factor of polynomial f(x) if f(a) = 0. 

Calculation:

put x = 1 in the polynomial x3 - 2x + 1

1 - 2 + 1 = 0 

⇒ x - 1 is the factor of x3 - 2x + 1.

put x = -1  in the polynomial x3 - 2x + 1

⇒ -1 -2(-1) + 1 ≠ 0 

⇒ x + 1 is not the factor of the polynomial x3 - 2x + 1.

put x = -2 in polynomial  -8 + 4 + 1 ≠ 0 

⇒ x + 2 is not the factor of polynomial 

put x = 2 in polynomial 8 - 4 + 1 ≠ 0 

⇒ x - 2 is not the factor of the polynomial 

∴ option 1 is correct 

Concept:

• The arithmetic mean is greater than the geometric mean.
• Arithmetic mean is equal to geometric mean if and only if, all the variables are equal

Calculation:

Let sin4α, 4 cos4 β, 1, 1 be four variables, '

then their arithmetic mean is greater than the geometric mean 

⇒ (sin4α + 4 cos4 β + 1 + 1)/4 ≥ 

⇒ (sin 4α + 4 cos4 β + 2) ≥ 4√2sin α.cos β

But given, sin4α + 4 cos4 β + 2 =  sin α cos β,

⇒  sin4α = 4 cos4 β = 1, 

⇒  sin4α = 1, cos4 β = 1/4,

⇒  α = π/2, β = π/4,

So, cos(α + β) - cos(α - β)

= cos() - cos($\frac{\pi }{2}-\frac{\pi }{4}$)

 = -√2 

∴ The correct answer is option (2).

CONCEPT:

Power Set:

Let A be a set, then the set of all the possible subsets of A is called the power set of A and is denoted by P(A).

Note: If A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2^m.

CALCULATION:

Given: A = {1, 3, 5}

Here, we have to find the cardinality of the power set of A i.e n (P(A))

As we know that if A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2m.

Here, we can see that, the given A has 3 elements i.e n(A) = 3

So, the cardinality of the given set is n(P(A)) = 2³ = 8

Hence, the correct option is 4.

Concept used: 

Limit Comparision test:

if an and bn are two positive series such that 

where c > 0 and finite then, either Both series converges or diverges together

P - Series test: 

∑  is convergent for p > 1 and divergent for p ≤  1 

By Limit comparison test both ∑u_n and ∑v_n converge or diverge, But p-series ∑v_n diverges (since p = 1);

Hence, ∑u_n diverges.

Solution:

Given:

Equation of the line : 

Equation of the plane : x - y + az + b = 0 

Also, the line lies in the plane.

Concept:

1. For a plane having equation ax + by + cz = d, (a, b, c) are direction ratios of normal to the plane.

So if the line lies in the plane then the direction ratios of the line will b perpendicular to the direction ratios of the normal to the plane i. e. a , b, c

⇒ The sum of the product of respective direction cosines will be 0.

2. Point through which the line passes must also satisfy the equation of the plane.

Calculation:

The direction ratios of the line are 2, -3, 4.

⇒ (2)(1) + (-3)(-1) + (4)(a) = 0

⇒ a = - 5/4

Also,

Line passes through (1, -2 , 3)

Hence (1, -2 , 3) must satisfy plane -

⇒ (1) - (-2) + (a)(3) + b = 0

Using a = -5/4, we get b = 3/4

∴ b - a = 2

Concept:

Formulas:

• 1 + tan²θ = sec²θ 
• secθ = 

Solution:

Given: cosθ = 2/3

1 + tan²θ

= sec²θ 

= 

= $\frac{1}{{\left(\frac{2}{3}\right)}^2}$

= 

= 9/4

∴ The correct option is (3)

Concept used:

Ratio test:

L = 

L > 1 the series is Divergent neither convergent or divergent 

L < 1 the series is Convergent 

L = 1 test fails Neither Convergent nor Divergent 
Calculations:

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

Chain Rule: 

​

Product Rule: 

Calculation:

Curve x - |y| = 0 and x - 2 = 0

The area of the region bounded by the curve is 

= 

= 

= 

= 

= 

= 

= 4

∴ The correct option is (3)

Given:

17P = 9Q = 7R

Calculation:

LCM (17, 9, 7) = 1071

17P = 9Q = 7R

⇒ 

⇒ 

Now, P ∶ Q ∶ R = 63 : 119 : 153

∴ P ∶ Q ∶ R is 63 : 119 : 153.

Concept:

General term of expansion of (a + b)ⁿ is given by:

Concept:

•  and 
• Hypotenuse ² = (adjacent side)² + (opposite side)²
• cos 2x = 2cos2x - 1
• cos 2x = 1- 2sin2x 

Calculation:

Given 

We have 

⇒ 

Since, tan x = - 4/3 we get a right-angled triangle as shown in the below image

∴ Using Pythagoras theorem we have 

Hypotenuse² = (- 4)² + 3² = 16 + 9 = 25

Hypotenuse = 5

From the figure, we have

Hence, the correct answer is option 1).

Shortcut Trick
Constructing this figure,

In this figure, 

∠AOB = 180° - 110° 

⇒ ∠AOB = 70°

In △OAB,

As, OA = OB (radius)

and ∠OAB = ∠OBA ( In a triangle, angles opposite to equal sides are equal)

Now,

∠OAB + ∠OBA + ∠AOB = 180° 

⇒ 2∠OAB = 110° 

∴ ∠OAB is 55°.

Alternate Method

Given: 

MA and MB are two tangents from a point M outside the circle with centre O.

And ∠AMB = 110°

Concept used:

Tangent to a circle is always perpendicular to its radius at the point of tangency.

Sum of all angles of a quadrilateral is 360° 

Sum of all angles of triangle is 180° 

In a triangle, angles opposite to equal sides are equal.

Calculation:

 As, MA and Mb are tangents,

Tangent to a circle is always perpendicular to its radius at the point of tangency.

⇒ ∠OAM = 90° 

⇒ ∠OBM = 90° 

Now In quadrilateral OAMB,

⇒ ∠OAM + ∠AMB + ∠OBM + ∠AOB = 360° 

⇒ 90° + 110° + 90° + ∠AOB =  360° 

⇒ ∠AOB = 360° - 290° 

⇒ ∠AOB = 70°

In △OAB,

As, OA = OB (radius)

and ∠OAB = ∠OBA ( In a triangle, angles opposite to equal sides are equal)

Now,

∠OAB + ∠OBA + ∠AOB = 180° 

⇒ ∠OAB + ∠OAB + 70° = 180° 

⇒ 2∠OAB = 110° 

⇒ ∠OAB = 55° 

∴ ∠OAB is 55°.

Concept:

1. $\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\mathrm{z}=\frac{{\mathrm{e}}^{\mathrm{z}}+{\mathrm{e}}^{-\mathrm{z}}}{2}$      ----(1)

2.       ----(2)

3. a² - b² = (a -b)(a + b)

Calculation:

cos h²z - sin h²z = (cos hz + sin hz)(cos hz - sin hz)

From equation (1) and (2)

⇒  

GIVEN:

x = {n; 2n² + 7n - 15 < 0}

CONCEPT:

Concept of sets and relation.

FORMULA USED:

No formula

CALCULATION:

x = {n; 2n² + 7n - 15 < 0}

⇒ 2n² + 7n - 15 < 0

⇒ 2n² + 10n - 3n - 15 < 0

⇒ 2n(n + 5) - 3(n + 5) < 0

⇒ (n + 5)(2n - 3) < 0

⇒ n = (-5, 3/2)

Calculation:

Concept:  

• A relation R in a set A is called
•   Reflexive, if (a, a) є R, for every a є A,

For example, a relation that implies a line is parallel to another line is a reflexive relation since each line is parallel to itself.

• Symmetric, if (a₁, a₂) є R implies that (a₂, a₁) є R for all a_1, a₂ є A,

For example, a relation that implies a line is parallel to another line is a symmetric relation. For example, if line 1 is parallel to line 2, then line 2 is parallel to line 1.

• Transitive, if (a₁, a₂) є R and (a₂, a₃) є R implies that (a₁, a₃) є R for all a_1, a_2, a₃ є A.

If a relation R is reflexive, symmetric, and transitive at once, then it is said to be an equivalence relation.

Calculation:

Given:

Relation R is defined by function R = {(a, b) | (a - b) = km for some fixed integer m and a, b, k ∈ z}.

1) For the same number a, R = a - a = 0 × m, where m is a fixed integer and 0 є z, hence the relation R is reflexive.

2) For two numbers (a, b) if R = a - b = km, where m is a fixed integer and k є z, then for (b, a), R = b - a = -(a - b) = -km, where -k є z. Hence relation R is symmetric.

3) Consider three numbers a, b, and c. If, for (a, b), R = a - b = km and for (b, c), R = b - c = lm, where m is a fixed integer and k, l є z, then for (a, c), R = a - c = a - b + b - c = km + lm = m(k + l), where (k + l) є z, since k, l є z. Hence relation R is transitive.

Hence, relation R is an equivalence relation.

Hence, the correct answer is option 4.

Calculation:

The function f(x) is define, if 

4 - x² ≠ 0

2² - x2 ≠ 0

⇒ (2 - x)(2 + x) ≠ 0

⇒ x ≠ 2, -2

⇒ Domain D = R - {-2, 2}

∴ The correct option is (2)