UP LT Grade Teacher Mathematics Mock Test - 5 Free Online Test 2026

In a democracy, the meeting where representatives discuss and make decisions about new laws is called Parliament.

The analogy is that the first word controls or operates the object mentioned as the second word. A 'pilot' is a person who controls a 'plane'; similarly, the one who controls a 'ship' is called the 'captain'.

The content explicitly states that the Mithun is listed as Vulnerable on the IUCN Red List and under Appendix I of CITES. Therefore, option C is the correct answer, and the others are incorrect.

Shortcut Trick

Alternate Method
Given:

Formula used:

tanθ = 

cos2θ = 2cos²θ  - 1 = 1 - 2sin²θ 

Calculation:

Given:

S2: x² +y² +z² + 2ux + 2vy + 2wz + d = 0

S1: x² +y² +z² +2u₁x + 2v₁y + 2w₁z + d₁=0

Calculations:

The conditions such that two spheres are orthogonal to each other is

2uu1 + 2vv1 + 2ww1 = d + d₁

Given:

Length of a chord is 30 cm

Chord is 8 cm away from the centre of the circle

Calculation:

Let AB be the chord and O be the center of the circle.

Also, OP is perpendicular to AB

In Δ OPB,

OB² = PB² + OP²

OB² = 15² + 8²

OB² = 225 + 64

OB² = 289

OB = √289 

OB = 17

Diameter = 2 × OB = 2 × 17 = 34

∴ The diameter of the circle is 34 cm.

Concept:

Let G be a non-empty set and 'o' be the binary operation, then an algebraic structure (G, o) is said to be a abelian group if the binary operation 'o' satisfies the following axioms:

(1) Closure Axiom: G is closed under the operation o, 

a o b ∈ G, for every a, b ∈ G.

(2) Associative Axiom: The binary operation o is associative.

(a o b)o c = a o(b o c), for every a, b, c ∈ G.

(3) Identity Axiom: For every element a ∈ G, there exists an element e ∈ G such that a o e = e o a = a, for every a ∈ G.

(4) Inverse Axiom: Each element of G possess inverse. 

For every a ∈ G, there exists an element b in G such that 

a o b = b o a = e

Thus the element 'b' is called the inverse of a and written as b = a^-1

(5) Commutative Axiom: G is commutative under the operation 'o'

a o b = b o a, for every a, b ∈ G,

Calculation:

We have, G = {e, a, b, c}

We have the following observations: 

(1) Closure Axiom: We see that all the entries in the composition table are elements of the set G. 

Thus closure axiom is satisfied.

(2) Associative Axiom: Associative law always hold in modulo system.

(3) Identity Axiom: The order of elements in first row and first column is the same as row and column makings.

Hence, e is identity element.

(4) Inverse Axiom: From table we have the inverses of e, a, b, c are e, a, b, c.

(5) Commutative Axiom: From the above table we have that each row is identical to corresponding columns.

Hence, G is abelian.

∴ G is an abelian group.

Calculation:

m = sin θ + cos θ 

⇒ m² = (sin θ + cos θ)²

⇒ m² = sin²θ + cos²θ + 2sinθ.cosθ 

⇒ m² = 1 + 2sinθcosθ 

⇒ 

n = sec θ + cosec θ 

⇒ n(m² - 1) = 2m

⇒ n(m + 1)(m - 1) = 2m

Solution:

The equation of tangent to the parabola x² = 4ay can be written as y = mx - am²

For the Parabola y = x², we can written of the equation of the tangent as 

Similarly, for the Parabola y = -(x - 2)², we can written of the equation of the tangent as 

As the tangent is common to both the parabola, both the equations should represent the same line.

Hence,

⇒ m(m - 4) = 0

⇒ m = 0, 4

Hence, the common tangent are y = 0 and y = 4x - 4

⇒ y = 4(x - 1)

∴ The correct option is (4)

Concept:

• If set A has n elements and set B has m elements, then A×B has n×m number of elements.
• A set having n elements can have 2ⁿ number of subsets.

Calculation:

Given, A has 4 elements and B has 2 elements.

The total number of elements is A × B is 2 × 4 = 8.

Hence, the total number of subsets possible of A × B is 2⁸.

A × B has one subset with zero elements, i.e. the null set (ϕ), ⁸C₁ number of subsets with 1 element, and 8C₂ number of subsets with 2 elements.

Hence, the number of subsets A × B has with three or more subsets is 2⁸ - (1 + 8C1 + 8C2) = 256 - ( 1 + 8 + 28) = 256 - 37 = 219

Given:

  is perpendicular on 

PB = 10 and PM = 8

Concept:

In a circle, perpendicular to a chord bisects it.

Calculation:

In ΔPMB, PB² = PM² + MB²

⇒  102 = 82 + MB2

⇒ MB = √36

⇒ MB = 6

By above concept,

AB = 2 × MB

⇒ AB = 2 × 6

⇒ AB = 12

Using Newton’s Second law of motion:

Concept:

The dot product of two vectors:

• The dot product or scalar product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
• Let the two vectors are A and B then dot the Product of the two vectors:

A⋅B = |A||B| cos θ

Where |A| = Magnitude of vectors A and |B| = Magnitude of vectors B and θ is the angle between A and B

Vector Product of Two Vectors:

• The vector product or cross product of two vectors is defined as a vector having a magnitude equal to the product of the magnitudes of two vectors with the sine of the angle between them, and direction perpendicular to the plane containing the two vectors in accordance with the right-hand screw rule.
• The symbol for the cross product is given by the cross sign (×).
• Let the two vectors be A and B then cross the Product of the two vectors:

A × B = |A||B| sin θ

Where |A| = Magnitude of vectors A and |B| = Magnitude of vectors B and θ is the angle between A and B

Explanation:

• From the above, it is clear that the dot product of two vectors is known as the scalar product and the angle between them is cosine or cos.

Concept:

Complex Conjugate: Let z = a + bi, where a and b are real numbers. The Complex Conjugate of z denoted , is a - bi.

The multiplicative inverse of a complex number: The multiplicative inverse of z is z^-1 or .

z^-1 = 1z=1x+iy

Dividing and Multiplying by (x - iy)

⇒ z−1=x−iyx2+y2

Concept:

tan 15° = 2 - √3

cot 15° = 2 + √3

Formula used:

(a + b)² = a² + 2ab + b²

(a - b)2 = a2 - 2ab + b2

Calculation:

Let the required value is y.

⇒ y = cot2 15° + tan2 15°

⇒ y = (2 + √3)² + (2 - √3)²

Using the formula given above

⇒ y = (4 + 4√3 + 3) + (4 - 4√3 + 3) 

⇒ y = 8 + 6 = 14

∴ cot2 15° + tan2 15° = 14

Given:

if an  is unbounded and bn is unbounded 

Calculations:

take a_n  and b_n both equal to n

then a_n + b_n  = n + n = 2n which is unbounded

⇒ 1st option is incorrect

take a_n  = n and b_n  = -n

then a_n + b_n  = n - n = 0 which is a bounded sequence contains all terms equal to 0

⇒ 2nd option is incorrect

∴ option 3 is correct 

Concept:

The slope of the tangent on a curve is given by 

And if any line is parallel to y-axis then the slope of its normal,  = 0

Calculation:

Given:

Curve:   = 1 

Differentiating both sides

We get   = 

Now, As tangents are parallel to the y-axis 

So,  = 0

 = 0

So, y = 0

then put y = 0 in the curve

we get,

x = 3, -3

So the points are (±3, 0)

Concept:

• When any two consecutive numbers in a series have the same difference, they are in Arithmetic Progression or AP. An example of AP is 1 , 3 , 5 ...
• The difference between these two consecutive numbers of the series is called the common difference and it is often denoted by d.
• The formula for the nth terms of an AP is  where a is the first number of the series.
• The formula for the sum of n numbers in an AP is 

Calculation:

Given, the ratio of the sum of m and n terms of an AP is m² : n²

According to the formula, the sum of m terms is  where  is the m^th term of the series.

Similarly, the sum of n terms is  where


Now, the ratio of the m^th and the n^th terms of the series are 

Given:

cos A =  (A is an acute angle)

Concept used:

Calculation:

cos A = 

⇒ cosA = cos30°

⇒ A = 30°

Shortcut Trick

Formula Used:

Average = (Sum of observations)/(Number of observations)

Last term = a + (n - 1)d

Calculation: 

The above series is in arithmetic progression so the middlemost term 8th term will be the average

⇒ 8^th term = 8 + (8 - 1) × 3 = 29

⇒ Sum of the series = 29 × 15 = 435

∴ The sum of the above series is 435  

Additional Information

We can avoid this above (29 × 15) multiplication by digit sum Method and option

The digit sum of 29 is  (2 + 9) ⇒ (11) ⇒ (2) and 15 is (1 + 5) = 6 

⇒ 2 × 6 = 12 ⇒ (1 + 2) ⇒ 3 

Now check the options whose digit sum will be 3 there is only option 2 whose Digit sum is 3 

∴ 435 is the right answer

Traditional Method: 

Given:

Arithmetic progression 8 + 11 + 14 + 17 upto 15 terms

Formula Used: 

Sum of arithmetic progression = n[2a + (n - 1)d]/2

Calculation:

Sum of 1st 15 terms = 15[2 × 8 + (15-1)3]/2

⇒ (15 × 58)/2

⇒ 435

∴ 435 is the right answer

Concept:

Unit vector along x-axis = î + 0ĵ +0k̂ = î → (1)

Unit vector along y-axis = ĵ

Unit vector along z-axis = k̂

And angle between two vectors 

Where

Concept:

• cos 2x = 2cos²x - 1 = 1 - 2sin²x
• 
• 1 + tan²x = sec²x
• ∫sec²ax dx = tan ax + C

∴ The correct answer is option (4).

CONCEPT:

• Projection of a vector  on other vector  is given by: 

CALCULATION:

Given:  and 

Here, we have to find the projection of a vector  on other vector  is given by: 

Homomorphism:

Suppose the composition in G and G' are multiplicative. A mapping f from a group G into (onto) a group G' is said to be a homomorphism of G (onto) G' if,

⇒ f(xy) = f(x) f(y) ∀ x,y ∈ G.

f(G) is called homomorph or holomorphic image of G.

If f is homomorphism onto, then we say that G is holomorphic to G'. The symbol G^~ - G' is read as G is holomorphic to G'.